# Wall properties and slip effects on the magnetohydrodynamic peristaltic motion of a viscous fluid with heat transfer and porous space.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 Published online 28 June 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI:10.1002/apj.470 Research article Wall properties and slip effects on the magnetohydrodynamic peristaltic motion of a viscous fluid with heat transfer and porous space T. Hayat1,2 * and Maryiam Javed1 1 2 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Mathematics, College of Sciences, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia Received 8 March 2010; Revised 29 April 2010; Accepted 23 May 2010 ABSTRACT: Peristaltic flows are important for understanding the transport of fluids, especially blood flow, in physiological systems and in biomedical instruments. Several studies have considered the peristaltic flows of Newtonian and non-Newtonian fluids with one or more simplifying assumptions, and for peristaltic flows in biomechanics applications such as magnetohydrodynamic (MHD) pumps and blood flow. This paper presents a theoretical study of the effects of slip conditions and wall properties on the MHD peristaltic flow of a viscous fluid filling the porous space in a channel. The heat transfer analysis is also presented. The analytical solutions of the stream function, velocity and temperature distributions are constructed in closed form under long wavelength and low Reynolds number approximations. The effects of various influential key parameters and groups are reported and discussed. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: magnetohydrodynamics; peristaltic flow; slip effects; wall properties; porous media; heat transfer INTRODUCTION During the past few decades, the peristaltic flows have drawn intensive attention from many mathematicians, engineers, modelers and computer scientists. Areas of applications of peristaltic flows cover widely the transport of fluids in physiological systems, in biomedical instruments, in industry, etc. There are several attempts dealing with the peristaltic flows of Newtonian and non-Newtonian fluids under one or more simplified assumptions.[1 – 15] Jaffrin and Shapiro[15] and Rath[16] have made a complete survey of peristaltic flows in the domain of biomechanics. Little has been said about the influence of heat transfer on the peristaltic transport of fluids. Vajravelu et al .[17] discussed the heat transfer effects on peristaltic motion in a vertical porous annulus. Mekheimer and Elmaboud[18] studied the heat transfer and MHD effects on the peristaltic transport of a viscous fluid in a vertical annulus. Srinivas and Kothandapani[19] *Correspondence to: T. Hayat, Department of Mathematics, Quaidi-Azam University, Islamabad 44000, Pakistan. E-mail: pensy t@yahoo.com 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology examined the heat transfer analysis for MHD peristaltic flow of a viscous fluid in an asymmetric channel. Very recently, Radhakrishnamacharya and Srinivasulu[20] discussed the influence of wall properties on the peristaltic transport with heat transfer in a uniform channel. Kothandapani and Srinivas[21] studied the effect of wall properties on the MHD peristaltic transport with heat transfer and porous medium. Note that in Refs [18–21] no-slip condition is taken into account. To date the slip, heat transfer and wall effects on the MHD