SQU Journal for Science, 2016, 21(2), 107-119 © 2016 Sultan Qaboos University 107 Combined Effects of Hall Current and Heat Transfer on Peristaltic Transport of a Nanofluid in a Vertical Tapered Channel Through a Porous Medium Ahmed I. Abdellateef and Syed Z. Ul Haque* Department of Mathematics and Statistics, Caledonian College of Engineering, Sultanate of Oman, Muscat. University of Glasgow, Scotland, UK. *Email: hqzahoor@gmail.com. ABSTRACT: The influences of Hall currents and heat transfer on peristaltic transport of a nanofluid in a vertical porous tapered channel through a porous medium are investigated theoretically and graphically under the assumptions of low Reynolds number and long wavelength and the flow investigated is in a wavy frame of reference. Analytical solutions are obtained for temperature, axial velocity, stream function and pressure gradient. Graphical results are sketched for various embedded parameters. Keywords: Peristaltic flow; Heat transfer; Nanofluid; Tapered channel; Hall currents and Porous medium. ي وسط مساميف هقناة رأسية مستطيل عبرالتاثير المشترك لكل من االنتقال الحراري وتيارات الهول على النقل التمعجي لمائع نيوتوني الحقيد زهورس و عبد اللطيف إبراهيم حمدأ قي وسط هقناة رأسية مستطيل عبرتم دراسة التاثير المشترك لكل من االنتقال الحراري وتيارات الهول كرنت على النقل التمعجي لمائع نيوتوني :ستخلصم اد اطار االسن عبروكبر الطول الموجي . تمت دراسة السريان عدد رينولد بناًء على فرضية صغر الدراسة نظريا واستنتجت الرسومات مسامي. تمت نتائج المستخلصة الموجي . وتم حساب دالة التوزيع الحراري والسرعة المحورية ودالة السريان وكذلك تدرج الضغط خالل السريان .وتم مناقشة واظهار ال المختلفة. من خالل الرسومات .مساميسط و وتيارات هول ،قناة متباعدة ،مائع نانوي ،انتقال حراري ،نقل تمعجي : كلمات مفتاحية 1. Introduction ass transfer and heat transfer from different geometries embedded in a porous medium have many geophysical and engineering applications such as thermal insulations, drying of porous solids, cooling of nuclear reactors, underground energy transport, and crude oil extraction. The study of fluid flow through a porous medium and heat transfer constitute a fundamental study of nature and have great practical importance in several physical problems; these include porous heat exchangers, sewage of water in river beds, filtration and purification processes, for example. Their industrial importance has led to many experimental and analytical studies of flow and heat transfer in a porous medium in the presence of a magnetic field. McWhirter et al. [1], and Geindreau and Auriault [2] have discussed magnetohydrodynamic (MHD) flow through a porous medium in great details. The tapered channel is specially designed to assist in the management of subglottic, tracheal, and glottic stenosis. The superior end conforms to the contours of the glottis, while the larger inferior portion simultaneously acts as a stent for the trachea. The design of the tapered end is such that it projects up beyond the true vocal cords to the level of the laryngeal ventricle. Peristaltic flow is a phenomenon found in several industrial and physiological processes. Within the body peristaltic flows occur widely in fluid processes such as in urine transport from kidney to bladder through the ureter, transport of lymph in the lymphatic vessels, swallowed food through the esophagus, spermatozoa transport in the human reproductive tract, vaso motion of small blood vessels, etc. In addition to that, peristaltic pumping occurs in many practical applications involving biomechanical systems. The mechanism of peristaltic flow can be best explained by comparing the experimental clinical data and mathematical results. A perfect mathematical study can be served to many peristaltic flows occuring in the human body. Many research attempts had been made to understand peristaltic action in different situations. The first attempt was made by Latham [3] and some of the interesting investigations in the same direction are given in references [4–7]. Moreover, the flow through a porous medium has also attracted the attention