Microsoft Word - 203-211 https://doi.org/10.30526/31.1.1840 Mathematics | 203 2018)عام 1العدد ( 31مجلة إبن الهيثم للعلوم الصرفة والتطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018  An Accurate MHD Flux Solutions of a Viscose Fluid and Generalized Burgers' Model fluxwithin an Annular Pipe Under Sinusoidal Pressure Hanan F. Qasim Dept. of Mathematics/ College of Education/ Pure Science Ibn AL-Haitham/ University of Baghdad Hanankasim83@gmail.com Received in:3 /October/2017, Accepted in:17/ December/2017 Abstract The aim of this work presents the analytical studies of both the magnetohydrodynamic (MHD) flux and flow of the non-magnetohydro dynamic (MHD) for a fluid of generalized Burgers’ (GB) withinan annular pipe submitted under Sinusoidal Pressure (SP)gradient. Closed beginning velocity's' solutions are taken by performing the finite Hankel transform (FHT) and Laplace transform (LT) of the successivefraction derivatives. Lastly, the figures were planned to exhibition the transformations effects of different fractional parameters (DFP) on the profile of velocity of both flows. Keywords: Generalized Burgers’, finite Hankel transform, Laplace transform, Sinusoidal Pressure gradient. https://doi.org/10.30526/31.1.1840 Mathematics | 204 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018 Preface Lately, many attentions have been devoted to the project of nonNewtonian fluids. In general, the foremostobject is t hat fluids (such as paints, the molten plastics, slurries, pulps, emulsions, the petroleum was drilled, blood and other identical entities), which do not follow the Newtonian assume that the stress tensor is immediately symmetricto the rate of turn of deformitytensor, and show characteristics of flow quite severalto those of Newtonian fluids, the models are usually distribution gas fluids of differential, average and integral types (Rajagopal, [1]; and Dunn and Rajagopal, [2]). Different studies were performed on a generalized Oldroyd-B(GO-B) fluid flux includes those from Zheng et al. [3], Khan et al. [4], and Sultan et al. [5]. Fetecau et al. [6], Kamranc et al [7] and Hyder Ali Muttaqi Shah [8] thought fulsome summary fluxes of (GO-B) fluid through two wall sides that perpendicular to a sheet. Zheng et al. [9], and Nazar et al. [10] talk overMaxwell fluid flux because of a plate, with fixed velocity. Mahmood et al. [11] investigated the unsteady flux of a non-Newtonian fluid between two infinite coaxial circular cylinders. Whereas Khan et al. [12], Khan et al. [13], with Khan and Shafie, [14] described the exact solutions for the flux of an MHD (GB) fluid. In this paper, our target is to study the unsteady viscoelastic fluid flow with the model of fractional (GB) fluid within an annular pipe under (SP), and compare with flow under MHD (SP). The accurate solution for the distribution of velocity is performed by implying the (FHT) Garg et al. [15] and discrete (LT) of the sequential fractional derivatives. Prevalent Equations The constituent equations for an incompressible fractional(GB)fluid are agreed through SIT  p , 1AS )D ~ +(1=)D ~ D ~ +(1 t3 2 tt 21    (1) anywhere T fixed by Cauchy stress, Ip is undefined spherical stress, S means the additional stress tensor, T LLA 1  is the first tensor of Rivlin- Ericksen with the gradient of velocity anywhere VL grad ,  showed the efficient viscosity of fluid, 1  and 3  (< 1  ) are the relaxation, and the obstruction times, respectively, 2  is the modern item parameter of (GB)fluid, α and β the (DFP) calculus like that 10   and p t D ~ the upper convicted fractional derivative which described through T ).