PERISTALTIC MOTION WITH HEAT AND MASS TRANSFER OF A DUSTY FLUID THROUGH A HORIZONTAL POROUS CHANNEL UNDER THE EFFECT OF WALL PROPERTIES

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1 PERISTALTIC MOTION WITH HEAT AND MASS TRANSFER OF A DUSTY FLUID THROUGH A HORIZONTAL POROUS CHANNEL UNDER THE EFFECT OF WALL PROPERTIES Nabil T. Eldabe 1, Kawther A. Kamel 2, Galila M. Abd-Allah 2 & Shaimaa F. Ramadan 2 1 Department of mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt 2 Department of mathematics, Faculty of Science(Girls), Al-Azhar University, Nasr-City, Cairo, Egypt ABSTRACT The objective of this analysis is to study the effect of wall properties on the peristaltic transport of a dusty fluid through porous channel with heat and mass transfer. This phenomenon is modulated mathematically by a system of partial differential equations which govern the motion of the fluid and solid particles. This system of equations is solved analytically in dimensionless form with the appropriate boundary conditions by using perturbation technique for small geometric parameter δ. The expressions of velocity, temperature and concentration of fluid and solid particles are obtained as functions of the physical parameters of the problem. The effects of the physical parameters of the problem on these distributions are discussed and illustrated graphically through a set of figures. Keywords: Peristaltic; Heat Transfer; Mass Transfer; Dusty; Porous. 1. INTRODUCTION The peristaltic transport through a channel has attracted considerable attention due to their wide applications in medical and engineering science. Physiological fluids in animal and human bodies are in general, pumped by the continuous contraction and expansion of the ducts. These contraction and expansion are expected to be caused by peristaltic waves that propagate along the walls of the ducts. In general, during peristaltic the fluid pumped from lower pressure to higher pressure. This mechanism may be involved in urine transport from kidney to bladder through the ureter, food mixing and chime movement in the intetire, movement of eggs in the fallopian tube, the transport of the spermatozoa in the cervical canal, transport of bile in the bile duct, transport of cilia, and circulation of blood in small blood vessels. This mechanism is used also in engineering devices like finger pumps and roller pumps. The investigation of the effects of wall properties on peristaltic transport of a dusty fluid was done by Srinivasacharya et. al. [1]. In this work the author deals only with the velocity distribution of the fluid and solid particles. Peristaltic MHD flow of a Bingham fluid through a porous medium in a channel has been studied by Suryanarayana et. al. [2]. Peristaltic flow of a Newtonian fluid through a porous medium in a vertical tube under the effect of magnetic field was studied by Vasudev et. al. [3]. Narahari and S. Sreenadh [4] discussed the peristaltic transport of a Bingam fluid in contact with a Newtonian fluid. The effect of porous medium and magnetic field on peristaltic transport of a Jeffrey fluid was studied and reported by Mahmoud et. al. [5]. Krishna et. al. [6] Considered the peristaltic pumping of a Jeffrey fluid in a porous tube. The peristaltic pumping of a non Newtonian fluid was analyzed by Medhavi [7]. Non linear peristaltic pumping of Johnson Segalman fluid in an asymmetric channel under effect of magnetic field was studied by Suryanarayana [8]. Peristaltic pumping of Williamson fluid through a porous medium in a horizontal channel with heat transfer was investigated by Casudeu et. al. [9]. Hayat et. al. [10] discussed the heat transfer analysis for peristaltic mechanism in variable viscosity fluid. The effect of heat transfer on peristaltic transport of a Newtonian fluid through a porous medium in an asymmetric vertical channel was discussed by Vasudev [11]. Heat and mass transfer of MHD unsteady Maxwell fluid flow through porous medium past a porous flat plate was analyzed by El-Dabe et. al. [12]. El-Dabe et. al. [13] discussed the dusty non-newtonian flow between two coaxial circular cylinders. In the present paper, we shall extent the Srinivasacharya et. al. [1] work to include the heat and mass transfer for both fluid and solid particles. Analytical solutions of the momentum equation, heat equation and concentration equations are obtained by using perturbation technique for both fluid and solid particles. The velocity, temperature, concentration and stream functions are obtained for fluid and solid particles. The effects of various parameters on these solutions are discussed and illustrated through a set of figures. 300

2 2. PROBLEM FORMULATION λ y 0o x a Fig. (1). Geometry of peristaltic transport of a dusty fluid in a symmetric porous channel We consider the peristaltic transport of an incompressible fluid that contains small solid particles, whose number density N (assumed to be constant) is large enough to define average properties of the dust particles at a point through a symmetric two-dimensional porous channel. Choose the Cartesian coordinates (x,y), where x is along the walls and y is perpendicular to it as illustrated in figure (1). The channel wall equation is given by: η = d + a sin 2π x ct (1) λ Where d is the half width of the channel, a is the amplitude of the wave, λ is the wavelength, t is the time and c is the wave velocity. The equations governing the two-dimensional transport of an incompressible dusty fluid through porous channel are: For fluid particles The continuity equation: V = 0 (2) V The momentum equation: + V V = 1 P + ρ ν 2 V + kn ν ρ k 1 V s V (3) The temperature equation: The concentration equation: ρc p T C + V T = Q + V s V F + k c T (4) + V C = D 1 2 C (5) For solid particles The continuity equation: V s = 0 (6) V The momentum equation: s + V s V s = k ν m k 1 V V s (7) The temperature equation: The concentration equation: Nc m T s C s + V s T s = Nc p τ T (8) + V s C s = D 2 2 C s (9) Where V = u, v is the velocity of the fluid, ρ is the density of the fluid, P is the pressure, ν is the kinematic viscosity of the fluid, k is the resistance coefficient of the solid particles, k 1 is the permeability of porous medium, V s = u s, v s is the velocity of solid particles, c p is the specific heat of the fluid, c m is the specific heat of solid particles, T is the temperature of the fluid, T s the temperature of solid particles, Q = N c p T s T τ T is the thermal interaction between fluid and solid particles, τ T is the thermal relaxation time of the solid particles, F = N V s V τ ν is the velocity relaxation time of the solid particles, k c is the thermal conductivity of the fluid, C is the concentration of the fluid, C s is the concentration of the solid particles, D 1 is the coefficient of mass diffusivity of the fluid, D 2 is the coefficient of mass diffusivity of solid particles, and m is the mass of the solid particles. The equation of motion of the flexible wall is given by Srinivasacharya et. al. [1]. L η = P P O (10) Where L is an operator that is used to represnt the motion of the stretched membrane with damping forces such that: L = T 2 + M 2 1 x C 2 1 (11) Where T 1 is the tension in the membrane, M 1 is the mass per unit area, C 1 is the coefficient of the damping force and P O is the pressure on the outside of the wall due to tension in the muscles, if we assume that P O = 0, then equations (2-11) in two dimension form can be written as: 301

