id sid tid token lemma pos american_scientific_journal-7448 1 1 143 143 NUM american_scientific_journal-7448 1 2 american american ADJ american_scientific_journal-7448 1 3 academic academic ADJ american_scientific_journal-7448 1 4 scientific scientific ADJ american_scientific_journal-7448 1 5 research research NOUN american_scientific_journal-7448 1 6 journal journal NOUN american_scientific_journal-7448 1 7 for for ADP american_scientific_journal-7448 1 8 engineering engineering NOUN american_scientific_journal-7448 1 9 , , PUNCT american_scientific_journal-7448 1 10 technology technology NOUN american_scientific_journal-7448 1 11 , , PUNCT american_scientific_journal-7448 1 12 and and CCONJ american_scientific_journal-7448 1 13 sciences sciences PROPN american_scientific_journal-7448 1 14 issn issn PROPN american_scientific_journal-7448 1 15 ( ( PUNCT american_scientific_journal-7448 1 16 print print NOUN american_scientific_journal-7448 1 17 ) ) PUNCT american_scientific_journal-7448 1 18 2313 2313 NUM american_scientific_journal-7448 1 19 - - SYM american_scientific_journal-7448 1 20 4410 4410 NUM american_scientific_journal-7448 1 21 , , PUNCT american_scientific_journal-7448 1 22 issn issn PROPN american_scientific_journal-7448 1 23 ( ( PUNCT american_scientific_journal-7448 1 24 online online ADJ american_scientific_journal-7448 1 25 ) ) PUNCT american_scientific_journal-7448 1 26 2313 2313 NUM american_scientific_journal-7448 1 27 - - SYM american_scientific_journal-7448 1 28 4402 4402 NUM american_scientific_journal-7448 1 29 http://asrjetsjournal.org/ http://asrjetsjournal.org/ NOUN american_scientific_journal-7448 1 30 an an DET american_scientific_journal-7448 1 31 enhanced enhanced ADJ american_scientific_journal-7448 1 32 model model NOUN american_scientific_journal-7448 1 33 for for ADP american_scientific_journal-7448 1 34 the the DET american_scientific_journal-7448 1 35 hydrodynamic hydrodynamic ADJ american_scientific_journal-7448 1 36 drag drag NOUN american_scientific_journal-7448 1 37 force force NOUN american_scientific_journal-7448 1 38 on on ADP american_scientific_journal-7448 1 39 a a DET american_scientific_journal-7448 1 40 steadily steadily ADV american_scientific_journal-7448 1 41 translating translate VERB american_scientific_journal-7448 1 42 circular circular ADJ american_scientific_journal-7448 1 43 cylinder cylinder NOUN american_scientific_journal-7448 1 44 monday monday PROPN american_scientific_journal-7448 1 45 ekhator ekhator NOUN american_scientific_journal-7448 1 46 a a PROPN american_scientific_journal-7448 1 47 * * X american_scientific_journal-7448 1 48 , , PUNCT american_scientific_journal-7448 1 49 vincent vincent PROPN american_scientific_journal-7448 1 50 ele ele PROPN american_scientific_journal-7448 1 51 asor asor PROPN american_scientific_journal-7448 1 52 b b PROPN american_scientific_journal-7448 1 53 , , PUNCT american_scientific_journal-7448 1 54 stephen stephen PROPN american_scientific_journal-7448 1 55 etebefia etebefia PROPN american_scientific_journal-7448 1 56 c c PROPN american_scientific_journal-7448 1 57 a a DET american_scientific_journal-7448 1 58 department department NOUN american_scientific_journal-7448 1 59 of of ADP american_scientific_journal-7448 1 60 mathematics mathematic NOUN american_scientific_journal-7448 1 61 and and CCONJ american_scientific_journal-7448 1 62 statistics statistic NOUN american_scientific_journal-7448 1 63 , , PUNCT american_scientific_journal-7448 1 64 alex alex PROPN american_scientific_journal-7448 1 65 ekwueme ekwueme PROPN american_scientific_journal-7448 1 66 federal federal PROPN american_scientific_journal-7448 1 67 university university PROPN american_scientific_journal-7448 1 68 , , PUNCT american_scientific_journal-7448 1 69 ndufualike ndufualike PROPN american_scientific_journal-7448 1 70 , , PUNCT american_scientific_journal-7448 1 71 nigeria nigeria PROPN american_scientific_journal-7448 1 72 b b PROPN american_scientific_journal-7448 1 73 department department PROPN american_scientific_journal-7448 1 74 of of ADP american_scientific_journal-7448 1 75 mathematics mathematic NOUN american_scientific_journal-7448 1 76 , , PUNCT american_scientific_journal-7448 1 77 michael michael PROPN american_scientific_journal-7448 1 78 okpara okpara PROPN american_scientific_journal-7448 1 79 university university PROPN american_scientific_journal-7448 1 80 of of ADP american_scientific_journal-7448 1 81 agriculture agriculture PROPN american_scientific_journal-7448 1 82 , , PUNCT american_scientific_journal-7448 1 83 umudike umudike PROPN american_scientific_journal-7448 1 84 , , PUNCT american_scientific_journal-7448 1 85 nigeria nigeria PROPN american_scientific_journal-7448 1 86 c c PROPN american_scientific_journal-7448 1 87 federal federal PROPN american_scientific_journal-7448 1 88 school school PROPN american_scientific_journal-7448 1 89 of of ADP american_scientific_journal-7448 1 90 statistics statistic NOUN american_scientific_journal-7448 1 91 , , PUNCT american_scientific_journal-7448 1 92 enugu enugu PROPN american_scientific_journal-7448 1 93 , , PUNCT american_scientific_journal-7448 1 94 nigeria nigeria PROPN american_scientific_journal-7448 1 95 a a DET american_scientific_journal-7448 1 96 email email NOUN american_scientific_journal-7448 1 97 : : PUNCT american_scientific_journal-7448 1 98 ekhatorogie@gmail.com ekhatorogie@gmail.com X american_scientific_journal-7448 1 99 , , PUNCT american_scientific_journal-7448 1 100 b b X american_scientific_journal-7448 1 101 email email NOUN american_scientific_journal-7448 1 102 : : PUNCT american_scientific_journal-7448 1 103 vincent.asor@gmail.com vincent.asor@gmail.com X american_scientific_journal-7448 1 104 , , PUNCT american_scientific_journal-7448 1 105 c c NOUN american_scientific_journal-7448 1 106 email email NOUN american_scientific_journal-7448 1 107 : : PUNCT american_scientific_journal-7448 1 108 steveoghene@gmail.com steveoghene@gmail.com PROPN american_scientific_journal-7448 1 109 abstract abstract VERB american_scientific_journal-7448 1 110 a a DET american_scientific_journal-7448 1 111 model model NOUN american_scientific_journal-7448 1 112 for for ADP american_scientific_journal-7448 1 113 the the DET american_scientific_journal-7448 1 114 hydrodynamic hydrodynamic ADJ american_scientific_journal-7448 1 115 drag drag NOUN american_scientific_journal-7448 1 116 force force NOUN american_scientific_journal-7448 1 117 on on ADP american_scientific_journal-7448 1 118 a a DET american_scientific_journal-7448 1 119 steadily steadily ADV american_scientific_journal-7448 1 120 translating translate VERB american_scientific_journal-7448 1 121 circular circular ADJ american_scientific_journal-7448 1 122 cylinder cylinder NOUN american_scientific_journal-7448 1 123 was be AUX american_scientific_journal-7448 1 124 studied study VERB american_scientific_journal-7448 1 125 for for ADP american_scientific_journal-7448 1 126 reynolds reynold NOUN american_scientific_journal-7448 1 127 number number PROPN american_scientific_journal-7448 1 128 , , PUNCT american_scientific_journal-7448 1 129 re re ADP american_scientific_journal-7448 1 130 < < X american_scientific_journal-7448 1 131 < < X american_scientific_journal-7448 1 132 1 1 NUM american_scientific_journal-7448 1 133 . . PUNCT american_scientific_journal-7448 1 134 though though SCONJ american_scientific_journal-7448 1 135 the the DET american_scientific_journal-7448 1 136 literature literature NOUN american_scientific_journal-7448 1 137 appears appear VERB american_scientific_journal-7448 1 138 vast vast ADJ american_scientific_journal-7448 1 139 especially especially ADV american_scientific_journal-7448 1 140 by by ADP american_scientific_journal-7448 1 141 method method NOUN american_scientific_journal-7448 1 142 of of ADP american_scientific_journal-7448 1 143 rigorous rigorous ADJ american_scientific_journal-7448 1 144 asymptotic asymptotic ADJ american_scientific_journal-7448 1 145 analysis analysis NOUN american_scientific_journal-7448 1 146 , , PUNCT american_scientific_journal-7448 1 147 this this DET american_scientific_journal-7448 1 148 work work NOUN american_scientific_journal-7448 1 149 attempts attempt VERB american_scientific_journal-7448 1 150 to to PART american_scientific_journal-7448 1 151 remove remove VERB american_scientific_journal-7448 1 152 the the DET american_scientific_journal-7448 1 153 mathematical mathematical ADJ american_scientific_journal-7448 1 154 complexity complexity NOUN american_scientific_journal-7448 1 155 of of ADP american_scientific_journal-7448 1 156 asymptotic asymptotic ADJ american_scientific_journal-7448 1 157 analysis analysis NOUN american_scientific_journal-7448 1 158 by by ADP american_scientific_journal-7448 1 159 solving solve VERB american_scientific_journal-7448 1 160 the the DET american_scientific_journal-7448 1 161 navier navier NOUN american_scientific_journal-7448 1 162 – – PUNCT american_scientific_journal-7448 1 163 stokes stoke VERB american_scientific_journal-7448 1 164 equations equation NOUN american_scientific_journal-7448 1 165 directly directly ADV american_scientific_journal-7448 1 166 to to PART american_scientific_journal-7448 1 167 obtain obtain VERB american_scientific_journal-7448 1 168 the the DET american_scientific_journal-7448 1 169 fluid fluid ADJ american_scientific_journal-7448 1 170 velocity velocity NOUN american_scientific_journal-7448 1 171 and and CCONJ american_scientific_journal-7448 1 172 then then ADV american_scientific_journal-7448 1 173 proceed proceed VERB american_scientific_journal-7448 1 174 to to PART american_scientific_journal-7448 1 175 obtain obtain VERB american_scientific_journal-7448 1 176 the the DET american_scientific_journal-7448 1 177 drag drag NOUN american_scientific_journal-7448 1 178 force force NOUN american_scientific_journal-7448 1 179 and and CCONJ american_scientific_journal-7448 1 180 finally finally ADV american_scientific_journal-7448 1 181 the the DET american_scientific_journal-7448 1 182 drag drag NOUN american_scientific_journal-7448 1 183 coefficient coefficient NOUN american_scientific_journal-7448 1 184 which which PRON american_scientific_journal-7448 1 185 is be AUX american_scientific_journal-7448 1 186 a a DET american_scientific_journal-7448 1 187 function function NOUN american_scientific_journal-7448 1 188 of of ADP american_scientific_journal-7448 1 189 the the DET american_scientific_journal-7448 1 190 reynolds reynolds PROPN american_scientific_journal-7448 1 191 number number PROPN american_scientific_journal-7448 1 192 . . PUNCT american_scientific_journal-7448 2 1 the the DET american_scientific_journal-7448 2 2 result result NOUN american_scientific_journal-7448 2 3 of of ADP american_scientific_journal-7448 2 4 this this DET american_scientific_journal-7448 2 5 work work NOUN american_scientific_journal-7448 2 6 is be AUX american_scientific_journal-7448 2 7 in in ADP american_scientific_journal-7448 2 8 complete complete ADJ american_scientific_journal-7448 2 9 agreement agreement NOUN american_scientific_journal-7448 2 10 with with ADP american_scientific_journal-7448 2 11 the the DET american_scientific_journal-7448 2 12 results result NOUN american_scientific_journal-7448 2 13 in in ADP american_scientific_journal-7448 2 14 literature literature NOUN american_scientific_journal-7448 2 15 . . PUNCT american_scientific_journal-7448 3 1 key key ADJ american_scientific_journal-7448 3 2 words word NOUN american_scientific_journal-7448 3 3 : : PUNCT american_scientific_journal-7448 3 4 drag drag NOUN american_scientific_journal-7448 3 5 force force NOUN american_scientific_journal-7448 3 6 ; ; PUNCT american_scientific_journal-7448 3 7 drag drag VERB american_scientific_journal-7448 3 8 coefficient coefficient NOUN american_scientific_journal-7448 3 9 ; ; PUNCT american_scientific_journal-7448 3 10 streamline streamline ADJ american_scientific_journal-7448 3 11 ; ; PUNCT american_scientific_journal-7448 3 12 stream stream NOUN american_scientific_journal-7448 3 13 function function NOUN american_scientific_journal-7448 3 14 ; ; PUNCT american_scientific_journal-7448 3 15 reynolds reynolds PROPN american_scientific_journal-7448 3 16 number number PROPN american_scientific_journal-7448 3 17 . . PUNCT american_scientific_journal-7448 4 1 1 1 X american_scientific_journal-7448 4 2 . . X american_scientific_journal-7448 4 3 introduction introduction NOUN american_scientific_journal-7448 4 4 fluid fluid NOUN american_scientific_journal-7448 4 5 mechanics mechanic NOUN american_scientific_journal-7448 4 6 provides provide VERB american_scientific_journal-7448 4 7 scholarship scholarship NOUN american_scientific_journal-7448 4 8 on on ADP american_scientific_journal-7448 4 9 how how SCONJ american_scientific_journal-7448 4 10 impermeable impermeable ADJ american_scientific_journal-7448 4 11 and and CCONJ american_scientific_journal-7448 4 12 rigid rigid ADJ american_scientific_journal-7448 4 13 surfaces surface NOUN american_scientific_journal-7448 4 14 such such ADJ american_scientific_journal-7448 4 15 as as ADP american_scientific_journal-7448 4 16 pipes pipe NOUN american_scientific_journal-7448 4 17 and and CCONJ american_scientific_journal-7448 4 18 particles particle NOUN american_scientific_journal-7448 4 19 affect affect VERB american_scientific_journal-7448 4 20 the the DET american_scientific_journal-7448 4 21 flow flow NOUN american_scientific_journal-7448 4 22 of of ADP american_scientific_journal-7448 4 23 a a DET american_scientific_journal-7448 4 24 fluid fluid NOUN american_scientific_journal-7448 4 25 and and CCONJ american_scientific_journal-7448 4 26 the the DET american_scientific_journal-7448 4 27 effect effect NOUN american_scientific_journal-7448 4 28 of of ADP american_scientific_journal-7448 4 29 the the DET american_scientific_journal-7448 4 30 fluid fluid NOUN american_scientific_journal-7448 4 31 on on ADP american_scientific_journal-7448 4 32 the the DET american_scientific_journal-7448 4 33 particles particle NOUN american_scientific_journal-7448 4 34 . . PUNCT american_scientific_journal-7448 5 1 one one NUM american_scientific_journal-7448 5 2 of of ADP american_scientific_journal-7448 5 3 the the DET american_scientific_journal-7448 5 4 effects effect NOUN american_scientific_journal-7448 5 5 of of ADP american_scientific_journal-7448 5 6 fluid fluid NOUN american_scientific_journal-7448 5 7 on on ADP american_scientific_journal-7448 5 8 particles particle NOUN american_scientific_journal-7448 5 9 in in ADP american_scientific_journal-7448 5 10 which which PRON american_scientific_journal-7448 5 11 they they PRON american_scientific_journal-7448 5 12 flow flow VERB american_scientific_journal-7448 5 13 is be AUX american_scientific_journal-7448 5 14 the the DET american_scientific_journal-7448 5 15 drag drag NOUN american_scientific_journal-7448 5 16 exerted exert VERB american_scientific_journal-7448 5 17 by by ADP american_scientific_journal-7448 5 18 the the DET american_scientific_journal-7448 5 19 fluid fluid NOUN american_scientific_journal-7448 5 20 on on ADP american_scientific_journal-7448 5 21 the the DET american_scientific_journal-7448 5 22 particles particle NOUN american_scientific_journal-7448 5 23 . . PUNCT american_scientific_journal-7448 6 1 drag drag VERB american_scientific_journal-7448 6 2 , , PUNCT american_scientific_journal-7448 6 3 a a DET american_scientific_journal-7448 6 4 frictional frictional ADJ american_scientific_journal-7448 6 5 force force NOUN american_scientific_journal-7448 6 6 also also ADV american_scientific_journal-7448 6 7 called call VERB american_scientific_journal-7448 6 8 fluid fluid ADJ american_scientific_journal-7448 6 9 resistance resistance NOUN american_scientific_journal-7448 6 10 which which PRON american_scientific_journal-7448 6 11 acts act VERB american_scientific_journal-7448 6 12 opposite opposite ADV american_scientific_journal-7448 6 13 to to ADP american_scientific_journal-7448 6 14 the the DET american_scientific_journal-7448 6 15 relative relative ADJ american_scientific_journal-7448 6 16 motion motion NOUN american_scientific_journal-7448 6 17 of of ADP american_scientific_journal-7448 6 18 any any DET american_scientific_journal-7448 6 19 object object NOUN american_scientific_journal-7448 6 20 moving move VERB american_scientific_journal-7448 6 21 with with ADP american_scientific_journal-7448 6 22 respect respect NOUN american_scientific_journal-7448 6 23 to to ADP american_scientific_journal-7448 6 24 a a DET american_scientific_journal-7448 6 25 surrounding surround VERB american_scientific_journal-7448 6 26 fluid fluid NOUN american_scientific_journal-7448 6 27 is be AUX american_scientific_journal-7448 6 28 an an DET american_scientific_journal-7448 6 29 enemy enemy NOUN american_scientific_journal-7448 6 30 of of ADP american_scientific_journal-7448 6 31 moving move VERB american_scientific_journal-7448 6 32 objects object NOUN american_scientific_journal-7448 6 33 in in ADP american_scientific_journal-7448 6 34 a a DET american_scientific_journal-7448 6 35 fluid fluid NOUN american_scientific_journal-7448 6 36 , , PUNCT american_scientific_journal-7448 6 37 e.g. e.g. ADV american_scientific_journal-7448 6 38 aircrafts aircraft NOUN american_scientific_journal-7448 6 39 , , PUNCT american_scientific_journal-7448 6 40 submarines submarine NOUN american_scientific_journal-7448 6 41 , , PUNCT american_scientific_journal-7448 6 42 water water NOUN american_scientific_journal-7448 6 43 droplets droplet NOUN american_scientific_journal-7448 6 44 , , PUNCT american_scientific_journal-7448 6 45 etc etc X american_scientific_journal-7448 6 46 . . X american_scientific_journal-7448 6 47 , , PUNCT american_scientific_journal-7448 6 48 in in ADP american_scientific_journal-7448 6 49 the the DET american_scientific_journal-7448 6 50 sense sense NOUN american_scientific_journal-7448 6 51 that that SCONJ american_scientific_journal-7448 6 52 it it PRON american_scientific_journal-7448 6 53 tends tend VERB american_scientific_journal-7448 6 54 to to PART american_scientific_journal-7448 6 55 reduce reduce VERB american_scientific_journal-7448 6 56 the the DET american_scientific_journal-7448 6 57 speed speed NOUN american_scientific_journal-7448 6 58 of of ADP american_scientific_journal-7448 6 59 the the DET american_scientific_journal-7448 6 60 objects object NOUN american_scientific_journal-7448 6 61 as as SCONJ american_scientific_journal-7448 6 62 they they PRON american_scientific_journal-7448 6 63 move move VERB american_scientific_journal-7448 6 64 in in ADP american_scientific_journal-7448 6 65 the the DET american_scientific_journal-7448 6 66 fluid fluid NOUN american_scientific_journal-7448 6 67 . . PUNCT american_scientific_journal-7448 7 1 its its PRON american_scientific_journal-7448 7 2 calculation calculation NOUN american_scientific_journal-7448 7 3 becomes become VERB american_scientific_journal-7448 7 4 imperative imperative ADJ american_scientific_journal-7448 7 5 in in ADP american_scientific_journal-7448 7 6 the the DET american_scientific_journal-7448 7 7 design design NOUN american_scientific_journal-7448 7 8 of of ADP american_scientific_journal-7448 7 9 aircrafts aircraft NOUN american_scientific_journal-7448 7 10 , , PUNCT american_scientific_journal-7448 7 11 ships ship NOUN american_scientific_journal-7448 7 12 , , PUNCT american_scientific_journal-7448 7 13 submarines submarine NOUN american_scientific_journal-7448 7 14 , , PUNCT american_scientific_journal-7448 7 15 automobiles automobile NOUN american_scientific_journal-7448 7 16 , , PUNCT american_scientific_journal-7448 7 17 towers tower NOUN american_scientific_journal-7448 7 18 and and CCONJ american_scientific_journal-7448 7 19 in in ADP american_scientific_journal-7448 7 20 the the DET american_scientific_journal-7448 7 21 calculation calculation NOUN american_scientific_journal-7448 7 22 of of ADP american_scientific_journal-7448 7 23 the the DET american_scientific_journal-7448 7 24 rate rate NOUN american_scientific_journal-7448 7 25 of of ADP american_scientific_journal-7448 7 26 sedimentation sedimentation NOUN american_scientific_journal-7448 7 27 of of ADP american_scientific_journal-7448 7 28 small small ADJ american_scientific_journal-7448 7 29 particles particle NOUN american_scientific_journal-7448 7 30 as as SCONJ american_scientific_journal-7448 7 31 they they PRON american_scientific_journal-7448 7 32 fall fall VERB american_scientific_journal-7448 7 33 through through ADP american_scientific_journal-7448 7 34 a a DET american_scientific_journal-7448 7 35 fluid fluid NOUN american_scientific_journal-7448 7 36 as as ADP american_scientific_journal-7448 7 37 in in ADP american_scientific_journal-7448 7 38 the the DET american_scientific_journal-7448 7 39 sedimentation sedimentation NOUN american_scientific_journal-7448 7 40 of of ADP american_scientific_journal-7448 7 41 blood blood NOUN american_scientific_journal-7448 7 42 cells cell NOUN american_scientific_journal-7448 7 43 . . PUNCT american_scientific_journal-7448 8 1 just just ADV american_scientific_journal-7448 8 2 as as SCONJ american_scientific_journal-7448 8 3 other other ADJ american_scientific_journal-7448 8 4 forms form NOUN american_scientific_journal-7448 8 5 of of ADP american_scientific_journal-7448 8 6 frictional frictional ADJ american_scientific_journal-7448 8 7 force force NOUN american_scientific_journal-7448 8 8 , , PUNCT american_scientific_journal-7448 8 9 drag drag PROPN american_scientific_journal-7448 8 10 can can AUX american_scientific_journal-7448 8 11 be be AUX american_scientific_journal-7448 8 12 greatly greatly ADV american_scientific_journal-7448 8 13 reduced reduce VERB american_scientific_journal-7448 8 14 by by ADP american_scientific_journal-7448 8 15 shaping shape VERB american_scientific_journal-7448 8 16 a a DET american_scientific_journal-7448 8 17 body body NOUN american_scientific_journal-7448 8 18 to to ADP american_scientific_journal-7448 8 19 the the DET american_scientific_journal-7448 8 20 streamlines streamline NOUN american_scientific_journal-7448 8 21 of of ADP american_scientific_journal-7448 8 22 the the DET american_scientific_journal-7448 8 23 fluid fluid NOUN american_scientific_journal-7448 8 24 through through ADP american_scientific_journal-7448 8 25 which which PRON american_scientific_journal-7448 8 26 it it PRON american_scientific_journal-7448 8 27 is be AUX american_scientific_journal-7448 8 28 moving move VERB american_scientific_journal-7448 8 29 , , PUNCT american_scientific_journal-7448 8 30 hence hence ADV american_scientific_journal-7448 8 31 modern modern ADJ american_scientific_journal-7448 8 32 cars car NOUN american_scientific_journal-7448 8 33 , , PUNCT american_scientific_journal-7448 8 34 ships ship NOUN american_scientific_journal-7448 8 35 and and CCONJ american_scientific_journal-7448 8 36 aircrafts aircraft NOUN american_scientific_journal-7448 8 37 are be AUX american_scientific_journal-7448 8 38 made make VERB american_scientific_journal-7448 8 39 in in ADP american_scientific_journal-7448 8 40 such such ADJ american_scientific_journal-7448 8 41 shapes shape NOUN american_scientific_journal-7448 8 42 as as SCONJ american_scientific_journal-7448 8 43 to to PART american_scientific_journal-7448 8 44 lessen lessen VERB american_scientific_journal-7448 8 45 the the DET american_scientific_journal-7448 8 46 drag drag NOUN american_scientific_journal-7448 8 47 between between ADP american_scientific_journal-7448 8 48 them they PRON american_scientific_journal-7448 8 49 and and CCONJ american_scientific_journal-7448 8 50 the the DET american_scientific_journal-7448 8 51 fluid fluid NOUN american_scientific_journal-7448 8 52 through through ADP american_scientific_journal-7448 8 53 which which PRON american_scientific_journal-7448 8 54 they they PRON american_scientific_journal-7448 8 55 move move VERB american_scientific_journal-7448 8 56 . . PUNCT american_scientific_journal-7448 9 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-7448 10 1 * * PUNCT american_scientific_journal-7448 10 2 corresponding correspond VERB american_scientific_journal-7448 10 3 author author NOUN american_scientific_journal-7448 10 4 . . PUNCT american_scientific_journal-7448 11 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-7448 11 2 american american PROPN american_scientific_journal-7448 11 3 academic academic PROPN american_scientific_journal-7448 11 4 scientific scientific ADJ american_scientific_journal-7448 11 5 research research NOUN american_scientific_journal-7448 11 6 journal journal NOUN american_scientific_journal-7448 11 7 for for ADP american_scientific_journal-7448 11 8 engineering engineering NOUN american_scientific_journal-7448 11 9 , , PUNCT american_scientific_journal-7448 11 10 technology technology NOUN american_scientific_journal-7448 11 11 , , PUNCT american_scientific_journal-7448 11 12 and and CCONJ american_scientific_journal-7448 11 13 sciences science NOUN american_scientific_journal-7448 11 14 ( ( PUNCT american_scientific_journal-7448 11 15 asrjets asrjet NOUN american_scientific_journal-7448 11 16 ) ) PUNCT american_scientific_journal-7448 11 17 ( ( PUNCT american_scientific_journal-7448 11 18 2022 2022 NUM american_scientific_journal-7448 11 19 ) ) PUNCT american_scientific_journal-7448 11 20 volume volume NOUN american_scientific_journal-7448 11 21 86 86 NUM american_scientific_journal-7448 11 22 , , PUNCT american_scientific_journal-7448 11 23 no no DET american_scientific_journal-7448 11 24 1 1 NUM american_scientific_journal-7448 11 25 , , PUNCT american_scientific_journal-7448 11 26 pp143 pp143 PROPN american_scientific_journal-7448 11 27 - - SYM american_scientific_journal-7448 11 28 156 156 NUM american_scientific_journal-7448 11 29 144 144 NUM american_scientific_journal-7448 11 30 stokes stoke NOUN american_scientific_journal-7448 12 1 [ [ X american_scientific_journal-7448 12 2 1 1 NUM american_scientific_journal-7448 12 3 ] ] PUNCT american_scientific_journal-7448 12 4 , , PUNCT american_scientific_journal-7448 12 5 in in ADP american_scientific_journal-7448 12 6 his his PRON american_scientific_journal-7448 12 7 pioneering pioneer VERB american_scientific_journal-7448 12 8 work work NOUN american_scientific_journal-7448 12 9 , , PUNCT american_scientific_journal-7448 12 10 found find VERB american_scientific_journal-7448 12 11 the the DET american_scientific_journal-7448 12 12 drag drag NOUN american_scientific_journal-7448 12 13 force force NOUN american_scientific_journal-7448 12 14 , , PUNCT american_scientific_journal-7448 12 15 fd fd ADP american_scientific_journal-7448 12 16 acting act VERB american_scientific_journal-7448 12 17 on on ADP american_scientific_journal-7448 12 18 a a DET american_scientific_journal-7448 12 19 sphere sphere NOUN american_scientific_journal-7448 12 20 as as ADP american_scientific_journal-7448 12 21 𝐹𝐷 𝐹𝐷 PROPN american_scientific_journal-7448 12 22 = = SYM american_scientific_journal-7448 12 23 6πœ‹πœ‡π‘Žπ‘ˆπ‘ 6πœ‹πœ‡π‘Žπ‘ˆπ‘ NUM american_scientific_journal-7448 12 24 ( ( PUNCT american_scientific_journal-7448 12 25 1 1 NUM american_scientific_journal-7448 12 26 ) ) PUNCT american_scientific_journal-7448 12 27 where where SCONJ american_scientific_journal-7448 12 28 a a PRON american_scientific_journal-7448 12 29 is be AUX american_scientific_journal-7448 12 30 the the DET american_scientific_journal-7448 12 31 radius radius NOUN american_scientific_journal-7448 12 32 of of ADP american_scientific_journal-7448 12 33 the the DET american_scientific_journal-7448 12 34 sphere sphere NOUN american_scientific_journal-7448 12 35 moving move VERB american_scientific_journal-7448 12 36 with with ADP american_scientific_journal-7448 12 37 speed speed NOUN american_scientific_journal-7448 12 38 , , PUNCT american_scientific_journal-7448 12 39 up up ADV american_scientific_journal-7448 12 40 . . PUNCT american_scientific_journal-7448 13 1 the the DET american_scientific_journal-7448 13 2 dimensionless dimensionless NOUN american_scientific_journal-7448 13 3 quantity quantity NOUN american_scientific_journal-7448 13 4 called call VERB american_scientific_journal-7448 13 5 drag drag NOUN american_scientific_journal-7448 13 6 coefficient coefficient NOUN american_scientific_journal-7448 13 7 cd cd PROPN american_scientific_journal-7448 13 8 was be AUX american_scientific_journal-7448 13 9 defined define VERB american_scientific_journal-7448 13 10 as as ADP american_scientific_journal-7448 13 11 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 13 12 = = SYM american_scientific_journal-7448 13 13 2𝐹𝐷 2𝐹𝐷 NUM american_scientific_journal-7448 13 14 πœŒπ‘ˆπ‘ πœŒπ‘ˆπ‘ X american_scientific_journal-7448 13 15 2πœ‹π‘Ž2 2πœ‹π‘Ž2 NUM american_scientific_journal-7448 13 16 = = SYM american_scientific_journal-7448 13 17 24 24 NUM american_scientific_journal-7448 