( ~ SLSLSVSS   tt DD (2) T ).( ~ LALAAVAA 11111   tt DD (3) in whose  t D and  t D are the (DFP)of order  and  be contingent on the definition of Riemann- Liouville, identify as 10, )( )( )1( 1 )]([ 0     Pdt F dt d P tFD t p p t    and )( 2 SS p t p t p t DDD  (4) When (.) is the Gamma function. The type diminished to the model of (GO-B) when 02  and if, addendum for that α = β = 1 the normal Oldroyd-B typeshall be earned. So, we suppose that shear stress and the field of velocity of the format https://doi.org/10.30526/31.1.1840 Mathematics | 205 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018 ),(,),( tretr z SSV   (5) When z e meant vector unit along the z-direction .Equation (5) substituted into (1) and takeover an account for the first condition S (r,0) = 0 (6) Obtain 2 3rt21rzzz 2 t2t1 3 2 21 )),((2),()D+(S2S)DD+(1 ),()1()(1 trtr trDSDD r rttt       (7) When 0SSSS rzrr   . Thereafter the being gradient of pressure at z-direction, the motion equation provided next scalar equation: )( 1 z p t rz Sr rrd d         (8) When  showed constant fluid density. The judge rz S amidst Eqs. (7), and (8), we earn the next fractional differential equation      ) 1 )(D+1( dz dp )DD+(1 1 t )DD+(1 2 2 t3 2 t2t1 2 t2t1 rrr v            (9) When   v indicated the kinematic viscosity. First Problem of the Non-magnetohydrodynamic Flow Regard that the fluxaffair of an incompressible (GB) fluid is firstly at rest in between two infinitely long coaxial cylinders of the radius 0R and 1R ( 0R ). At time  0t fluid is generated because of(SP) gradient which acts on liquid in z-direction. Pointing to Eq. (9), the coinciding (DFP) equation that define such flux has the way             ) 1 )(D+1( )2()( +1)(cos t )DD+(1 2 2 t3 21 2 1 10 2 t2t1 rrr v tt ut                    (10) When dz dp  1 (ut)cos 0  showed the (SP) gradient The condition of initial and boundary relations are described as form 102 2 ,0)0,()0,()0,( RrRr t r t r         (11) 0,0),(),( 10  ttRtR  (12) For producing the accurate analytical solution of the previous problems (10)- (12), First, we perform(LT) rule Garg et al. [15] through respect to t, we got )+(1 s sP ) 1 )(+1()+(1s 22122 0 2 2 3 2 21   ss urrr svss          (13) 0,0),(),( 0)0,( 10   tsRsR r   (14) When ),( sr denoted of function image of ),( tr and s denoted the parameter transform. We imply the (FHT) Garg et al. [15], described as form drkrBr i R R H )(0 1 0   ,  321i (15) While its inverse as https://doi.org/10.30526/31.1.1840 Mathematics | 206 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018      1 1 2 00 2 0 1 2 00 22 )()( )()( 2 i ii iiHi kRJkRJ kRJrkBk   (16) Where ik are the positive roots of equation 0)( 10 ikRB and )()()()()( 0000000 iiiii kRJrkYkRYrkJrkB  When (.) 0 J while (.)0Y are the functions of Bessel of the first and second types of zero order. Here using (FHT) to Eqs. (13)-(14) through respect to r, we take )1(k)+s(1 )+(1 s s 3 2 i 2 21 2 2122 0      sss ss u H    (17) Currently, lettering Eq. (17) In the form of a chain as 1 1 2 1 32 1 1 212 0,,,,,,0 2 210 !!!!!!!! )()()()()( !)1()+(1                   k i zqrlnnkmq i nkhkjfedcba jfedcbak k H k shjfedcba svku kss         (18) Where zqrlnmhqlk    )222(2231 . Whileits discrete inverse (LT) Garg et al. (2007) will yield the form                                         1 1 2 1,12 21 1 2 1,11 1 1 2 1,1 0,, 1)1()1( 32 1 1 212 0 0 !!!!!!!! )()()()()( !)1(               t vk Ett vk Ett vk E t hjfedcba ku k ikikik kcba cba kzqrlnnkmq i nkh k k H (19) When       0 , )(! !)!