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Hydromagnetic Peristaltic Transport of Variable Viscosity Fluid with Heat Transfer and Porous Medium

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Hydromagnetic Peristaltic Transport of Variable Viscosity Fluid with Heat Transfer and Porous Medium Appl. Math. Inf. Sci. 10, No. 6, 2173-2181 (2016) 2173 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/100619 Hydromagnetic Peristaltic Transport of Variable Viscosity Fluid with Heat Transfer and Porous Medium F. M. Abbasi1,2,∗, T. Hayat2,3 and B. Ahmad3. 1 Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan. 2 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. 3 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. Received: 9 Jul. 2016, Revised: 11 Sep. 2016, Accepted: 17 Sep. 2016 Published online: 1 Nov. 2016 Abstract: This article examines the peristaltic transport of variable viscosity fluid in a planar channel with heat transfer. The fluid viscosity is taken temperature dependent. The fluid is electrically conducting in the presence of a constant applied magnetic field. Both channel and magnetic field possess the inclined considerations. An incompressible fluid saturates the porous space. Heat transfer analysis is carried out in the presence of viscous dissipation and Joule heating. The resulting problems are solved numerically. A parametric study is performed to predict the impact of embedded variables. Results indicate that the fluid with variable viscosity has higher value of pressure gradient compared with that of constant viscosity fluid. Size of the trapped bolus increases with an increase in viscosity and permeability parameters. Constant viscosity fluid possesses lower velocity near the center of channel when compared with variable viscosity fluid. Effects of porous medium on pressure gradient and velocity are qualitatively similar. Keywords: Peristaltic transport; Temperature dependent viscosity; Magnetohydrodynamics (MHD); Porous medium; Joule heating. 1 Introduction and we here only refer few recent studies in this direction [6,7,8,9,10,11,12,13,14]. The peristaltic transport of fluid caused by a progressive The porous medium and heat transfer effects are quite wave of area contraction/expansion along the length of a important in the biological tissue. Especially such distensible tube has received growing interest of the considerations are significant in blood flow simulation recent researchers. To a large extent, such interest is related to tumors and muscles, drugs transport, stimulated by the fact that peristaltic activity is involved production of osteoinductive material, nutrients to brain in practical applications of physiological and industrial cells etc. Moreover, the flow models with magnetic field processes. These include the locomotion of worms, urine have numerous applications in cross field accelerators, passage from kidney to bladder, food swallowing by the pumps, flow meters and MHD generators. In such esophagus, vasomotion of small blood vessels, sanitary devices, the flow is subjected to heat that dissipated and corrosive fluids transport, movement of bio fluids internally by viscous/Joule heating or that produced due (chyme in gastrointestinal tract, bile in the bile duct, to electric currents in the walls. The peristaltic transport ovum in the fallopian tube and spermatozoa in cervical of MHD fluid has interaction with problems of movement canal) and in heart lung machine and roller and finger of the conductive physiological fluids, e.g. the blood and pumps. Latham [1] was the first who studied blood pump machines. The peristalsis with heat transfer is experimentally the mechanism of peristalsis for flow of further prominent in oxygenation and hemodialysis viscous fluids. Afterwards, several researchers examined processes. With this importance in mind, Srinivas and the peristalsis subject to various assumptions through Muthuraj [15] addressed the effects of heat transfer in theoretical and experimental attempts. Available peristaltic flow of viscous fluid saturating porous information on the topic for peristalsis of viscous and medium. Srinivas and Kothandapani [16] addressed the non-Newtonian fluids [2,3,4,5] at present is impressive heat transfer analysis in peristaltic flow by complaint ∗ Corresponding author e-mail: [email protected] c 2016 NSP Natural Sciences Publishing Cor. 2174 F. Abbasi et al.: Hydromagnetic peristaltic transport of... walls. Abd elmaboud and Mekheimer [17] discussed the components and pressure in the wave frame (x,y). The peristaltic transport of second order fluid in a porous transformations between laboratory and wave frames are space. Mekheimer [18] investigated the nonlinear peristaltic transport through a porous medium in an x = X −ct, y =Y, u =U −c, v =V, p(x,y)= P(X,Y ,t). inclined planar channel. The effects of magnetic field and (1) space