14 1 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 14 2 ( ( PUNCT american_scientific_journal-7448 14 3 2 2 NUM american_scientific_journal-7448 14 4 ) ) PUNCT american_scientific_journal-7448 14 5 where where SCONJ american_scientific_journal-7448 14 6 the the DET american_scientific_journal-7448 14 7 reynolds reynolds PROPN american_scientific_journal-7448 14 8 number number NOUN american_scientific_journal-7448 14 9 , , PUNCT american_scientific_journal-7448 14 10 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 14 11 is be AUX american_scientific_journal-7448 14 12 defined define VERB american_scientific_journal-7448 14 13 as as ADP american_scientific_journal-7448 14 14 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 14 15 = = SYM american_scientific_journal-7448 14 16 2π‘Žπ‘ˆπ‘πœŒ 2π‘Žπ‘ˆπ‘πœŒ NUM american_scientific_journal-7448 14 17 πœ‡ πœ‡ ADP american_scientific_journal-7448 14 18 . . PUNCT american_scientific_journal-7448 15 1 ( ( PUNCT american_scientific_journal-7448 15 2 3 3 X american_scientific_journal-7448 15 3 ) ) PUNCT american_scientific_journal-7448 15 4 the the DET american_scientific_journal-7448 15 5 result result NOUN american_scientific_journal-7448 15 6 of of ADP american_scientific_journal-7448 15 7 [ [ X american_scientific_journal-7448 15 8 1 1 NUM american_scientific_journal-7448 15 9 ] ] PUNCT american_scientific_journal-7448 15 10 was be AUX american_scientific_journal-7448 15 11 useful useful ADJ american_scientific_journal-7448 15 12 for for ADP american_scientific_journal-7448 15 13 robert robert PROPN american_scientific_journal-7448 15 14 millikan millikan PROPN american_scientific_journal-7448 16 1 [ [ X american_scientific_journal-7448 16 2 2,3 2,3 NUM american_scientific_journal-7448 16 3 ] ] PUNCT american_scientific_journal-7448 16 4 who who PRON american_scientific_journal-7448 16 5 measured measure VERB american_scientific_journal-7448 16 6 the the DET american_scientific_journal-7448 16 7 electric electric ADJ american_scientific_journal-7448 16 8 charge charge NOUN american_scientific_journal-7448 16 9 on on ADP american_scientific_journal-7448 16 10 a a DET american_scientific_journal-7448 16 11 droplet droplet ADJ american_scientific_journal-7448 16 12 and and CCONJ american_scientific_journal-7448 16 13 showed show VERB american_scientific_journal-7448 16 14 that that SCONJ american_scientific_journal-7448 16 15 charge charge NOUN american_scientific_journal-7448 16 16 is be AUX american_scientific_journal-7448 16 17 quantised quantise VERB american_scientific_journal-7448 16 18 . . PUNCT american_scientific_journal-7448 17 1 it it PRON american_scientific_journal-7448 17 2 seemed seem VERB american_scientific_journal-7448 17 3 [ [ X american_scientific_journal-7448 17 4 1 1 X american_scientific_journal-7448 17 5 ] ] PUNCT american_scientific_journal-7448 17 6 was be AUX american_scientific_journal-7448 17 7 probably probably ADV american_scientific_journal-7448 17 8 not not PART american_scientific_journal-7448 17 9 interested interested ADJ american_scientific_journal-7448 17 10 in in ADP american_scientific_journal-7448 17 11 the the DET american_scientific_journal-7448 17 12 force force NOUN american_scientific_journal-7448 17 13 that that PRON american_scientific_journal-7448 17 14 was be AUX american_scientific_journal-7448 17 15 acting act VERB american_scientific_journal-7448 17 16 on on ADP american_scientific_journal-7448 17 17 the the DET american_scientific_journal-7448 17 18 cylinder cylinder NOUN american_scientific_journal-7448 17 19 and and CCONJ american_scientific_journal-7448 17 20 so so ADV american_scientific_journal-7448 17 21 did do AUX american_scientific_journal-7448 17 22 not not PART american_scientific_journal-7448 17 23 determine determine VERB american_scientific_journal-7448 17 24 it it PRON american_scientific_journal-7448 17 25 . . PUNCT american_scientific_journal-7448 18 1 a a DET american_scientific_journal-7448 18 2 mathematical mathematical ADJ american_scientific_journal-7448 18 3 problem problem NOUN american_scientific_journal-7448 18 4 thus thus ADV american_scientific_journal-7448 18 5 arose arise VERB american_scientific_journal-7448 18 6 , , PUNCT american_scientific_journal-7448 18 7 because because SCONJ american_scientific_journal-7448 18 8 the the DET american_scientific_journal-7448 18 9 flow flow NOUN american_scientific_journal-7448 18 10 tended tend VERB american_scientific_journal-7448 18 11 to to PART american_scientific_journal-7448 18 12 decrease decrease VERB american_scientific_journal-7448 18 13 so so ADV american_scientific_journal-7448 18 14 slowly slowly ADV american_scientific_journal-7448 18 15 that that SCONJ american_scientific_journal-7448 18 16 the the DET american_scientific_journal-7448 18 17 far far ADJ american_scientific_journal-7448 18 18 field field NOUN american_scientific_journal-7448 18 19 boundary boundary ADJ american_scientific_journal-7448 18 20 conditions condition NOUN american_scientific_journal-7448 18 21 could could AUX american_scientific_journal-7448 18 22 not not PART american_scientific_journal-7448 18 23 be be AUX american_scientific_journal-7448 18 24 satisfied satisfied ADJ american_scientific_journal-7448 18 25 . . PUNCT american_scientific_journal-7448 19 1 the the DET american_scientific_journal-7448 19 2 problem problem NOUN american_scientific_journal-7448 19 3 of of ADP american_scientific_journal-7448 19 4 viscous viscous ADJ american_scientific_journal-7448 19 5 flow flow NOUN american_scientific_journal-7448 19 6 past past ADP american_scientific_journal-7448 19 7 a a DET american_scientific_journal-7448 19 8 cylinder cylinder NOUN american_scientific_journal-7448 19 9 was be AUX american_scientific_journal-7448 19 10 thus thus ADV american_scientific_journal-7448 19 11 created create VERB american_scientific_journal-7448 19 12 and and CCONJ american_scientific_journal-7448 19 13 remained remain VERB american_scientific_journal-7448 19 14 unsolved unsolved ADJ american_scientific_journal-7448 19 15 until until ADP american_scientific_journal-7448 19 16 sixty sixty NUM american_scientific_journal-7448 19 17 years year NOUN american_scientific_journal-7448 19 18 later later ADV american_scientific_journal-7448 19 19 when when SCONJ american_scientific_journal-7448 19 20 lamb lamb PROPN american_scientific_journal-7448 19 21 [ [ X american_scientific_journal-7448 19 22 4 4 X american_scientific_journal-7448 19 23 ] ] PUNCT american_scientific_journal-7448 19 24 suggested suggest VERB american_scientific_journal-7448 19 25 a a DET american_scientific_journal-7448 19 26 solution solution NOUN american_scientific_journal-7448 19 27 , , PUNCT american_scientific_journal-7448 19 28 using use VERB american_scientific_journal-7448 19 29 the the DET american_scientific_journal-7448 19 30 equations equation NOUN american_scientific_journal-7448 19 31 of of ADP american_scientific_journal-7448 19 32 oseen oseen NOUN american_scientific_journal-7448 20 1 [ [ X american_scientific_journal-7448 20 2 5 5 NUM american_scientific_journal-7448 20 3 ] ] PUNCT american_scientific_journal-7448 20 4 to to PART american_scientific_journal-7448 20 5 derive derive VERB american_scientific_journal-7448 20 6 the the DET american_scientific_journal-7448 20 7 drag drag NOUN american_scientific_journal-7448 20 8 force force NOUN american_scientific_journal-7448 20 9 on on ADP american_scientific_journal-7448 20 10 a a DET american_scientific_journal-7448 20 11 cylinder cylinder NOUN american_scientific_journal-7448 20 12 for for ADP american_scientific_journal-7448 20 13 re re VERB american_scientific_journal-7448 20 14 < < X american_scientific_journal-7448 20 15 < < X american_scientific_journal-7448 20 16 1 1 NUM american_scientific_journal-7448 20 17 . . PUNCT american_scientific_journal-7448 21 1 he he PRON american_scientific_journal-7448 21 2 showed show VERB american_scientific_journal-7448 21 3 that that SCONJ american_scientific_journal-7448 21 4 : : PUNCT american_scientific_journal-7448 21 5 𝐹𝐷 𝐹𝐷 PROPN american_scientific_journal-7448 21 6 = = SYM american_scientific_journal-7448 21 7 2πœ‹πœ‡πΆ 2πœ‹πœ‡πΆ PROPN american_scientific_journal-7448 21 8 = = SYM american_scientific_journal-7448 21 9 4πœ‹πœ‡π‘ˆ 4πœ‹πœ‡π‘ˆ NOUN american_scientific_journal-7448 21 10 1 1 NUM american_scientific_journal-7448 21 11 2 2 NUM american_scientific_journal-7448 21 12 βˆ’π›Ύβˆ’log βˆ’π›Ύβˆ’log NOUN american_scientific_journal-7448 21 13 ( ( PUNCT american_scientific_journal-7448 21 14 1 1 NUM american_scientific_journal-7448 21 15 2 2 NUM american_scientific_journal-7448 21 16 π‘˜π‘Ž π‘˜π‘Ž NOUN american_scientific_journal-7448 21 17 ) ) PUNCT american_scientific_journal-7448 22 1 β‰ˆ β‰ˆ PROPN american_scientific_journal-7448 22 2 4πœ‹πœ‡π‘ˆ 4πœ‹πœ‡π‘ˆ PROPN american_scientific_journal-7448 22 3 ln(7.4 ln(7.4 ADJ american_scientific_journal-7448 22 4 𝑅𝑒⁄ 𝑅𝑒⁄ X american_scientific_journal-7448 22 5 ) ) PUNCT american_scientific_journal-7448 22 6 ( ( PUNCT american_scientific_journal-7448 22 7 4 4 NUM american_scientific_journal-7448 22 8 ) ) PUNCT american_scientific_journal-7448 22 9 and and CCONJ american_scientific_journal-7448 22 10 the the DET american_scientific_journal-7448 22 11 drag drag NOUN american_scientific_journal-7448 22 12 coefficient coefficient NOUN american_scientific_journal-7448 22 13 is be AUX american_scientific_journal-7448 22 14 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 22 15 = = SYM american_scientific_journal-7448 22 16 2𝐹𝐷 2𝐹𝐷 NUM american_scientific_journal-7448 22 17 2πœŒπ‘Žπ‘ˆπ‘ 2πœŒπ‘Žπ‘ˆπ‘ NUM american_scientific_journal-7448 22 18 2 2 NUM american_scientific_journal-7448 22 19 = = SYM american_scientific_journal-7448 22 20 8πœ‹ 8πœ‹ NUM american_scientific_journal-7448 23 1 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 23 2 ln(7.4 ln(7.4 ADJ american_scientific_journal-7448 23 3 𝑅𝑒⁄ 𝑅𝑒⁄ X american_scientific_journal-7448 23 4 ) ) PUNCT american_scientific_journal-7448 23 5 ( ( PUNCT american_scientific_journal-7448 23 6 5 5 X american_scientific_journal-7448 23 7 ) ) PUNCT american_scientific_journal-7448 23 8 the the DET american_scientific_journal-7448 23 9 problem problem NOUN american_scientific_journal-7448 23 10 on on ADP american_scientific_journal-7448 23 11 the the DET american_scientific_journal-7448 23 12 drag drag NOUN american_scientific_journal-7448 23 13 force force NOUN american_scientific_journal-7448 23 14 acting act VERB american_scientific_journal-7448 23 15 on on ADP american_scientific_journal-7448 23 16 a a DET american_scientific_journal-7448 23 17 circular circular ADJ american_scientific_journal-7448 23 18 cylinder cylinder NOUN american_scientific_journal-7448 23 19 has have AUX american_scientific_journal-7448 23 20 always always ADV american_scientific_journal-7448 23 21 been be AUX american_scientific_journal-7448 23 22 the the DET american_scientific_journal-7448 23 23 logarithm logarithm NOUN american_scientific_journal-7448 23 24 term term NOUN american_scientific_journal-7448 23 25 in in ADP american_scientific_journal-7448 23 26 the the DET american_scientific_journal-7448 23 27 velocity velocity NOUN american_scientific_journal-7448 23 28 distribution distribution NOUN american_scientific_journal-7448 23 29 of of ADP american_scientific_journal-7448 23 30 the the DET american_scientific_journal-7448 23 31 fluid fluid ADJ american_scientific_journal-7448 23 32 velocity velocity NOUN american_scientific_journal-7448 23 33 which which PRON american_scientific_journal-7448 23 34 diverges diverge VERB american_scientific_journal-7448 23 35 at at ADP american_scientific_journal-7448 23 36 large large ADJ american_scientific_journal-7448 23 37 distances distance NOUN american_scientific_journal-7448 23 38 from from ADP american_scientific_journal-7448 23 39 the the DET american_scientific_journal-7448 23 40 cylinder cylinder NOUN american_scientific_journal-7448 23 41 , , PUNCT american_scientific_journal-7448 23 42 thereby thereby ADV american_scientific_journal-7448 23 43 making make VERB american_scientific_journal-7448 23 44 the the DET american_scientific_journal-7448 23 45 velocity velocity NOUN american_scientific_journal-7448 23 46 of of ADP american_scientific_journal-7448 23 47 the the DET american_scientific_journal-7448 23 48 fluid fluid NOUN american_scientific_journal-7448 23 49 at at ADP american_scientific_journal-7448 23 50 large large ADJ american_scientific_journal-7448 23 51 distances distance NOUN american_scientific_journal-7448 23 52 not not PART american_scientific_journal-7448 23 53 finite finite VERB american_scientific_journal-7448 23 54 . . PUNCT american_scientific_journal-7448 23 55 to to PART american_scientific_journal-7448 23 56 solve solve VERB american_scientific_journal-7448 23 57 this this DET american_scientific_journal-7448 23 58 problem problem NOUN american_scientific_journal-7448 23 59 many many ADJ american_scientific_journal-7448 23 60 authors author NOUN american_scientific_journal-7448 23 61 have have AUX american_scientific_journal-7448 23 62 adopted adopt VERB american_scientific_journal-7448 23 63 the the DET american_scientific_journal-7448 23 64 asymptotic asymptotic ADJ american_scientific_journal-7448 23 65 method method NOUN american_scientific_journal-7448 23 66 . . PUNCT american_scientific_journal-7448 24 1 in in ADP american_scientific_journal-7448 24 2 this this DET american_scientific_journal-7448 24 3 method method NOUN american_scientific_journal-7448 24 4 , , PUNCT american_scientific_journal-7448 24 5 solutions solution NOUN american_scientific_journal-7448 24 6 are be AUX american_scientific_journal-7448 24 7 obtained obtain VERB american_scientific_journal-7448 24 8 at at ADP american_scientific_journal-7448 24 9 both both CCONJ american_scientific_journal-7448 24 10 the the DET american_scientific_journal-7448 24 11 vicinity vicinity NOUN american_scientific_journal-7448 24 12 of of ADP american_scientific_journal-7448 24 13 the the DET american_scientific_journal-7448 24 14 cylinder cylinder NOUN american_scientific_journal-7448 24 15 and and CCONJ american_scientific_journal-7448 24 16 far far ADV american_scientific_journal-7448 24 17 from from ADP american_scientific_journal-7448 24 18 the the DET american_scientific_journal-7448 24 19 cylinder cylinder NOUN american_scientific_journal-7448 24 20 . . PUNCT american_scientific_journal-7448 25 1 both both DET american_scientific_journal-7448 25 2 solutions solution NOUN american_scientific_journal-7448 25 3 are be AUX american_scientific_journal-7448 25 4 then then ADV american_scientific_journal-7448 25 5 matched match VERB american_scientific_journal-7448 25 6 to to PART american_scientific_journal-7448 25 7 give give VERB american_scientific_journal-7448 25 8 a a DET american_scientific_journal-7448 25 9 solution solution NOUN american_scientific_journal-7448 25 10 at at ADP american_scientific_journal-7448 25 11 any any DET american_scientific_journal-7448 25 12 point point NOUN american_scientific_journal-7448 25 13 in in ADP american_scientific_journal-7448 25 14 the the DET american_scientific_journal-7448 25 15 flow flow NOUN american_scientific_journal-7448 25 16 field field NOUN american_scientific_journal-7448 25 17 , , PUNCT american_scientific_journal-7448 25 18 the the DET american_scientific_journal-7448 25 19 matched match VERB american_scientific_journal-7448 25 20 solution solution NOUN american_scientific_journal-7448 25 21 of of ADP american_scientific_journal-7448 25 22 the the DET american_scientific_journal-7448 25 23 fluid fluid ADJ american_scientific_journal-7448 25 24 velocity velocity NOUN american_scientific_journal-7448 25 25 and and CCONJ american_scientific_journal-7448 25 26 stream stream NOUN american_scientific_journal-7448 25 27 - - PUNCT american_scientific_journal-7448 25 28 function function NOUN american_scientific_journal-7448 25 29 is be AUX american_scientific_journal-7448 25 30 then then ADV american_scientific_journal-7448 25 31 used use VERB american_scientific_journal-7448 25 32 to to PART american_scientific_journal-7448 25 33 compute compute VERB american_scientific_journal-7448 25 34 for for ADP american_scientific_journal-7448 25 35 the the DET american_scientific_journal-7448 25 36 drag drag NOUN american_scientific_journal-7448 25 37 force force NOUN american_scientific_journal-7448 25 38 . . PUNCT american_scientific_journal-7448 26 1 while while SCONJ american_scientific_journal-7448 26 2 [ [ X american_scientific_journal-7448 26 3 4 4 NUM american_scientific_journal-7448 26 4 ] ] PUNCT american_scientific_journal-7448 26 5 analysis analysis NOUN american_scientific_journal-7448 26 6 could could AUX american_scientific_journal-7448 26 7 not not PART american_scientific_journal-7448 26 8 be be AUX american_scientific_journal-7448 26 9 described describe VERB american_scientific_journal-7448 26 10 as as ADP american_scientific_journal-7448 26 11 an an DET american_scientific_journal-7448 26 12 asymptotic asymptotic ADJ american_scientific_journal-7448 26 13 analysis analysis NOUN american_scientific_journal-7448 26 14 , , PUNCT american_scientific_journal-7448 26 15 it it PRON american_scientific_journal-7448 26 16 does do AUX american_scientific_journal-7448 26 17 involve involve VERB american_scientific_journal-7448 26 18 the the DET american_scientific_journal-7448 26 19 idea idea NOUN american_scientific_journal-7448 26 20 of of ADP american_scientific_journal-7448 26 21 smallness smallness NOUN american_scientific_journal-7448 26 22 and and CCONJ american_scientific_journal-7448 26 23 expansion expansion NOUN american_scientific_journal-7448 26 24 . . PUNCT american_scientific_journal-7448 27 1 mutlu mutlu PROPN american_scientific_journal-7448 27 2 sumer sumer PROPN american_scientific_journal-7448 27 3 and and CCONJ american_scientific_journal-7448 27 4 jorgen jorgen PROPN american_scientific_journal-7448 27 5 fredsoe fredsoe PROPN american_scientific_journal-7448 28 1 [ [ X american_scientific_journal-7448 28 2 6 6 NUM american_scientific_journal-7448 28 3 ] ] PUNCT american_scientific_journal-7448 28 4 calculated calculate VERB american_scientific_journal-7448 28 5 analytically analytically ADV american_scientific_journal-7448 28 6 , , PUNCT american_scientific_journal-7448 28 7 the the DET american_scientific_journal-7448 28 8 drag drag NOUN american_scientific_journal-7448 28 9 force force NOUN american_scientific_journal-7448 28 10 by by ADP american_scientific_journal-7448 28 11 a a DET american_scientific_journal-7448 28 12 fluid fluid NOUN american_scientific_journal-7448 28 13 , , PUNCT american_scientific_journal-7448 28 14 flowing flow VERB american_scientific_journal-7448 28 15 past past ADP american_scientific_journal-7448 28 16 a a DET american_scientific_journal-7448 28 17 stationary stationary ADJ american_scientific_journal-7448 28 18 circular circular ADJ american_scientific_journal-7448 28 19 cylinder cylinder NOUN american_scientific_journal-7448 28 20 . . PUNCT american_scientific_journal-7448 29 1 eames eame NOUN american_scientific_journal-7448 29 2 and and CCONJ american_scientific_journal-7448 29 3 klettner klettner NOUN american_scientific_journal-7448 29 4 [ [ X american_scientific_journal-7448 29 5 7 7 NUM american_scientific_journal-7448 29 6 ] ] PUNCT american_scientific_journal-7448 29 7 found find VERB american_scientific_journal-7448 29 8 the the DET american_scientific_journal-7448 29 9 drag drag NOUN american_scientific_journal-7448 29 10 force force NOUN american_scientific_journal-7448 29 11 on on ADP american_scientific_journal-7448 29 12 a a DET american_scientific_journal-7448 29 13 circular circular ADJ american_scientific_journal-7448 29 14 cylinder cylinder NOUN american_scientific_journal-7448 29 15 by by ADP american_scientific_journal-7448 29 16 solving solve VERB american_scientific_journal-7448 29 17 the the DET american_scientific_journal-7448 29 18 american american ADJ american_scientific_journal-7448 29 19 academic academic ADJ american_scientific_journal-7448 29 20 scientific scientific ADJ american_scientific_journal-7448 29 21 research research NOUN american_scientific_journal-7448 29 22 journal journal NOUN american_scientific_journal-7448 29 23 for for ADP american_scientific_journal-7448 29 24 engineering engineering NOUN american_scientific_journal-7448 29 25 , , PUNCT american_scientific_journal-7448 29 26 technology technology NOUN american_scientific_journal-7448 29 27 , , PUNCT american_scientific_journal-7448 29 28 and and CCONJ american_scientific_journal-7448 29 29 sciences science NOUN american_scientific_journal-7448 29 30 ( ( PUNCT american_scientific_journal-7448 29 31 asrjets asrjet NOUN american_scientific_journal-7448 29 32 ) ) PUNCT american_scientific_journal-7448 29 33 ( ( PUNCT american_scientific_journal-7448 29 34 2022 2022 NUM american_scientific_journal-7448 29 35 ) ) PUNCT american_scientific_journal-7448 29 36 volume volume NOUN american_scientific_journal-7448 29 37 86 86 NUM american_scientific_journal-7448 29 38 , , PUNCT american_scientific_journal-7448 29 39 no no DET american_scientific_journal-7448 29 40 1 1 NUM american_scientific_journal-7448 29 41 , , PUNCT american_scientific_journal-7448 29 42 pp143 pp143 PROPN american_scientific_journal-7448 29 43 - - PUNCT american_scientific_journal-7448 29 44 156 156 NUM american_scientific_journal-7448 29 45 145 145 NUM american_scientific_journal-7448 29 46 navier navier NOUN american_scientific_journal-7448 29 47 - - PUNCT american_scientific_journal-7448 29 48 stokes stoke NOUN american_scientific_journal-7448 29 49 equations equation NOUN american_scientific_journal-7448 29 50 directly directly ADV american_scientific_journal-7448 29 51 as as SCONJ american_scientific_journal-7448 29 52 follows follow VERB american_scientific_journal-7448 29 53 : : PUNCT american_scientific_journal-7448 29 54 non non ADJ american_scientific_journal-7448 29 55 - - NOUN american_scientific_journal-7448 29 56 dimensionalization dimensionalization NOUN american_scientific_journal-7448 29 57 of of ADP american_scientific_journal-7448 29 58 the the DET american_scientific_journal-7448 29 59 equation equation NOUN american_scientific_journal-7448 29 60 of of ADP american_scientific_journal-7448 29 61 motion motion NOUN american_scientific_journal-7448 29 62 gives give VERB american_scientific_journal-7448 29 63 βˆ‡Μƒ. βˆ‡Μƒ. PROPN american_scientific_journal-7448 29 64 οΏ½ οΏ½ PROPN american_scientific_journal-7448 29 65 Μƒ Μƒ PROPN american_scientific_journal-7448 29 66 οΏ½ οΏ½ NOUN american_scientific_journal-7448 29 67 = = SYM american_scientific_journal-7448 29 68 0 0 PROPN american_scientific_journal-7448 29 69 . . PUNCT american_scientific_journal-7448 30 1 ( ( PUNCT american_scientific_journal-7448 30 2 6 6 NUM american_scientific_journal-7448 30 3 ) ) PUNCT american_scientific_journal-7448 30 4 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 30 5 2 2 NUM american_scientific_journal-7448 30 6 ( ( PUNCT american_scientific_journal-7448 30 7 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 8 Μƒ Μƒ PROPN american_scientific_journal-7448 30 9 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 10 . . PUNCT american_scientific_journal-7448 30 11 βˆ‡Μƒ) βˆ‡Μƒ) NUM american_scientific_journal-7448 30 12 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 13 Μƒ Μƒ PROPN american_scientific_journal-7448 30 14 οΏ½ οΏ½ NOUN american_scientific_journal-7448 30 15 = = SYM american_scientific_journal-7448 30 16 βˆ’βˆ‡Μƒπ‘ βˆ’βˆ‡Μƒπ‘ NUM american_scientific_journal-7448 30 17 + + CCONJ american_scientific_journal-7448 30 18 βˆ‡Μƒ2 βˆ‡Μƒ2 PROPN american_scientific_journal-7448 30 19 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 20 Μƒ Μƒ PROPN american_scientific_journal-7448 30 21 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 22 ( ( PUNCT american_scientific_journal-7448 30 23 7 7 NUM american_scientific_journal-7448 30 24 ) ) PUNCT american_scientific_journal-7448 30 25 the the DET american_scientific_journal-7448 30 26 x x NOUN american_scientific_journal-7448 30 27 - - NOUN american_scientific_journal-7448 30 28 component component NOUN american_scientific_journal-7448 30 29 of of ADP american_scientific_journal-7448 30 30 the the DET american_scientific_journal-7448 30 31 inertial inertial ADJ american_scientific_journal-7448 30 32 term term NOUN american_scientific_journal-7448 30 33 of of ADP american_scientific_journal-7448 30 34 ( ( PUNCT american_scientific_journal-7448 30 35 7 7 NUM american_scientific_journal-7448 30 36 ) ) PUNCT american_scientific_journal-7448 30 37 is be AUX american_scientific_journal-7448 30 38 ( ( PUNCT american_scientific_journal-7448 30 39 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 40 Μƒ Μƒ PROPN american_scientific_journal-7448 30 41 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 42 . . PUNCT american_scientific_journal-7448 30 43 βˆ‡Μƒ) βˆ‡Μƒ) NUM american_scientific_journal-7448 30 44 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 45 Μƒ Μƒ PROPN american_scientific_journal-7448 30 46 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 47 β‰ˆ β‰ˆ PROPN american_scientific_journal-7448 30 48 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-7448 30 49 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-7448 30 50 Μƒ Μƒ PROPN american_scientific_journal-7448 30 51 ( ( PUNCT american_scientific_journal-7448 30 52 8) 8) NUM american_scientific_journal-7448 30 53 substituting substituting NOUN american_scientific_journal-7448 30 54 ( ( PUNCT american_scientific_journal-7448 30 55 8) 8) NUM american_scientific_journal-7448 30 56 in in ADP american_scientific_journal-7448 30 57 ( ( PUNCT american_scientific_journal-7448 30 58 7 7 NUM american_scientific_journal-7448 30 59 ) ) PUNCT american_scientific_journal-7448 30 60 gives give VERB american_scientific_journal-7448 30 61 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-7448 30 62 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-7448 30 63 Μƒ Μƒ NOUN american_scientific_journal-7448 30 64 = = SYM american_scientific_journal-7448 30 65 βˆ’ βˆ’ PROPN american_scientific_journal-7448 30 66 2 2 NUM american_scientific_journal-7448 30 67 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 30 68 βˆ‡Μƒπ‘ βˆ‡Μƒπ‘ NOUN american_scientific_journal-7448 30 69 + + CCONJ american_scientific_journal-7448 30 70 2 2 NUM american_scientific_journal-7448 30 71 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 30 72 βˆ‡Μƒ2 βˆ‡Μƒ2 NOUN american_scientific_journal-7448 30 73 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 74 Μƒ Μƒ PROPN american_scientific_journal-7448 30 75 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 76 ( ( PUNCT american_scientific_journal-7448 30 77 9 9 X american_scientific_journal-7448 30 78 ) ) PUNCT american_scientific_journal-7448 30 79 taking take VERB american_scientific_journal-7448 30 80 the the DET american_scientific_journal-7448 30 81 curl curl NOUN american_scientific_journal-7448 30 82 of of ADP american_scientific_journal-7448 30 83 each each DET american_scientific_journal-7448 30 84 term term NOUN american_scientific_journal-7448 30 85 of of ADP american_scientific_journal-7448 30 86 ( ( PUNCT american_scientific_journal-7448 30 87 9 9 NUM american_scientific_journal-7448 30 88 ) ) PUNCT american_scientific_journal-7448 30 89 gives gives AUX american_scientific_journal-7448 30 90 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 30 91 2 2 NUM american_scientific_journal-7448 30 92 πœ• πœ• PROPN american_scientific_journal-7448 30 93 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 94 Μƒ Μƒ PROPN american_scientific_journal-7448 30 95 οΏ½ οΏ½ PROPN american_scientific_journal-7448 30 96 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-7448 30 97 Μƒ Μƒ PROPN american_scientific_journal-7448 31 1 = = SYM american_scientific_journal-7448 31 2 βˆ‡Μƒ2 βˆ‡Μƒ2 PROPN american_scientific_journal-7448 31 3 οΏ½ οΏ½ PROPN american_scientific_journal-7448 31 4 Μƒ Μƒ PROPN american_scientific_journal-7448 31 5 οΏ½ οΏ½ PROPN american_scientific_journal-7448 31 6 , , PUNCT american_scientific_journal-7448 31 7 ( ( PUNCT american_scientific_journal-7448 31 8 10 10 NUM american_scientific_journal-7448 31 9 ) ) PUNCT american_scientific_journal-7448 31 10 where where SCONJ american_scientific_journal-7448 31 11 οΏ½ οΏ½ PROPN american_scientific_journal-7448 31 12 Μƒ Μƒ PROPN american_scientific_journal-7448 31 13 οΏ½ οΏ½ PROPN american_scientific_journal-7448 31 14 is be AUX american_scientific_journal-7448 31 15 the the DET american_scientific_journal-7448 31 16 vorticity vorticity NOUN american_scientific_journal-7448 31 17 of of ADP american_scientific_journal-7448 31 18 flow flow NOUN american_scientific_journal-7448 31 19 . . PUNCT american_scientific_journal-7448 32 1 solving solve VERB american_scientific_journal-7448 32 2 ( ( PUNCT american_scientific_journal-7448 32 3 10 