( )( j m mjj zmj zE  showed generalized Mittag- Leffler function Garg et al. [15] andto earn Eq. (19), the following property of inverse (LT) is used (20) )( )( ! , 11      ctEt cs sm L mm            ,         1 )Re( cs (20) eventually, the inverse (FHT) obtains the analytic solution of velocity classification                                                             1 1 2 1,12 21 1 2 1,11 1 1 2 1,1 0,,,,,, 1)1()1(32 1 1 212 0 1 2 00 2 0 0 22 0 !!!!!!!! )()()()()( !)1( )()( )()( 2 ),(                 t vk Ett vk Ett vk E t dfjehcba ku k kRJkRJ kRJrkBk tr ikikik kjdefcba jjedcba k zqrlnnkmq i qlkh k k i iii iiii (21) The Special Cases Working the limits of Eq.(21) where 0 , 02  (b=0) , we obtain the distribution of velocity for a (GO-B) fluid. So the field of velocity decreases to                                                  1 1 2 1,1 11 1 2 1,1 0,,,,, 1)1()1(3 1 1 212 01 1 2 00 2 0 1 2 00 22 0 !!!!!! )()()()( !)1( )()( )()( 2 ),(             t vk E t t vk E t cdefba ku k kRJkRJ kRJrkBk tr ikik cdefba kcdefba k hnkml i fkn k k i ii iii (22) Where hfhlnfhlmk   3)2(13 . Second Problem of the Magnetohydrodynamic (MHD) Flow Moreover, it believes that showing fluid is prevailed by imposing magnetic field 0[0, H , 0]H which work in positive z-direction. In the calculation of the low-magnetic Reynolds number, https://doi.org/10.30526/31.1.1840 Mathematics | 207 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018 the magnetic power of the body is considered as 2 0H w , when indicated electrical accessibility of fluid.Now, by adding magnetic field to Eq.(8) we get an Eq. (23): 2 0 p 1 (r S ) Hrz dw w dt z r r           (23) the judge rzS among Eqs. (7) and (23), we make the next fractional differential equation        )DD+(1M) 1 )(D+1( dz dp )DD+(1 1 t )DD+(1 2 t2t12 2 t3 2 t2t1 2 t2t1              rrr v (24) Anywhere   v denoted the kinetic viscosity while 2 0HM    denoted the dimensionless magnetic number. In the same way as calculating the flux of the first problem we find Eq. (24), the according fractional partial differential equation that term such fluxhas the shape             )1() 1 )(D+1( )2()( +1)cos( t )DD+(1 2 212 2 t3 12 2 1 10 2 t2t1 Tt DDM rrr v tt ut                       (25) Where dz dp utP  1 )cos(0  indicated the continual pressure gradient To earn the accurate analytical solution of the previous problems (25)- (12), First, we perform (LT) rule Garg et al. [15] through respect to t, we obtain   )1() 1 )(+1()+(1 s s )+(1s 2 212 2 3 2 21220 2 21 SSM rrr svss u ss          (26) Now using (FHT) to Eqs. (26) -(14) through respect to r, we obtain )1(k)+M)(1(s )+(1 s s 3 2 i 2 21 2 2122 0      sss ss u H    (27) Now, inscription Eq. (27) in sequence form as 1 1 2 1 33210 321 22 0,,,,,,,,,,0 2 210 !!!!!!!!!!!!! 6*)(5*)(4*)(3*)(2*)(1*)( !)1()+(1 321 32101                k i i kcccoigedcba wwwwtjfhnzqlk k H k Mswccccoigedcba sMvku kss         (28) Where 3111* wwfk  , 12* w , 213* wtfz  , 0114* wthnqlk  , 1015* wwtjfhzq  , 26* w , 3201 2)(2)21()21()21()(31 wwwjtfzhnqlk   . While its discrete inverse (LT) Garg et al [15] will yield the form                                         1 1 2 1 1,12 21 1 2 1 1,11 1 1 2 1 1,1 ..... 0,.....,, 1)1()1( 6 3 5 2 4 1 32212 0 0 ()()( !!!!!!!! *)(*)(*)(*)(*)(*)( !)1(                t vkM Ett vkM Ett vkM E t hjfedcba Mku k ikikik kcba zqj ki k k H (29) the following property of inverse (LT) is used (20). finally, the inverse (FHT) gets the analytic solution of velocity classification                                                            1 1 2 1 1,12 21 1 2 1 1,11 1 1 2 1 1,1 .... 