of many M AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 108 researchers in the past few decades because of its important practical applications, as it occurs in sandstone, limestone, filtration of fluids and seepage of water in river beds, wood, and in human lungs, the bile duct, the gall bladder with stones, and in small blood vessels etc. Shehawey et al. [8] formulated a mathematical model for the peristaltic transport of a viscous incompressible fluid through a porous medium bounded by two porous plates and El Shehawey et al. [9] studied the peristaltic transport in an asymmetric channel through a porous medium. Vajravelu et al. [10] examined peristaltic flow through vertical porous tube. Srinvas and Kothandapani [11] investigated the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Srinvas and Gayathri [12] continued this work and discussed the peristaltic transport of Newtonian fluid in an asymmetric channel of a porous medium with heat transfer. The Hall effect is important when the Hall parameter which is the ratio between the electron-cyclotron frequency and the electron-atom-collision frequency is high. This can occur if the collision frequency is low or when the magnetic field is high. This is a current trend in MHD because of the important influence of the electromagnetic force. There has been considerable study of Hall effects and heat transfer effects on flow to determine the efficiency of certain devices such as heat exchangers and power generators. Attia [13] examined unsteady Hartmann flow with heat transfer of a viscoelastic fluid taking the Hall effect into consideration. Asghar et al. [14] studied the effects of Hall current and heat transfer on flow due to the pull of an eccentric rotating disk. Hayat et al. [15] studied the Hall effects on peristaltic flow of a Maxwell fluid in a porous medium. Abo-Eldahab et al. [16, 17] investigated the effects of Hall and ion-slip currents on MHD peristaltic transport and couple stress fluid. For further information the references [18–25] provide additional source. The aim of this paper is to investigate the effects of Hall current and heat transfer on peristaltic transport of a Newtonian nanofluid in a tapered channel through a porous medium. Nomenclature a1, b1 Wave amplitudes eR Reynolds number d1+d2 Two-dimensional channel width  Dimensionless wave number λ Wavelength of the peristaltic wave 1 m Hall current parameter 1 c Wave speed M Magnetic field parameter ϕ Phase difference k̂ Thermal conductivity f  Density of the fluid tN Thermophoresis parameter p  Density of the particle bN Brownian motion parameter q Velocity vector rG Thermal Grashof number s T Fluid temperature in static condition rB Local nanoparticle Grashof number D/Dt Material derivative p Pressure rise p Pressure F Friction force C Nanoparticle volume fraction K Porosity parameter B D Brownian diffusion coefficient θ Temperature distribution T D Thermophoretic diffusion coefficient  Dimensionless nanoparticle volume fraction d Amplitude ratio 2. Formulation of the problem Consider the peristaltic transport of a nanofluid in a two-dimensional channel of width 1 2( )d d . The flow is induced by propagating a sinusoidal wave with constant speed ‘c’ along the channel walls (as shown in Fig. 1). We denote the channel walls by 1 H and 2 .H 1 1 1 1 1 Right wall 2 ( , ) = cos[ ( )] H X t d K x a X c t      (1) 2 2 2 1 1 eft wall 2 ( , ) = cos[ ( ) ] LH X t d K x b X c t         (2) where 1 a and 2 a are the wave’s amplitudes,  is the wave length, 1 c is the wave speed, and  )(0   is the phase difference. It should be noted that 0= corresponds to a symmetric channel with waves out of phase and that COMBINED EFFECTS OF HALL CURRENT AND HEAT TRANSFER 109 for  = the waves are in phase. Further, 1121 ,,, badd and  satisfy the following inequality 2 2111 2 1 2 1 )(cos2 ddbaba   . The walls 1 = HY and 2 = HY , are maintained at temperatures 1 T and 2 T respectively. The governing equations for the flow and heat transfer of an incompressible