10 NUM american_scientific_journal-7448 32 4 ) ) PUNCT american_scientific_journal-7448 32 5 using use VERB american_scientific_journal-7448 32 6 separation separation NOUN american_scientific_journal-7448 32 7 of of ADP american_scientific_journal-7448 32 8 variables variable NOUN american_scientific_journal-7448 32 9 gives give VERB american_scientific_journal-7448 32 10 οΏ½ οΏ½ PROPN american_scientific_journal-7448 32 11 Μƒ Μƒ PROPN american_scientific_journal-7448 32 12 οΏ½ οΏ½ NOUN american_scientific_journal-7448 32 13 = = PUNCT american_scientific_journal-7448 32 14 𝑃𝑒(𝑅𝑒 𝑃𝑒(𝑅𝑒 PROPN american_scientific_journal-7448 32 15 4⁄ 4⁄ NUM american_scientific_journal-7448 32 16 ) ) PUNCT american_scientific_journal-7448 33 1 π‘₯ π‘₯ PUNCT american_scientific_journal-7448 33 2 Μƒ Μƒ NOUN american_scientific_journal-7448 33 3 = = PUNCT american_scientific_journal-7448 33 4 𝑃𝑒(𝑅𝑒 𝑃𝑒(𝑅𝑒 X american_scientific_journal-7448 33 5 4⁄ 4⁄ NUM american_scientific_journal-7448 33 6 ) ) PUNCT american_scientific_journal-7448 33 7 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 33 8 Μƒ Μƒ ADP american_scientific_journal-7448 33 9 cos cos ADP american_scientific_journal-7448 33 10 πœƒ πœƒ X american_scientific_journal-7448 33 11 ( ( PUNCT american_scientific_journal-7448 33 12 11 11 NUM american_scientific_journal-7448 33 13 ) ) PUNCT american_scientific_journal-7448 33 14 substituting substitute VERB american_scientific_journal-7448 33 15 ( ( PUNCT american_scientific_journal-7448 33 16 11 11 NUM american_scientific_journal-7448 33 17 ) ) PUNCT american_scientific_journal-7448 33 18 in in ADP american_scientific_journal-7448 33 19 ( ( PUNCT american_scientific_journal-7448 33 20 10 10 NUM american_scientific_journal-7448 33 21 ) ) PUNCT american_scientific_journal-7448 33 22 and and CCONJ american_scientific_journal-7448 33 23 evaluating evaluating NOUN american_scientific_journal-7448 33 24 gives give VERB american_scientific_journal-7448 33 25 ( ( PUNCT american_scientific_journal-7448 33 26 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 33 27 4⁄ 4⁄ NUM american_scientific_journal-7448 33 28 ) ) PUNCT american_scientific_journal-7448 33 29 2𝑃 2𝑃 NOUN american_scientific_journal-7448 33 30 = = SYM american_scientific_journal-7448 33 31 βˆ‡Μƒ2𝑃 βˆ‡Μƒ2𝑃 NOUN american_scientific_journal-7448 33 32 ( ( PUNCT american_scientific_journal-7448 33 33 12 12 NUM american_scientific_journal-7448 33 34 ) ) PUNCT american_scientific_journal-7448 33 35 at at ADP american_scientific_journal-7448 33 36 low low ADJ american_scientific_journal-7448 33 37 reynolds reynolds PROPN american_scientific_journal-7448 33 38 numbers number NOUN american_scientific_journal-7448 33 39 , , PUNCT american_scientific_journal-7448 33 40 the the DET american_scientific_journal-7448 33 41 flow flow NOUN american_scientific_journal-7448 33 42 is be AUX american_scientific_journal-7448 33 43 to to ADP american_scientific_journal-7448 33 44 leading lead VERB american_scientific_journal-7448 33 45 order order NOUN american_scientific_journal-7448 33 46 symmetric symmetric NOUN american_scientific_journal-7448 33 47 and and CCONJ american_scientific_journal-7448 33 48 a a DET american_scientific_journal-7448 33 49 solution solution NOUN american_scientific_journal-7448 33 50 to to ADP american_scientific_journal-7448 33 51 ( ( PUNCT american_scientific_journal-7448 33 52 12 12 NUM american_scientific_journal-7448 33 53 ) ) PUNCT american_scientific_journal-7448 33 54 is be AUX american_scientific_journal-7448 33 55 of of ADP american_scientific_journal-7448 33 56 the the DET american_scientific_journal-7448 33 57 form form NOUN american_scientific_journal-7448 33 58 , , PUNCT american_scientific_journal-7448 33 59 𝑃 𝑃 NOUN american_scientific_journal-7448 33 60 = = PUNCT american_scientific_journal-7448 33 61 𝑃1( 𝑃1( NOUN american_scientific_journal-7448 33 62 οΏ½ οΏ½ PROPN american_scientific_journal-7448 33 63 Μƒ Μƒ PROPN american_scientific_journal-7448 33 64 οΏ½ οΏ½ PROPN american_scientific_journal-7448 33 65 ) ) PUNCT american_scientific_journal-7448 33 66 sin sin NOUN american_scientific_journal-7448 33 67 πœƒ πœƒ X american_scientific_journal-7448 33 68 ( ( PUNCT american_scientific_journal-7448 33 69 13 13 NUM american_scientific_journal-7448 33 70 ) ) PUNCT american_scientific_journal-7448 33 71 substituting substitute VERB american_scientific_journal-7448 33 72 ( ( PUNCT american_scientific_journal-7448 33 73 13 13 NUM american_scientific_journal-7448 33 74 ) ) PUNCT american_scientific_journal-7448 33 75 in in ADP american_scientific_journal-7448 33 76 ( ( PUNCT american_scientific_journal-7448 33 77 12 12 NUM american_scientific_journal-7448 33 78 ) ) PUNCT american_scientific_journal-7448 33 79 gives give VERB american_scientific_journal-7448 33 80 an an DET american_scientific_journal-7448 33 81 ode ode ADJ american_scientific_journal-7448 33 82 𝑃1 𝑃1 NOUN american_scientific_journal-7448 33 83 β€²β€² β€²β€² PROPN american_scientific_journal-7448 33 84 + + CCONJ american_scientific_journal-7448 33 85 1 1 NUM american_scientific_journal-7448 33 86 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 33 87 Μƒ Μƒ NOUN american_scientific_journal-7448 33 88 𝑃1 𝑃1 NOUN american_scientific_journal-7448 33 89 β€² β€² NUM american_scientific_journal-7448 33 90 βˆ’ βˆ’ PROPN american_scientific_journal-7448 33 91 ( ( PUNCT american_scientific_journal-7448 33 92 ( ( PUNCT american_scientific_journal-7448 33 93 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 33 94 4 4 NUM american_scientific_journal-7448 33 95 ) ) SYM american_scientific_journal-7448 33 96 2 2 NUM american_scientific_journal-7448 33 97 + + SYM american_scientific_journal-7448 33 98 1 1 NUM american_scientific_journal-7448 33 99 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 33 100 Μƒ2 Μƒ2 NUM american_scientific_journal-7448 33 101 ) ) PUNCT american_scientific_journal-7448 33 102 𝑃1 𝑃1 NOUN american_scientific_journal-7448 33 103 = = SYM american_scientific_journal-7448 33 104 0 0 NUM american_scientific_journal-7448 34 1 ( ( PUNCT american_scientific_journal-7448 34 2 14 14 NUM american_scientific_journal-7448 34 3 ) ) PUNCT american_scientific_journal-7448 34 4 american american ADJ american_scientific_journal-7448 34 5 academic academic ADJ american_scientific_journal-7448 34 6 scientific scientific ADJ american_scientific_journal-7448 34 7 research research NOUN american_scientific_journal-7448 34 8 journal journal NOUN american_scientific_journal-7448 34 9 for for ADP american_scientific_journal-7448 34 10 engineering engineering NOUN american_scientific_journal-7448 34 11 , , PUNCT american_scientific_journal-7448 34 12 technology technology NOUN american_scientific_journal-7448 34 13 , , PUNCT american_scientific_journal-7448 34 14 and and CCONJ american_scientific_journal-7448 34 15 sciences science NOUN american_scientific_journal-7448 34 16 ( ( PUNCT american_scientific_journal-7448 34 17 asrjets asrjet NOUN american_scientific_journal-7448 34 18 ) ) PUNCT american_scientific_journal-7448 34 19 ( ( PUNCT american_scientific_journal-7448 34 20 2022 2022 NUM american_scientific_journal-7448 34 21 ) ) PUNCT american_scientific_journal-7448 34 22 volume volume NOUN american_scientific_journal-7448 34 23 86 86 NUM american_scientific_journal-7448 34 24 , , PUNCT american_scientific_journal-7448 34 25 no no DET american_scientific_journal-7448 34 26 1 1 NUM american_scientific_journal-7448 34 27 , , PUNCT american_scientific_journal-7448 34 28 pp143 pp143 PROPN american_scientific_journal-7448 34 29 - - SYM american_scientific_journal-7448 34 30 156 156 NUM american_scientific_journal-7448 34 31 146 146 NUM american_scientific_journal-7448 34 32 the the DET american_scientific_journal-7448 34 33 solution solution NOUN american_scientific_journal-7448 34 34 to to ADP american_scientific_journal-7448 34 35 ( ( PUNCT american_scientific_journal-7448 34 36 14 14 NUM american_scientific_journal-7448 34 37 ) ) PUNCT american_scientific_journal-7448 34 38 that that PRON american_scientific_journal-7448 34 39 satisfies satisfy VERB american_scientific_journal-7448 34 40 𝑃1 𝑃1 NOUN american_scientific_journal-7448 34 41 β†’ β†’ SYM american_scientific_journal-7448 34 42 0 0 NUM american_scientific_journal-7448 34 43 as as ADP american_scientific_journal-7448 34 44 οΏ½ οΏ½ PROPN american_scientific_journal-7448 34 45 Μƒ Μƒ PROPN american_scientific_journal-7448 34 46 οΏ½ οΏ½ PROPN american_scientific_journal-7448 34 47 β†’ β†’ SYM american_scientific_journal-7448 34 48 ∞ ∞ PROPN american_scientific_journal-7448 34 49 is be AUX american_scientific_journal-7448 34 50 𝑃1 𝑃1 NOUN american_scientific_journal-7448 34 51 = = SYM american_scientific_journal-7448 34 52 𝐢1𝐾1 𝐢1𝐾1 PROPN american_scientific_journal-7448 34 53 ( ( PUNCT american_scientific_journal-7448 34 54 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 34 55 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 34 56 Μƒ Μƒ NOUN american_scientific_journal-7448 34 57 4 4 NUM american_scientific_journal-7448 34 58 ) ) PUNCT american_scientific_journal-7448 34 59 , , PUNCT american_scientific_journal-7448 34 60 ( ( PUNCT american_scientific_journal-7448 34 61 15 15 NUM american_scientific_journal-7448 34 62 ) ) PUNCT american_scientific_journal-7448 34 63 where where SCONJ american_scientific_journal-7448 34 64 𝐾1 𝐾1 PROPN american_scientific_journal-7448 34 65 is be AUX american_scientific_journal-7448 34 66 the the DET american_scientific_journal-7448 34 67 modified modify VERB american_scientific_journal-7448 34 68 bessel bessel NOUN american_scientific_journal-7448 34 69 function function NOUN american_scientific_journal-7448 34 70 of of ADP american_scientific_journal-7448 34 71 the the DET american_scientific_journal-7448 34 72 second second ADJ american_scientific_journal-7448 34 73 kind kind NOUN american_scientific_journal-7448 34 74 . . PUNCT american_scientific_journal-7448 35 1 the the DET american_scientific_journal-7448 35 2 stream stream NOUN american_scientific_journal-7448 35 3 function function NOUN american_scientific_journal-7448 35 4 is be AUX american_scientific_journal-7448 35 5 the the DET american_scientific_journal-7448 35 6 sum sum NOUN american_scientific_journal-7448 35 7 of of ADP american_scientific_journal-7448 35 8 the the DET american_scientific_journal-7448 35 9 known know VERB american_scientific_journal-7448 35 10 free free ADJ american_scientific_journal-7448 35 11 stream stream NOUN american_scientific_journal-7448 35 12 component component NOUN american_scientific_journal-7448 35 13 ( ( PUNCT american_scientific_journal-7448 35 14 first first ADJ american_scientific_journal-7448 35 15 term term NOUN american_scientific_journal-7448 35 16 ) ) PUNCT american_scientific_journal-7448 35 17 and and CCONJ american_scientific_journal-7448 35 18 a a DET american_scientific_journal-7448 35 19 component component NOUN american_scientific_journal-7448 35 20 to to PART american_scientific_journal-7448 35 21 be be AUX american_scientific_journal-7448 35 22 determined determine VERB american_scientific_journal-7448 35 23 ( ( PUNCT american_scientific_journal-7448 35 24 second second ADJ american_scientific_journal-7448 35 25 term term NOUN american_scientific_journal-7448 35 26 ) ) PUNCT american_scientific_journal-7448 35 27 . . PUNCT american_scientific_journal-7448 36 1 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 2 Μƒ Μƒ PROPN american_scientific_journal-7448 36 3 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 4 = = SYM american_scientific_journal-7448 36 5 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 6 Μƒ Μƒ PROPN american_scientific_journal-7448 36 7 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 8 sin sin NOUN american_scientific_journal-7448 36 9 πœƒ πœƒ NOUN american_scientific_journal-7448 36 10 + + CCONJ american_scientific_journal-7448 36 11 𝑓1 𝑓1 ADJ american_scientific_journal-7448 36 12 ( ( PUNCT american_scientific_journal-7448 36 13 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 14 Μƒ Μƒ PROPN american_scientific_journal-7448 36 15 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 16 ) ) PUNCT american_scientific_journal-7448 36 17 sin sin NOUN american_scientific_journal-7448 36 18 πœƒ πœƒ X american_scientific_journal-7448 36 19 ( ( PUNCT american_scientific_journal-7448 36 20 16 16 NUM american_scientific_journal-7448 36 21 ) ) PUNCT american_scientific_journal-7448 36 22 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 23 Μƒ Μƒ PROPN american_scientific_journal-7448 36 24 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 25 = = SYM american_scientific_journal-7448 36 26 βˆ’ βˆ’ PROPN american_scientific_journal-7448 36 27 πœ•2 πœ•2 NOUN american_scientific_journal-7448 36 28 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 29 Μƒ Μƒ PROPN american_scientific_journal-7448 36 30 οΏ½ οΏ½ PROPN american_scientific_journal-7448 36 31 πœ•π‘Ÿ πœ•π‘Ÿ CCONJ american_scientific_journal-7448 36 32 Μƒ2 Μƒ2 PUNCT american_scientific_journal-7448 37 1 βˆ’ βˆ’ PROPN american_scientific_journal-7448 37 2 1 1 NUM american_scientific_journal-7448 37 3 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 37 4 Μƒ Μƒ PROPN american_scientific_journal-7448 37 5 πœ• πœ• PROPN american_scientific_journal-7448 37 6 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 7 Μƒ Μƒ PROPN american_scientific_journal-7448 37 8 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 9 πœ•π‘Ÿ πœ•π‘Ÿ ADP american_scientific_journal-7448 37 10 Μƒ Μƒ PROPN american_scientific_journal-7448 37 11 βˆ’ βˆ’ NUM american_scientific_journal-7448 37 12 1 1 NUM american_scientific_journal-7448 37 13 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 37 14 Μƒ2 Μƒ2 SYM american_scientific_journal-7448 37 15 πœ•2 πœ•2 NOUN american_scientific_journal-7448 37 16 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 17 Μƒ Μƒ PROPN american_scientific_journal-7448 37 18 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 19 πœ•πœƒ2 πœ•πœƒ2 X american_scientific_journal-7448 37 20 ( ( PUNCT american_scientific_journal-7448 37 21 17 17 NUM american_scientific_journal-7448 37 22 ) ) PUNCT american_scientific_journal-7448 37 23 by by ADP american_scientific_journal-7448 37 24 substituting substitute VERB american_scientific_journal-7448 37 25 ( ( PUNCT american_scientific_journal-7448 37 26 16 16 NUM american_scientific_journal-7448 37 27 ) ) PUNCT american_scientific_journal-7448 37 28 in in ADP american_scientific_journal-7448 37 29 ( ( PUNCT american_scientific_journal-7448 37 30 17 17 NUM american_scientific_journal-7448 37 31 ) ) PUNCT american_scientific_journal-7448 37 32 and and CCONJ american_scientific_journal-7448 37 33 considering consider VERB american_scientific_journal-7448 37 34 the the DET american_scientific_journal-7448 37 35 lowest low ADJ american_scientific_journal-7448 37 36 order order NOUN american_scientific_journal-7448 37 37 symmetric symmetric ADJ american_scientific_journal-7448 37 38 solution solution NOUN american_scientific_journal-7448 37 39 ( ( PUNCT american_scientific_journal-7448 37 40 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 41 Μƒ Μƒ PROPN american_scientific_journal-7448 37 42 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 43 β‰… β‰… PROPN american_scientific_journal-7448 37 44 𝑃1( 𝑃1( PROPN american_scientific_journal-7448 37 45 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 46 Μƒ Μƒ PROPN american_scientific_journal-7448 37 47 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 48 ) ) PUNCT american_scientific_journal-7448 37 49 sin sin NOUN american_scientific_journal-7448 37 50 πœƒ πœƒ NOUN american_scientific_journal-7448 37 51 ) ) PUNCT american_scientific_journal-7448 37 52 another another DET american_scientific_journal-7448 37 53 ode ode PROPN american_scientific_journal-7448 37 54 emerges emerge VERB american_scientific_journal-7448 37 55 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 56 Μƒ Μƒ NOUN american_scientific_journal-7448 37 57 οΏ½ οΏ½ NOUN american_scientific_journal-7448 37 58 2𝑓1 2𝑓1 NUM american_scientific_journal-7448 37 59 β€²β€² β€²β€² PROPN american_scientific_journal-7448 37 60 + + CCONJ american_scientific_journal-7448 37 61 οΏ½ οΏ½ PROPN american_scientific_journal-7448 37 62 Μƒ Μƒ PROPN american_scientific_journal-7448 37 63 οΏ½ οΏ½ NOUN american_scientific_journal-7448 37 64 𝑓1 𝑓1 ADJ american_scientific_journal-7448 37 65 β€² β€² NUM american_scientific_journal-7448 38 1 βˆ’ βˆ’ PROPN american_scientific_journal-7448 38 2 𝑓1 𝑓1 PROPN american_scientific_journal-7448 38 3 = = SYM american_scientific_journal-7448 38 4 βˆ’πΆ1𝐾1 βˆ’πΆ1𝐾1 PROPN american_scientific_journal-7448 38 5 οΏ½ οΏ½ PROPN american_scientific_journal-7448 38 6 Μƒ Μƒ PROPN american_scientific_journal-7448 38 7 οΏ½ οΏ½ NOUN american_scientific_journal-7448 38 8 2 2 NUM american_scientific_journal-7448 38 9 ( ( PUNCT american_scientific_journal-7448 38 10 18 18 NUM american_scientific_journal-7448 38 11 ) ) PUNCT american_scientific_journal-7448 38 12 solving solving NOUN american_scientific_journal-7448 38 13 ( ( PUNCT american_scientific_journal-7448 38 14 18 18 NUM american_scientific_journal-7448 38 15 ) ) PUNCT american_scientific_journal-7448 38 16 subject subject NOUN american_scientific_journal-7448 38 17 to to ADP american_scientific_journal-7448 38 18 the the DET american_scientific_journal-7448 38 19 boundary boundary ADJ american_scientific_journal-7448 38 20 conditions condition NOUN american_scientific_journal-7448 38 21 𝑓1(1 𝑓1(1 ADJ american_scientific_journal-7448 38 22 ) ) PUNCT american_scientific_journal-7448 38 23 = = SYM american_scientific_journal-7448 38 24 βˆ’1 βˆ’1 NOUN american_scientific_journal-7448 38 25 and and CCONJ american_scientific_journal-7448 38 26 𝑓1 𝑓1 PROPN american_scientific_journal-7448 38 27 β€²(1 β€²(1 PROPN american_scientific_journal-7448 38 28 ) ) PUNCT american_scientific_journal-7448 39 1 = = SYM american_scientific_journal-7448 39 2 βˆ’1 βˆ’1 NOUN american_scientific_journal-7448 39 3 gives give VERB american_scientific_journal-7448 39 4 𝑓1( 𝑓1( PROPN american_scientific_journal-7448 39 5 οΏ½ οΏ½ PROPN american_scientific_journal-7448 39 6 Μƒ Μƒ PROPN american_scientific_journal-7448 39 7 οΏ½ οΏ½ PROPN american_scientific_journal-7448 39 8 ) ) PUNCT american_scientific_journal-7448 39 9 = = PUNCT american_scientific_journal-7448 40 1 βˆ’ βˆ’ NUM american_scientific_journal-7448 40 2 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 3 Μƒ Μƒ NOUN american_scientific_journal-7448 40 4 οΏ½ οΏ½ NOUN american_scientific_journal-7448 40 5 βˆ’ βˆ’ NUM american_scientific_journal-7448 40 6 𝐢1 𝐢1 PROPN american_scientific_journal-7448 40 7 2 2 NUM american_scientific_journal-7448 40 8 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 9 Μƒ Μƒ PROPN american_scientific_journal-7448 40 10 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 11 ∫ ∫ PROPN american_scientific_journal-7448 40 12 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 13 Μƒ Μƒ PROPN american_scientific_journal-7448 40 14 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 15 ∞ ∞ PROPN american_scientific_journal-7448 40 16 1 1 NUM american_scientific_journal-7448 40 17 𝐾1𝑑 𝐾1𝑑 PROPN american_scientific_journal-7448 40 18 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 19 Μƒ Μƒ PROPN american_scientific_journal-7448 40 20 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 21 + + CCONJ american_scientific_journal-7448 40 22 𝐢1 𝐢1 PROPN american_scientific_journal-7448 40 23 2 2 NUM american_scientific_journal-7448 40 24 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 25 Μƒ Μƒ PROPN american_scientific_journal-7448 40 26 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 27 ∫ ∫ PROPN american_scientific_journal-7448 40 28 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 29 Μƒ Μƒ PROPN american_scientific_journal-7448 40 30 οΏ½ οΏ½ NOUN american_scientific_journal-7448 40 31 2∞ 2∞ NUM american_scientific_journal-7448 40 32 1 1 NUM american_scientific_journal-7448 40 33 𝐾1𝑑 𝐾1𝑑 PROPN american_scientific_journal-7448 40 34 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 35 Μƒ Μƒ PROPN american_scientific_journal-7448 40 36 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 37 ( ( PUNCT american_scientific_journal-7448 40 38 19 19 NUM american_scientific_journal-7448 40 39 ) ) PUNCT american_scientific_journal-7448 40 40 the the DET american_scientific_journal-7448 40 41 far far ADJ american_scientific_journal-7448 40 42 field field NOUN american_scientific_journal-7448 40 43 condition condition NOUN american_scientific_journal-7448 40 44 , , PUNCT american_scientific_journal-7448 40 45 𝑓1 𝑓1 PROPN american_scientific_journal-7448 40 46 β€² β€² NUM american_scientific_journal-7448 40 47 β†’ β†’ SYM american_scientific_journal-7448 40 48 0 0 NUM american_scientific_journal-7448 40 49 , , PUNCT american_scientific_journal-7448 40 50 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 51 Μƒ Μƒ PROPN american_scientific_journal-7448 40 52 οΏ½ οΏ½ PROPN american_scientific_journal-7448 40 53 β†’ β†’ SYM american_scientific_journal-7448 40 54 ∞ ∞ PROPN american_scientific_journal-7448 40 55 determines determine VERB american_scientific_journal-7448 40 56 𝐢1 𝐢1 PRON american_scientific_journal-7448 40 57 , , PUNCT american_scientific_journal-7448 40 58 𝐢1 𝐢1 NOUN american_scientific_journal-7448 40 59 = = SYM american_scientific_journal-7448 41 1 2 2 NUM american_scientific_journal-7448 41 2 βˆ’ βˆ’ NOUN american_scientific_journal-7448 41 3 ∫ ∫ NOUN american_scientific_journal-7448 41 4 𝐾1(𝑅𝑒 𝐾1(𝑅𝑒 NOUN american_scientific_journal-7448 41 5 π‘Ÿ π‘Ÿ SYM american_scientific_journal-7448 41 6 Μƒ Μƒ NOUN american_scientific_journal-7448 41 7 4 4 NUM american_scientific_journal-7448 41 8 ) ) PUNCT american_scientific_journal-7448 41 9 𝑑 𝑑 PROPN american_scientific_journal-7448 41 10 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 11 Μƒ Μƒ PROPN american_scientific_journal-7448 41 12 οΏ½ οΏ½ NOUN american_scientific_journal-7448 41 13 ∞ ∞ ADJ american_scientific_journal-7448 41 14 1 1 NUM american_scientific_journal-7448 41 15 β‰… β‰… NUM american_scientific_journal-7448 41 16 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 41 17 2𝐾0 2𝐾0 NUM american_scientific_journal-7448 41 18 ( ( PUNCT american_scientific_journal-7448 41 19 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 41 20 4 4 NUM american_scientific_journal-7448 41 21 ) ) PUNCT american_scientific_journal-7448 41 22 ( ( PUNCT american_scientific_journal-7448 41 23 20 20 X american_scientific_journal-7448 41 24 ) ) PUNCT american_scientific_journal-7448 41 25 substituting substitute VERB american_scientific_journal-7448 41 26 ( ( PUNCT american_scientific_journal-7448 41 27 20 20 NUM american_scientific_journal-7448 41 28 ) ) PUNCT american_scientific_journal-7448 41 29 in in ADP american_scientific_journal-7448 41 30 ( ( PUNCT american_scientific_journal-7448 41 31 19 19 NUM american_scientific_journal-7448 41 32 ) ) PUNCT american_scientific_journal-7448 41 33 and and CCONJ american_scientific_journal-7448 41 34 putting put VERB american_scientific_journal-7448 41 35 the the DET american_scientific_journal-7448 41 36 result result NOUN american_scientific_journal-7448 41 37 in in ADP american_scientific_journal-7448 41 38 ( ( PUNCT american_scientific_journal-7448 41 39 16 16 NUM american_scientific_journal-7448 41 40 ) ) PUNCT american_scientific_journal-7448 41 41 gives give VERB american_scientific_journal-7448 41 42 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 43 Μƒ Μƒ PROPN american_scientific_journal-7448 41 44 οΏ½ οΏ½ NOUN american_scientific_journal-7448 41 45 = = SYM american_scientific_journal-7448 41 46 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 41 47 4𝐾0 4𝐾0 NUM american_scientific_journal-7448 41 48 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 49 Μƒ Μƒ PROPN american_scientific_journal-7448 41 50 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 51 sin sin NOUN american_scientific_journal-7448 41 52 πœƒ πœƒ X american_scientific_journal-7448 41 53 ∫ ∫ PROPN american_scientific_journal-7448 41 54 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 55 Μƒ Μƒ PROPN american_scientific_journal-7448 41 56 οΏ½ οΏ½ NOUN american_scientific_journal-7448 41 57 2∞ 2∞ NUM american_scientific_journal-7448 41 58 1 1 NUM american_scientific_journal-7448 41 59 𝐾1𝑑 𝐾1𝑑 PROPN american_scientific_journal-7448 41 60 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 61 Μƒ Μƒ PROPN american_scientific_journal-7448 41 62 οΏ½ οΏ½ NOUN american_scientific_journal-7448 41 63 βˆ’ βˆ’ ADP american_scientific_journal-7448 41 64 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 41 65 4𝐾0 4𝐾0 NUM american_scientific_journal-7448 41 66 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 67 Μƒ Μƒ PROPN american_scientific_journal-7448 41 68 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 69 sin sin NOUN american_scientific_journal-7448 41 70 πœƒ πœƒ X american_scientific_journal-7448 41 71 ∫ ∫ PROPN american_scientific_journal-7448 41 72 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 73 Μƒ Μƒ PROPN american_scientific_journal-7448 41 74 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 75 ∞ ∞ PROPN american_scientific_journal-7448 41 76 1 1 NUM american_scientific_journal-7448 41 77 𝐾1𝑑 𝐾1𝑑 PROPN american_scientific_journal-7448 41 78 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 79 Μƒ Μƒ PROPN american_scientific_journal-7448 41 80 οΏ½ οΏ½ PROPN american_scientific_journal-7448 41 81 ( ( PUNCT american_scientific_journal-7448 41 82 21 21 NUM american_scientific_journal-7448 41 83 ) ) PUNCT american_scientific_journal-7448 41 84 the the DET american_scientific_journal-7448 41 85 drag drag NOUN american_scientific_journal-7448 41 86 coefficient coefficient NOUN american_scientific_journal-7448 41 87 is be AUX american_scientific_journal-7448 41 88 given give VERB american_scientific_journal-7448 41 89 by by ADP american_scientific_journal-7448 41 90 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 41 91 = = SYM american_scientific_journal-7448 41 92 ∫ ∫ PROPN american_scientific_journal-7448 42 1 2 2 NUM american_scientific_journal-7448 42 2 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 42 3 πœ• πœ• PROPN american_scientific_journal-7448 42 4 οΏ½ οΏ½ PROPN american_scientific_journal-7448 42 5 Μƒ Μƒ PROPN american_scientific_journal-7448 42 6 οΏ½ οΏ½ PROPN american_scientific_journal-7448 42 7 πœ• πœ• PROPN american_scientific_journal-7448 42 8 οΏ½ οΏ½ PROPN american_scientific_journal-7448 42 9 Μƒ Μƒ PROPN american_scientific_journal-7448 42 10 οΏ½ οΏ½ PROPN american_scientific_journal-7448 42 11 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 42 12 0 0 NUM american_scientific_journal-7448 42 13 sin sin NOUN american_scientific_journal-7448 42 14 πœƒ πœƒ NOUN american_scientific_journal-7448 42 15 π‘‘πœƒ π‘‘πœƒ PROPN american_scientific_journal-7448 42 16 βˆ’ βˆ’ PROPN american_scientific_journal-7448 42 17 ∫ ∫ NOUN american_scientific_journal-7448 42 18 2 2 NUM american_scientific_journal-7448 43 1 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 43 2 οΏ½ οΏ½ PROPN american_scientific_journal-7448 43 3 Μƒ Μƒ PROPN american_scientific_journal-7448 43 4 οΏ½ οΏ½ PROPN american_scientific_journal-7448 43 5 𝑠 𝑠 ADJ american_scientific_journal-7448 43 6 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 43 7 0 0 PUNCT american_scientific_journal-7448 43 8 sin sin PROPN american_scientific_journal-7448 43 9 πœƒπ‘‘πœƒ πœƒπ‘‘πœƒ X american_scientific_journal-7448 43 10 ( ( PUNCT american_scientific_journal-7448 43 11 22 22 NUM american_scientific_journal-7448 43 12 ) ) PUNCT american_scientific_journal-7448 43 13 american american ADJ american_scientific_journal-7448 43 14 academic academic ADJ american_scientific_journal-7448 43 15 scientific scientific ADJ american_scientific_journal-7448 43 16 research research NOUN american_scientific_journal-7448 43 17 journal journal NOUN american_scientific_journal-7448 43 18 for for ADP american_scientific_journal-7448 43 19 engineering engineering NOUN american_scientific_journal-7448 43 20 , , PUNCT american_scientific_journal-7448 43 21 technology technology NOUN american_scientific_journal-7448 43 22 , , PUNCT american_scientific_journal-7448 43 23 and and CCONJ american_scientific_journal-7448 43 24 sciences science NOUN american_scientific_journal-7448 43 25 ( ( PUNCT american_scientific_journal-7448 43 26 asrjets asrjet NOUN american_scientific_journal-7448 43 27 ) ) PUNCT american_scientific_journal-7448 43 28 ( ( PUNCT american_scientific_journal-7448 43 29 2022 2022 NUM american_scientific_journal-7448 43 30 ) ) PUNCT american_scientific_journal-7448 43 31 volume volume NOUN american_scientific_journal-7448 43 32 86 86 NUM american_scientific_journal-7448 43 33 , , PUNCT american_scientific_journal-7448 43 34 no no DET american_scientific_journal-7448 43 35 1 1 NUM american_scientific_journal-7448 43 36 , , PUNCT