peristaltic transport in a porous space has not been analyzed to the author’s knowledge. The goal is to develop a model that discusses the slip and heat transfer effects on the peristaltic flow in a channel with compliant walls. It is worth mentioning to note that such analysis for hydrodynamic fluid is even not available yet in the existing literature. The velocity slip condition is imposed in terms of the shear stress. The MHD consideration is important especially in MHD pumps and blood flow. In the present study, the analytic solutions to the axial velocity, stream function, temperature and heat transfer coefficient are presented. Moreover, some flow quantities of interest are analyzed through graphical results. T. HAYAT AND M. JAVED Asia-Pacific Journal of Chemical Engineering (b) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 M = 1.5 M = 2.0 M = 2.5 M = 3.0 0 1 y y (a) 2 3 4 5 6 7 8 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 9 K = 0.2 K = 0.5 K = 2.0 K = inf 0 1 2 3 u 4 5 7 6 u (c) (d) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -0.5 β = 0.0 β = 0.1 β = 0.2 β = 0.3 0 y y 0.5 u 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1.5 ε = 0.10 ε = 0.15 ε = 0.20 ε = 0.25 0 1 2 3 4 5 6 7 8 u (e) y 650 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 E1 = 1, E2 = 0.5, E3 = 0.1 E1 = 2, E2 = 0.3, E3 = 0.2 E1 = 1, E2 = 1, E3 = 0.1 E1 = 2, E2 = 0.5, E3 = 0 0 2 4 6 u 8 10 12 Figure 1. Variations of the longitudinal velocity u for different values of Hartman number M, porosity parameter K, velocity slip parameter β, amplitude ratio ε and wall parameters. The other parameters chosen are E1 = 0.5, E2 = 0.2, E3 = 0.1, ε = 0.2, K = 5, β = 0.2, x = 0.2 and t = 0.2 (a); E1 = 1.0, E2 = 0.5, E3 = 0.1, ε = 0.1, M = 2, β = 0.2, x = 0.2 and t = 0.2 (b); E1 = 1, E2 = 0.5, E3 = 0.5, ε = 0.1, K = 5, M = 2, x = 0.3 and t = 0.1 (c); E1 = 0.5, E2 = 0.2, E3 = 0.1, M = 2, K = 5, β = 0.2, x = 0.2 and t = 0.2 (d) and ε = 0.1, M = 2, K = 5, β = 0.2, x = 0.2 and t = 0.2 (e). MATHEMATICAL MODEL We consider a channel of width 2d. The channel is filled with an incompressible and magnetohydrodynamic (MHD) viscous fluid. A uniform magnetic field 2010 Curtin University of Technology and John Wiley & Sons, Ltd. B = (0, B0 , 0) is applied in the y-direction. The induced magnetic field is negligible for small magnetic Reynolds number. The fluid fills the porous space. The temperatures of the upper and lower walls of the channel are maintained at T0 and T1 , respectively. The imposed Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID (a) (b) 1 1 0.5 M = 1.0 M = 1.5 M = 2.0 M = 2.5 0 y y 0.5 -0.5 -1 -1 0.5 1 θ 1.5 2 0 1 2 3 θ 4 5 6 (d) (c) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1 0.5 β = 0.0 γ = 0.0 γ = 0.02 γ = 0.05 γ = 0.10 β = 0.1 β = 0.2 y y 0 -0.5 0 K = 0.2 K = 0.5 K = 2.0 K = inf β = 0.3 0 -0.5 -1 0 0.5 1 θ 1.5 0 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 θ Figure 2. Variations of temperature distribution for different values of Hartman number M, porosity parameter K, velocity slip parameter β, thermal slip parameter γ , Brinkman number Br , amplitude ratio ε and wall parameters. The other parameters chosen are E1 = 0.4, E2 = 0.2, E3 = 0.1, ε = 0.1, K = 5, β = 0.2, γ = 0.1, x = 0.4, Br = 1 and t = 0.1 (a); E1 = 1, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 2, β = 0.2, γ = 0.1, x = 0.4, Br = 2 and t = 0.2 (b); E1 = 0.8, E2 = 0.2, E3 = 0.1, ε = 0.1, M = 2, K = 5, γ = 0.1, x = 0.3, Br = 2 and t = 0.1 (c); E1 = 0.8, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 2, K = 5, x = 0.3, Br = 2 and t = 0.08 (d); E1 = 0.5, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 2, K = 5, β = 0.2, γ = 0.1, x = 0.3 and t = 0.1 (e); E1 = 0.5, E2 = 0.2, E3 = 0.1, M = 2, K = 5, β = 0.2, γ = 0.1, Br = 2, x = 0.1 and t = 0.3 (f) and ε = 0.1, M = 2, K = 5, β = 0.2, γ = 0.1, Br = 2, x = 0.3 and t = 0.1 (g). traveling waves of small amplitudes on the insulating walls of the channel are: 2π (x − ct) y = ±η = ± d + a sin λ +µ (2) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. ∂2 ∂x 2 + ∂2 ∂y 2 u − σ B02 u − ∂ ∂ ∂p ∂ +u +v v =− ρ ∂t ∂x ∂y ∂y 2 2 µ ∂ ∂ + 2 v− v +µ 2 k ∂x ∂y ∂ ∂ κ ∂ +u +v T = ∇ 2T ζ ∂t ∂x ∂y ρ 2 2 ∂u + ∂v ∂x ∂y + ν 2 + ∂u + ∂v 2 ∂x ∂y (1) where λ is the wavelength, c is the wave speed, a is the wave amplitude and x and y are the Cartesian coordinates with x measured in the direction of the wave propagation and y measured in the direction normal to the mean position of the channel walls. The equations which can govern the motion for the present flow are: ∂v ∂u + =0 ∂x ∂y ∂ ∂ ∂ ∂p ρ +u +v u=− ∂t ∂x ∂y ∂x µ u k (3) (4) (5) subjected to the boundary conditions u ±α ∂u = 0 aty = ±η ∂y Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj 651 T. HAYAT AND M. JAVED (e) Asia-Pacific Journal of Chemical Engineering (f) Br = 0.0 1 Br = 1.0 0.4 Br = 3.0 0.5 0.8 0.6 Br = 2.0 ε = 0.10 ε = 0.15 ε = 0.18 ε = 0.20 0.2 0 y y 0 -0.2 -0.5 -0.4 -0.6 -1 -0.8 0 0.5 1 1.5 θ 2 2.5 0 3 0.5 1 θ 1.5 2 (g) y 652 E1 = 0.5, E2 = 0.1, E3 = 0.3 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 E1 = 1, E2 = 0.1, E3 = 0.3 E1 = 0.6, E2 = 0.1, E3 = 0 E1 = 0.7, E2 = 0.3, E3 = 0.3 0 0.5 1 1.5 θ Figure 2. (Continued). 2π = ± d + a sin (x − ct) (6) λ ∂p ∂ 2u ∂ 2u ∂ L(η) = =µ + 2 ∂x ∂x ∂y 2 ∂y ∂u ∂u ∂u +u +v −ρ ∂t ∂x ∂y µ − σ B02 u − u at y = ±η (7) k ∂T T −ξ = T0 on y = −η (8) ∂y ∂T = T1 on y = η (9) T +ξ ∂y where u and v are the velocity components in x and y directions, respectively, and ρ, t, p, µ, ν, σ , ζ , κ and T are the fluid density, the time, the pressure, the dynamic viscosity, the kinematic viscosity, the electrical conductivity, the specific heat, the thermal conductivity and the temperature, respectively. Furthermore, L = −τ ∂ 2 ∂x 2 + m∂ 2 ∂t 2 + C ∂ ∂t, τ is the elastic tension in the membrane, m is the mass per unit area and C is the coefficient of viscous damping. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. On setting u= ∂ ∂ , v =− ∂y ∂x (10) and defining the following nondimensional variables and parameters: x y , y ∗ = , ∗ = , λ d cd ct η d 2p t ∗ = , η∗ = , p ∗ = λ d µcλ d k a K∗ = 2, ε = , δ = , M2 d λ d σ B02 d 2 τd3 , E1 = − 3 = µ λ µc x∗ = E2 = mcd 3 λ3 µ , E3 = Cd 3 λ2 µ (12) , Pr c2 ρνζ , E= κ ζ (T1 − T0 ) T − T0 cd , θ= R= ν T1 − T0 = (11) (13) (14) Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering (a) 6 (b) M=0 M=2 M=4 4 3 K = 1.0 2 1 0 0 -2 -1 -4 -2 -6 -3 0 0.2 K = 0.2 2 Z Z MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID 0.4 0.6 0.8 K = inf 0 1 0.2 0.4 (c) (d) 0.5 β = 0.0 β = 0.1 β = 0.2 6 0.6 0.8 1 0.6 0.8 1 x x γ = 0.0 γ = 0.1 γ = 0.2 0.4 0.3 4 0.2 0.1 0 0 Z Z 2 -0.1 -2 -0.2 -4 -0.3 -6 -0.4 -0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 x x (e) 2.5 Br = 1 2 Br = 2 (f) 6 4 Br = 3 1.5 1 2 Z Z 0.5 0 -0.5 0 -2 -1 E1 = 0.5, E2 = 0.1, E3 = 0.5 E1 = 1, E2 = 0.1, E3 = 0.5 -1.5 -4 E1 = 0.5, E2 = 0.5, E3 = 0.5 -2 E1 = 1.0, E2 = 0.1, E3 = 0.0 -6 -2.