nanofluid in the absence of body force are: 0,=q (3) ,= 2 fqp Dt qD f   (4) 2 0 ( ) = ( ) [ ( ) ( ) ( ) ( )], ( ) T f p B DDT c k T c D C T T T Dt T           (5) , )( = 2 0 2 T T D CD Dt CD T B  (6) where f is the density of the fluid, p is the density of the particle, q is the velocity vector, f is the body force, s T is the fluid temperature in static condition, D Dt represents the material derivative, p is the pressure , C is the nanoparticle volume fraction, B D is the Brownian diffusion coefficient and T D is the thermophoretic diffusion coefficient. We now introduce a wavy frame ),( yx moving with velocity 1 c away from the fixed frame ),( YX by the transformation ),,(=)(,=,=,=,= 11 tXPxpVvcUuYytcXx  (7) Figure 1. The simplified model geometry of the problem. where ),( vu are the velocity components in the wavy frame ),( yx , p and P are pressures in wavy and fixed frame respectively. The pressure p remains constant across any axial station of the channel under the assumption that the wavelength is large and the curvature effects are negligible. After using these transformations, the governing equations in steady case become: 0,= y v x u      (8) AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 110 )( )(1 )(=][ 12 1 2 1 2 2 2 2 vum m B v ky v x v y p y v v x v u o                    (9) 2 2 2 2 [ ] ( ) u u p u u u v x y x x y                  2 * 0 0 12 1 1 ( ) ( ) ( ) (1 ) o B g T T g C C u m v u mk             (10) 2 2 2 2 2 2 0 [ ] = ( ) [ {( ) ( )} {( ) ( ) }] ( ) r B T T T T T C T C T u v D x y y y x xx y D T T T y x                            (11) )( )( )(=][ 2 2 2 2 0 2 2 2 2 y T x T T D y C x C D y C v x C u T B                 (12) where f p r c c )( )( =    is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid. The corresponding no-slip boundary conditions are: 1 1 1 1 1 2 2 2 = , = , = at = , = , = , = at = . u c C C T T y H u c C C T T y H   (13) Consider the following non-dimensional variables and parameters: 2 1 1 1 2 1 2 1 1 1 1 1 1 12 2 1 1 1 1 1 1 1 00 0 = , = , = , = , = , = , = , = , = , ( ) ( ) = , = , = , = , = , = , = , ( ) ˆ( )( ) = , = , = , ( ) ( ) p T ss r e t s f p B b f f d d p H Hu v x y ct u v x y p t h h c c d c d d c D T TT Td a a c d d b a P R N d d d T T c c D CC C k N G c cC                      2 1 1 ( ) = ,s r g d T T c    * 2 2 2 11 0 0 1 1 2 1 22 1 1 = , = , = , = , = r g d C B d K Kk B k M k k c a d d     (14) where sT - the fluid temperature in static condition, d - the amplitude ratio, eR - the Reynolds number,  - the dimensionless wave number, 1 m - the Hall current parameter, M - the magnetic field parameter, k̂ -the thermal conductivity, and bN , tN , rB and rG are the Brownian motion parameter, thermophoresis parameter, local nanoparticle Grashof number, and local thermal Grashof number respectively. With the help of Eq. (2.14) in Eqs.( 2.8)–(2.13) under the assumptions of long wavelength and low Reynold number approximation takes the form 0,= y v x u      (15) 0= y p   (16) u m M k BG y u x p rr ) )(1 1 (= 2 11 2 2        (17) 2 2 2 )()(=0 y N yy N y tb            (18) COMBINED EFFECTS OF HALL CURRENT AND HEAT TRANSFER 111 2 2 2 2 =0 yN N y b t       (19) The corresponding boundary conditions become: ,=0,=,=1,= ,=0,=1,=1,= 20 1 hymu hyu     (20) where ))/((= 120 ss TTTTm  , the wall temperature ratio. The solution of Eqs. (2.18,2.19), valid in 12 hyh  satisfying the corresponding boundary conditions (20) is given by  bNhsybNs eeA 1 1 11 1=   (21) )()(1= 11 hys N N b t  (22) By substituting Eqs. (2.21)-(2.22) in Eq. (2.17) and solving it analytically as a second order ordinary differential equation, we get a solution in the form, y b NsyAyA ecececyccxu 1 6 2 5 2 432 =)(   (23) 3. Rate of volume flow The instantaneous volume flow rate in the fixed frame is given by ,),,(= 1 2 YdtYXUQ H H (3.1) where 1 H and 2 H are functions of tX , . The rate of volume flow in the wave frame is given by ,),(= 1 2 ydyxuq H H (3.2) where 1 h and 2 h are the functions of x alone. If we substitute equation (1.7) into (2.1) and make use of (2.2), it is found that the related rates of volume flow is: .