american_scientific_journal-7448 43 37 pp143 pp143 PROPN american_scientific_journal-7448 43 38 - - SYM american_scientific_journal-7448 43 39 156 156 NUM american_scientific_journal-7448 43 40 147 147 NUM american_scientific_journal-7448 43 41 the the DET american_scientific_journal-7448 43 42 integral integral ADJ american_scientific_journal-7448 43 43 in in ADP american_scientific_journal-7448 43 44 ( ( PUNCT american_scientific_journal-7448 43 45 22 22 NUM american_scientific_journal-7448 43 46 ) ) PUNCT american_scientific_journal-7448 43 47 is be AUX american_scientific_journal-7448 43 48 the the DET american_scientific_journal-7448 43 49 pressure pressure NOUN american_scientific_journal-7448 43 50 stress stress NOUN american_scientific_journal-7448 43 51 evaluated evaluate VERB american_scientific_journal-7448 43 52 at at ADP american_scientific_journal-7448 43 53 οΏ½ οΏ½ PROPN american_scientific_journal-7448 43 54 Μƒ Μƒ NOUN american_scientific_journal-7448 43 55 οΏ½ οΏ½ NOUN american_scientific_journal-7448 43 56 = = SYM american_scientific_journal-7448 43 57 1 1 NUM american_scientific_journal-7448 43 58 while while SCONJ american_scientific_journal-7448 43 59 the the DET american_scientific_journal-7448 43 60 second second ADJ american_scientific_journal-7448 43 61 integral integral NOUN american_scientific_journal-7448 43 62 is be AUX american_scientific_journal-7448 43 63 the the DET american_scientific_journal-7448 43 64 viscous viscous ADJ american_scientific_journal-7448 43 65 stress stress NOUN american_scientific_journal-7448 43 66 and and CCONJ american_scientific_journal-7448 43 67 both both DET american_scientific_journal-7448 43 68 stresses stress NOUN american_scientific_journal-7448 43 69 contribute contribute VERB american_scientific_journal-7448 43 70 equally equally ADV american_scientific_journal-7448 43 71 to to ADP american_scientific_journal-7448 43 72 the the DET american_scientific_journal-7448 43 73 drag drag NOUN american_scientific_journal-7448 43 74 force force NOUN american_scientific_journal-7448 43 75 . . PUNCT american_scientific_journal-7448 44 1 substituting substitute VERB american_scientific_journal-7448 44 2 ( ( PUNCT american_scientific_journal-7448 44 3 21 21 NUM american_scientific_journal-7448 44 4 ) ) PUNCT american_scientific_journal-7448 44 5 in in ADP american_scientific_journal-7448 44 6 ( ( PUNCT american_scientific_journal-7448 44 7 17 17 NUM american_scientific_journal-7448 44 8 ) ) PUNCT american_scientific_journal-7448 44 9 and and CCONJ american_scientific_journal-7448 44 10 putting put VERB american_scientific_journal-7448 44 11 the the DET american_scientific_journal-7448 44 12 result result NOUN american_scientific_journal-7448 44 13 in in ADP american_scientific_journal-7448 44 14 ( ( PUNCT american_scientific_journal-7448 44 15 22 22 NUM american_scientific_journal-7448 44 16 ) ) PUNCT american_scientific_journal-7448 44 17 gives give VERB american_scientific_journal-7448 44 18 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 44 19 = = SYM american_scientific_journal-7448 44 20 2πœ‹πΆ1 2πœ‹πΆ1 NUM american_scientific_journal-7448 45 1 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 45 2 ( ( PUNCT american_scientific_journal-7448 45 3 𝑑𝐾1 𝑑𝐾1 PROPN american_scientific_journal-7448 45 4 𝑑𝑧 𝑑𝑧 PROPN american_scientific_journal-7448 45 5 βˆ’ βˆ’ PROPN american_scientific_journal-7448 45 6 𝐾1 𝐾1 PROPN american_scientific_journal-7448 45 7 ) ) PUNCT american_scientific_journal-7448 45 8 ( ( PUNCT american_scientific_journal-7448 45 9 23 23 NUM american_scientific_journal-7448 45 10 ) ) PUNCT american_scientific_journal-7448 45 11 using use VERB american_scientific_journal-7448 45 12 the the DET american_scientific_journal-7448 45 13 expansion expansion NOUN american_scientific_journal-7448 45 14 𝐾1(𝑧 𝐾1(𝑧 NUM american_scientific_journal-7448 45 15 ) ) PUNCT american_scientific_journal-7448 45 16 = = PRON american_scientific_journal-7448 45 17 { { PUNCT american_scientific_journal-7448 45 18 1 1 NUM american_scientific_journal-7448 45 19 𝑧 𝑧 NOUN american_scientific_journal-7448 45 20 + + NOUN american_scientific_journal-7448 45 21 𝑧 𝑧 DET american_scientific_journal-7448 45 22 2 2 NUM american_scientific_journal-7448 45 23 log log NOUN american_scientific_journal-7448 45 24 ( ( PUNCT american_scientific_journal-7448 45 25 𝑧 𝑧 PROPN american_scientific_journal-7448 45 26 2 2 NUM american_scientific_journal-7448 45 27 ) ) PUNCT american_scientific_journal-7448 46 1 + + CCONJ american_scientific_journal-7448 46 2 β‹― β‹― PROPN american_scientific_journal-7448 46 3 , , PUNCT american_scientific_journal-7448 46 4 𝑧 𝑧 ADJ american_scientific_journal-7448 46 5 β‰ͺ β‰ͺ ADJ american_scientific_journal-7448 46 6 1 1 NUM american_scientific_journal-7448 46 7 ( ( PUNCT american_scientific_journal-7448 46 8 πœ‹ πœ‹ NOUN american_scientific_journal-7448 46 9 2𝑧 2𝑧 NOUN american_scientific_journal-7448 46 10 ) ) PUNCT american_scientific_journal-7448 46 11 1 1 NUM american_scientific_journal-7448 46 12 2π‘’βˆ’π‘§ 2π‘’βˆ’π‘§ NUM american_scientific_journal-7448 46 13 , , PUNCT american_scientific_journal-7448 46 14 𝑧 𝑧 PROPN american_scientific_journal-7448 46 15 ≫ ≫ PROPN american_scientific_journal-7448 46 16 1 1 NUM american_scientific_journal-7448 46 17 ( ( PUNCT american_scientific_journal-7448 46 18 23 23 NUM american_scientific_journal-7448 46 19 ) ) PUNCT american_scientific_journal-7448 46 20 becomes become VERB american_scientific_journal-7448 46 21 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 46 22 β‰… β‰… PROPN american_scientific_journal-7448 46 23 8πœ‹ 8πœ‹ PROPN american_scientific_journal-7448 46 24 π‘…π‘’π‘™π‘œπ‘” π‘…π‘’π‘™π‘œπ‘” PROPN american_scientific_journal-7448 46 25 ( ( PUNCT american_scientific_journal-7448 46 26 7.4 7.4 NUM american_scientific_journal-7448 46 27 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 46 28 ) ) PUNCT american_scientific_journal-7448 46 29 ( ( PUNCT american_scientific_journal-7448 46 30 24 24 NUM american_scientific_journal-7448 46 31 ) ) SYM american_scientific_journal-7448 46 32 2 2 NUM american_scientific_journal-7448 46 33 . . X american_scientific_journal-7448 46 34 mathematical mathematical ADJ american_scientific_journal-7448 46 35 formulation formulation NOUN american_scientific_journal-7448 46 36 consider consider VERB american_scientific_journal-7448 46 37 the the DET american_scientific_journal-7448 46 38 two two NUM american_scientific_journal-7448 46 39 dimensional dimensional ADJ american_scientific_journal-7448 46 40 steady steady ADJ american_scientific_journal-7448 46 41 incompressible incompressible ADJ american_scientific_journal-7448 46 42 viscous viscous ADJ american_scientific_journal-7448 46 43 flow flow NOUN american_scientific_journal-7448 46 44 of of ADP american_scientific_journal-7448 46 45 a a DET american_scientific_journal-7448 46 46 uniform uniform ADJ american_scientific_journal-7448 46 47 stream stream NOUN american_scientific_journal-7448 46 48 𝑒 𝑒 NOUN american_scientific_journal-7448 46 49 in in ADP american_scientific_journal-7448 46 50 which which PRON american_scientific_journal-7448 46 51 a a DET american_scientific_journal-7448 46 52 rigid rigid ADJ american_scientific_journal-7448 46 53 circular circular ADJ american_scientific_journal-7448 46 54 cylinder cylinder NOUN american_scientific_journal-7448 46 55 of of ADP american_scientific_journal-7448 46 56 radius radius NOUN american_scientific_journal-7448 46 57 π‘Ÿπ‘œ π‘Ÿπ‘œ PROPN american_scientific_journal-7448 46 58 , , PUNCT american_scientific_journal-7448 46 59 centre centre PROPN american_scientific_journal-7448 46 60 π‘œ π‘œ PRON american_scientific_journal-7448 46 61 , , PUNCT american_scientific_journal-7448 46 62 is be AUX american_scientific_journal-7448 46 63 translating translate VERB american_scientific_journal-7448 46 64 with with ADP american_scientific_journal-7448 46 65 a a DET american_scientific_journal-7448 46 66 constant constant ADJ american_scientific_journal-7448 46 67 velocity velocity NOUN american_scientific_journal-7448 46 68 π‘ˆ. π‘ˆ. NOUN american_scientific_journal-7448 46 69 figure figure NOUN american_scientific_journal-7448 46 70 1 1 NUM american_scientific_journal-7448 46 71 : : PUNCT american_scientific_journal-7448 46 72 flow flow VERB american_scientific_journal-7448 46 73 around around ADP american_scientific_journal-7448 46 74 a a DET american_scientific_journal-7448 46 75 translating translate VERB american_scientific_journal-7448 46 76 circular circular ADJ american_scientific_journal-7448 46 77 cylinder cylinder NOUN american_scientific_journal-7448 46 78 . . PUNCT american_scientific_journal-7448 47 1 it it PRON american_scientific_journal-7448 47 2 is be AUX american_scientific_journal-7448 47 3 well well ADV american_scientific_journal-7448 47 4 established establish VERB american_scientific_journal-7448 47 5 that that SCONJ american_scientific_journal-7448 47 6 the the DET american_scientific_journal-7448 47 7 motion motion NOUN american_scientific_journal-7448 47 8 of of ADP american_scientific_journal-7448 47 9 fluids fluid NOUN american_scientific_journal-7448 47 10 is be AUX american_scientific_journal-7448 47 11 governed govern VERB american_scientific_journal-7448 47 12 by by ADP american_scientific_journal-7448 47 13 the the DET american_scientific_journal-7448 47 14 famous famous ADJ american_scientific_journal-7448 47 15 navier navier NOUN american_scientific_journal-7448 47 16 - - PUNCT american_scientific_journal-7448 47 17 stokes stoke NOUN american_scientific_journal-7448 47 18 equations equation NOUN american_scientific_journal-7448 47 19 . . PUNCT american_scientific_journal-7448 48 1 for for ADP american_scientific_journal-7448 48 2 a a DET american_scientific_journal-7448 48 3 compressible compressible ADJ american_scientific_journal-7448 48 4 newtonian newtonian ADJ american_scientific_journal-7448 48 5 fluid fluid NOUN american_scientific_journal-7448 48 6 , , PUNCT american_scientific_journal-7448 48 7 this this PRON american_scientific_journal-7448 48 8 yields yield VERB american_scientific_journal-7448 48 9 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 48 10 πœƒ πœƒ X american_scientific_journal-7448 48 11 π‘œ π‘œ VERB american_scientific_journal-7448 48 12 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 48 13 π‘Ÿπ‘œ π‘Ÿπ‘œ INTJ american_scientific_journal-7448 48 14 π‘œ π‘œ NOUN american_scientific_journal-7448 48 15 π‘ˆ π‘ˆ PROPN american_scientific_journal-7448 48 16 πœƒ πœƒ NUM american_scientific_journal-7448 48 17 american american ADJ american_scientific_journal-7448 48 18 academic academic ADJ american_scientific_journal-7448 48 19 scientific scientific ADJ american_scientific_journal-7448 48 20 research research NOUN american_scientific_journal-7448 48 21 journal journal NOUN american_scientific_journal-7448 48 22 for for ADP american_scientific_journal-7448 48 23 engineering engineering NOUN american_scientific_journal-7448 48 24 , , PUNCT american_scientific_journal-7448 48 25 technology technology NOUN american_scientific_journal-7448 48 26 , , PUNCT american_scientific_journal-7448 48 27 and and CCONJ american_scientific_journal-7448 48 28 sciences science NOUN american_scientific_journal-7448 48 29 ( ( PUNCT american_scientific_journal-7448 48 30 asrjets asrjet NOUN american_scientific_journal-7448 48 31 ) ) PUNCT american_scientific_journal-7448 48 32 ( ( PUNCT american_scientific_journal-7448 48 33 2022 2022 NUM american_scientific_journal-7448 48 34 ) ) PUNCT american_scientific_journal-7448 48 35 volume volume NOUN american_scientific_journal-7448 48 36 86 86 NUM american_scientific_journal-7448 48 37 , , PUNCT american_scientific_journal-7448 48 38 no no DET american_scientific_journal-7448 48 39 1 1 NUM american_scientific_journal-7448 48 40 , , PUNCT american_scientific_journal-7448 48 41 pp143 pp143 PROPN american_scientific_journal-7448 48 42 - - SYM american_scientific_journal-7448 48 43 156 156 NUM american_scientific_journal-7448 48 44 148 148 NUM american_scientific_journal-7448 48 45 𝜌 𝜌 NOUN american_scientific_journal-7448 48 46 ( ( PUNCT american_scientific_journal-7448 48 47 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-7448 48 48 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-7448 48 49 + + CCONJ american_scientific_journal-7448 48 50 ( ( PUNCT american_scientific_journal-7448 48 51 𝑒 𝑒 X american_scientific_journal-7448 48 52 βˆ™ βˆ™ PROPN american_scientific_journal-7448 48 53 βˆ‡)𝑒 βˆ‡)𝑒 PROPN american_scientific_journal-7448 48 54 ) ) PUNCT american_scientific_journal-7448 48 55 = = SYM american_scientific_journal-7448 49 1 βˆ’βˆ‡π‘ βˆ’βˆ‡π‘ PROPN american_scientific_journal-7448 49 2 + + CCONJ american_scientific_journal-7448 49 3 πœ‡βˆ‡2𝑒 πœ‡βˆ‡2𝑒 PROPN american_scientific_journal-7448 49 4 + + ADJ american_scientific_journal-7448 49 5 𝜌𝐹 𝜌𝐹 PROPN american_scientific_journal-7448 49 6 ( ( PUNCT american_scientific_journal-7448 49 7 25 25 NUM american_scientific_journal-7448 49 8 ) ) PUNCT american_scientific_journal-7448 49 9 where where SCONJ american_scientific_journal-7448 49 10 𝑒 𝑒 PRON american_scientific_journal-7448 49 11 is be AUX american_scientific_journal-7448 49 12 the the DET american_scientific_journal-7448 49 13 fluid fluid ADJ american_scientific_journal-7448 49 14 velocity velocity NOUN american_scientific_journal-7448 49 15 , , PUNCT american_scientific_journal-7448 49 16 𝑝 𝑝 NOUN american_scientific_journal-7448 49 17 is be AUX american_scientific_journal-7448 49 18 the the DET american_scientific_journal-7448 49 19 fluid fluid ADJ american_scientific_journal-7448 49 20 pressure pressure NOUN american_scientific_journal-7448 49 21 , , PUNCT american_scientific_journal-7448 49 22 𝜌 𝜌 X american_scientific_journal-7448 49 23 is be AUX american_scientific_journal-7448 49 24 the the DET american_scientific_journal-7448 49 25 fluid fluid ADJ american_scientific_journal-7448 49 26 density density NOUN american_scientific_journal-7448 49 27 , , PUNCT american_scientific_journal-7448 49 28 and and CCONJ american_scientific_journal-7448 49 29 πœ‡ πœ‡ X american_scientific_journal-7448 49 30 is be AUX american_scientific_journal-7448 49 31 the the DET american_scientific_journal-7448 49 32 fluid fluid ADJ american_scientific_journal-7448 49 33 dynamic dynamic ADJ american_scientific_journal-7448 49 34 viscosity viscosity NOUN american_scientific_journal-7448 49 35 . . PUNCT american_scientific_journal-7448 50 1 the the DET american_scientific_journal-7448 50 2 terms term NOUN american_scientific_journal-7448 50 3 on on ADP american_scientific_journal-7448 50 4 the the DET american_scientific_journal-7448 50 5 left left NOUN american_scientific_journal-7448 50 6 correspond correspond NOUN american_scientific_journal-7448 50 7 to to ADP american_scientific_journal-7448 50 8 the the DET american_scientific_journal-7448 50 9 inertial inertial ADJ american_scientific_journal-7448 50 10 forces force NOUN american_scientific_journal-7448 50 11 , , PUNCT american_scientific_journal-7448 50 12 the the DET american_scientific_journal-7448 50 13 first first ADJ american_scientific_journal-7448 50 14 term term NOUN american_scientific_journal-7448 50 15 on on ADP american_scientific_journal-7448 50 16 the the DET american_scientific_journal-7448 50 17 right right NOUN american_scientific_journal-7448 50 18 is be AUX american_scientific_journal-7448 50 19 the the DET american_scientific_journal-7448 50 20 pressure pressure NOUN american_scientific_journal-7448 50 21 forces force NOUN american_scientific_journal-7448 50 22 , , PUNCT american_scientific_journal-7448 50 23 the the DET american_scientific_journal-7448 50 24 second second ADJ american_scientific_journal-7448 50 25 term term NOUN american_scientific_journal-7448 50 26 is be AUX american_scientific_journal-7448 50 27 the the DET american_scientific_journal-7448 50 28 viscous viscous ADJ american_scientific_journal-7448 50 29 forces force NOUN american_scientific_journal-7448 50 30 , , PUNCT american_scientific_journal-7448 50 31 and and CCONJ american_scientific_journal-7448 50 32 the the DET american_scientific_journal-7448 50 33 last last ADJ american_scientific_journal-7448 50 34 term term NOUN american_scientific_journal-7448 50 35 is be AUX american_scientific_journal-7448 50 36 the the DET american_scientific_journal-7448 50 37 applied apply VERB american_scientific_journal-7448 50 38 external external ADJ american_scientific_journal-7448 50 39 force force NOUN american_scientific_journal-7448 50 40 on on ADP american_scientific_journal-7448 50 41 the the DET american_scientific_journal-7448 50 42 fluid fluid NOUN american_scientific_journal-7448 50 43 . . PUNCT american_scientific_journal-7448 51 1 when when SCONJ american_scientific_journal-7448 51 2 there there PRON american_scientific_journal-7448 51 3 are be VERB american_scientific_journal-7448 51 4 no no DET american_scientific_journal-7448 51 5 applied applied ADJ american_scientific_journal-7448 51 6 forces force NOUN american_scientific_journal-7448 51 7 on on ADP american_scientific_journal-7448 51 8 the the DET american_scientific_journal-7448 51 9 fluid fluid NOUN american_scientific_journal-7448 51 10 , , PUNCT american_scientific_journal-7448 51 11 and and CCONJ american_scientific_journal-7448 51 12 the the DET american_scientific_journal-7448 51 13 flow flow NOUN american_scientific_journal-7448 51 14 being be AUX american_scientific_journal-7448 51 15 steady steady ADJ american_scientific_journal-7448 51 16 , , PUNCT american_scientific_journal-7448 51 17 𝐹 𝐹 PROPN american_scientific_journal-7448 51 18 = = NOUN american_scientific_journal-7448 51 19 0 0 NUM american_scientific_journal-7448 51 20 , , PUNCT american_scientific_journal-7448 51 21 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-7448 51 22 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-7448 51 23 = = NOUN american_scientific_journal-7448 51 24 0 0 NUM american_scientific_journal-7448 51 25 , , PUNCT american_scientific_journal-7448 51 26 then then ADV american_scientific_journal-7448 51 27 , , PUNCT american_scientific_journal-7448 51 28 equation equation NOUN american_scientific_journal-7448 51 29 ( ( PUNCT american_scientific_journal-7448 51 30 25 25 NUM american_scientific_journal-7448 51 31 ) ) PUNCT american_scientific_journal-7448 51 32 becomes become VERB american_scientific_journal-7448 51 33 𝜌(𝑒 𝜌(𝑒 PROPN american_scientific_journal-7448 51 34 βˆ™ βˆ™ PROPN american_scientific_journal-7448 51 35 βˆ‡)𝑒 βˆ‡)𝑒 PROPN american_scientific_journal-7448 51 36 = = SYM american_scientific_journal-7448 51 37 βˆ’βˆ‡π‘ βˆ’βˆ‡π‘ PROPN american_scientific_journal-7448 51 38 + + CCONJ american_scientific_journal-7448 52 1 πœ‡βˆ‡2𝑒 πœ‡βˆ‡2𝑒 PROPN american_scientific_journal-7448 52 2 ( ( PUNCT american_scientific_journal-7448 52 3 26 26 NUM american_scientific_journal-7448 52 4 ) ) PUNCT american_scientific_journal-7448 52 5 these these DET american_scientific_journal-7448 52 6 equations equation NOUN american_scientific_journal-7448 52 7 are be AUX american_scientific_journal-7448 52 8 always always ADV american_scientific_journal-7448 52 9 solved solve VERB american_scientific_journal-7448 52 10 together together ADV american_scientific_journal-7448 52 11 with with ADP american_scientific_journal-7448 52 12 the the DET american_scientific_journal-7448 52 13 continuity continuity NOUN american_scientific_journal-7448 52 14 equation equation NOUN american_scientific_journal-7448 52 15 πœ•πœŒ πœ•πœŒ NOUN american_scientific_journal-7448 52 16 πœ•π‘‘ πœ•π‘‘ PROPN american_scientific_journal-7448 53 1 + + NOUN american_scientific_journal-7448 53 2 βˆ‡. βˆ‡. ADJ american_scientific_journal-7448 53 3 ( ( PUNCT american_scientific_journal-7448 53 4 πœŒπ‘’ πœŒπ‘’ NOUN american_scientific_journal-7448 53 5 ) ) PUNCT american_scientific_journal-7448 53 6 = = SYM american_scientific_journal-7448 53 7 0 0 NUM american_scientific_journal-7448 53 8 ( ( PUNCT american_scientific_journal-7448 53 9 27 27 NUM american_scientific_journal-7448 53 10 ) ) PUNCT american_scientific_journal-7448 53 11 since since SCONJ american_scientific_journal-7448 53 12 the the DET american_scientific_journal-7448 53 13 fluid fluid NOUN american_scientific_journal-7448 53 14 is be AUX american_scientific_journal-7448 53 15 incompressible incompressible ADJ american_scientific_journal-7448 53 16 , , PUNCT american_scientific_journal-7448 53 17 πœ•πœŒ πœ•πœŒ NOUN american_scientific_journal-7448 53 18 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-7448 53 19 = = NOUN american_scientific_journal-7448 53 20 0 0 NUM american_scientific_journal-7448 53 21 and and CCONJ american_scientific_journal-7448 53 22 upon upon SCONJ american_scientific_journal-7448 53 23 dividing divide VERB american_scientific_journal-7448 53 24 by by ADP american_scientific_journal-7448 53 25 𝜌 𝜌 ADP american_scientific_journal-7448 53 26 , , PUNCT american_scientific_journal-7448 53 27 equation equation NOUN american_scientific_journal-7448 53 28 ( ( PUNCT american_scientific_journal-7448 53 29 27 27 NUM american_scientific_journal-7448 53 30 ) ) PUNCT american_scientific_journal-7448 53 31 reduces reduce VERB american_scientific_journal-7448 53 32 to to ADP american_scientific_journal-7448 53 33 βˆ‡. βˆ‡. VERB american_scientific_journal-7448 53 34 𝑒 𝑒 X american_scientific_journal-7448 53 35 = = SYM american_scientific_journal-7448 53 36 0 0 NUM american_scientific_journal-7448 53 37 ( ( PUNCT american_scientific_journal-7448 53 38 28 28 NUM american_scientific_journal-7448 53 39 ) ) PUNCT american_scientific_journal-7448 53 40 which which PRON american_scientific_journal-7448 53 41 when when SCONJ american_scientific_journal-7448 53 42 taken take VERB american_scientific_journal-7448 53 43 together together ADV american_scientific_journal-7448 53 44 leaves leave VERB american_scientific_journal-7448 53 45 the the DET american_scientific_journal-7448 53 46 navier navier NOUN american_scientific_journal-7448 53 47 - - PUNCT american_scientific_journal-7448 53 48 stokes stoke NOUN american_scientific_journal-7448 53 49 equations equation NOUN american_scientific_journal-7448 53 50 to to PART american_scientific_journal-7448 53 51 represent represent VERB american_scientific_journal-7448 53 52 the the DET american_scientific_journal-7448 53 53 conservation conservation NOUN american_scientific_journal-7448 53 54 of of ADP american_scientific_journal-7448 53 55 momentum momentum NOUN american_scientific_journal-7448 53 56 and and CCONJ american_scientific_journal-7448 53 57 the the DET american_scientific_journal-7448 53 58 continuity continuity NOUN american_scientific_journal-7448 53 59 equation equation NOUN american_scientific_journal-7448 53 60 to to PART american_scientific_journal-7448 53 61 represent represent VERB american_scientific_journal-7448 53 62 the the DET american_scientific_journal-7448 53 63 conservation conservation NOUN american_scientific_journal-7448 53 64 of of ADP american_scientific_journal-7448 53 65 mass mass PROPN american_scientific_journal-7448 53 66 . . PUNCT american_scientific_journal-7448 54 1 the the DET american_scientific_journal-7448 54 2 boundary boundary ADJ american_scientific_journal-7448 54 3 conditions condition NOUN american_scientific_journal-7448 54 4 for for ADP american_scientific_journal-7448 54 5 the the DET american_scientific_journal-7448 54 6 solution solution NOUN american_scientific_journal-7448 54 7 of of ADP american_scientific_journal-7448 54 8 ( ( PUNCT american_scientific_journal-7448 54 9 26 26 NUM american_scientific_journal-7448 54 10 ) ) PUNCT american_scientific_journal-7448 54 11 and and CCONJ american_scientific_journal-7448 54 12 ( ( PUNCT american_scientific_journal-7448 54 13 28 28 NUM american_scientific_journal-7448 54 14 ) ) PUNCT american_scientific_journal-7448 54 15 are be AUX american_scientific_journal-7448 54 16 𝑒(π‘Ÿ 𝑒(π‘Ÿ NOUN american_scientific_journal-7448 54 17 , , PUNCT american_scientific_journal-7448 54 18 πœƒ πœƒ NOUN american_scientific_journal-7448 54 19 ) ) PUNCT american_scientific_journal-7448 54 20 = = SYM american_scientific_journal-7448 54 21 βˆ’π‘ˆ βˆ’π‘ˆ NUM american_scientific_journal-7448 54 22 , , PUNCT american_scientific_journal-7448 54 23 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 54 24 = = SYM american_scientific_journal-7448 54 25 π‘Ÿ0 π‘Ÿ0 PROPN american_scientific_journal-7448 54 26 𝑒(π‘Ÿ 𝑒(π‘Ÿ NOUN american_scientific_journal-7448 54 27 , , PUNCT american_scientific_journal-7448 54 28 πœƒ πœƒ NOUN american_scientific_journal-7448 54 29 ) ) PUNCT american_scientific_journal-7448 54 30 = = SYM american_scientific_journal-7448 54 31 0 0 NUM american_scientific_journal-7448 54 32 , , PUNCT american_scientific_journal-7448 54 33 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 54 34 β†’ β†’ SYM american_scientific_journal-7448 54 35 ∞ ∞ PROPN american_scientific_journal-7448 54 36 the the DET american_scientific_journal-7448 54 37 stream stream NOUN american_scientific_journal-7448 54 38 function function NOUN american_scientific_journal-7448 54 39 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 54 40 , , PUNCT american_scientific_journal-7448 54 41 πœƒ πœƒ NOUN american_scientific_journal-7448 54 42 ) ) PUNCT american_scientific_journal-7448 54 43 for for ADP american_scientific_journal-7448 54 44 the the DET american_scientific_journal-7448 54 45 flow flow NOUN american_scientific_journal-7448 54 46 is be AUX american_scientific_journal-7448 54 47 of of ADP american_scientific_journal-7448 54 48 the the DET american_scientific_journal-7448 54 49 form form NOUN american_scientific_journal-7448 54 50 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 54 51 , , PUNCT american_scientific_journal-7448 54 52 πœƒ πœƒ NOUN american_scientific_journal-7448 54 53 ) ) PUNCT american_scientific_journal-7448 54 54 = = PUNCT american_scientific_journal-7448 55 1 βˆ’π‘“(π‘Ÿ)π‘ˆ βˆ’π‘“(π‘Ÿ)π‘ˆ ADP american_scientific_journal-7448 55 2 sin sin NOUN american_scientific_journal-7448 55 3 πœƒ πœƒ X american_scientific_journal-7448 55 4 ( ( PUNCT american_scientific_journal-7448 55 5 29 29 NUM american_scientific_journal-7448 55 6 ) ) PUNCT american_scientific_journal-7448 55 7 where where SCONJ american_scientific_journal-7448 55 8 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 55 9 ) ) PUNCT american_scientific_journal-7448 55 10 is be AUX american_scientific_journal-7448 55 11 an an DET american_scientific_journal-7448 55 12 unknown unknown ADJ american_scientific_journal-7448 55 13 function function NOUN american_scientific_journal-7448 55 14 . . PUNCT american_scientific_journal-7448 56 1 in in ADP american_scientific_journal-7448 56 2 polar polar ADJ american_scientific_journal-7448 56 3 coordinates coordinate NOUN american_scientific_journal-7448 56 4 american american PROPN american_scientific_journal-7448 56 5 academic academic ADJ american_scientific_journal-7448 56 6 scientific scientific ADJ american_scientific_journal-7448 56 7 research research NOUN american_scientific_journal-7448 56 8 journal journal NOUN american_scientific_journal-7448 56 9 for for ADP american_scientific_journal-7448 56 10 engineering engineering NOUN american_scientific_journal-7448 56 11 , , PUNCT american_scientific_journal-7448 56 12 technology technology NOUN american_scientific_journal-7448 56 13 , , PUNCT american_scientific_journal-7448 56 14 and and CCONJ american_scientific_journal-7448 56 15 sciences science NOUN american_scientific_journal-7448 56 16 ( ( PUNCT american_scientific_journal-7448 56 17 asrjets asrjet NOUN american_scientific_journal-7448 56 18 ) ) PUNCT american_scientific_journal-7448 56 19 ( ( PUNCT american_scientific_journal-7448 56 20 2022 2022 NUM american_scientific_journal-7448 56 21 ) ) PUNCT american_scientific_journal-7448 56 22 volume volume NOUN american_scientific_journal-7448 56 23 86 86 NUM american_scientific_journal-7448 56 24 , , PUNCT american_scientific_journal-7448 56 25 no no DET american_scientific_journal-7448 56 26 1 1 NUM american_scientific_journal-7448 56 27 , , PUNCT american_scientific_journal-7448 56 28 pp143 pp143 PROPN american_scientific_journal-7448 56 29 - - PUNCT american_scientific_journal-7448 56 30 156 156 NUM american_scientific_journal-7448 56 31 149 149 NUM american_scientific_journal-7448 56 32 βˆ‡2≑ βˆ‡2≑ NOUN american_scientific_journal-7448 56 33 πœ•2 πœ•2 NOUN american_scientific_journal-7448 56 34 πœ•π‘Ÿ2 πœ•π‘Ÿ2 NOUN american_scientific_journal-7448 56 35 + + CCONJ american_scientific_journal-7448 56 36 1 1 NUM american_scientific_journal-7448 56 37 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 56 38 πœ• πœ• PROPN american_scientific_journal-7448 56 39 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 56 40 + + CCONJ american_scientific_journal-7448 56 41 1 1 NUM american_scientific_journal-7448 56 42 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 56 43 πœ•2 πœ•2 NOUN american_scientific_journal-7448 