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x Figure 3. Variations of heat transfer coefficient Z for different values of Hartman number M, porosity parameter K, velocity slip parameter β, thermal slip parameter γ , Brinkman number Br and wall parameters. The other parameters chosen are E1 = 1, E2 = 0.2, E3 = 0.1, ε = 0.2, K = 0.05, β = 0.2, γ = 0.1, Br = 1 and t = 0.2 (a); E1 = 0.3, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 4, β = 0.2, γ = 0.1, Br = 2 and t = 0.1 (b); E1 = 0.3, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 4, K = 0.2, γ = 0.1, Br = 2 and t = 0.2 (c); E1 = 0.8, E2 = 0.1, E3 = 0.1, ε = 0.2, M = 2, K = 0.02, β = 0.2, Br = 2 and t = 0.2 (d); E1 = 0.3, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 4, K = 0.2, β = 0.2, γ = 0.1 and t = 0.2 (e) and ε = 0.15, M = 1, K = 0.05, Br = 3, β = 0.2, γ = 0.1 and t = 0.25 (f). Equation (2) is automatically satisfied and Eqns (3)– (9) under long wavelength and low Reynolds number are reduced to: ∂ 3 ∂p − =0 −N 3 ∂y ∂x ∂y 2 ∂ (15) 2010 Curtin University of Technology and John Wiley & Sons, Ltd. 1 Pr ∂y 2 ∂ 3 ∂y ∂ 2θ 3 − N2 +E ∂ 2 ∂y 2 2 =0 (16) ∂ ∂y Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj 653 654 T. HAYAT AND M. JAVED Asia-Pacific Journal of Chemical Engineering (a) (b) 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 (c) 1.5 1 0.5 0 -0.5 -1 -1.5 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 4. Streamlines for (a) M = 0, (b) M = 4, (c) M = 6. The other parameters chosen are (ε = 0.2, β = 0.1, E1 = 0.3, E2 = 0.1, E3 = 0.1, K = 0.04 and t = 0). ∂3 ∂3 ∂2 = E1 3 + E2 + E 3 ∂x ∂t ∂x ∂x ∂t 2 η at y = ±η SOLUTION OF THE PROBLEM The closed form solutions of Eqns (15)–(20) are (17) 2 =L ∂ ∂ ± β 2 = 0 at y = ±η ∂y ∂y = ±[1 + ε sin 2π(x − t)] ∂θ θ −γ = 0 at y = −η ∂y ∂θ θ +γ = 1 at y = η ∂y 2 (18) (19) (20) in which asterisks have been suppressed for simplicity, ∂p/∂y = 0, is a stream function, N = M 2 + 1/K , M is the Hartman number, K is the porosity parameter, Pr is the Prandtl number and E is the Eckert number. Furthermore, Eqns (18)–(20) are the slip conditions for the velocity and temperature, respectively, and β = α/d, γ = ξ/d (α and ξ are the dimensional slip parameters corresponding to velocity and temperature). 2010 Curtin University of Technology and John Wiley & Sons, Ltd. sinh Ny −y N (cosh N η + βN sinh N η) Br L4 N 2 8(cosh N η + βN sinh N η)2 cosh 2N η − cosh 2Ny +2(y 2 − η2 ) 2 N sin 2N η +4γ −η 2N y +η+γ + 2(η + γ ) 8επ 3 E3 sin 2π(x − t) 2 L = 2π N2 (21) θ= −(E1 + E2 ) cos 2π(x − t) (22) (23) Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID (a) (b) 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.2 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 (c) 1.5 1 0.5 0 -0.5 -1 -1.5 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 5. Streamlines for (a) K = 0.05, (b) K = 0.2, (c) K = ∞. The other parameters chosen are (ε = 0.17, β = 0.1, E1 = 0.3, E2 = 0.1, E3 = 0.1, M = 4 and t = 0). where Br = Pr E is the Brinkman number and the longitudinal velocity is ∂ ∂y 2 =L u= cosh Ny −1 (cosh N η + βN sinh N η) (24) The heat transfer coefficient at the upper wall is Z = ηx θ y = ηx Br L4 N 2 8(cosh N η + βN sinh N η)2 ηx −2 sinh N η + 4η + N 2(η + γ ) (25) GRAPHICAL RESULTS AND DISCUSSION The behavior of the longitudinal velocity, stream function, temperature and heat transfer coefficient is