= 2111 HcHcqQ  (3.3) The time-mean flow over a period T at a fixed position X is defined as: , 1 = 0 * Qdt T Q T  (3.4) Substituting (2.3) into (2.4), and integrating, we get: .= 2111 * dcdcqQ  (3.5) On defining the dimensionless time-mean flows Q and q in the fixed and wavy frame respectively as: ,=and= 1111 * dc q q dc Q Q (3.6) One finds that (2.5) may be written as: .1= dqQ  (3.7) From Eq. (2.23), with the help of Eqs. (3.1)-(3.7), we get the pressure gradient. The pressure rise p and the friction force F (at the walls) for a channel of length L , in their non-dimensional forms, are given by ,= 1 0 dx x p p     (3.8) .)(= 2 1 1 0 1 dx x p hF h    (3.9) .)(= 2 2 1 0 2 dx x p hF h    (3.10) The integrals in equations (3.9)-(3.10) are not integrable in closed form, they are evaluated numerically using a ready package of Nintegrate Library from Mathematica 7. AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 112 Nintegrate uses an algorithm called ‘integration strategies’ that attempts to compute an integral which satisfy user specified precision or accuracy goals. The corresponding stream functions ( y u   = , x v     = ) are evaluated with boundary condition 0= at 1 2 1 2 = ( )z h h . 4. Graphical results and discussion Here we discuss the quantitative effects of sundry parameters used in analysis with the help of graphs and results. First, let us discuss the variations observed for the pressure rise distributions versus the time average flux ( Q ), for different values of the local nanoparticle Grashof number ( r B ), local thermal Grashof number ( rG ), porosity parameter ( K ), thermophoresis parameter ( tN ), non-uniform parameters ( 1k ) and ( 2k ), Hall parameter ( M ). It is noticed in Figure 2(a) that with an increase in the local nanoparticle Grashof Number ( r B ), pressure rise reduces but Figure 2(b) shows the opposite behavior and pressure rise increases with the increase in the thermal Grashof number ( rG ). At the same time Figure 2(c) shows the pressure rise plunges with the increase in porosity parameter ( K ), showing anticlockwise rotation in the graph. Figure 2(d) shows that with an increase in the thermophoresis number ( tN ) pressure rise reduces. Figures 2(e) and 2(f) give the effect of the non-uniform parameters ( 1k ) and ( 2k ) on the pressure rise in the flow, and show that it plunges with the increase of the non-uniform parameters ( 1k ) and ( 2k ) showing anticlockwise rotation in the graph. In the retrograde pumping region the pressure rise decreases with an increase in non-uniform parameters ( 1k ) and ( 2k ). Finally, in Figure 2(g) the Hall parameter ( M ) gives the opposite effect to the porosity parameter ( K ), where the pressure rise increases in the retrograde quadrant with an increase in Hall parameter ( M ) and goes in clockwise rotation. Figure 2. Variation of pressure rise over the length versus .Q a b c d COMBINED EFFECTS OF HALL CURRENT AND HEAT TRANSFER 113 Figure 2. Contd. A fluid particle that moves with a velocity that is different from its neighbours will be subjected to a frictional force, such forces also exist between a particle and a rigid boundary. The friction forces 1 F  and 2 F  on the right and left walls respectively can be obtained by integral evaluations of the analytic results gained for the pressure gradient dx dp . The average of the right friction force (on the right flexible wall) as well as the average of the left friction force (on the left flexible wall) are plotted in Figure 3. Figure 3. Variation of frictional forces. e f g a b AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 114 Figure 3. Contd. We notice from these figures that the right and left wall friction forces have the opposite behavior as compared to the pressure rise. The right friction force behaves similarly to the left friction force for the same values of the parameters; however, the absolute value of the right friction force is greater than that of