56 44 πœ•πœƒ2 πœ•πœƒ2 ADJ american_scientific_journal-7448 56 45 and and CCONJ american_scientific_journal-7448 56 46 in in ADP american_scientific_journal-7448 56 47 cartesian cartesian ADJ american_scientific_journal-7448 56 48 coordinates coordinate NOUN american_scientific_journal-7448 56 49 ( ( PUNCT american_scientific_journal-7448 56 50 two two NUM american_scientific_journal-7448 56 51 dimensional dimensional ADJ american_scientific_journal-7448 56 52 ) ) PUNCT american_scientific_journal-7448 56 53 βˆ‡2≑ βˆ‡2≑ NOUN american_scientific_journal-7448 56 54 πœ•2 πœ•2 NOUN american_scientific_journal-7448 56 55 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-7448 56 56 + + CCONJ american_scientific_journal-7448 56 57 πœ•2 πœ•2 NOUN american_scientific_journal-7448 56 58 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-7448 56 59 3 3 NUM american_scientific_journal-7448 56 60 . . NOUN american_scientific_journal-7448 56 61 method method NOUN american_scientific_journal-7448 56 62 of of ADP american_scientific_journal-7448 56 63 solution solution NOUN american_scientific_journal-7448 56 64 since since SCONJ american_scientific_journal-7448 56 65 the the DET american_scientific_journal-7448 56 66 motion motion NOUN american_scientific_journal-7448 56 67 is be AUX american_scientific_journal-7448 56 68 extremely extremely ADV american_scientific_journal-7448 56 69 slow slow ADJ american_scientific_journal-7448 56 70 so so SCONJ american_scientific_journal-7448 56 71 that that SCONJ american_scientific_journal-7448 56 72 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 56 73 β‰ͺ β‰ͺ ADJ american_scientific_journal-7448 56 74 1 1 NUM american_scientific_journal-7448 56 75 , , PUNCT american_scientific_journal-7448 56 76 the the DET american_scientific_journal-7448 56 77 inertia inertia NOUN american_scientific_journal-7448 56 78 forces force NOUN american_scientific_journal-7448 56 79 will will AUX american_scientific_journal-7448 56 80 be be AUX american_scientific_journal-7448 56 81 small small ADJ american_scientific_journal-7448 56 82 compared compare VERB american_scientific_journal-7448 56 83 with with ADP american_scientific_journal-7448 56 84 the the DET american_scientific_journal-7448 56 85 viscous viscous ADJ american_scientific_journal-7448 56 86 forces force NOUN american_scientific_journal-7448 56 87 , , PUNCT american_scientific_journal-7448 56 88 so so SCONJ american_scientific_journal-7448 56 89 the the DET american_scientific_journal-7448 56 90 inertia inertia NOUN american_scientific_journal-7448 56 91 forces force NOUN american_scientific_journal-7448 56 92 can can AUX american_scientific_journal-7448 56 93 be be AUX american_scientific_journal-7448 56 94 ignored ignore VERB american_scientific_journal-7448 56 95 at at ADP american_scientific_journal-7448 56 96 the the DET american_scientific_journal-7448 56 97 vicinity vicinity NOUN american_scientific_journal-7448 56 98 of of ADP american_scientific_journal-7448 56 99 the the DET american_scientific_journal-7448 56 100 cylinder cylinder NOUN american_scientific_journal-7448 57 1 [ [ X american_scientific_journal-7448 57 2 1 1 NUM american_scientific_journal-7448 57 3 ] ] PUNCT american_scientific_journal-7448 57 4 . . PUNCT american_scientific_journal-7448 58 1 equation equation NOUN american_scientific_journal-7448 58 2 ( ( PUNCT american_scientific_journal-7448 58 3 26 26 NUM american_scientific_journal-7448 58 4 ) ) PUNCT american_scientific_journal-7448 58 5 becomes become VERB american_scientific_journal-7448 58 6 0 0 NUM american_scientific_journal-7448 58 7 = = SYM american_scientific_journal-7448 58 8 βˆ’βˆ‡π‘ βˆ’βˆ‡π‘ PROPN american_scientific_journal-7448 58 9 + + CCONJ american_scientific_journal-7448 59 1 πœ‡βˆ‡2𝑒 πœ‡βˆ‡2𝑒 PROPN american_scientific_journal-7448 59 2 ( ( PUNCT american_scientific_journal-7448 59 3 30 30 NUM american_scientific_journal-7448 59 4 ) ) PUNCT american_scientific_journal-7448 59 5 by by ADP american_scientific_journal-7448 59 6 taking take VERB american_scientific_journal-7448 59 7 the the DET american_scientific_journal-7448 59 8 divergence divergence NOUN american_scientific_journal-7448 59 9 of of ADP american_scientific_journal-7448 59 10 each each DET american_scientific_journal-7448 59 11 term term NOUN american_scientific_journal-7448 59 12 of of ADP american_scientific_journal-7448 59 13 ( ( PUNCT american_scientific_journal-7448 59 14 30 30 NUM american_scientific_journal-7448 59 15 ) ) PUNCT american_scientific_journal-7448 59 16 , , PUNCT american_scientific_journal-7448 59 17 we we PRON american_scientific_journal-7448 59 18 have have VERB american_scientific_journal-7448 59 19 0 0 NUM american_scientific_journal-7448 59 20 = = SYM american_scientific_journal-7448 59 21 βˆ’βˆ‡. βˆ’βˆ‡. PROPN american_scientific_journal-7448 59 22 ( ( PUNCT american_scientific_journal-7448 59 23 βˆ‡π‘ βˆ‡π‘ NOUN american_scientific_journal-7448 59 24 ) ) PUNCT american_scientific_journal-7448 59 25 + + NUM american_scientific_journal-7448 59 26 πœ‡βˆ‡2(βˆ‡. πœ‡βˆ‡2(βˆ‡. NUM american_scientific_journal-7448 59 27 𝑒 𝑒 NOUN american_scientific_journal-7448 59 28 ) ) PUNCT american_scientific_journal-7448 59 29 ( ( PUNCT american_scientific_journal-7448 59 30 31 31 NUM american_scientific_journal-7448 59 31 ) ) PUNCT american_scientific_journal-7448 59 32 since since SCONJ american_scientific_journal-7448 59 33 βˆ‡. βˆ‡. NOUN american_scientific_journal-7448 59 34 𝑒 𝑒 X american_scientific_journal-7448 59 35 = = SYM american_scientific_journal-7448 59 36 0 0 NUM american_scientific_journal-7448 59 37 , , PUNCT american_scientific_journal-7448 59 38 ( ( PUNCT american_scientific_journal-7448 59 39 31 31 NUM american_scientific_journal-7448 59 40 ) ) PUNCT american_scientific_journal-7448 59 41 becomes become VERB american_scientific_journal-7448 59 42 βˆ‡2𝑝 βˆ‡2𝑝 NOUN american_scientific_journal-7448 59 43 = = SYM american_scientific_journal-7448 59 44 0 0 NUM american_scientific_journal-7448 59 45 ( ( PUNCT american_scientific_journal-7448 59 46 32 32 NUM american_scientific_journal-7448 59 47 ) ) PUNCT american_scientific_journal-7448 59 48 equation equation NOUN american_scientific_journal-7448 59 49 ( ( PUNCT american_scientific_journal-7448 59 50 32 32 NUM american_scientific_journal-7448 59 51 ) ) PUNCT american_scientific_journal-7448 59 52 shows show VERB american_scientific_journal-7448 59 53 that that SCONJ american_scientific_journal-7448 59 54 for for ADP american_scientific_journal-7448 59 55 very very ADV american_scientific_journal-7448 59 56 slow slow ADJ american_scientific_journal-7448 59 57 motion motion NOUN american_scientific_journal-7448 59 58 the the DET american_scientific_journal-7448 59 59 pressure pressure NOUN american_scientific_journal-7448 59 60 𝑝 𝑝 NOUN american_scientific_journal-7448 59 61 satisfies satisfy VERB american_scientific_journal-7448 59 62 the the DET american_scientific_journal-7448 59 63 laplace laplace NOUN american_scientific_journal-7448 59 64 's 's PART american_scientific_journal-7448 59 65 equation equation NOUN american_scientific_journal-7448 59 66 and and CCONJ american_scientific_journal-7448 59 67 it it PRON american_scientific_journal-7448 59 68 is be AUX american_scientific_journal-7448 59 69 therefore therefore ADV american_scientific_journal-7448 59 70 a a DET american_scientific_journal-7448 59 71 harmonic harmonic ADJ american_scientific_journal-7448 59 72 function function NOUN american_scientific_journal-7448 59 73 . . PUNCT american_scientific_journal-7448 60 1 the the DET american_scientific_journal-7448 60 2 solutions solution NOUN american_scientific_journal-7448 60 3 of of ADP american_scientific_journal-7448 60 4 ( ( PUNCT american_scientific_journal-7448 60 5 32 32 NUM american_scientific_journal-7448 60 6 ) ) PUNCT american_scientific_journal-7448 60 7 are be AUX american_scientific_journal-7448 60 8 series series NOUN american_scientific_journal-7448 60 9 termed term VERB american_scientific_journal-7448 60 10 circular circular ADJ american_scientific_journal-7448 60 11 harmonics harmonic NOUN american_scientific_journal-7448 60 12 of of ADP american_scientific_journal-7448 60 13 integral integral ADJ american_scientific_journal-7448 60 14 degree degree NOUN american_scientific_journal-7448 60 15 . . PUNCT american_scientific_journal-7448 61 1 hence hence ADV american_scientific_journal-7448 61 2 the the DET american_scientific_journal-7448 61 3 solution solution NOUN american_scientific_journal-7448 61 4 of of ADP american_scientific_journal-7448 61 5 equation equation NOUN american_scientific_journal-7448 61 6 32 32 NUM american_scientific_journal-7448 61 7 , , PUNCT american_scientific_journal-7448 61 8 batchelor batchelor PROPN american_scientific_journal-7448 62 1 [ [ X american_scientific_journal-7448 62 2 8,9 8,9 NUM american_scientific_journal-7448 62 3 ] ] PUNCT american_scientific_journal-7448 62 4 is be AUX american_scientific_journal-7448 62 5 of of ADP american_scientific_journal-7448 62 6 the the DET american_scientific_journal-7448 62 7 form form NOUN american_scientific_journal-7448 62 8 𝑝 𝑝 NOUN american_scientific_journal-7448 62 9 = = PUNCT american_scientific_journal-7448 62 10 βˆ’ βˆ’ PROPN american_scientific_journal-7448 62 11 1 1 NUM american_scientific_journal-7448 62 12 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 62 13 π›Όπ‘ˆ π›Όπ‘ˆ PROPN american_scientific_journal-7448 62 14 cos cos NOUN american_scientific_journal-7448 62 15 πœƒ πœƒ X american_scientific_journal-7448 62 16 ( ( PUNCT american_scientific_journal-7448 62 17 33 33 NUM american_scientific_journal-7448 62 18 ) ) PUNCT american_scientific_journal-7448 62 19 here here ADV american_scientific_journal-7448 62 20 𝛼 𝛼 NOUN american_scientific_journal-7448 62 21 is be AUX american_scientific_journal-7448 62 22 a a DET american_scientific_journal-7448 62 23 constant constant ADJ american_scientific_journal-7448 62 24 . . PUNCT american_scientific_journal-7448 63 1 consider consider VERB american_scientific_journal-7448 63 2 the the DET american_scientific_journal-7448 63 3 x x NOUN american_scientific_journal-7448 63 4 - - NOUN american_scientific_journal-7448 63 5 component component NOUN american_scientific_journal-7448 63 6 of of ADP american_scientific_journal-7448 63 7 ( ( PUNCT american_scientific_journal-7448 63 8 30 30 NUM american_scientific_journal-7448 63 9 ) ) PUNCT american_scientific_journal-7448 63 10 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 63 11 πœ•π‘₯ πœ•π‘₯ NOUN american_scientific_journal-7448 63 12 = = SYM american_scientific_journal-7448 63 13 πœ‡ πœ‡ X american_scientific_journal-7448 63 14 ( ( PUNCT american_scientific_journal-7448 63 15 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-7448 63 16 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-7448 63 17 + + CCONJ american_scientific_journal-7448 63 18 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-7448 63 19 πœ•π‘¦2 πœ•π‘¦2 NUM american_scientific_journal-7448 63 20 ) ) PUNCT american_scientific_journal-7448 63 21 ( ( PUNCT american_scientific_journal-7448 63 22 34 34 NUM american_scientific_journal-7448 63 23 ) ) PUNCT american_scientific_journal-7448 63 24 american american ADJ american_scientific_journal-7448 63 25 academic academic ADJ american_scientific_journal-7448 63 26 scientific scientific ADJ american_scientific_journal-7448 63 27 research research NOUN american_scientific_journal-7448 63 28 journal journal NOUN american_scientific_journal-7448 63 29 for for ADP american_scientific_journal-7448 63 30 engineering engineering NOUN american_scientific_journal-7448 63 31 , , PUNCT american_scientific_journal-7448 63 32 technology technology NOUN american_scientific_journal-7448 63 33 , , PUNCT american_scientific_journal-7448 63 34 and and CCONJ american_scientific_journal-7448 63 35 sciences science NOUN american_scientific_journal-7448 63 36 ( ( PUNCT american_scientific_journal-7448 63 37 asrjets asrjet NOUN american_scientific_journal-7448 63 38 ) ) PUNCT american_scientific_journal-7448 63 39 ( ( PUNCT american_scientific_journal-7448 63 40 2022 2022 NUM american_scientific_journal-7448 63 41 ) ) PUNCT american_scientific_journal-7448 63 42 volume volume NOUN american_scientific_journal-7448 63 43 86 86 NUM american_scientific_journal-7448 63 44 , , PUNCT american_scientific_journal-7448 63 45 no no DET american_scientific_journal-7448 63 46 1 1 NUM american_scientific_journal-7448 63 47 , , PUNCT american_scientific_journal-7448 63 48 pp143 pp143 PROPN american_scientific_journal-7448 63 49 - - PUNCT american_scientific_journal-7448 63 50 156 156 NUM american_scientific_journal-7448 63 51 150 150 NUM american_scientific_journal-7448 63 52 resolving resolving NOUN american_scientific_journal-7448 63 53 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 63 54 πœ•π‘₯ πœ•π‘₯ NOUN american_scientific_journal-7448 63 55 into into ADP american_scientific_journal-7448 63 56 its its PRON american_scientific_journal-7448 63 57 components component NOUN american_scientific_journal-7448 63 58 in in ADP american_scientific_journal-7448 63 59 polar polar ADJ american_scientific_journal-7448 63 60 form form NOUN american_scientific_journal-7448 63 61 ( ( PUNCT american_scientific_journal-7448 63 62 see see VERB american_scientific_journal-7448 63 63 figure figure NOUN american_scientific_journal-7448 63 64 1 1 NUM american_scientific_journal-7448 63 65 ) ) PUNCT american_scientific_journal-7448 63 66 , , PUNCT american_scientific_journal-7448 63 67 we we PRON american_scientific_journal-7448 63 68 have have VERB american_scientific_journal-7448 63 69 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 63 70 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-7448 63 71 = = SYM american_scientific_journal-7448 63 72 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 63 73 πœ•π‘Ÿ πœ•π‘Ÿ PUNCT american_scientific_journal-7448 64 1 cos cos NOUN american_scientific_journal-7448 64 2 πœƒ πœƒ NOUN american_scientific_journal-7448 64 3 βˆ’ βˆ’ NUM american_scientific_journal-7448 64 4 1 1 NUM american_scientific_journal-7448 65 1 π‘Ÿ π‘Ÿ PRON american_scientific_journal-7448 65 2 πœ•π‘ πœ•π‘ VERB american_scientific_journal-7448 65 3 πœ•πœƒ πœ•πœƒ ADJ american_scientific_journal-7448 65 4 sin sin NOUN american_scientific_journal-7448 65 5 πœƒ πœƒ X american_scientific_journal-7448 65 6 ( ( PUNCT american_scientific_journal-7448 65 7 35 35 NUM american_scientific_journal-7448 65 8 ) ) PUNCT american_scientific_journal-7448 65 9 differentiating differentiating NOUN american_scientific_journal-7448 65 10 ( ( PUNCT american_scientific_journal-7448 65 11 33 33 NUM american_scientific_journal-7448 65 12 ) ) PUNCT american_scientific_journal-7448 65 13 with with ADP american_scientific_journal-7448 65 14 respect respect NOUN american_scientific_journal-7448 65 15 to to ADP american_scientific_journal-7448 65 16 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 65 17 and and CCONJ american_scientific_journal-7448 65 18 πœƒ πœƒ PRON american_scientific_journal-7448 65 19 respectively respectively ADV american_scientific_journal-7448 65 20 , , PUNCT american_scientific_journal-7448 65 21 we we PRON american_scientific_journal-7448 65 22 have have VERB american_scientific_journal-7448 65 23 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 65 24 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 65 25 = = SYM american_scientific_journal-7448 65 26 1 1 NUM american_scientific_journal-7448 65 27 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 65 28 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 65 29 cos cos NOUN american_scientific_journal-7448 65 30 πœƒ πœƒ NOUN american_scientific_journal-7448 65 31 and and CCONJ american_scientific_journal-7448 65 32 πœ•π‘ πœ•π‘ VERB american_scientific_journal-7448 65 33 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 65 34 = = SYM american_scientific_journal-7448 65 35 1 1 NUM american_scientific_journal-7448 65 36 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 65 37 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 65 38 sin sin NOUN american_scientific_journal-7448 65 39 πœƒ πœƒ X american_scientific_journal-7448 65 40 ( ( PUNCT american_scientific_journal-7448 65 41 36 36 NUM american_scientific_journal-7448 65 42 ) ) PUNCT american_scientific_journal-7448 65 43 so so ADV american_scientific_journal-7448 65 44 , , PUNCT american_scientific_journal-7448 65 45 πœ•π‘ πœ•π‘ NOUN american_scientific_journal-7448 65 46 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-7448 65 47 = = SYM american_scientific_journal-7448 65 48 1 1 NUM american_scientific_journal-7448 65 49 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 65 50 π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ NOUN american_scientific_journal-7448 65 51 βˆ’ βˆ’ PROPN american_scientific_journal-7448 65 52 1 1 NUM american_scientific_journal-7448 65 53 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 65 54 π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ NOUN american_scientific_journal-7448 65 55 ( ( PUNCT american_scientific_journal-7448 65 56 37 37 NUM american_scientific_journal-7448 65 57 ) ) PUNCT american_scientific_journal-7448 65 58 now now ADV american_scientific_journal-7448 65 59 , , PUNCT american_scientific_journal-7448 65 60 resolving resolve VERB american_scientific_journal-7448 65 61 𝑒 𝑒 NOUN american_scientific_journal-7448 65 62 into into ADP american_scientific_journal-7448 65 63 its its PRON american_scientific_journal-7448 65 64 components component NOUN american_scientific_journal-7448 65 65 , , PUNCT american_scientific_journal-7448 65 66 in in ADP american_scientific_journal-7448 65 67 polar polar ADJ american_scientific_journal-7448 65 68 form form NOUN american_scientific_journal-7448 65 69 ( ( PUNCT american_scientific_journal-7448 65 70 see see VERB american_scientific_journal-7448 65 71 figure figure NOUN american_scientific_journal-7448 65 72 1 1 NUM american_scientific_journal-7448 65 73 ) ) PUNCT american_scientific_journal-7448 65 74 , , PUNCT american_scientific_journal-7448 65 75 we we PRON american_scientific_journal-7448 65 76 obtain obtain VERB american_scientific_journal-7448 65 77 𝑒 𝑒 NOUN american_scientific_journal-7448 65 78 = = PUNCT american_scientific_journal-7448 65 79 π‘’π‘Ÿ π‘’π‘Ÿ ADP american_scientific_journal-7448 65 80 cos cos PROPN american_scientific_journal-7448 65 81 πœƒ πœƒ X american_scientific_journal-7448 65 82 βˆ’ βˆ’ PROPN american_scientific_journal-7448 65 83 π‘’πœƒ π‘’πœƒ NOUN american_scientific_journal-7448 65 84 sin sin NOUN american_scientific_journal-7448 65 85 πœƒ πœƒ PROPN american_scientific_journal-7448 65 86 ( ( PUNCT american_scientific_journal-7448 65 87 38 38 NUM american_scientific_journal-7448 65 88 ) ) PUNCT american_scientific_journal-7448 66 1 π‘’π‘Ÿ π‘’π‘Ÿ ADV american_scientific_journal-7448 66 2 = = SYM american_scientific_journal-7448 66 3 βˆ’ βˆ’ PROPN american_scientific_journal-7448 66 4 1 1 NUM american_scientific_journal-7448 67 1 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 67 2 πœ•πœ‘ πœ•πœ‘ NOUN american_scientific_journal-7448 67 3 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 67 4 and and CCONJ american_scientific_journal-7448 67 5 π‘’πœƒ π‘’πœƒ NOUN american_scientific_journal-7448 67 6 = = NOUN american_scientific_journal-7448 67 7 πœ•πœ‘ πœ•πœ‘ NOUN american_scientific_journal-7448 67 8 πœ•π‘Ÿ πœ•π‘Ÿ PROPN american_scientific_journal-7448 67 9 ( ( PUNCT american_scientific_journal-7448 67 10 39 39 NUM american_scientific_journal-7448 67 11 ) ) PUNCT american_scientific_journal-7448 67 12 by by ADP american_scientific_journal-7448 67 13 equation equation NOUN american_scientific_journal-7448 67 14 ( ( PUNCT american_scientific_journal-7448 67 15 29 29 NUM american_scientific_journal-7448 67 16 ) ) PUNCT american_scientific_journal-7448 67 17 , , PUNCT american_scientific_journal-7448 67 18 πœ•πœ‘ πœ•πœ‘ NOUN american_scientific_journal-7448 67 19 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 67 20 = = PUNCT american_scientific_journal-7448 67 21 βˆ’π‘“(π‘Ÿ)π‘ˆ βˆ’π‘“(π‘Ÿ)π‘ˆ PROPN american_scientific_journal-7448 67 22 cos cos PROPN american_scientific_journal-7448 67 23 πœƒ πœƒ PROPN american_scientific_journal-7448 67 24 and and CCONJ american_scientific_journal-7448 67 25 πœ•πœ‘ πœ•πœ‘ NOUN american_scientific_journal-7448 67 26 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 67 27 = = SYM american_scientific_journal-7448 67 28 βˆ’π‘“β€²(π‘Ÿ)π‘ˆ βˆ’π‘“β€²(π‘Ÿ)π‘ˆ ADJ american_scientific_journal-7448 67 29 sin sin NOUN american_scientific_journal-7448 67 30 πœƒ πœƒ X american_scientific_journal-7448 67 31 ( ( PUNCT american_scientific_journal-7448 67 32 40 40 NUM american_scientific_journal-7448 67 33 ) ) PUNCT american_scientific_journal-7448 67 34 american american ADJ american_scientific_journal-7448 67 35 academic academic ADJ american_scientific_journal-7448 67 36 scientific scientific ADJ american_scientific_journal-7448 67 37 research research NOUN american_scientific_journal-7448 67 38 journal journal NOUN american_scientific_journal-7448 67 39 for for ADP american_scientific_journal-7448 67 40 engineering engineering NOUN american_scientific_journal-7448 67 41 , , PUNCT american_scientific_journal-7448 67 42 technology technology NOUN american_scientific_journal-7448 67 43 , , PUNCT american_scientific_journal-7448 67 44 and and CCONJ american_scientific_journal-7448 67 45 sciences science NOUN american_scientific_journal-7448 67 46 ( ( PUNCT american_scientific_journal-7448 67 47 asrjets asrjet NOUN american_scientific_journal-7448 67 48 ) ) PUNCT american_scientific_journal-7448 67 49 ( ( PUNCT american_scientific_journal-7448 67 50 2022 2022 NUM american_scientific_journal-7448 67 51 ) ) PUNCT american_scientific_journal-7448 67 52 volume volume NOUN american_scientific_journal-7448 67 53 86 86 NUM american_scientific_journal-7448 67 54 , , PUNCT american_scientific_journal-7448 67 55 no no DET american_scientific_journal-7448 67 56 1 1 NUM american_scientific_journal-7448 67 57 , , PUNCT american_scientific_journal-7448 67 58 pp143 pp143 PROPN american_scientific_journal-7448 67 59 - - SYM american_scientific_journal-7448 67 60 156 156 NUM american_scientific_journal-7448 67 61 151 151 NUM american_scientific_journal-7448 67 62 so so ADV american_scientific_journal-7448 67 63 , , PUNCT american_scientific_journal-7448 67 64 𝑒 𝑒 X american_scientific_journal-7448 67 65 = = SYM american_scientific_journal-7448 67 66 βˆ’ βˆ’ PROPN american_scientific_journal-7448 67 67 1 1 NUM american_scientific_journal-7448 67 68 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 67 69 πœ•πœ‘ πœ•πœ‘ VERB american_scientific_journal-7448 67 70 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 67 71 cos cos PROPN american_scientific_journal-7448 67 72 πœƒ πœƒ PRON american_scientific_journal-7448 67 73 βˆ’ βˆ’ NOUN american_scientific_journal-7448 67 74 πœ•πœ‘ πœ•πœ‘ NOUN american_scientific_journal-7448 67 75 πœ•π‘Ÿ πœ•π‘Ÿ ADV american_scientific_journal-7448 67 76 sin sin VERB american_scientific_journal-7448 67 77 πœƒ πœƒ NOUN american_scientific_journal-7448 67 78 𝑒 𝑒 NOUN american_scientific_journal-7448 67 79 = = SYM american_scientific_journal-7448 67 80 1 1 NUM american_scientific_journal-7448 67 81 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 67 82 𝑓(π‘Ÿ)π‘ˆπ‘π‘œπ‘ 2πœƒ 𝑓(π‘Ÿ)π‘ˆπ‘π‘œπ‘ 2πœƒ ADJ american_scientific_journal-7448 67 83 + + NUM american_scientific_journal-7448 67 84 𝑓′(π‘Ÿ)π‘ˆπ‘ π‘–π‘›2πœƒ 𝑓′(π‘Ÿ)π‘ˆπ‘ π‘–π‘›2πœƒ NUM american_scientific_journal-7448 67 85 ( ( PUNCT american_scientific_journal-7448 67 86 41 41 NUM american_scientific_journal-7448 67 87 ) ) PUNCT american_scientific_journal-7448 67 88 expressing express VERB american_scientific_journal-7448 67 89 equation equation NOUN american_scientific_journal-7448 67 90 ( ( PUNCT american_scientific_journal-7448 67 91 34 34 NUM american_scientific_journal-7448 67 92 ) ) PUNCT american_scientific_journal-7448 67 93 in in ADP american_scientific_journal-7448 67 94 polar polar ADJ american_scientific_journal-7448 67 95 form form NOUN american_scientific_journal-7448 67 96 , , PUNCT american_scientific_journal-7448 67 97 we we PRON american_scientific_journal-7448 67 98 have have VERB american_scientific_journal-7448 67 99 1 1 NUM american_scientific_journal-7448 67 100 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 67 101 π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ NOUN american_scientific_journal-7448 67 102 βˆ’ βˆ’ PROPN american_scientific_journal-7448 67 103 1 1 NUM american_scientific_journal-7448 67 104 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 67 105 π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ NOUN american_scientific_journal-7448 67 106 = = PUNCT american_scientific_journal-7448 67 107 πœ‡ πœ‡ X american_scientific_journal-7448 67 108 ( ( PUNCT american_scientific_journal-7448 67 109 πœ•2 πœ•2 NOUN american_scientific_journal-7448 67 110 πœ•π‘Ÿ2 πœ•π‘Ÿ2 NOUN american_scientific_journal-7448 67 111 + + CCONJ american_scientific_journal-7448 67 112 1 1 NUM american_scientific_journal-7448 67 113 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 67 114 πœ• πœ• PROPN american_scientific_journal-7448 67 115 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 67 116 + + CCONJ american_scientific_journal-7448 67 117 1 1 NUM american_scientific_journal-7448 67 118 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 67 119 πœ•2 πœ•2 ADV american_scientific_journal-7448 67 120 πœ•πœƒ2 πœ•πœƒ2 VERB american_scientific_journal-7448 67 121 ) ) PUNCT american_scientific_journal-7448 67 122 1 1 NUM american_scientific_journal-7448 67 123 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 67 124 𝑓(π‘Ÿ)π‘ˆπ‘π‘œπ‘ 2πœƒ 𝑓(π‘Ÿ)π‘ˆπ‘π‘œπ‘ 2πœƒ ADJ american_scientific_journal-7448 67 125 + + NUM american_scientific_journal-7448 68 1 𝑓′(π‘Ÿ)π‘ˆπ‘ π‘–π‘›2πœƒ 𝑓′(π‘Ÿ)π‘ˆπ‘ π‘–π‘›2πœƒ NUM american_scientific_journal-7448 68 2 ( ( PUNCT american_scientific_journal-7448 68 3 42 42 NUM american_scientific_journal-7448 68 4 ) ) SYM american_scientific_journal-7448 68 5 1 1 NUM american_scientific_journal-7448 68 6 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 68 7 π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ π›Όπ‘ˆπ‘π‘œπ‘ 2πœƒ NOUN american_scientific_journal-7448 68 8 βˆ’ βˆ’ PROPN american_scientific_journal-7448 68 9 1 1 NUM american_scientific_journal-7448 68 10 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 68 11 π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ π›Όπ‘ˆπ‘ π‘–π‘›2πœƒ NOUN american_scientific_journal-7448 68 12 = = PUNCT american_scientific_journal-7448 68 13 πœ‡ πœ‡ X american_scientific_journal-7448 68 14 ( ( PUNCT american_scientific_journal-7448 68 15 βˆ’ βˆ’ PROPN american_scientific_journal-7448 68 16 1 1 NUM american_scientific_journal-7448 68 17 π‘Ÿ3 π‘Ÿ3 NOUN american_scientific_journal-7448 68 18 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 68 19 ) ) PUNCT american_scientific_journal-7448 68 20 + + CCONJ american_scientific_journal-7448 68 21 1 1 NUM american_scientific_journal-7448 68 22 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 68 23 𝑓′(π‘Ÿ 𝑓′(π‘Ÿ NUM american_scientific_journal-7448 68 24 ) ) PUNCT american_scientific_journal-7448 68 25 + + CCONJ american_scientific_journal-7448 68 26 1 1 NUM american_scientific_journal-7448 68 27 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 68 28 𝑓′′(π‘Ÿ 𝑓′′(π‘Ÿ NUM american_scientific_journal-7448 68 29 ) ) PUNCT american_scientific_journal-7448 68 30 ) ) PUNCT american_scientific_journal-7448 68 31 π‘ˆπ‘π‘œπ‘ 2πœƒ π‘ˆπ‘π‘œπ‘ 2πœƒ NOUN american_scientific_journal-7448 69 1 + + CCONJ american_scientific_journal-7448 69 2 πœ‡ πœ‡ X american_scientific_journal-7448 69 3 ( ( PUNCT american_scientific_journal-7448 69 4 2 2 NUM american_scientific_journal-7448 69 5 π‘Ÿ3 π‘Ÿ3 NOUN american_scientific_journal-7448 69 6 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 69 7 ) ) PUNCT american_scientific_journal-7448 69 8 βˆ’ βˆ’ PROPN american_scientific_journal-7448 69 9 2 2 NUM american_scientific_journal-7448 69 10 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-7448 69 11 𝑓′(π‘Ÿ 𝑓′(π‘Ÿ NUM american_scientific_journal-7448 69 12 ) ) PUNCT american_scientific_journal-7448 69 13 + + CCONJ american_scientific_journal-7448 69 14 1 1 NUM american_scientific_journal-7448 69 15 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 69 16 𝑓′′(π‘Ÿ 𝑓′′(π‘Ÿ NUM american_scientific_journal-7448 69 17 ) ) PUNCT american_scientific_journal-7448 69 18 + + CCONJ american_scientific_journal-7448 69 19 𝑓′′′(π‘Ÿ 𝑓′′′(π‘Ÿ NUM american_scientific_journal-7448 69 20 ) ) PUNCT american_scientific_journal-7448 69 21 ) ) PUNCT american_scientific_journal-7448 70 1 π‘ˆπ‘ π‘–π‘›2πœƒ π‘ˆπ‘ π‘–π‘›2πœƒ PROPN american_scientific_journal-7448 70 2 ( ( PUNCT american_scientific_journal-7448 70 3 43 43 NUM american_scientific_journal-7448 70 4 ) ) PUNCT american_scientific_journal-7448 70 5 equating