porosity on compressible Maxwell fluid were The governing equations in the wave frame are given by studied by Mekheimer et al. [19]. Tripathi [20] analyzed ∂u ∂v + = 0, (2) the transient peristaltic heat flow in a finite porous ∂x ∂y channel. Influence of heat transfer on peristaltic flow of ∂ ∂ ∂ p ∂ ∂ ∂ ∂ ∂ ρ (u + c) + v (u + c) = − + 2 µ (T) u + µ (T) v + u an electrically conducting fluid through porous medium ∂x ∂y ∂x ∂x ∂x ∂y ∂x ∂y µ(T )(u+c) has been addressed by Hayat et al. [21]. The interaction −σB2Cosβ ((u + c)Cosβ − vSinβ)+ ρgα∗ T − T Sinα + ρgSinα − , (3) 0 0 k of peristaltic motion with heat transfer in planar and curved flow configurations has been explored in the ∂ ∂ ∂ p ∂ ∂ ∂ ∂ ∂ ρ (u + c) + v (v) = − + 2 µ (T ) v + µ (T ) v + u ∂x ∂y ∂y ∂y ∂y ∂x ∂x ∂y studies [21,22,23,24,25,26,27]. µ(T)(v) −σB2Sinβ ((u+ c)Cosβ − vSinβ)− ρgα∗ T − T Cosα − ρgCosα − , (4) Existing information on the topic reveals that no 0 0 k attention has been given so far to the peristaltic flow of ∂u 2 ∂v 2 ∂v ∂u 2 ρCp (u + c)T + vTy = K T + T + µ (T ) 2 + + + variable viscosity fluid through an inclined channel and x xx yy ∂x ∂y ∂x ∂y " ( ) # h i inclined applied magnetic field. The objective of present µ (T)(u+ c)2 +σB2 ((u + c)Cosβ − vSinβ)2 + , (5) communication is to address this problem in the presence 0 k of viscous dissipation and Joule heating effects. The in which ρ is the density of fluid, µ is the dynamic problem is modeled when an incompressible fluid fills the viscosity, g is the acceleration due to gravity, k is porous medium. Numerical solution is given after permeability of the porous medium parameter, β is the utilizing the long wavelength and low Reynolds number inclination of applied magnetic field and α∗ is the thermal concept. Physical quantities of interest are analyzed for expansion coefficient. The additional viscous dissipation the pertinent parameters entering into problem statement. term in the energy equation accounts effect in limits of small and large permeability of porous medium. We also denote Cp as the specific heat, K the thermal conductivity 2 Mathematical analysis and T the temperature of the fluid. The long wave length approximation is widely used in Let us examine the peristaltic transport of an the analysis of peristaltic flows [29]. Such approximation incompressible viscous fluid in a symmetric channel. The makes use of the fact that the wavelength of peristaltic channel is taken inclined at an angle α to the vertical. The wave is considerably large when compared with the half fluid is electrically conducting in the presence of an width of the channel/tube. These considerations are relevant for the case of chyme transport through small inclined magnetic field with constant strength B0. λ Applied/induced electric fields are absent. Further the intestine [6] where a = 1.25cm and = 8.01cm. Clearly half width of the intestine is small in comparison to the effects of induced magnetic fields are ignored subject to λ low magnetic Reynolds number approximation. The flow wavelength of peristaltic wave i.e. a/ = 0.156. Further, generated is due to peristaltic waves travelling along the Lew et al. [7] concluded that Reynolds number for the channel walls. Schematic diagram of the geometry is fluid mechanics in small intestine is small. Making use of given in Fig. A. the following dimensionless quantities. Defining the dimensionless quantities Mathematically the wall geometry is chosen in the 2 µ x y u v δ a H b a p υ 0 form x = λ , y = , u = , v = δ , = λ , h = , d = , p = λµ , = ρ , a c c a a c 0 µ(T) σ 2 ρca ct T − T ρgα∗T a2 µ(θ) = , M2 = B2a2, Re = , t = , θ = 0 , Gr = 0 , µ µ 0 µ λ T µ c H(X,t)= a + ε, 0 0 0 0 0 2 2 µ c k c 0Cp Fr = , k = , Br = PrE, E = , Pr = , u = ψy, v = −ψx , (6) where a is the half channel width and ε is the disturbance ga a2 CpT0 K produced due to propagation of peristaltic waves at the and adopting long wavelength and low Reynolds number walls. This disturbance can be written in the form approach, the equations in terms of stream function ψ are 2π ∂ p ∂ ∂ 2ψ µ (θ) ∂ψ ε = µ (θ) − M2Cos2β + + 1 + = bcos λ (X − ct) , ∂x ∂y ∂y2 k ∂y ReSinα GrθSinα + , (7) in which t is time, b is the amplitude of the peristaltic Fr wave, c and λ are the speed and wavelength of the waves ∂ p respectively. Appropriate velocity field for this problem is = 0, (8) V = U(X,Y ,t),V(X,Y ,t),0 . Here U, V and P are the ∂y velocity components and pressure in the laboratory frame ∂ 2θ µ (θ) ∂ψ 2 + Brµ (θ)(ψ )2 + Br M2Cos2β + + 1 = 0, (9) (X,Y,t). We further adopt (u,v) and p as the velocity ∂y2 yy k ∂y c 2016 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 10, No. 6, 2173-2181 (2016) / www.naturalspublishing.com/Journals.asp 2175 - T =T0 at Y = H Y axis b a Uniform porous medium X -axis B B0 B0 B0 0 Y - T =T0 at Y = - H Figure A: Geometry of the problem.
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