the left friction force at the same values of the parameters. Trapping is the development and transport of an internally circulating bolus of fluid. In this bolus a fluid travels along the wave as it propagates. The effect of the Hall parameter ( M ) on the trapping can be observed through Figure 4 where the size of the bolus reduces with an increase in Hall parameter ( M ). The influences of phase c d e f g h COMBINED EFFECTS OF HALL CURRENT AND HEAT TRANSFER 115 difference ( ), porosity parameter ( K ), wall temperature ratio 0m , non-uniform parameters ( 1k ) and ( 2k ), Hall parameter ( M ) and thermophoresis parameter ( tN ) are analyzed through Figures (4) and (5). We notice that the size of the trapped bolus increases with an increase in porosity parameter ( K ) and wall temperature ratio 0m but it reduces with an increase in thermophoresis parameter ( tN ) and the Hall parameter ( M ). We can also observe that streamlines get shifted due to the change in the values of phase difference ( ) and wall temperature ratio 0m , and non-uniform parameter ( 1k ) and ( 2k ). Figure 4. Streamlines for different values of , k, m0 when 0.2 ,tN  0.4,bN  0.4,Q  0.4,rB  0.6,rG  0,m  1 0,k  2 0.k  a b  = c d  =0 k = 0.6 m0 = -1 AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 116 Figure 6 depicts the variations of temperature profile ( ) and nanoparticle volume fraction ( ) for different values of the thermophoresis parameter ( tN ), wall temperature ratio 0m and non-uniform parameters ( 1k ) and ( 2k ). It is seen in Figures 6(c), 6(e) and 6(g) that the heat distribution ( ) increases with an increase in non-uniform parameter ( 2k ), thermophoresis parameter ( tN ) and temperature ratio 0m . On the other hand, Figures 6(b), 6(d), 6(f), and 6(h) show two different behaviour of the nanoparticles volume fraction ( ) profiles. It decreases with the increase of the thermophoresis parameter when 0tN  , wall temperature ratio 0m and non-uniform parameters ( 1k ) and ( 2k ). An opposite trend is observed when 0tN  . 1 0.6k  2 0.6k  7M  Nt = 0.9 Figure 5. Streamlines for the different values of 1 ,k 2 ,k M, tN when 0.4 ,bN  0.4,Q  0.4,rB  0.6,rG  0.2,k  0 2.m  e f h g COMBINED EFFECTS OF HALL CURRENT AND HEAT TRANSFER 117 Figure 6. Concentration profile  and heat distribution . a b c d e f AHMED I. ABDELLATEEF and SYED. Z. UL HAQUE 118 Figure 6. Contd. 6. Conclusion The pressure rise reduces with an increase in the Hartmann number ( M ) and Thermophoresis parameter ( tN ). It increases with the increase of the local nanoparticle Grashof number ( rB ), local temperature Grashof number ( rG ) and . b N All these behaviors are in agreement with those reported by Kothandapani and Prakash [25]. The temperature distribution profile θ increases whereas nanoparticle concentration decreases with the increase of tN and .bN References 1. Mc Whirter, J.D., Crawford, M.E. and Klein, D.E. Magnetohydrodynamic flows in porous media II: experimental results. Fusion Techno., 1998, 34, 187-197. 2. Geindreau, C. and Auriault, J.L. 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Appendix   0 1 2 2 1 1 1 2 1 1 1 = , = 1s N h s N hb b m M A A k me e      ,     21 0 1 1 = hhN mN s b t   1 1 1 1 72 1 3 2 2 s = , = s N h b r t r r b B N A Be c c G A c A N A                                          2 1 122 12 11 2 11 84 1= A pG N NBA GA NsA e A e cc r b tr r b b Nhs b Nhs                              2 21111 122 12 21 2 11 9 1 A phsBhsBG N NBA GA NsA e A e c rrr b tr r b b Nhs b Nhs ,                                2 1 122 12 11 2 11 12 5 1= A pG N NBA GA NsA e A e ec r b tr r b b Nhs b Nhs hA                                   2 1 122 72 11 2 11 22 7 1 A pG N NBA GA NsA e A e ec r b tr r b b Nhs b Nhs hA                                  2 21111 122 12 21 2 11 12 1 A phsBhsBG N NBA GA NsA e A e e rrr b tr r b b Nhs b Nhs hA ,           b tr r b N ANB AG AsN c 1 1 2 2 1 26 1 = , 2 11 7 = A hsBGp c rr  , ( 2 ) 3 2 1 2 1 2 7 78 9 = , = h h A h A c e c c e c  Received 7 February 2016 Accepted 4 September 2016