equate VERB american_scientific_journal-7448 70 6 coefficients coefficient NOUN american_scientific_journal-7448 70 7 of of ADP american_scientific_journal-7448 70 8 π‘ˆπ‘π‘œπ‘ 2πœƒ π‘ˆπ‘π‘œπ‘ 2πœƒ NOUN american_scientific_journal-7448 70 9 and and CCONJ american_scientific_journal-7448 70 10 π‘ˆπ‘ π‘–π‘›2πœƒ π‘ˆπ‘ π‘–π‘›2πœƒ NUM american_scientific_journal-7448 70 11 , , PUNCT american_scientific_journal-7448 70 12 we we PRON american_scientific_journal-7448 70 13 have have VERB american_scientific_journal-7448 70 14 the the DET american_scientific_journal-7448 70 15 following follow VERB american_scientific_journal-7448 70 16 odes ode NOUN american_scientific_journal-7448 70 17 π‘Ÿ2𝑓′′(π‘Ÿ π‘Ÿ2𝑓′′(π‘Ÿ NOUN american_scientific_journal-7448 70 18 ) ) PUNCT american_scientific_journal-7448 71 1 + + CCONJ american_scientific_journal-7448 71 2 π‘Ÿπ‘“β€²(π‘Ÿ π‘Ÿπ‘“β€²(π‘Ÿ X american_scientific_journal-7448 71 3 ) ) PUNCT american_scientific_journal-7448 71 4 βˆ’ βˆ’ ADP american_scientific_journal-7448 71 5 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 71 6 ) ) PUNCT american_scientific_journal-7448 71 7 = = SYM american_scientific_journal-7448 71 8 𝛼 𝛼 NOUN american_scientific_journal-7448 71 9 πœ‡ πœ‡ ADP american_scientific_journal-7448 71 10 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 71 11 ( ( PUNCT american_scientific_journal-7448 71 12 44a 44a ADJ american_scientific_journal-7448 71 13 ) ) PUNCT american_scientific_journal-7448 71 14 π‘Ÿ3𝑓′′′(π‘Ÿ π‘Ÿ3𝑓′′′(π‘Ÿ NOUN american_scientific_journal-7448 71 15 ) ) PUNCT american_scientific_journal-7448 72 1 + + SYM american_scientific_journal-7448 72 2 π‘Ÿ2𝑓′′(π‘Ÿ π‘Ÿ2𝑓′′(π‘Ÿ NOUN american_scientific_journal-7448 72 3 ) ) PUNCT american_scientific_journal-7448 72 4 βˆ’ βˆ’ ADP american_scientific_journal-7448 72 5 2π‘Ÿπ‘“β€²(π‘Ÿ 2π‘Ÿπ‘“β€²(π‘Ÿ NUM american_scientific_journal-7448 72 6 ) ) PUNCT american_scientific_journal-7448 72 7 + + CCONJ american_scientific_journal-7448 72 8 2𝑓(π‘Ÿ 2𝑓(π‘Ÿ NOUN american_scientific_journal-7448 72 9 ) ) PUNCT american_scientific_journal-7448 73 1 = = SYM american_scientific_journal-7448 73 2 βˆ’ βˆ’ NOUN american_scientific_journal-7448 73 3 𝛼 𝛼 NOUN american_scientific_journal-7448 73 4 πœ‡ πœ‡ ADP american_scientific_journal-7448 73 5 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 73 6 ( ( PUNCT american_scientific_journal-7448 73 7 44b 44b NOUN american_scientific_journal-7448 73 8 ) ) PUNCT american_scientific_journal-7448 73 9 solving solve VERB american_scientific_journal-7448 73 10 the the DET american_scientific_journal-7448 73 11 non non ADJ american_scientific_journal-7448 73 12 - - ADJ american_scientific_journal-7448 73 13 homogeneous homogeneous ADJ american_scientific_journal-7448 73 14 euler euler PROPN american_scientific_journal-7448 73 15 - - PUNCT american_scientific_journal-7448 73 16 cauchy cauchy ADJ american_scientific_journal-7448 73 17 equation equation NOUN american_scientific_journal-7448 73 18 ( ( PUNCT american_scientific_journal-7448 73 19 44a 44a X american_scientific_journal-7448 73 20 ) ) PUNCT american_scientific_journal-7448 73 21 subject subject NOUN american_scientific_journal-7448 73 22 to to ADP american_scientific_journal-7448 73 23 the the DET american_scientific_journal-7448 73 24 boundary boundary ADJ american_scientific_journal-7448 73 25 conditions condition NOUN american_scientific_journal-7448 73 26 𝑓 𝑓 PRON american_scientific_journal-7448 73 27 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 73 28 ( ( PUNCT american_scientific_journal-7448 73 29 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 73 30 ) ) PUNCT american_scientific_journal-7448 73 31 = = SYM american_scientific_journal-7448 73 32 βˆ’1 βˆ’1 NOUN american_scientific_journal-7448 73 33 and and CCONJ american_scientific_journal-7448 73 34 𝑓′(π‘Ÿ0 𝑓′(π‘Ÿ0 NOUN american_scientific_journal-7448 73 35 ) ) PUNCT american_scientific_journal-7448 73 36 = = SYM american_scientific_journal-7448 73 37 βˆ’1 βˆ’1 NOUN american_scientific_journal-7448 73 38 , , PUNCT american_scientific_journal-7448 73 39 gives give VERB american_scientific_journal-7448 73 40 american american ADJ american_scientific_journal-7448 73 41 academic academic ADJ american_scientific_journal-7448 73 42 scientific scientific ADJ american_scientific_journal-7448 73 43 research research NOUN american_scientific_journal-7448 73 44 journal journal NOUN american_scientific_journal-7448 73 45 for for ADP american_scientific_journal-7448 73 46 engineering engineering NOUN american_scientific_journal-7448 73 47 , , PUNCT american_scientific_journal-7448 73 48 technology technology NOUN american_scientific_journal-7448 73 49 , , PUNCT american_scientific_journal-7448 73 50 and and CCONJ american_scientific_journal-7448 73 51 sciences science NOUN american_scientific_journal-7448 73 52 ( ( PUNCT american_scientific_journal-7448 73 53 asrjets asrjet NOUN american_scientific_journal-7448 73 54 ) ) PUNCT american_scientific_journal-7448 73 55 ( ( PUNCT american_scientific_journal-7448 73 56 2022 2022 NUM american_scientific_journal-7448 73 57 ) ) PUNCT american_scientific_journal-7448 73 58 volume volume NOUN american_scientific_journal-7448 73 59 86 86 NUM american_scientific_journal-7448 73 60 , , PUNCT american_scientific_journal-7448 73 61 no no DET american_scientific_journal-7448 73 62 1 1 NUM american_scientific_journal-7448 73 63 , , PUNCT american_scientific_journal-7448 73 64 pp143 pp143 PROPN american_scientific_journal-7448 73 65 - - PUNCT american_scientific_journal-7448 73 66 156 156 NUM american_scientific_journal-7448 73 67 152 152 NUM american_scientific_journal-7448 73 68 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 73 69 ) ) PUNCT american_scientific_journal-7448 73 70 = = SYM american_scientific_journal-7448 73 71 ( ( PUNCT american_scientific_journal-7448 73 72 βˆ’1 βˆ’1 NOUN american_scientific_journal-7448 73 73 βˆ’ βˆ’ NOUN american_scientific_journal-7448 73 74 𝛼 𝛼 NOUN american_scientific_journal-7448 73 75 2πœ‡ 2πœ‡ NOUN american_scientific_journal-7448 73 76 ln ln ADJ american_scientific_journal-7448 73 77 π‘Ÿ0)π‘Ÿ π‘Ÿ0)π‘Ÿ NOUN american_scientific_journal-7448 74 1 + + CCONJ american_scientific_journal-7448 74 2 𝛼 𝛼 PRON american_scientific_journal-7448 74 3 4πœ‡ 4πœ‡ NUM american_scientific_journal-7448 74 4 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 74 5 2 2 NUM american_scientific_journal-7448 74 6 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 74 7 + + NOUN american_scientific_journal-7448 74 8 𝛼 𝛼 PRON american_scientific_journal-7448 74 9 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 74 10 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 74 11 ln ln NOUN american_scientific_journal-7448 74 12 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 74 13 βˆ’ βˆ’ NOUN american_scientific_journal-7448 74 14 𝛼 𝛼 PRON american_scientific_journal-7448 74 15 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 74 16 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 74 17 ( ( PUNCT american_scientific_journal-7448 74 18 45 45 NUM american_scientific_journal-7448 74 19 ) ) PUNCT american_scientific_journal-7448 74 20 now now ADV american_scientific_journal-7448 74 21 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 74 22 ) ) PUNCT american_scientific_journal-7448 74 23 also also ADV american_scientific_journal-7448 74 24 satisfy satisfy VERB american_scientific_journal-7448 74 25 the the DET american_scientific_journal-7448 74 26 third third ADJ american_scientific_journal-7448 74 27 order order NOUN american_scientific_journal-7448 74 28 ode ode X american_scientific_journal-7448 74 29 ( ( PUNCT american_scientific_journal-7448 74 30 44b 44b NOUN american_scientific_journal-7448 74 31 ) ) PUNCT american_scientific_journal-7448 74 32 . . PUNCT american_scientific_journal-7448 75 1 substituting substitute VERB american_scientific_journal-7448 75 2 for for ADP american_scientific_journal-7448 75 3 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN american_scientific_journal-7448 75 4 ) ) PUNCT american_scientific_journal-7448 75 5 in in ADP american_scientific_journal-7448 75 6 equation equation NOUN american_scientific_journal-7448 75 7 ( ( PUNCT american_scientific_journal-7448 75 8 29 29 NUM american_scientific_journal-7448 75 9 ) ) PUNCT american_scientific_journal-7448 75 10 , , PUNCT american_scientific_journal-7448 75 11 the the DET american_scientific_journal-7448 75 12 expression expression NOUN american_scientific_journal-7448 75 13 for for ADP american_scientific_journal-7448 75 14 the the DET american_scientific_journal-7448 75 15 stream stream NOUN american_scientific_journal-7448 75 16 function function NOUN american_scientific_journal-7448 75 17 ( ( PUNCT american_scientific_journal-7448 75 18 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 75 19 , , PUNCT american_scientific_journal-7448 75 20 πœƒ πœƒ NOUN american_scientific_journal-7448 75 21 ) ) PUNCT american_scientific_journal-7448 75 22 ) ) PUNCT american_scientific_journal-7448 75 23 becomes become VERB american_scientific_journal-7448 75 24 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 75 25 , , PUNCT american_scientific_journal-7448 75 26 πœƒ πœƒ NOUN american_scientific_journal-7448 75 27 ) ) PUNCT american_scientific_journal-7448 75 28 = = SYM american_scientific_journal-7448 76 1 ( ( PUNCT american_scientific_journal-7448 76 2 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 76 3 + + CCONJ american_scientific_journal-7448 76 4 𝛼 𝛼 PRON american_scientific_journal-7448 76 5 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 76 6 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 7 ln ln ADJ american_scientific_journal-7448 76 8 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 76 9 + + CCONJ american_scientific_journal-7448 76 10 𝛼 𝛼 PRON american_scientific_journal-7448 76 11 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 76 12 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 13 βˆ’ βˆ’ NOUN american_scientific_journal-7448 76 14 𝛼 𝛼 PRON american_scientific_journal-7448 76 15 4πœ‡ 4πœ‡ NUM american_scientific_journal-7448 76 16 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 76 17 2 2 NUM american_scientific_journal-7448 76 18 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 19 βˆ’ βˆ’ NOUN american_scientific_journal-7448 76 20 𝛼 𝛼 PRON american_scientific_journal-7448 76 21 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 76 22 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 23 ln ln ADJ american_scientific_journal-7448 76 24 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 76 25 ) ) PUNCT american_scientific_journal-7448 76 26 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 76 27 sin sin NOUN american_scientific_journal-7448 76 28 πœƒ πœƒ NOUN american_scientific_journal-7448 76 29 ( ( PUNCT american_scientific_journal-7448 76 30 46 46 NUM american_scientific_journal-7448 76 31 ) ) PUNCT american_scientific_journal-7448 76 32 πœ•πœ‘(π‘Ÿ πœ•πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 76 33 , , PUNCT american_scientific_journal-7448 76 34 πœƒ πœƒ NOUN american_scientific_journal-7448 76 35 ) ) PUNCT american_scientific_journal-7448 76 36 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 76 37 = = SYM american_scientific_journal-7448 76 38 ( ( PUNCT american_scientific_journal-7448 76 39 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 40 + + CCONJ american_scientific_journal-7448 76 41 𝛼 𝛼 PRON american_scientific_journal-7448 76 42 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 76 43 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 44 ln ln ADJ american_scientific_journal-7448 76 45 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 76 46 + + CCONJ american_scientific_journal-7448 76 47 𝛼 𝛼 PRON american_scientific_journal-7448 76 48 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 76 49 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 50 βˆ’ βˆ’ NOUN american_scientific_journal-7448 76 51 𝛼 𝛼 PRON american_scientific_journal-7448 76 52 4πœ‡ 4πœ‡ NUM american_scientific_journal-7448 76 53 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 76 54 2 2 NUM american_scientific_journal-7448 76 55 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 56 βˆ’ βˆ’ NOUN american_scientific_journal-7448 76 57 𝛼 𝛼 PRON american_scientific_journal-7448 76 58 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 76 59 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 76 60 ln ln ADJ american_scientific_journal-7448 76 61 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 76 62 ) ) PUNCT american_scientific_journal-7448 76 63 π‘ˆ π‘ˆ PROPN american_scientific_journal-7448 76 64 cos cos PROPN american_scientific_journal-7448 76 65 πœƒ πœƒ X american_scientific_journal-7448 76 66 πœ•πœ‘(π‘Ÿ πœ•πœ‘(π‘Ÿ ADJ american_scientific_journal-7448 76 67 , , PUNCT american_scientific_journal-7448 76 68 πœƒ πœƒ NOUN american_scientific_journal-7448 76 69 ) ) PUNCT american_scientific_journal-7448 76 70 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 76 71 = = SYM american_scientific_journal-7448 77 1 ( ( PUNCT american_scientific_journal-7448 77 2 1 1 NUM american_scientific_journal-7448 77 3 + + NUM american_scientific_journal-7448 77 4 𝛼 𝛼 PRON american_scientific_journal-7448 77 5 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 77 6 ln ln ADJ american_scientific_journal-7448 77 7 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 77 8 + + CCONJ american_scientific_journal-7448 77 9 𝛼 𝛼 PRON american_scientific_journal-7448 77 10 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 77 11 ( ( PUNCT american_scientific_journal-7448 77 12 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 77 13 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 77 14 ) ) PUNCT american_scientific_journal-7448 77 15 2 2 NUM american_scientific_journal-7448 77 16 βˆ’ βˆ’ PROPN american_scientific_journal-7448 77 17 𝛼 𝛼 PRON american_scientific_journal-7448 77 18 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 77 19 ln ln ADP american_scientific_journal-7448 78 1 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 78 2 βˆ’ βˆ’ NOUN american_scientific_journal-7448 78 3 𝛼 𝛼 PRON american_scientific_journal-7448 78 4 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 78 5 ) ) PUNCT american_scientific_journal-7448 78 6 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 78 7 sin sin NOUN american_scientific_journal-7448 78 8 πœƒ πœƒ NOUN american_scientific_journal-7448 78 9 π‘’π‘Ÿ π‘’π‘Ÿ ADV american_scientific_journal-7448 78 10 = = SYM american_scientific_journal-7448 78 11 βˆ’ βˆ’ PROPN american_scientific_journal-7448 79 1 1 1 NUM american_scientific_journal-7448 79 2 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 79 3 πœ•πœ‘(π‘Ÿ πœ•πœ‘(π‘Ÿ ADP american_scientific_journal-7448 79 4 , , PUNCT american_scientific_journal-7448 79 5 πœƒ πœƒ NOUN american_scientific_journal-7448 79 6 ) ) PUNCT american_scientific_journal-7448 79 7 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 79 8 = = PUNCT american_scientific_journal-7448 80 1 βˆ’(1 βˆ’(1 NOUN american_scientific_journal-7448 80 2 + + CCONJ american_scientific_journal-7448 80 3 𝛼 𝛼 PRON american_scientific_journal-7448 80 4 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 80 5 ln ln ADJ american_scientific_journal-7448 80 6 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 80 7 + + CCONJ american_scientific_journal-7448 80 8 𝛼 𝛼 PRON american_scientific_journal-7448 80 9 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 80 10 βˆ’ βˆ’ NOUN american_scientific_journal-7448 80 11 𝛼 𝛼 PRON american_scientific_journal-7448 80 12 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 80 13 ( ( PUNCT american_scientific_journal-7448 80 14 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 80 15 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 80 16 ) ) PUNCT american_scientific_journal-7448 80 17 2 2 NUM american_scientific_journal-7448 80 18 βˆ’ βˆ’ PROPN american_scientific_journal-7448 80 19 𝛼 𝛼 PRON american_scientific_journal-7448 80 20 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 80 21 ln ln ADJ american_scientific_journal-7448 80 22 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 80 23 ) ) PUNCT american_scientific_journal-7448 80 24 π‘ˆ π‘ˆ PROPN american_scientific_journal-7448 80 25 cos cos PROPN american_scientific_journal-7448 80 26 πœƒ πœƒ PROPN american_scientific_journal-7448 80 27 ( ( PUNCT american_scientific_journal-7448 80 28 47 47 NUM american_scientific_journal-7448 80 29 ) ) PUNCT american_scientific_journal-7448 80 30 π‘’πœƒ π‘’πœƒ NOUN american_scientific_journal-7448 80 31 = = PUNCT american_scientific_journal-7448 80 32 πœ•πœ‘(π‘Ÿ πœ•πœ‘(π‘Ÿ PROPN american_scientific_journal-7448 80 33 , , PUNCT american_scientific_journal-7448 80 34 πœƒ πœƒ NOUN american_scientific_journal-7448 80 35 ) ) PUNCT american_scientific_journal-7448 80 36 πœ•πœƒ πœ•πœƒ NOUN american_scientific_journal-7448 80 37 = = PUNCT american_scientific_journal-7448 80 38 ( ( PUNCT american_scientific_journal-7448 80 39 1 1 NUM american_scientific_journal-7448 80 40 + + NUM american_scientific_journal-7448 80 41 𝛼 𝛼 PRON american_scientific_journal-7448 80 42 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 80 43 ln ln ADJ american_scientific_journal-7448 80 44 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 81 1 + + CCONJ american_scientific_journal-7448 81 2 𝛼 𝛼 PRON american_scientific_journal-7448 81 3 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 81 4 ( ( PUNCT american_scientific_journal-7448 81 5 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 81 6 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 81 7 ) ) PUNCT american_scientific_journal-7448 81 8 2 2 NUM american_scientific_journal-7448 81 9 βˆ’ βˆ’ PROPN american_scientific_journal-7448 81 10 𝛼 𝛼 PRON american_scientific_journal-7448 81 11 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 81 12 ln ln ADP american_scientific_journal-7448 81 13 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 81 14 βˆ’ βˆ’ NOUN american_scientific_journal-7448 81 15 𝛼 𝛼 PRON american_scientific_journal-7448 81 16 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 81 17 ) ) PUNCT american_scientific_journal-7448 81 18 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 81 19 sin sin NOUN american_scientific_journal-7448 81 20 πœƒ πœƒ X american_scientific_journal-7448 81 21 ( ( PUNCT american_scientific_journal-7448 81 22 48 48 NUM american_scientific_journal-7448 81 23 ) ) PUNCT american_scientific_journal-7448 81 24 recall recall VERB american_scientific_journal-7448 81 25 that that SCONJ american_scientific_journal-7448 81 26 american american PROPN american_scientific_journal-7448 81 27 academic academic ADJ american_scientific_journal-7448 81 28 scientific scientific ADJ american_scientific_journal-7448 81 29 research research NOUN american_scientific_journal-7448 81 30 journal journal NOUN american_scientific_journal-7448 81 31 for for ADP american_scientific_journal-7448 81 32 engineering engineering NOUN american_scientific_journal-7448 81 33 , , PUNCT american_scientific_journal-7448 81 34 technology technology NOUN american_scientific_journal-7448 81 35 , , PUNCT american_scientific_journal-7448 81 36 and and CCONJ american_scientific_journal-7448 81 37 sciences science NOUN american_scientific_journal-7448 81 38 ( ( PUNCT american_scientific_journal-7448 81 39 asrjets asrjet NOUN american_scientific_journal-7448 81 40 ) ) PUNCT american_scientific_journal-7448 81 41 ( ( PUNCT american_scientific_journal-7448 81 42 2022 2022 NUM american_scientific_journal-7448 81 43 ) ) PUNCT american_scientific_journal-7448 81 44 volume volume NOUN american_scientific_journal-7448 81 45 86 86 NUM american_scientific_journal-7448 81 46 , , PUNCT american_scientific_journal-7448 81 47 no no DET american_scientific_journal-7448 81 48 1 1 NUM american_scientific_journal-7448 81 49 , , PUNCT american_scientific_journal-7448 81 50 pp143 pp143 PROPN american_scientific_journal-7448 81 51 - - PUNCT american_scientific_journal-7448 81 52 156 156 NUM american_scientific_journal-7448 81 53 153 153 NUM american_scientific_journal-7448 81 54 𝑒(π‘Ÿ 𝑒(π‘Ÿ X american_scientific_journal-7448 81 55 , , PUNCT american_scientific_journal-7448 81 56 πœƒ πœƒ NOUN american_scientific_journal-7448 81 57 ) ) PUNCT american_scientific_journal-7448 81 58 = = SYM american_scientific_journal-7448 82 1 π‘’π‘Ÿ π‘’π‘Ÿ ADP american_scientific_journal-7448 82 2 cos cos PROPN american_scientific_journal-7448 82 3 πœƒ πœƒ X american_scientific_journal-7448 82 4 βˆ’ βˆ’ PROPN american_scientific_journal-7448 82 5 π‘’πœƒ π‘’πœƒ NOUN american_scientific_journal-7448 82 6 sin sin NOUN american_scientific_journal-7448 82 7 πœƒ πœƒ X american_scientific_journal-7448 82 8 ( ( PUNCT american_scientific_journal-7448 82 9 49 49 NUM american_scientific_journal-7448 82 10 ) ) PUNCT american_scientific_journal-7448 82 11 substituting substitute VERB american_scientific_journal-7448 82 12 ( ( PUNCT american_scientific_journal-7448 82 13 47 47 NUM american_scientific_journal-7448 82 14 ) ) PUNCT american_scientific_journal-7448 82 15 and and CCONJ american_scientific_journal-7448 82 16 ( ( PUNCT american_scientific_journal-7448 82 17 48 48 NUM american_scientific_journal-7448 82 18 ) ) PUNCT american_scientific_journal-7448 82 19 into into ADP american_scientific_journal-7448 82 20 ( ( PUNCT american_scientific_journal-7448 82 21 49 49 NUM american_scientific_journal-7448 82 22 ) ) PUNCT american_scientific_journal-7448 82 23 and and CCONJ american_scientific_journal-7448 82 24 simplifying simplify VERB american_scientific_journal-7448 82 25 , , PUNCT american_scientific_journal-7448 82 26 we we PRON american_scientific_journal-7448 82 27 have have VERB american_scientific_journal-7448 82 28 𝑒(π‘Ÿ 𝑒(π‘Ÿ NOUN american_scientific_journal-7448 82 29 , , PUNCT american_scientific_journal-7448 82 30 πœƒ πœƒ NOUN american_scientific_journal-7448 82 31 ) ) PUNCT american_scientific_journal-7448 82 32 = = SYM american_scientific_journal-7448 82 33 βˆ’π‘ˆ βˆ’π‘ˆ NOUN american_scientific_journal-7448 82 34 + + CCONJ american_scientific_journal-7448 82 35 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 82 36 2πœ‡ 2πœ‡ NOUN american_scientific_journal-7448 82 37 ( ( PUNCT american_scientific_journal-7448 82 38 ln ln NOUN american_scientific_journal-7448 82 39 π‘Ÿ π‘Ÿ X american_scientific_journal-7448 82 40 βˆ’ βˆ’ X american_scientific_journal-7448 82 41 ln ln ADJ american_scientific_journal-7448 82 42 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 82 43 ) ) PUNCT american_scientific_journal-7448 83 1 + + CCONJ american_scientific_journal-7448 83 2 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 83 3 4πœ‡ 4πœ‡ NOUN american_scientific_journal-7448 83 4 ( ( PUNCT american_scientific_journal-7448 83 5 ( ( PUNCT american_scientific_journal-7448 83 6 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 83 7 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 83 8 ) ) PUNCT american_scientific_journal-7448 83 9 2 2 NUM american_scientific_journal-7448 83 10 βˆ’ βˆ’ NUM american_scientific_journal-7448 83 11 1 1 NUM american_scientific_journal-7448 83 12 ) ) PUNCT american_scientific_journal-7448 83 13 cos cos ADP american_scientific_journal-7448 83 14 2πœƒ 2πœƒ NUM american_scientific_journal-7448 83 15 ( ( PUNCT american_scientific_journal-7448 83 16 50 50 NUM american_scientific_journal-7448 83 17 ) ) PUNCT american_scientific_journal-7448 83 18 4 4 NUM american_scientific_journal-7448 83 19 . . PUNCT american_scientific_journal-7448 84 1 the the DET american_scientific_journal-7448 84 2 drag drag NOUN american_scientific_journal-7448 84 3 force force NOUN american_scientific_journal-7448 84 4 the the DET american_scientific_journal-7448 84 5 force force NOUN american_scientific_journal-7448 84 6 on on ADP american_scientific_journal-7448 84 7 the the DET american_scientific_journal-7448 84 8 cylinder cylinder NOUN american_scientific_journal-7448 84 9 due due ADP american_scientific_journal-7448 84 10 to to ADP american_scientific_journal-7448 84 11 pressure pressure NOUN american_scientific_journal-7448 84 12 is be AUX american_scientific_journal-7448 84 13 𝐹𝑝 𝐹𝑝 PROPN american_scientific_journal-7448 84 14 = = PUNCT american_scientific_journal-7448 85 1 βˆ’ βˆ’ PROPN american_scientific_journal-7448 85 2 ∫ ∫ PROPN american_scientific_journal-7448 85 3 𝑝(π‘Ÿ0 𝑝(π‘Ÿ0 NUM american_scientific_journal-7448 85 4 π‘‘πœƒ π‘‘πœƒ NOUN american_scientific_journal-7448 85 5 ) ) PUNCT american_scientific_journal-7448 85 6 cos cos PROPN american_scientific_journal-7448 85 7 πœƒ πœƒ PRON american_scientific_journal-7448 85 8 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 85 9 0 0 NUM american_scientific_journal-7448 85 10 using use VERB american_scientific_journal-7448 85 11 equation equation NOUN american_scientific_journal-7448 85 12 ( ( PUNCT american_scientific_journal-7448 85 13 15 15 NUM american_scientific_journal-7448 85 14 ) ) PUNCT american_scientific_journal-7448 85 15 and and CCONJ american_scientific_journal-7448 85 16 evaluating evaluate VERB american_scientific_journal-7448 85 17 at at ADP american_scientific_journal-7448 85 18 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 85 19 = = SYM american_scientific_journal-7448 85 20 π‘Ÿ0 π‘Ÿ0 PROPN american_scientific_journal-7448 85 21 𝐹𝑝 𝐹𝑝 PROPN american_scientific_journal-7448 85 22 = = NOUN american_scientific_journal-7448 85 23 1 1 NUM american_scientific_journal-7448 85 24 2 2 NUM american_scientific_journal-7448 85 25 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 85 26 ∫ ∫ NOUN american_scientific_journal-7448 85 27 ( ( PUNCT american_scientific_journal-7448 85 28 1 1 NUM american_scientific_journal-7448 85 29 + + NUM american_scientific_journal-7448 85 30 2πœ‹ 2πœ‹ NUM american_scientific_journal-7448 85 31 0 0 PUNCT american_scientific_journal-7448 86 1 cos cos PROPN american_scientific_journal-7448 86 2 2πœƒ)π‘‘πœƒ 2πœƒ)π‘‘πœƒ NUM american_scientific_journal-7448 86 3 = = SYM american_scientific_journal-7448 86 4 1 1 NUM american_scientific_journal-7448 86 5 2 2 NUM american_scientific_journal-7448 86 6 π›Όπ‘ˆ(πœƒ π›Όπ‘ˆ(πœƒ NUM american_scientific_journal-7448 86 7 + + NUM american_scientific_journal-7448 86 8 sin sin VERB american_scientific_journal-7448 86 9 2πœƒ 2πœƒ NUM american_scientific_journal-7448 86 10 2 2 NUM american_scientific_journal-7448 86 11 ) ) PUNCT american_scientific_journal-7448 86 12 0 0 NUM american_scientific_journal-7448 86 13 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 86 14 = = PUNCT american_scientific_journal-7448 86 15 πœ‹π›Όπ‘ˆ πœ‹π›Όπ‘ˆ NOUN american_scientific_journal-7448 86 16 ( ( PUNCT american_scientific_journal-7448 86 17 51 51 NUM american_scientific_journal-7448 86 18 ) ) PUNCT american_scientific_journal-7448 86 19 the the DET american_scientific_journal-7448 86 20 force force NOUN american_scientific_journal-7448 86 21 on on ADP american_scientific_journal-7448 86 22 the the DET american_scientific_journal-7448 86 23 cylinder cylinder NOUN american_scientific_journal-7448 86 24 due due ADP american_scientific_journal-7448 86 25 to to ADP american_scientific_journal-7448 86 26 friction friction NOUN american_scientific_journal-7448 86 27 or or CCONJ american_scientific_journal-7448 86 28 shear shear NOUN american_scientific_journal-7448 86 29 stress stress NOUN american_scientific_journal-7448 86 30 is be AUX american_scientific_journal-7448 86 31 πΉπœπ‘Ÿπœƒ πΉπœπ‘Ÿπœƒ PROPN american_scientific_journal-7448 86 32 = = SYM american_scientific_journal-7448 87 1 βˆ’ βˆ’ PROPN american_scientific_journal-7448 87 2 ∫ ∫ PROPN american_scientific_journal-7448 87 3 πœπ‘Ÿπœƒ(π‘Ÿ0 πœπ‘Ÿπœƒ(π‘Ÿ0 PROPN american_scientific_journal-7448 87 4 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 87 5 0 0 NUM american_scientific_journal-7448 88 1 π‘‘πœƒ π‘‘πœƒ PROPN american_scientific_journal-7448 88 2 ) ) PUNCT american_scientific_journal-7448 89 1 sin sin NOUN american_scientific_journal-7448 89 2 πœƒ πœƒ SYM american_scientific_journal-7448 89 3 ( ( PUNCT american_scientific_journal-7448 89 4 52 52 NUM american_scientific_journal-7448 89 5 ) ) PUNCT american_scientific_journal-7448 89 6 πœπ‘Ÿπœƒ πœπ‘Ÿπœƒ NOUN american_scientific_journal-7448 89 7 = = PUNCT american_scientific_journal-7448 89 8 πœ‡ πœ‡ X american_scientific_journal-7448 89 9 πœ•π‘’πœƒ πœ•π‘’πœƒ PUNCT american_scientific_journal-7448 89 10 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-7448 89 11 = = SYM american_scientific_journal-7448 89 12 βˆ’ βˆ’ PROPN american_scientific_journal-7448 89 13 𝛼 𝛼 SYM american_scientific_journal-7448 89 14 2 2 NUM american_scientific_journal-7448 89 15 ( ( PUNCT american_scientific_journal-7448 89 16 1 1 NUM american_scientific_journal-7448 89 17 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 89 18 + + NUM american_scientific_journal-7448 89 19 π‘Ÿ0 π‘Ÿ0 NOUN american_scientific_journal-7448 