exhibited in this section. For this purpose, Figs (1–7) are 2010 Curtin University of Technology and John Wiley & Sons, Ltd. displayed. Figure 1 is plotted to examine the effect of various parameters on the longitudinal velocity u. One observes from Fig. 1(b) that the longitudinal velocity u increases when the porosity parameter K increases, whereas it decreases when the Hartman number M is increased Fig. 1(a). Figure 1(c) and (d) records the behavior of velocity slip parameter β and the occlusion parameter ε on the velocity, respectively. These figures depict an increase in the velocity when M and ε are increased. The effect of the elastic parameters E1 , E2 and E3 are evident in Fig. 1(e). It may be of interest to note from this figure that with increasing elastic parameters, the velocity increases. It is also interesting to note that the velocity profile is parabolic for fixed values of the parameters and its magnitude is maximum near the center of the channel. Moreover, it is observed that the elastic tension E1 has a significant effect on the axial velocity as compared with the mass characterizing parameter E2 and the damping nature of the wall E3 . In Fig. 2, the nature of the temperature profile is also parabolic. Here, the temperature decreases by increasing the Hartman number M (Fig. 2(a)) and the velocity Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj 655 656 T. HAYAT AND M. JAVED Asia-Pacific Journal of Chemical Engineering (a) (b) 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.2 -0.1 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 (c) 1.5 1 0.5 0 -0.5 -1 -1.5 -0.2 -0.1 0 0.1 0.2 0.5 0.6 Figure 6. Streamlines for (a) β = 0, (b) β = 0.1, (c) β = 0.2. The other parameters chosen are (ε = 0.17, E1 = 0.3, E2 = 0.1, E3 = 0.1, M = 4, K = 0.05 and t = 0). slip parameter β (Fig. 2(c)). Note that the temperature decreases in the downstream. However, Fig. 2(b) and (d) illustrates that the temperature increases by increasing the permeability parameter K and the thermal slip parameter γ . The variations of the Brinkman number Br and the occlusion parameter ε on the temperature are sketched in Fig. 2(e) and (f). It is noted from these figures that temperature is an increasing function of Br and ε. Figure 2(g) elucidates the effect of the elastic parameters E1 , E2 and E3 on the temperature. This figure reveals that the amplitude of temperature increases with an increase in E1 , E2 and E3 . It is further observed that the effect of E1 on temperature is quite significant. The results presented in Fig. 3 indicate the behavior of M , K , β, γ , Br , E1 , E2 and E3 on the heat transfer coefficient Z . This figure shows the typical oscillatory behavior of heat transfer which may be due to the peristaltic phenomena. Figure 3(b), (d)–(f) depict that the absolute value of the heat transfer coefficient increases by increasing K , γ , Br , E1 , E2 and E3 , respectively, while the behavior is quite opposite in the case of M and β (Fig. 3(a) and (c)). The damping nature of the wall E3 has a very insignificant effect on the heat transfer. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. The formation of an internally circulating bolus of fluid by closed streamlines is shown in Figs 4–7. Figure 4 displays the effect of the Hartman number M on the streamlines for fixed value of the other parameters. This figure shows that the size of the trapping bolus decreases with an increase in the Hartman number M , whereas the behavior is quite opposite in the case of the permeability parameter K (Fig. 5). Figure 6 reveals the behavior of velocity for slip parameter β on the streamlines. Here, we observed that the number of the streamlines increases by increasing β. The effect of the elastic parameters on the streamlines is plotted in Fig. 7 . The number of the trapped bolus increases with an increase in E1 , E2 and E3 . We also note that the damping seems less effective in the trapping phenomenon. CONCLUSIONS We have presented theoretical study for the slip effects on the MHD peristaltic flow of a viscous fluid through Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID (a) (b) 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.1 (c) (d) 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 7. Streamlines for (ε = 0.2, β = 0.1, K = 0.04, M = 5 and t = 0.1). (a) E1 = 0.5, E2 = 0.3, E3 = 0.1. (b) E1 = 0.8, E2 = 0.3, E3 = 0.1. (c) E1 = 0.5, E2 = 0.5, E3 = 0.1. (d) E1 = 0.5, E2 = 0.3, E3 = 0.7. a porous medium. The influence of heat transfer is also seen. Closed form solutions are derived for the velocity, stream function, temperature and heat transfer coefficient. The salient features of the presented attempt are as follows. • The longitudinal velocity increases in the neighborhood of the walls and increases near the center of the channel when the velocity slip parameter is increased. • An increase in velocity slip parameter yields a decrease in the temperature. • There is an increase in the temperature when the thermal slip parameter increases. • The effects of Brinkman number and velocity slip parameter on the temperature are quite opposite. However, the variations of Brinkman number and thermal slip parameter on the temperature are similar in a qualitative sense. • There is a decrease in the heat transfer coefficient at the upper wall when velocity slip parameter is increased. • The influence of Hartman number on the heat transfer coefficient at the upper wall is quite opposite to that of a Brinkman number. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. • The absolute value of the heat transfer coefficient increases in the upper part of the channel when an elastic parameters increase. Acknowledgements The authors are grateful to the Higher Education Commission (HEC) of Pakistan for the financial support. REFERENCES [1] K.h.S. Mekheimer, Y.A. Elmaboud. Physica A, 2008; 387, 2403–2415. [2] K.h.S. Mekheimer. Arab. J. Sci. Eng., 2005; 30, 69–83. [3] K.h.S. Mekheimer. Appl. Math. Comput., 2004; 153, 763–777. [4] K. Vajravelu, S. Sreenadh, V.R. Babu. Int. J. Nonlinear Mech., 2005; 40, 83–90. [5] M.H. Haroun. 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Fluid Mech., 1971; 3, 13–36. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering [16] H.J. Rath. Peristaltische Stromungen, Springer: Germany, 1980. [17] K. Vajravelu, G. Radhakrishnamacharya, V. Radhakrishnamurty. Int. J. Nonlinear Mech., 2007; 42, 754–759. [18] K.h.S. Mekheimer, Y.A.bd. Elmaboud. Phys. Lett. A, 2008; 372, 1657–1665. [19] S. Srinivas, M. Kothandapani. Int. Commun. Heat Mass Transfer, 2008; 35, 514–522. [20] G. Radhakrishnamacharya, C. Srinivasulu. C. R. Mec., 2007; 335, 369–373. [21] M. Kothandapani, S. Srinivas. Phys. Lett. A, 2008; 372, 4586–4591. Asia-Pac. J. Chem. Eng. 2011; 6: 649–658 DOI: 10.1002/apj

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