0,....,, 1)1()1(321 22 0 1 2 00 2 0 0 22 0 )()()( !!!!!!!! 6*)(5*)(4*)(3*)(2*)(1*)( !)1( )()( )()( 2 ),(                 t vkM Ett vkM Ett vkM E t hjfedcba Mku k kRJkRJ kRJrkBk tr ikikik kcba zql ki k k i iii iiii (30) https://doi.org/10.30526/31.1.1840 Mathematics | 208 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018 The Special cases Working the limitsfor Eq.(30) where 0 , 02  (b=0) while 0M (c=d=0) , we obtain the distribution of velocity for (GO-B) fluid. So the field of velocity decreases to                                                    1 1 2 1,1 11 1 2 1,1 0,,,,, 1)1()1( 10 3 1 1 212 01 1 2 00 2 0 1 2 00 22 0 !!!!!!!! )()()()( !)1( )()( )()( 2 ),(             t vk E t t vk EM t hhfedcba ku k kRJkRJ kRJrkBk tr ikikywiq efcdba kfdecba k yzjkyiwnlji i zk k k i ii iii (31) Where )()(31 yzjwzqywinqlk   . Results Discussion In the present study, we have been discussed MHD flux of (GB) fluid that passed an annular pipe. The accurate solution for the field of velocity u is gotten by performing the (LT) and (FHT). Furthermore, figures were plotted to show the behavior of diverse parameters included the velocity expressions u . A comparison between the effect of magnetic parameter (M≠0) (Panel (a)) and the effect of non-magnetic parameter (M=0) (Panel (b)) were also done graphically in figures (1-6). figures (1) and (2) the velocity is increased with the increasing of the with both cases (M=0 & M≠0), while it increased with  (M≠0) more than with  (M=0). figures (3), (4) and (5) showed the relaxation parameter effect 1 on the fields of velocity. Velocity is decreased for the incensement of 1 for (M≠0), and it did not affected with the increase of 1 for (M=0). Velocity is increased with the incensement of 2 when (M=0), and it oscillated with the increase of 2 for (M≠0).The velocity is decreased with the incensement of 3 when (M=0), and decreased more with the incensement of 3 for (M≠0). figure (6) has shown the effect of the magnetic parameter M inshort as well as in long time. It is detected that the velocity profile is increased with the increase of t = 0.5 – 1.2 for (M≠0) more than for (M=0). Comparison displays that velocity sketch with the effect of magnetic field is greater when compared with velocity sketch without the effect of magnetic field. The result is demonstrated in long time. Figure (1): Shows velocity for various values of  while remaining another parameters constant (a) M 3 , and (b) M 0 https://doi.org/10.30526/31.1.1840 Mathematics | 209 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018   Figure (2): Shows velocity for various values of  while remaining another parameters constant (a) M 3 and (b) M 0   Figure (3) Shows velocity for various values of 1 while remaining another parameters constant (a) M 3 and (b) M 0 Figure (4): Shows velocity for various values of 2 while remaining another parameters constant (a) M 3 and (b) M 0 https://doi.org/10.30526/31.1.1840 Mathematics | 210 2018)عام 1العدد ( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة والتطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (1) 2018 Figure (5): Shows velocity for various values of 3 while remaining another parameters constant Figure (6): Shows velocity for various values of M while remaining another parameters constant (a) 0.1t  and (b) 0.5t  References [1] J. Dunn; and K. Rajagopal. Fluids of differential type-critical review and thermodynamic analysis. International Journal of Engineering Science 33 ,689-729, 1995. [2] K. Rajagopal. 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