89 20 2 2 NUM american_scientific_journal-7448 89 21 π‘Ÿ3 π‘Ÿ3 NOUN american_scientific_journal-7448 89 22 ) ) PUNCT american_scientific_journal-7448 90 1 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 90 2 sin sin NOUN american_scientific_journal-7448 90 3 πœƒ πœƒ X american_scientific_journal-7448 90 4 ( ( PUNCT american_scientific_journal-7448 90 5 53 53 NUM american_scientific_journal-7448 90 6 ) ) PUNCT american_scientific_journal-7448 90 7 substituting substitute VERB american_scientific_journal-7448 90 8 ( ( PUNCT american_scientific_journal-7448 90 9 53 53 NUM american_scientific_journal-7448 90 10 ) ) PUNCT american_scientific_journal-7448 90 11 into into ADP american_scientific_journal-7448 90 12 ( ( PUNCT american_scientific_journal-7448 90 13 52 52 NUM american_scientific_journal-7448 90 14 ) ) PUNCT american_scientific_journal-7448 90 15 and and CCONJ american_scientific_journal-7448 90 16 evaluating evaluate VERB american_scientific_journal-7448 90 17 at at ADP american_scientific_journal-7448 90 18 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 90 19 = = SYM american_scientific_journal-7448 90 20 π‘Ÿ0 π‘Ÿ0 PROPN american_scientific_journal-7448 90 21 , , PUNCT american_scientific_journal-7448 90 22 we we PRON american_scientific_journal-7448 90 23 have have VERB american_scientific_journal-7448 90 24 american american ADJ american_scientific_journal-7448 90 25 academic academic ADJ american_scientific_journal-7448 90 26 scientific scientific ADJ american_scientific_journal-7448 90 27 research research NOUN american_scientific_journal-7448 90 28 journal journal NOUN american_scientific_journal-7448 90 29 for for ADP american_scientific_journal-7448 90 30 engineering engineering NOUN american_scientific_journal-7448 90 31 , , PUNCT american_scientific_journal-7448 90 32 technology technology NOUN american_scientific_journal-7448 90 33 , , PUNCT american_scientific_journal-7448 90 34 and and CCONJ american_scientific_journal-7448 90 35 sciences science NOUN american_scientific_journal-7448 90 36 ( ( PUNCT american_scientific_journal-7448 90 37 asrjets asrjet NOUN american_scientific_journal-7448 90 38 ) ) PUNCT american_scientific_journal-7448 90 39 ( ( PUNCT american_scientific_journal-7448 90 40 2022 2022 NUM american_scientific_journal-7448 90 41 ) ) PUNCT american_scientific_journal-7448 90 42 volume volume NOUN american_scientific_journal-7448 90 43 86 86 NUM american_scientific_journal-7448 90 44 , , PUNCT american_scientific_journal-7448 90 45 no no DET american_scientific_journal-7448 90 46 1 1 NUM american_scientific_journal-7448 90 47 , , PUNCT american_scientific_journal-7448 90 48 pp143 pp143 PROPN american_scientific_journal-7448 90 49 - - SYM american_scientific_journal-7448 90 50 156 156 NUM american_scientific_journal-7448 90 51 154 154 NUM american_scientific_journal-7448 90 52 πΉπœπ‘Ÿπœƒ πΉπœπ‘Ÿπœƒ PROPN american_scientific_journal-7448 90 53 = = NOUN american_scientific_journal-7448 90 54 1 1 NUM american_scientific_journal-7448 90 55 2 2 NUM american_scientific_journal-7448 90 56 π›Όπ‘ˆ π›Όπ‘ˆ ADJ american_scientific_journal-7448 90 57 ∫ ∫ NOUN american_scientific_journal-7448 90 58 ( ( PUNCT american_scientific_journal-7448 90 59 1 1 NUM american_scientific_journal-7448 90 60 βˆ’ βˆ’ PROPN american_scientific_journal-7448 90 61 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 90 62 0 0 PUNCT american_scientific_journal-7448 91 1 cos cos PROPN american_scientific_journal-7448 91 2 2πœƒ)π‘‘πœƒ 2πœƒ)π‘‘πœƒ NUM american_scientific_journal-7448 91 3 = = SYM american_scientific_journal-7448 91 4 1 1 NUM american_scientific_journal-7448 91 5 2 2 NUM american_scientific_journal-7448 91 6 π›Όπ‘ˆ(πœƒ π›Όπ‘ˆ(πœƒ NUM american_scientific_journal-7448 91 7 βˆ’ βˆ’ NOUN american_scientific_journal-7448 91 8 sin sin NOUN american_scientific_journal-7448 91 9 2πœƒ 2πœƒ NUM american_scientific_journal-7448 91 10 2 2 NUM american_scientific_journal-7448 91 11 ) ) PUNCT american_scientific_journal-7448 91 12 0 0 NUM american_scientific_journal-7448 91 13 2πœ‹ 2πœ‹ NOUN american_scientific_journal-7448 91 14 = = PUNCT american_scientific_journal-7448 91 15 πœ‹π›Όπ‘ˆ πœ‹π›Όπ‘ˆ NOUN american_scientific_journal-7448 91 16 ( ( PUNCT american_scientific_journal-7448 91 17 54 54 NUM american_scientific_journal-7448 91 18 ) ) PUNCT american_scientific_journal-7448 91 19 the the DET american_scientific_journal-7448 91 20 drag drag NOUN american_scientific_journal-7448 91 21 force force NOUN american_scientific_journal-7448 91 22 is be AUX american_scientific_journal-7448 91 23 𝐹𝐷 𝐹𝐷 NOUN american_scientific_journal-7448 91 24 = = SYM american_scientific_journal-7448 91 25 𝐹𝑝 𝐹𝑝 PROPN american_scientific_journal-7448 91 26 + + CCONJ american_scientific_journal-7448 91 27 πΉπœπ‘Ÿπœƒ πΉπœπ‘Ÿπœƒ PROPN american_scientific_journal-7448 91 28 = = PUNCT american_scientific_journal-7448 91 29 πœ‹π›Όπ‘ˆ πœ‹π›Όπ‘ˆ PROPN american_scientific_journal-7448 91 30 + + CCONJ american_scientific_journal-7448 91 31 πœ‹π›Όπ‘ˆ πœ‹π›Όπ‘ˆ NOUN american_scientific_journal-7448 91 32 = = SYM american_scientific_journal-7448 91 33 2πœ‹π›Όπ‘ˆ 2πœ‹π›Όπ‘ˆ NUM american_scientific_journal-7448 91 34 ( ( PUNCT american_scientific_journal-7448 91 35 55 55 NUM american_scientific_journal-7448 91 36 ) ) PUNCT american_scientific_journal-7448 91 37 the the DET american_scientific_journal-7448 91 38 arbitrary arbitrary ADJ american_scientific_journal-7448 91 39 constant constant ADJ american_scientific_journal-7448 91 40 𝛼 𝛼 NOUN american_scientific_journal-7448 91 41 has have VERB american_scientific_journal-7448 91 42 to to PART american_scientific_journal-7448 91 43 be be AUX american_scientific_journal-7448 91 44 obtained obtain VERB american_scientific_journal-7448 91 45 from from ADP american_scientific_journal-7448 91 46 the the DET american_scientific_journal-7448 91 47 far far ADJ american_scientific_journal-7448 91 48 - - PUNCT american_scientific_journal-7448 91 49 field field NOUN american_scientific_journal-7448 91 50 boundary boundary ADJ american_scientific_journal-7448 91 51 condition condition NOUN american_scientific_journal-7448 91 52 . . PUNCT american_scientific_journal-7448 92 1 however however ADV american_scientific_journal-7448 92 2 , , PUNCT american_scientific_journal-7448 92 3 in in ADP american_scientific_journal-7448 92 4 equation equation NOUN american_scientific_journal-7448 92 5 ( ( PUNCT american_scientific_journal-7448 92 6 50 50 NUM american_scientific_journal-7448 92 7 ) ) PUNCT american_scientific_journal-7448 92 8 , , PUNCT american_scientific_journal-7448 92 9 no no DET american_scientific_journal-7448 92 10 value value NOUN american_scientific_journal-7448 92 11 of of ADP american_scientific_journal-7448 92 12 𝛼 𝛼 PRON american_scientific_journal-7448 92 13 will will AUX american_scientific_journal-7448 92 14 make make VERB american_scientific_journal-7448 92 15 𝑒(π‘Ÿ 𝑒(π‘Ÿ NOUN american_scientific_journal-7448 92 16 , , PUNCT american_scientific_journal-7448 92 17 πœƒ πœƒ X american_scientific_journal-7448 92 18 ) ) PUNCT american_scientific_journal-7448 92 19 go go VERB american_scientific_journal-7448 92 20 to to ADP american_scientific_journal-7448 92 21 the the DET american_scientific_journal-7448 92 22 constant constant ADJ american_scientific_journal-7448 92 23 value value NOUN american_scientific_journal-7448 92 24 corresponding correspond VERB american_scientific_journal-7448 92 25 to to ADP american_scientific_journal-7448 92 26 the the DET american_scientific_journal-7448 92 27 undisturbed undisturbed ADJ american_scientific_journal-7448 92 28 flow flow NOUN american_scientific_journal-7448 92 29 since since SCONJ american_scientific_journal-7448 92 30 ln ln NOUN american_scientific_journal-7448 92 31 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 92 32 β†’ β†’ SYM american_scientific_journal-7448 92 33 ∞ ∞ PROPN american_scientific_journal-7448 92 34 as as ADP american_scientific_journal-7448 92 35 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 92 36 β†’ β†’ SYM american_scientific_journal-7448 92 37 ∞. ∞. PROPN american_scientific_journal-7448 92 38 the the DET american_scientific_journal-7448 92 39 inertia inertia NOUN american_scientific_journal-7448 92 40 forces force NOUN american_scientific_journal-7448 92 41 are be AUX american_scientific_journal-7448 92 42 as as ADV american_scientific_journal-7448 92 43 significant significant ADJ american_scientific_journal-7448 92 44 as as ADP american_scientific_journal-7448 92 45 the the DET american_scientific_journal-7448 92 46 viscous viscous ADJ american_scientific_journal-7448 92 47 forces force NOUN american_scientific_journal-7448 92 48 at at ADP american_scientific_journal-7448 92 49 large large ADJ american_scientific_journal-7448 92 50 distances distance NOUN american_scientific_journal-7448 92 51 from from ADP american_scientific_journal-7448 92 52 the the DET american_scientific_journal-7448 92 53 cylinder cylinder NOUN american_scientific_journal-7448 92 54 and and CCONJ american_scientific_journal-7448 92 55 equation equation NOUN american_scientific_journal-7448 92 56 ( ( PUNCT american_scientific_journal-7448 92 57 50 50 NUM american_scientific_journal-7448 92 58 ) ) PUNCT american_scientific_journal-7448 92 59 thus thus ADV american_scientific_journal-7448 92 60 represent represent VERB american_scientific_journal-7448 92 61 the the DET american_scientific_journal-7448 92 62 solution solution NOUN american_scientific_journal-7448 92 63 of of ADP american_scientific_journal-7448 92 64 the the DET american_scientific_journal-7448 92 65 flow flow NOUN american_scientific_journal-7448 92 66 field field NOUN american_scientific_journal-7448 92 67 at at ADP american_scientific_journal-7448 92 68 the the DET american_scientific_journal-7448 92 69 vicinity vicinity NOUN american_scientific_journal-7448 92 70 of of ADP american_scientific_journal-7448 92 71 the the DET american_scientific_journal-7448 92 72 cylinder cylinder NOUN american_scientific_journal-7448 92 73 and and CCONJ american_scientific_journal-7448 92 74 not not PART american_scientific_journal-7448 92 75 at at ADP american_scientific_journal-7448 92 76 large large ADJ american_scientific_journal-7448 92 77 values value NOUN american_scientific_journal-7448 92 78 of of ADP american_scientific_journal-7448 92 79 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 92 80 ( ( PUNCT american_scientific_journal-7448 92 81 oseen oseen PROPN american_scientific_journal-7448 92 82 ’s ’s PART american_scientific_journal-7448 92 83 paradox paradox NOUN american_scientific_journal-7448 92 84 ) ) PUNCT american_scientific_journal-7448 92 85 . . PUNCT american_scientific_journal-7448 93 1 clearly clearly ADV american_scientific_journal-7448 93 2 , , PUNCT american_scientific_journal-7448 93 3 some some DET american_scientific_journal-7448 93 4 approximation approximation NOUN american_scientific_journal-7448 93 5 to to ADP american_scientific_journal-7448 93 6 the the DET american_scientific_journal-7448 93 7 equation equation NOUN american_scientific_journal-7448 93 8 of of ADP american_scientific_journal-7448 93 9 motion motion NOUN american_scientific_journal-7448 93 10 is be AUX american_scientific_journal-7448 93 11 needed need VERB american_scientific_journal-7448 93 12 for for ADP american_scientific_journal-7448 93 13 large large ADJ american_scientific_journal-7448 93 14 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-7448 93 15 and and CCONJ american_scientific_journal-7448 93 16 equation equation NOUN american_scientific_journal-7448 93 17 ( ( PUNCT american_scientific_journal-7448 93 18 50 50 NUM american_scientific_journal-7448 93 19 ) ) PUNCT american_scientific_journal-7448 93 20 must must AUX american_scientific_journal-7448 93 21 match match VERB american_scientific_journal-7448 93 22 with with ADP american_scientific_journal-7448 93 23 this this DET american_scientific_journal-7448 93 24 approximate approximate ADJ american_scientific_journal-7448 93 25 solution solution NOUN american_scientific_journal-7448 93 26 at at ADP american_scientific_journal-7448 93 27 large large ADJ american_scientific_journal-7448 93 28 distance distance NOUN american_scientific_journal-7448 93 29 from from ADP american_scientific_journal-7448 93 30 the the DET american_scientific_journal-7448 93 31 cylinder cylinder NOUN american_scientific_journal-7448 93 32 . . PUNCT american_scientific_journal-7448 94 1 the the DET american_scientific_journal-7448 94 2 author author NOUN american_scientific_journal-7448 94 3 [ [ X american_scientific_journal-7448 94 4 5 5 NUM american_scientific_journal-7448 94 5 ] ] PUNCT american_scientific_journal-7448 94 6 made make VERB american_scientific_journal-7448 94 7 an an DET american_scientific_journal-7448 94 8 approximation approximation NOUN american_scientific_journal-7448 94 9 to to ADP american_scientific_journal-7448 94 10 the the DET american_scientific_journal-7448 94 11 equation equation NOUN american_scientific_journal-7448 94 12 of of ADP american_scientific_journal-7448 94 13 motion motion NOUN american_scientific_journal-7448 94 14 , , PUNCT american_scientific_journal-7448 94 15 which which PRON american_scientific_journal-7448 94 16 is be AUX american_scientific_journal-7448 94 17 𝜌(βˆ’π‘ˆ. 𝜌(βˆ’π‘ˆ. PROPN american_scientific_journal-7448 94 18 βˆ‡π‘’ βˆ‡π‘’ PROPN american_scientific_journal-7448 94 19 + + CCONJ american_scientific_journal-7448 94 20 𝑒. 𝑒. PROPN american_scientific_journal-7448 94 21 βˆ‡π‘’ βˆ‡π‘’ PROPN american_scientific_journal-7448 94 22 ) ) PUNCT american_scientific_journal-7448 95 1 = = SYM american_scientific_journal-7448 95 2 βˆ’βˆ‡π‘ βˆ’βˆ‡π‘ PROPN american_scientific_journal-7448 95 3 + + CCONJ american_scientific_journal-7448 95 4 πœ‡βˆ‡2𝑒 πœ‡βˆ‡2𝑒 PROPN american_scientific_journal-7448 95 5 ( ( PUNCT american_scientific_journal-7448 95 6 56 56 NUM american_scientific_journal-7448 95 7 ) ) PUNCT american_scientific_journal-7448 95 8 the the DET american_scientific_journal-7448 95 9 two two NUM american_scientific_journal-7448 95 10 terms term NOUN american_scientific_journal-7448 95 11 in in ADP american_scientific_journal-7448 95 12 the the DET american_scientific_journal-7448 95 13 inertial inertial ADJ american_scientific_journal-7448 95 14 force force NOUN american_scientific_journal-7448 95 15 of of ADP american_scientific_journal-7448 95 16 equation equation NOUN american_scientific_journal-7448 95 17 ( ( PUNCT american_scientific_journal-7448 95 18 56 56 NUM american_scientific_journal-7448 95 19 ) ) PUNCT american_scientific_journal-7448 95 20 are be AUX american_scientific_journal-7448 95 21 of of ADP american_scientific_journal-7448 95 22 the the DET american_scientific_journal-7448 95 23 same same ADJ american_scientific_journal-7448 95 24 order order NOUN american_scientific_journal-7448 95 25 near near ADP american_scientific_journal-7448 95 26 the the DET american_scientific_journal-7448 95 27 cylinder cylinder NOUN american_scientific_journal-7448 95 28 , , PUNCT american_scientific_journal-7448 95 29 but but CCONJ american_scientific_journal-7448 95 30 the the DET american_scientific_journal-7448 95 31 first first ADJ american_scientific_journal-7448 95 32 term term NOUN american_scientific_journal-7448 95 33 is be AUX american_scientific_journal-7448 95 34 dominant dominant ADJ american_scientific_journal-7448 95 35 far far ADV american_scientific_journal-7448 95 36 from from ADP american_scientific_journal-7448 95 37 the the DET american_scientific_journal-7448 95 38 cylinder cylinder NOUN american_scientific_journal-7448 95 39 . . PUNCT american_scientific_journal-7448 96 1 so so ADV american_scientific_journal-7448 96 2 , , PUNCT american_scientific_journal-7448 96 3 far far ADV american_scientific_journal-7448 96 4 from from ADP american_scientific_journal-7448 96 5 the the DET american_scientific_journal-7448 96 6 cylinder cylinder NOUN american_scientific_journal-7448 96 7 equation equation NOUN american_scientific_journal-7448 96 8 ( ( PUNCT american_scientific_journal-7448 96 9 56 56 NUM american_scientific_journal-7448 96 10 ) ) PUNCT american_scientific_journal-7448 96 11 becomes become VERB american_scientific_journal-7448 96 12 𝜌(βˆ’π‘ˆ. 𝜌(βˆ’π‘ˆ. ADP american_scientific_journal-7448 96 13 βˆ‡π‘’ βˆ‡π‘’ PROPN american_scientific_journal-7448 96 14 ) ) PUNCT american_scientific_journal-7448 96 15 = = SYM american_scientific_journal-7448 97 1 βˆ’βˆ‡π‘ βˆ’βˆ‡π‘ PROPN american_scientific_journal-7448 97 2 + + CCONJ american_scientific_journal-7448 97 3 πœ‡βˆ‡2𝑒 πœ‡βˆ‡2𝑒 PROPN american_scientific_journal-7448 97 4 ( ( PUNCT american_scientific_journal-7448 97 5 57 57 NUM american_scientific_journal-7448 97 6 ) ) PUNCT american_scientific_journal-7448 97 7 equation equation NOUN american_scientific_journal-7448 97 8 ( ( PUNCT american_scientific_journal-7448 97 9 57 57 NUM american_scientific_journal-7448 97 10 ) ) PUNCT american_scientific_journal-7448 97 11 and and CCONJ american_scientific_journal-7448 97 12 the the DET american_scientific_journal-7448 97 13 continuity continuity NOUN american_scientific_journal-7448 97 14 equation equation NOUN american_scientific_journal-7448 97 15 are be AUX american_scientific_journal-7448 97 16 known know VERB american_scientific_journal-7448 97 17 as as ADP american_scientific_journal-7448 97 18 the the DET american_scientific_journal-7448 97 19 oseen oseen PROPN american_scientific_journal-7448 97 20 equations equation NOUN american_scientific_journal-7448 97 21 . . PUNCT american_scientific_journal-7448 98 1 the the DET american_scientific_journal-7448 98 2 author author NOUN american_scientific_journal-7448 98 3 [ [ X american_scientific_journal-7448 98 4 4 4 X american_scientific_journal-7448 98 5 ] ] PUNCT american_scientific_journal-7448 98 6 showed show VERB american_scientific_journal-7448 98 7 that that SCONJ american_scientific_journal-7448 98 8 equation equation NOUN american_scientific_journal-7448 98 9 ( ( PUNCT american_scientific_journal-7448 98 10 57 57 NUM american_scientific_journal-7448 98 11 ) ) PUNCT american_scientific_journal-7448 98 12 has have VERB american_scientific_journal-7448 98 13 a a DET american_scientific_journal-7448 98 14 solution solution NOUN american_scientific_journal-7448 98 15 which which PRON american_scientific_journal-7448 98 16 approximates approximate VERB american_scientific_journal-7448 98 17 to to ADP american_scientific_journal-7448 98 18 equation equation NOUN american_scientific_journal-7448 98 19 ( ( PUNCT american_scientific_journal-7448 98 20 50 50 NUM american_scientific_journal-7448 98 21 ) ) PUNCT american_scientific_journal-7448 98 22 near near ADP american_scientific_journal-7448 98 23 the the DET american_scientific_journal-7448 98 24 cylinder cylinder NOUN american_scientific_journal-7448 98 25 provided provide VERB american_scientific_journal-7448 98 26 the the DET american_scientific_journal-7448 98 27 constant constant ADJ american_scientific_journal-7448 98 28 in in ADP american_scientific_journal-7448 98 29 equation equation NOUN american_scientific_journal-7448 98 30 ( ( PUNCT american_scientific_journal-7448 98 31 50 50 NUM american_scientific_journal-7448 98 32 ) ) PUNCT american_scientific_journal-7448 98 33 is be AUX american_scientific_journal-7448 98 34 chosen choose VERB american_scientific_journal-7448 98 35 as as ADP american_scientific_journal-7448 98 36 𝛼 𝛼 NOUN american_scientific_journal-7448 98 37 = = SYM american_scientific_journal-7448 98 38 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 98 39 ln(7.4 ln(7.4 PROPN american_scientific_journal-7448 98 40 𝑅𝑒)⁄ 𝑅𝑒)⁄ SYM american_scientific_journal-7448 98 41 ( ( PUNCT american_scientific_journal-7448 98 42 58 58 NUM american_scientific_journal-7448 98 43 ) ) PUNCT american_scientific_journal-7448 98 44 american american ADJ american_scientific_journal-7448 98 45 academic academic ADJ american_scientific_journal-7448 98 46 scientific scientific ADJ american_scientific_journal-7448 98 47 research research NOUN american_scientific_journal-7448 98 48 journal journal NOUN american_scientific_journal-7448 98 49 for for ADP american_scientific_journal-7448 98 50 engineering engineering NOUN american_scientific_journal-7448 98 51 , , PUNCT american_scientific_journal-7448 98 52 technology technology NOUN american_scientific_journal-7448 98 53 , , PUNCT american_scientific_journal-7448 98 54 and and CCONJ american_scientific_journal-7448 98 55 sciences science NOUN american_scientific_journal-7448 98 56 ( ( PUNCT american_scientific_journal-7448 98 57 asrjets asrjet NOUN american_scientific_journal-7448 98 58 ) ) PUNCT american_scientific_journal-7448 98 59 ( ( PUNCT american_scientific_journal-7448 98 60 2022 2022 NUM american_scientific_journal-7448 98 61 ) ) PUNCT american_scientific_journal-7448 98 62 volume volume NOUN american_scientific_journal-7448 98 63 86 86 NUM american_scientific_journal-7448 98 64 , , PUNCT american_scientific_journal-7448 98 65 no no DET american_scientific_journal-7448 98 66 1 1 NUM american_scientific_journal-7448 98 67 , , PUNCT american_scientific_journal-7448 98 68 pp143 pp143 PROPN american_scientific_journal-7448 98 69 - - PUNCT american_scientific_journal-7448 98 70 156 156 NUM american_scientific_journal-7448 98 71 155 155 NUM american_scientific_journal-7448 98 72 substituting substitute VERB american_scientific_journal-7448 98 73 ( ( PUNCT american_scientific_journal-7448 98 74 58 58 NUM american_scientific_journal-7448 98 75 ) ) PUNCT american_scientific_journal-7448 98 76 into into ADP american_scientific_journal-7448 98 77 ( ( PUNCT american_scientific_journal-7448 98 78 55 55 NUM american_scientific_journal-7448 98 79 ) ) PUNCT american_scientific_journal-7448 98 80 , , PUNCT american_scientific_journal-7448 98 81 we we PRON american_scientific_journal-7448 98 82 have have VERB american_scientific_journal-7448 98 83 𝐹𝐷 𝐹𝐷 PROPN american_scientific_journal-7448 98 84 = = SYM american_scientific_journal-7448 98 85 2πœ‹π‘ˆ 2πœ‹π‘ˆ PROPN american_scientific_journal-7448 98 86 ( ( PUNCT american_scientific_journal-7448 98 87 2πœ‡ 2πœ‡ NUM american_scientific_journal-7448 98 88 ln(7.4 ln(7.4 PROPN american_scientific_journal-7448 98 89 𝑅𝑒)⁄ 𝑅𝑒)⁄ X american_scientific_journal-7448 98 90 ) ) PUNCT american_scientific_journal-7448 99 1 = = SYM american_scientific_journal-7448 99 2 4πœ‹π‘ˆπœ‡ 4πœ‹π‘ˆπœ‡ PROPN american_scientific_journal-7448 99 3 ln(7.4 ln(7.4 PROPN american_scientific_journal-7448 99 4 𝑅𝑒)⁄ 𝑅𝑒)⁄ SYM american_scientific_journal-7448 99 5 ( ( PUNCT american_scientific_journal-7448 99 6 59 59 NUM american_scientific_journal-7448 99 7 ) ) PUNCT american_scientific_journal-7448 99 8 ( ( PUNCT american_scientific_journal-7448 99 9 59 59 NUM american_scientific_journal-7448 99 10 ) ) PUNCT american_scientific_journal-7448 99 11 is be AUX american_scientific_journal-7448 99 12 the the DET american_scientific_journal-7448 99 13 drag drag NOUN american_scientific_journal-7448 99 14 force force NOUN american_scientific_journal-7448 99 15 acting act VERB american_scientific_journal-7448 99 16 on on ADP american_scientific_journal-7448 99 17 the the DET american_scientific_journal-7448 99 18 circular circular ADJ american_scientific_journal-7448 99 19 cylinder cylinder NOUN american_scientific_journal-7448 99 20 . . PUNCT american_scientific_journal-7448 100 1 the the DET american_scientific_journal-7448 100 2 drag drag NOUN american_scientific_journal-7448 100 3 coefficient coefficient NOUN american_scientific_journal-7448 100 4 ( ( PUNCT american_scientific_journal-7448 100 5 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 100 6 ) ) PUNCT american_scientific_journal-7448 100 7 which which PRON american_scientific_journal-7448 100 8 is be AUX american_scientific_journal-7448 100 9 a a DET american_scientific_journal-7448 100 10 function function NOUN american_scientific_journal-7448 100 11 of of ADP american_scientific_journal-7448 100 12 the the DET american_scientific_journal-7448 100 13 reynolds reynolds PROPN american_scientific_journal-7448 100 14 number number NOUN american_scientific_journal-7448 100 15 ( ( PUNCT american_scientific_journal-7448 100 16 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 100 17 ) ) PUNCT american_scientific_journal-7448 100 18 is be AUX american_scientific_journal-7448 100 19 defined define VERB american_scientific_journal-7448 100 20 generally generally ADV american_scientific_journal-7448 100 21 as as ADP american_scientific_journal-7448 100 22 𝐢𝐷 𝐢𝐷 NOUN american_scientific_journal-7448 100 23 = = SYM american_scientific_journal-7448 100 24 2𝐹𝐷 2𝐹𝐷 NUM american_scientific_journal-7448 100 25 πœŒπ‘ˆ2𝐴 πœŒπ‘ˆ2𝐴 NOUN american_scientific_journal-7448 100 26 ( ( PUNCT american_scientific_journal-7448 100 27 60 60 NUM american_scientific_journal-7448 100 28 ) ) PUNCT american_scientific_journal-7448 100 29 where where SCONJ american_scientific_journal-7448 100 30 𝐴 𝐴 PROPN american_scientific_journal-7448 100 31 is be AUX american_scientific_journal-7448 100 32 the the DET american_scientific_journal-7448 100 33 reference reference NOUN american_scientific_journal-7448 100 34 area area NOUN american_scientific_journal-7448 100 35 of of ADP american_scientific_journal-7448 100 36 the the DET american_scientific_journal-7448 100 37 object object NOUN american_scientific_journal-7448 100 38 in in ADP american_scientific_journal-7448 100 39 the the DET american_scientific_journal-7448 100 40 fluid fluid NOUN american_scientific_journal-7448 100 41 . . PUNCT american_scientific_journal-7448 101 1 for for ADP american_scientific_journal-7448 101 2 this this DET american_scientific_journal-7448 101 3 work work NOUN american_scientific_journal-7448 101 4 𝐢𝐷 𝐢𝐷 NOUN american_scientific_journal-7448 101 5 = = SYM american_scientific_journal-7448 101 6 2𝐹𝐷 2𝐹𝐷 NUM american_scientific_journal-7448 101 7 2π‘Ÿ0πœŒπ‘ˆ2 2π‘Ÿ0πœŒπ‘ˆ2 PROPN american_scientific_journal-7448 101 8 ( ( PUNCT american_scientific_journal-7448 101 9 61 61 NUM american_scientific_journal-7448 101 10 ) ) PUNCT american_scientific_journal-7448 101 11 where where SCONJ american_scientific_journal-7448 101 12 𝐴 𝐴 PROPN american_scientific_journal-7448 101 13 = = SYM american_scientific_journal-7448 101 14 2π‘Ÿ0 2π‘Ÿ0 NUM american_scientific_journal-7448 101 15 substituting substitute VERB american_scientific_journal-7448 101 16 ( ( PUNCT american_scientific_journal-7448 101 17 59 59 NUM american_scientific_journal-7448 101 18 ) ) PUNCT american_scientific_journal-7448 101 19 into into ADP american_scientific_journal-7448 101 20 ( ( PUNCT american_scientific_journal-7448 101 21 61 61 NUM american_scientific_journal-7448 101 22 ) ) PUNCT american_scientific_journal-7448 101 23 , , PUNCT american_scientific_journal-7448 101 24 we we PRON american_scientific_journal-7448 101 25 have have VERB american_scientific_journal-7448 101 26 𝐢𝐷 𝐢𝐷 NOUN american_scientific_journal-7448 101 27 = = SYM american_scientific_journal-7448 101 28 8πœ‹ 8πœ‹ NUM american_scientific_journal-7448 102 1 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 102 2 ln(7.4 ln(7.4 PROPN american_scientific_journal-7448 102 3 𝑅𝑒)⁄ 𝑅𝑒)⁄ X american_scientific_journal-7448 102 4 , , PUNCT american_scientific_journal-7448 102 5 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 102 6 β‰ͺ β‰ͺ ADJ american_scientific_journal-7448 102 7 1 1 NUM american_scientific_journal-7448 102 8 ( ( PUNCT american_scientific_journal-7448 102 9 62 62 NUM american_scientific_journal-7448 102 10 ) ) PUNCT american_scientific_journal-7448 102 11 5 5 NUM american_scientific_journal-7448 102 12 . . PUNCT american_scientific_journal-7448 102 13 discussion discussion NOUN american_scientific_journal-7448 102 14 equation equation NOUN american_scientific_journal-7448 102 15 ( ( PUNCT american_scientific_journal-7448 102 16 62 62 NUM american_scientific_journal-7448 102 17 ) ) PUNCT american_scientific_journal-7448 102 18 is be AUX american_scientific_journal-7448 102 19 the the DET american_scientific_journal-7448 102 20 drag drag NOUN american_scientific_journal-7448 102 21 coefficient coefficient NOUN american_scientific_journal-7448 102 22 ( ( PUNCT american_scientific_journal-7448 102 23 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 102 24 ) ) PUNCT american_scientific_journal-7448 102 25 of of ADP american_scientific_journal-7448 102 26 a a DET american_scientific_journal-7448 102 27 fluid fluid NOUN american_scientific_journal-7448 102 28 on on ADP american_scientific_journal-7448 102 29 a a DET american_scientific_journal-7448 102 30 translating translate VERB american_scientific_journal-7448 102 31 circular circular ADJ american_scientific_journal-7448 102 32 cylinder cylinder NOUN american_scientific_journal-7448 102 33 and and CCONJ american_scientific_journal-7448 102 34 this this DET american_scientific_journal-7448 102 35 result result NOUN american_scientific_journal-7448 102 36 agrees agree VERB american_scientific_journal-7448 102 37 with with ADP american_scientific_journal-7448 102 38 the the DET american_scientific_journal-7448 102 39 result result NOUN american_scientific_journal-7448 102 40 in in ADP american_scientific_journal-7448 102 41 literature literature NOUN american_scientific_journal-7448 102 42 . . PUNCT american_scientific_journal-7448 103 1 𝐢𝐷 𝐢𝐷 PROPN american_scientific_journal-7448 103 2 = = SYM american_scientific_journal-7448 103 3 8πœ‹ 8πœ‹ NUM american_scientific_journal-7448 103 4 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 103 5 ln(7.4 ln(7.4 PROPN american_scientific_journal-7448 103 6 𝑅𝑒)⁄ 𝑅𝑒)⁄ X american_scientific_journal-7448 103 7 , , PUNCT american_scientific_journal-7448 103 8 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 103 9 β‰ͺ β‰ͺ ADJ american_scientific_journal-7448 103 10 1 1 NUM american_scientific_journal-7448 103 11 american american ADJ american_scientific_journal-7448 103 12 academic academic ADJ american_scientific_journal-7448 103 13 scientific scientific ADJ american_scientific_journal-7448 103 14 research research NOUN american_scientific_journal-7448 103 15 journal journal NOUN american_scientific_journal-7448 103 16 for for ADP american_scientific_journal-7448 103 17 engineering engineering NOUN american_scientific_journal-7448 103 18 , , PUNCT american_scientific_journal-7448 103 19 technology technology NOUN american_scientific_journal-7448 103 20 , , PUNCT american_scientific_journal-7448 103 21 and and CCONJ american_scientific_journal-7448 103 22 sciences science NOUN american_scientific_journal-7448 103 23 ( ( PUNCT american_scientific_journal-7448 103 24 asrjets asrjet NOUN american_scientific_journal-7448 103 25 ) ) PUNCT american_scientific_journal-7448 103 26 ( ( PUNCT american_scientific_journal-7448 103 27 2022 2022 NUM american_scientific_journal-7448 103 28 ) ) PUNCT american_scientific_journal-7448 103 29 volume volume NOUN american_scientific_journal-7448 103 30 86 86 NUM american_scientific_journal-7448 103 31 , , PUNCT american_scientific_journal-7448 103 32 no no DET american_scientific_journal-7448 103 33 1 1 NUM american_scientific_journal-7448 103 34 , , PUNCT american_scientific_journal-7448 103 35 pp143 pp143 PROPN american_scientific_journal-7448 103 36 - - SYM american_scientific_journal-7448 103 37 156 156 NUM american_scientific_journal-7448 103 38 156 156 NUM american_scientific_journal-7448 103 39 the the DET american_scientific_journal-7448 103 40 description description NOUN american_scientific_journal-7448 103 41 of of ADP american_scientific_journal-7448 103 42 the the DET american_scientific_journal-7448 103 43 flow flow NOUN american_scientific_journal-7448 103 44 field field NOUN american_scientific_journal-7448 103 45 of of ADP american_scientific_journal-7448 103 46 a a DET american_scientific_journal-7448 103 47 rigid rigid ADJ american_scientific_journal-7448 103 48 body body NOUN american_scientific_journal-7448 103 49 translating translate VERB american_scientific_journal-7448 103 50 steadily steadily ADV american_scientific_journal-7448 103 51 and and CCONJ american_scientific_journal-7448 103 52 with with ADP american_scientific_journal-7448 103 53 a a DET american_scientific_journal-7448 103 54 speed speed NOUN american_scientific_journal-7448 103 55 π‘ˆ π‘ˆ NOUN american_scientific_journal-7448 103 56 through through ADP american_scientific_journal-7448 103 57 an an DET american_scientific_journal-7448 103 58 infinite infinite ADJ american_scientific_journal-7448 103 59 body body NOUN american_scientific_journal-7448 103 60 which which PRON american_scientific_journal-7448 103 61 is be AUX american_scientific_journal-7448 103 62 otherwise otherwise ADV american_scientific_journal-7448 103 63 undisturbed undisturbed ADJ american_scientific_journal-7448 103 64 depends depend VERB american_scientific_journal-7448 103 65 only only ADV american_scientific_journal-7448 103 66 on on ADP american_scientific_journal-7448 103 67 a a DET american_scientific_journal-7448 103 68 dimensionless dimensionless NOUN american_scientific_journal-7448 103 69 quantity quantity NOUN american_scientific_journal-7448 103 70 called call VERB american_scientific_journal-7448 103 71 the the DET american_scientific_journal-7448 103 72 reynolds reynolds PROPN american_scientific_journal-7448 103 73 number number NOUN american_scientific_journal-7448 103 74 ( ( PUNCT american_scientific_journal-7448 103 75 𝑅𝑒 𝑅𝑒 PROPN american_scientific_journal-7448 103 76 ) ) PUNCT american_scientific_journal-7448 103 77 . . PUNCT american_scientific_journal-7448 104 1 for for ADP american_scientific_journal-7448 104 2 𝑅𝑒 𝑅𝑒 VERB american_scientific_journal-7448 104 3 β‰ͺ β‰ͺ ADJ american_scientific_journal-7448 104 4 1 1 NUM american_scientific_journal-7448 104 5 , , PUNCT american_scientific_journal-7448 104 6 the the DET american_scientific_journal-7448 104 7 flow flow NOUN american_scientific_journal-7448 104 8 is be AUX american_scientific_journal-7448 104 9 laminar laminar ADJ american_scientific_journal-7448 104 10 and and CCONJ american_scientific_journal-7448 104 11 the the DET american_scientific_journal-7448 104 12 drag drag NOUN american_scientific_journal-7448 104 13 coefficient coefficient NOUN american_scientific_journal-7448 104 14 varies vary VERB american_scientific_journal-7448 104 15 inversely inversely ADV american_scientific_journal-7448 104 16 as as ADP american_scientific_journal-7448 104 17 the the DET american_scientific_journal-7448 104 18 reynolds reynolds PROPN american_scientific_journal-7448 104 19 number number PROPN american_scientific_journal-7448 104 20 . . PUNCT american_scientific_journal-7448 105 1 this this PRON american_scientific_journal-7448 105 2 means mean VERB american_scientific_journal-7448 105 3 that that SCONJ american_scientific_journal-7448 105 4 as as SCONJ american_scientific_journal-7448 105 5 the the DET american_scientific_journal-7448 105 6 drag drag NOUN american_scientific_journal-7448 105 7 coefficient coefficient NOUN american_scientific_journal-7448 105 8 decreases decrease VERB american_scientific_journal-7448 105 9 , , PUNCT american_scientific_journal-7448 105 10 the the DET american_scientific_journal-7448 105 11 speed speed NOUN american_scientific_journal-7448 105 12 of of ADP american_scientific_journal-7448 105 13 the the DET american_scientific_journal-7448 105 14 circular circular ADJ american_scientific_journal-7448 105 15 cylinder cylinder NOUN american_scientific_journal-7448 105 16 increases increase NOUN american_scientific_journal-7448 105 17 in in ADP american_scientific_journal-7448 105 18 the the DET american_scientific_journal-7448 105 19 fluid fluid NOUN american_scientific_journal-7448 105 20 . . PUNCT american_scientific_journal-7448 106 1 6 6 NUM american_scientific_journal-7448 106 2 . . X american_scientific_journal-7448 106 3 conclusion conclusion NOUN american_scientific_journal-7448 106 4 this this DET american_scientific_journal-7448 106 5 paper paper NOUN american_scientific_journal-7448 106 6 addressed address VERB american_scientific_journal-7448 106 7 the the DET american_scientific_journal-7448 106 8 analytic analytic ADJ american_scientific_journal-7448 106 9 method method NOUN american_scientific_journal-7448 106 10 of of ADP american_scientific_journal-7448 106 11 obtaining obtain VERB american_scientific_journal-7448 106 12 the the DET american_scientific_journal-7448 106 13 drag drag NOUN american_scientific_journal-7448 106 14 coefficient coefficient NOUN american_scientific_journal-7448 106 15 of of ADP american_scientific_journal-7448 106 16 a a DET american_scientific_journal-7448 106 17 fluid fluid NOUN american_scientific_journal-7448 106 18 on on ADP american_scientific_journal-7448 106 19 a a DET american_scientific_journal-7448 106 20 translating translate VERB american_scientific_journal-7448 106 21 circular circular ADJ american_scientific_journal-7448 106 22 cylinder cylinder NOUN american_scientific_journal-7448 106 23 by by ADP american_scientific_journal-7448 106 24 solving solve VERB american_scientific_journal-7448 106 25 the the DET american_scientific_journal-7448 106 26 navier navier NOUN american_scientific_journal-7448 106 27 - - PUNCT american_scientific_journal-7448 106 28 stokes stokes PROPN american_scientific_journal-7448 106 29 equations equation NOUN american_scientific_journal-7448 106 30 directly directly ADV american_scientific_journal-7448 106 31 . . PUNCT american_scientific_journal-7448 107 1 the the DET american_scientific_journal-7448 107 2 closure closure NOUN american_scientific_journal-7448 107 3 method method NOUN american_scientific_journal-7448 107 4 of of ADP american_scientific_journal-7448 107 5 [ [ X american_scientific_journal-7448 107 6 5 5 NUM american_scientific_journal-7448 107 7 ] ] PUNCT american_scientific_journal-7448 107 8 applied apply VERB american_scientific_journal-7448 107 9 by by ADP american_scientific_journal-7448 107 10 [ [ X american_scientific_journal-7448 107 11 4 4 NUM american_scientific_journal-7448 107 12 ] ] PUNCT american_scientific_journal-7448 107 13 and and CCONJ american_scientific_journal-7448 107 14 the the DET american_scientific_journal-7448 107 15 asymptotic asymptotic ADJ american_scientific_journal-7448 107 16 method method NOUN american_scientific_journal-7448 107 17 used use VERB american_scientific_journal-7448 107 18 by by ADP american_scientific_journal-7448 107 19 some some DET american_scientific_journal-7448 107 20 authors author NOUN american_scientific_journal-7448 107 21 in in ADP american_scientific_journal-7448 107 22 finding find VERB american_scientific_journal-7448 107 23 the the DET american_scientific_journal-7448 107 24 drag drag NOUN american_scientific_journal-7448 107 25 coefficient coefficient NOUN american_scientific_journal-7448 107 26 on on ADP american_scientific_journal-7448 107 27 a a DET american_scientific_journal-7448 107 28 cylinder cylinder NOUN american_scientific_journal-7448 107 29 are be AUX american_scientific_journal-7448 107 30 also also ADV american_scientific_journal-7448 107 31 important important ADJ american_scientific_journal-7448 107 32 technique technique NOUN american_scientific_journal-7448 107 33 . . PUNCT american_scientific_journal-7448 108 1 inspired inspire VERB american_scientific_journal-7448 108 2 by by ADP american_scientific_journal-7448 108 3 [ [ X american_scientific_journal-7448 108 4 7 7 NUM american_scientific_journal-7448 108 5 ] ] PUNCT american_scientific_journal-7448 108 6 , , PUNCT american_scientific_journal-7448 108 7 this this DET american_scientific_journal-7448 108 8 paper paper NOUN american_scientific_journal-7448 108 9 can can AUX american_scientific_journal-7448 108 10 inspire inspire VERB american_scientific_journal-7448 108 11 more more ADJ american_scientific_journal-7448 108 12 researchers researcher NOUN american_scientific_journal-7448 108 13 to to PART american_scientific_journal-7448 108 14 develop develop VERB american_scientific_journal-7448 108 15 analytic analytic ADJ american_scientific_journal-7448 108 16 methods method NOUN american_scientific_journal-7448 108 17 in in ADP american_scientific_journal-7448 108 18 obtaining obtain VERB american_scientific_journal-7448 108 19 drag drag NOUN american_scientific_journal-7448 108 20 coefficient coefficient NOUN american_scientific_journal-7448 108 21 on on ADP american_scientific_journal-7448 108 22 objects object NOUN american_scientific_journal-7448 108 23 . . PUNCT american_scientific_journal-7448 109 1 references reference NOUN american_scientific_journal-7448 109 2 [ [ X american_scientific_journal-7448 109 3 1 1 NUM american_scientific_journal-7448 109 4 ] ] X american_scientific_journal-7448 109 5 g. g. PROPN american_scientific_journal-7448 109 6 g. g. PROPN american_scientific_journal-7448 109 7 stokes stokes PROPN american_scientific_journal-7448 109 8 . . PUNCT american_scientific_journal-7448 110 1 β€œ " PUNCT american_scientific_journal-7448 110 2 on on ADP american_scientific_journal-7448 110 3 the the DET american_scientific_journal-7448 110 4 effect effect NOUN american_scientific_journal-7448 110 5 of of ADP american_scientific_journal-7448 110 6 the the DET american_scientific_journal-7448 110 7 internal internal ADJ american_scientific_journal-7448 110 8 friction friction NOUN american_scientific_journal-7448 110 9 of of ADP american_scientific_journal-7448 110 10 fluids fluid NOUN american_scientific_journal-7448 110 11 on on ADP american_scientific_journal-7448 110 12 the the DET american_scientific_journal-7448 110 13 motion motion NOUN american_scientific_journal-7448 110 14 of of ADP american_scientific_journal-7448 110 15 pendulums pendulum NOUN american_scientific_journal-7448 110 16 . . PUNCT american_scientific_journal-7448 110 17 ” " PUNCT american_scientific_journal-7448 111 1 cambridge cambridge PROPN american_scientific_journal-7448 111 2 philosophical philosophical ADJ american_scientific_journal-7448 111 3 society society NOUN american_scientific_journal-7448 111 4 , , PUNCT american_scientific_journal-7448 111 5 vol vol NOUN american_scientific_journal-7448 111 6 . . PROPN american_scientific_journal-7448 111 7 9 9 NUM american_scientific_journal-7448 111 8 , , PUNCT american_scientific_journal-7448 111 9 pp pp ADJ american_scientific_journal-7448 111 10 . . PUNCT american_scientific_journal-7448 112 1 8 8 NUM american_scientific_journal-7448 112 2 - - SYM american_scientific_journal-7448 112 3 106 106 NUM american_scientific_journal-7448 112 4 , , PUNCT american_scientific_journal-7448 112 5 1851 1851 NUM american_scientific_journal-7448 112 6 . . PUNCT american_scientific_journal-7448 113 1 [ [ X american_scientific_journal-7448 113 2 2 2 NUM american_scientific_journal-7448 113 3 ] ] PUNCT american_scientific_journal-7448 113 4 r. r. PROPN american_scientific_journal-7448 113 5 a. a. PROPN american_scientific_journal-7448 113 6 millikan millikan PROPN american_scientific_journal-7448 113 7 . . PUNCT american_scientific_journal-7448 114 1 β€œ " PUNCT american_scientific_journal-7448 114 2 the the DET american_scientific_journal-7448 114 3 isolation isolation NOUN american_scientific_journal-7448 114 4 of of ADP american_scientific_journal-7448 114 5 an an DET american_scientific_journal-7448 114 6 ion ion NOUN american_scientific_journal-7448 114 7 , , PUNCT american_scientific_journal-7448 114 8 a a DET american_scientific_journal-7448 114 9 precision precision NOUN american_scientific_journal-7448 114 10 instrument instrument NOUN american_scientific_journal-7448 114 11 of of ADP american_scientific_journal-7448 114 12 its its PRON american_scientific_journal-7448 114 13 charge charge NOUN american_scientific_journal-7448 114 14 and and CCONJ american_scientific_journal-7448 114 15 the the DET american_scientific_journal-7448 114 16 correction correction NOUN american_scientific_journal-7448 114 17 of of ADP american_scientific_journal-7448 114 18 stokes stokes PROPN american_scientific_journal-7448 114 19 ' ' PART american_scientific_journal-7448 114 20 law law NOUN american_scientific_journal-7448 114 21 . . PUNCT american_scientific_journal-7448 114 22 ” " PUNCT american_scientific_journal-7448 115 1 phys phy NOUN american_scientific_journal-7448 115 2 . . PUNCT american_scientific_journal-7448 116 1 rev rev PROPN american_scientific_journal-7448 116 2 , , PUNCT american_scientific_journal-7448 116 3 , , PUNCT american_scientific_journal-7448 116 4 vol vol NOUN american_scientific_journal-7448 116 5 . . PROPN american_scientific_journal-7448 117 1 32 32 NUM american_scientific_journal-7448 117 2 , , PUNCT american_scientific_journal-7448 117 3 pp pp ADJ american_scientific_journal-7448 117 4 . . PUNCT american_scientific_journal-7448 118 1 349 349 NUM american_scientific_journal-7448 118 2 - - SYM american_scientific_journal-7448 118 3 970 970 NUM american_scientific_journal-7448 118 4 , , PUNCT american_scientific_journal-7448 118 5 1911 1911 NUM american_scientific_journal-7448 118 6 . . PUNCT american_scientific_journal-7448 119 1 [ [ X american_scientific_journal-7448 119 2 3 3 X american_scientific_journal-7448 119 3 ] ] X american_scientific_journal-7448 119 4 r. r. PROPN american_scientific_journal-7448 119 5 a. a. PROPN american_scientific_journal-7448 119 6 millikan millikan PROPN american_scientific_journal-7448 119 7 . . PUNCT american_scientific_journal-7448 120 1 β€œ " PUNCT american_scientific_journal-7448 120 2 on on ADP american_scientific_journal-7448 120 3 the the DET american_scientific_journal-7448 120 4 elementary elementary ADJ american_scientific_journal-7448 120 5 electrical electrical ADJ american_scientific_journal-7448 120 6 charge charge NOUN american_scientific_journal-7448 120 7 and and CCONJ american_scientific_journal-7448 120 8 the the DET american_scientific_journal-7448 120 9 avogadro avogadro NOUN american_scientific_journal-7448 120 10 constant constant ADJ american_scientific_journal-7448 120 11 . . PUNCT american_scientific_journal-7448 120 12 ” " PUNCT american_scientific_journal-7448 121 1 phys phy NOUN american_scientific_journal-7448 121 2 . . PUNCT american_scientific_journal-7448 122 1 rev rev PROPN american_scientific_journal-7448 122 2 . . PROPN american_scientific_journal-7448 122 3 , , PUNCT american_scientific_journal-7448 122 4 vol vol NOUN american_scientific_journal-7448 122 5 . . PROPN american_scientific_journal-7448 122 6 2 2 NUM american_scientific_journal-7448 122 7 , , PUNCT american_scientific_journal-7448 122 8 pp pp ADJ american_scientific_journal-7448 122 9 . . PUNCT american_scientific_journal-7448 123 1 109 109 NUM american_scientific_journal-7448 123 2 - - SYM american_scientific_journal-7448 123 3 430 430 NUM american_scientific_journal-7448 123 4 , , PUNCT american_scientific_journal-7448 123 5 1913 1913 NUM american_scientific_journal-7448 123 6 . . PUNCT american_scientific_journal-7448 124 1 [ [ X american_scientific_journal-7448 124 2 4 4 X american_scientific_journal-7448 124 3 ] ] X american_scientific_journal-7448 124 4 h. h. PROPN american_scientific_journal-7448 124 5 lamb lamb PROPN american_scientific_journal-7448 124 6 . . PUNCT american_scientific_journal-7448 125 1 β€œ " PUNCT american_scientific_journal-7448 125 2 on on ADP american_scientific_journal-7448 125 3 the the DET american_scientific_journal-7448 125 4 uniform uniform ADJ american_scientific_journal-7448 125 5 motion motion NOUN american_scientific_journal-7448 125 6 of of ADP american_scientific_journal-7448 125 7 a a DET american_scientific_journal-7448 125 8 sphere sphere NOUN american_scientific_journal-7448 125 9 through through ADP american_scientific_journal-7448 125 10 a a DET american_scientific_journal-7448 125 11 viscous viscous ADJ american_scientific_journal-7448 125 12 fluid fluid NOUN american_scientific_journal-7448 125 13 . . PUNCT american_scientific_journal-7448 125 14 ” " PUNCT american_scientific_journal-7448 125 15 philosophical philosophical ADJ american_scientific_journal-7448 125 16 magazine magazine NOUN american_scientific_journal-7448 125 17 , , PUNCT american_scientific_journal-7448 125 18 vol vol NOUN american_scientific_journal-7448 125 19 . . PROPN american_scientific_journal-7448 125 20 21 21 NUM american_scientific_journal-7448 125 21 , , PUNCT american_scientific_journal-7448 125 22 pp pp ADJ american_scientific_journal-7448 125 23 . . PUNCT american_scientific_journal-7448 125 24 112 112 NUM american_scientific_journal-7448 125 25 - - SYM american_scientific_journal-7448 125 26 121 121 NUM american_scientific_journal-7448 125 27 , , PUNCT american_scientific_journal-7448 125 28 1911 1911 NUM american_scientific_journal-7448 125 29 . . PUNCT american_scientific_journal-7448 126 1 [ [ X american_scientific_journal-7448 126 2 5 5 X american_scientific_journal-7448 126 3 ] ] PUNCT american_scientific_journal-7448 126 4 c. c. PROPN american_scientific_journal-7448 126 5 w. w. PROPN american_scientific_journal-7448 126 6 oseen oseen PROPN american_scientific_journal-7448 126 7 . . PUNCT american_scientific_journal-7448 127 1 β€œ " PUNCT american_scientific_journal-7448 127 2 ueber ueber PROPN american_scientific_journal-7448 127 3 die die VERB american_scientific_journal-7448 127 4 stokes'sche stokes'sche PROPN american_scientific_journal-7448 127 5 formel formel X american_scientific_journal-7448 127 6 und und PROPN american_scientific_journal-7448 127 7 ueber ueber PROPN american_scientific_journal-7448 127 8 einer einer PROPN american_scientific_journal-7448 127 9 verwandte verwandte PROPN american_scientific_journal-7448 127 10 aufgabe aufgabe PROPN american_scientific_journal-7448 127 11 in in ADP american_scientific_journal-7448 127 12 der der ADJ american_scientific_journal-7448 127 13 hydrodynamik hydrodynamik ADJ american_scientific_journal-7448 127 14 . . PUNCT american_scientific_journal-7448 127 15 ” " PUNCT american_scientific_journal-7448 128 1 ark ark PROPN american_scientific_journal-7448 128 2 . . PUNCT american_scientific_journal-7448 129 1 f. f. PROPN american_scientific_journal-7448 129 2 mat mat PROPN american_scientific_journal-7448 129 3 . . PROPN american_scientific_journal-7448 129 4 astr astr PROPN american_scientific_journal-7448 129 5 . . PUNCT american_scientific_journal-7448 130 1 och och PROPN american_scientific_journal-7448 130 2 . . PUNCT american_scientific_journal-7448 131 1 fys fys PROPN american_scientific_journal-7448 131 2 . . PROPN american_scientific_journal-7448 131 3 , , PUNCT american_scientific_journal-7448 131 4 vol vol NOUN american_scientific_journal-7448 131 5 . . PROPN american_scientific_journal-7448 132 1 6 6 NUM american_scientific_journal-7448 132 2 , , PUNCT american_scientific_journal-7448 132 3 pp pp ADJ american_scientific_journal-7448 132 4 . . PUNCT american_scientific_journal-7448 133 1 1 1 NUM american_scientific_journal-7448 133 2 - - SYM american_scientific_journal-7448 133 3 20 20 NUM american_scientific_journal-7448 133 4 , , PUNCT american_scientific_journal-7448 133 5 1910 1910 NUM american_scientific_journal-7448 133 6 . . PUNCT american_scientific_journal-7448 134 1 [ [ X american_scientific_journal-7448 134 2 6 6 NUM american_scientific_journal-7448 134 3 ] ] PUNCT american_scientific_journal-7448 134 4 b. b. PROPN american_scientific_journal-7448 134 5 mutlu mutlu PROPN american_scientific_journal-7448 134 6 sumer sumer PROPN american_scientific_journal-7448 134 7 , , PUNCT american_scientific_journal-7448 134 8 and and CCONJ american_scientific_journal-7448 134 9 jorgen jorgen PROPN american_scientific_journal-7448 134 10 fredsoe fredsoe PROPN american_scientific_journal-7448 134 11 . . PUNCT american_scientific_journal-7448 135 1 hydrodynamics hydrodynamic NOUN american_scientific_journal-7448 135 2 around around ADP american_scientific_journal-7448 135 3 cylindrical cylindrical ADJ american_scientific_journal-7448 135 4 structures structure NOUN american_scientific_journal-7448 135 5 . . PUNCT american_scientific_journal-7448 136 1 world world PROPN american_scientific_journal-7448 136 2 scientific scientific PROPN american_scientific_journal-7448 136 3 co. co. PROPN american_scientific_journal-7448 136 4 pte pte PROPN american_scientific_journal-7448 136 5 . . PROPN american_scientific_journal-7448 136 6 ltd ltd PROPN american_scientific_journal-7448 136 7 . . PROPN american_scientific_journal-7448 136 8 ,revised ,revised PUNCT american_scientific_journal-7448 136 9 edition edition PROPN american_scientific_journal-7448 136 10 , , PUNCT american_scientific_journal-7448 136 11 2006 2006 NUM american_scientific_journal-7448 136 12 . . PUNCT american_scientific_journal-7448 137 1 [ [ X american_scientific_journal-7448 137 2 7 7 X american_scientific_journal-7448 137 3 ] ] X american_scientific_journal-7448 137 4 i. i. NOUN american_scientific_journal-7448 137 5 eames eames PROPN american_scientific_journal-7448 137 6 , , PUNCT american_scientific_journal-7448 137 7 and and CCONJ american_scientific_journal-7448 137 8 c. c. PROPN american_scientific_journal-7448 137 9 a. a. NOUN american_scientific_journal-7448 137 10 klettner klettner NOUN american_scientific_journal-7448 137 11 . . PUNCT american_scientific_journal-7448 138 1 β€œ " PUNCT american_scientific_journal-7448 138 2 stokes stokes PROPN american_scientific_journal-7448 138 3 ’ ' PUNCT american_scientific_journal-7448 138 4 and and CCONJ american_scientific_journal-7448 138 5 lamb lamb PROPN american_scientific_journal-7448 138 6 ’s ’s PART american_scientific_journal-7448 138 7 viscous viscous ADJ american_scientific_journal-7448 138 8 drag drag NOUN american_scientific_journal-7448 138 9 laws law NOUN american_scientific_journal-7448 138 10 . . PUNCT american_scientific_journal-7448 138 11 ” " PUNCT american_scientific_journal-7448 139 1 european european PROPN american_scientific_journal-7448 139 2 journal journal PROPN american_scientific_journal-7448 139 3 of of ADP american_scientific_journal-7448 139 4 physics physics PROPN american_scientific_journal-7448 139 5 , , PUNCT american_scientific_journal-7448 139 6 vol vol NOUN american_scientific_journal-7448 139 7 . . PROPN american_scientific_journal-7448 140 1 38 38 NUM american_scientific_journal-7448 140 2 , , PUNCT american_scientific_journal-7448 140 3 pp pp ADJ american_scientific_journal-7448 140 4 . . PUNCT american_scientific_journal-7448 141 1 11 11 NUM american_scientific_journal-7448 141 2 , , PUNCT american_scientific_journal-7448 141 3 2017 2017 NUM american_scientific_journal-7448 141 4 . . PUNCT american_scientific_journal-7448 142 1 [ [ X american_scientific_journal-7448 142 2 8 8 NUM american_scientific_journal-7448 142 3 ] ] X american_scientific_journal-7448 142 4 g. g. PROPN american_scientific_journal-7448 142 5 k. k. PROPN american_scientific_journal-7448 142 6 batchelor batchelor PROPN american_scientific_journal-7448 142 7 . . PUNCT american_scientific_journal-7448 143 1 an an DET american_scientific_journal-7448 143 2 introduction introduction NOUN american_scientific_journal-7448 143 3 to to ADP american_scientific_journal-7448 143 4 fluid fluid ADJ american_scientific_journal-7448 143 5 dynamics dynamic NOUN american_scientific_journal-7448 143 6 . . PUNCT american_scientific_journal-7448 144 1 cambridge cambridge PROPN american_scientific_journal-7448 144 2 : : PUNCT american_scientific_journal-7448 144 3 cambridge cambridge PROPN american_scientific_journal-7448 144 4 university university PROPN american_scientific_journal-7448 144 5 press press PROPN american_scientific_journal-7448 144 6 , , PUNCT american_scientific_journal-7448 144 7 1st 1st PROPN american_scientific_journal-7448 144 8 edition edition NOUN american_scientific_journal-7448 144 9 , , PUNCT american_scientific_journal-7448 144 10 1967 1967 NUM american_scientific_journal-7448 144 11 . . PUNCT american_scientific_journal-7448 145 1 [ [ X american_scientific_journal-7448 145 2 9 9 NUM american_scientific_journal-7448 145 3 ] ] PUNCT american_scientific_journal-7448 145 4 g. g. PROPN american_scientific_journal-7448 145 5 k. k. PROPN american_scientific_journal-7448 145 6 batchelor batchelor PROPN american_scientific_journal-7448 145 7 . . PUNCT american_scientific_journal-7448 146 1 β€œ " PUNCT american_scientific_journal-7448 146 2 slender slender ADJ american_scientific_journal-7448 146 3 - - PUNCT american_scientific_journal-7448 146 4 body body NOUN american_scientific_journal-7448 146 5 theory theory NOUN american_scientific_journal-7448 146 6 for for ADP american_scientific_journal-7448 146 7 particles particle NOUN american_scientific_journal-7448 146 8 of of ADP american_scientific_journal-7448 146 9 arbitrary arbitrary ADJ american_scientific_journal-7448 146 10 cross cross NOUN american_scientific_journal-7448 146 11 - - NOUN american_scientific_journal-7448 146 12 section section NOUN american_scientific_journal-7448 146 13 in in ADP american_scientific_journal-7448 146 14 stokes stokes PROPN american_scientific_journal-7448 146 15 flow flow NOUN american_scientific_journal-7448 146 16 . . PUNCT american_scientific_journal-7448 146 17 ” " PUNCT american_scientific_journal-7448 147 1 journal journal NOUN american_scientific_journal-7448 147 2 of of ADP american_scientific_journal-7448 147 3 fluid fluid ADJ american_scientific_journal-7448 147 4 mechanics mechanic NOUN american_scientific_journal-7448 147 5 , , PUNCT american_scientific_journal-7448 147 6 vol vol NOUN american_scientific_journal-7448 147 7 . . PROPN american_scientific_journal-7448 147 8 44 44 NUM american_scientific_journal-7448 147 9 , , PUNCT american_scientific_journal-7448 147 10 pp pp ADJ american_scientific_journal-7448 147 11 . . PUNCT american_scientific_journal-7448 148 1 419 419 NUM american_scientific_journal-7448 148 2 - - SYM american_scientific_journal-7448 148 3 440 440 NUM american_scientific_journal-7448 148 4 , , PUNCT american_scientific_journal-7448 148 5 1970 1970 NUM american_scientific_journal-7448 148 6 . . PUNCT