3 P u u x x 2 2 x + kn μ k 1 u s u (12) P v v x 2 2 x + kn μ k 1 v s v Fluid particles (13) ρc T p = Nc p x τ T T s T + N τ ν u s u 2 + v s v 2 + k 2 T T c x 2 2 (14) C C x 1 C x 2 2 (15) u s v s + u s u s x + v s + u s v s x + v s u s v s Nc T s T m + u s + v s s C s + u s C s x x + v s C s = k ν m k 1 u u s (16) = k ν m T s = D 2 2 C s x 2 k 1 v v s Dust particles (17) = Nc p τ T T s T (18) + 2 C s 2 (19) With the appropriate boundary conditions: u = u s = 0, T = T 1, T s = T s1, C = C 1 and C s = C s1 at y = η u = u s = 0, T = T 2, T s = T s2, C = C 2 and C s = C s2 at y = η (20) P u u x x x 2 2 x + kn μ k 1 u s u Flexible wall (21) Introducing the stream functions ψ and φ such that: u = ψ, v = ψ, u x s = φ, v s = φ x (22) Using the non-dimensional variables: x = x, λ y = y, d η = η, d t = tν, λd ψ = ψ, ν φ = φm, θ = T T 1, kd 2 T 2 T 1 θ s = T s T s 1,C = C C 1,C T s2 T s 1 C 2 C s = C s C s1. (23) 1 C s 2 C s1 In view of the above non-dimensional variables, equations (12-22) are reduced to the following non-dimensional form after dropping the star mark and eliminating the pressure term. δ 1 2 ψ + ψ x 1 2 ψ ψ x 1 2 ψ = 4 1 ψ + A 1 1 R 1 2 φ 2 1 ψ (24) δ θ + ψ θ ψ θ x x ψ C x x δ C + ψ C δ R 1 2 φ + φ δ θ s δ C s = Nl 1 τ T = 1 S c1 x + 1 φ θ s 1 φ θ s R x R x + 1 φ C s 1 φ C s R x R x x 1 2 φ φ θ s θ + Nl 1E c 1 φ ψ τ ν R δ 2 2 C + 2 C x δ 2 1 R φ ψ x x P r δ 2 2 θ x θ 2 (25) 2 (26) 1 2 φ = A 2 R 1 2 ψ 1 2 φ (27) = l 2 θ s θ (28) = 1 S c2 Where 1 2 = δ 2 2 x , δ = d λ d 2 δ 2 2 C s x C s 2 (29) is geometric parameter, A 1 = knd2 ν 2 ρν d2 k 1, A 2 = 1 mν kk 1, R = mν kd 2, l 1 = and l μ 2 = d2 c p are non-dimentional parameters, E νc m τ c = Eckert number, P T d 2 c p T 2 T r = μc p Prandtl number 1 k c, S c1 = ν is the Schmidt number of the fluid and S D c2 = ν is the Schmidt number of solid particles. 1 D 2 With the boundary conditions: ψ φ = 0, s = 0, C = 0, C s = 0 at y = η ψ φ = 0, s = 1, C = 1, C s = 1 at y = η (30) 2 ψ ψ ψ ψ φ 1 x x 1 = E 3 3 R 1 x x 2 3 x η (31) Where E 1 = T 1d 3 ρν 2 λ 3 2 = M 1d ρλ 3 3 = C 1 d 2 ρνλ 2 is the damping parameter. 302

4 3. METHOD OF SOLUTION To have a solution of the system of equations (24-29) subject to the boundary conditions (30) and (31) we assume the following perturbation method for small geometric parameter i. e δ 1 as: ψ = ψ o + δψ 1 + δ 2 ψ 2 + (32) θ = θ o + δθ 1 + δ 2 θ 2 + (33) C = C o + δc 1 + δ 2 C 2 + (34) φ = φ o + δφ 1 + δ 2 φ 2 + (35) θ s = θ so + δθ s1 + δ 2 θ s2 + (36) C s = C so + δc s1 + δ 2 C s2 + (37) Substituting from equations (32-37) in equations (24-29) and collecting the coefficient of various power of δ on both sides, we obtain the flowing sets of equations: Zero order of δ 4 ψ o 4 + A 1 1 R 1 2 θ o + Nl 1 P r 2 τ T 2 φ o 2 2 ψ o = 0 (38) 2 2 = 0 (39) θ so θ o + Nl 1E c 1 φ o ψ o τ ν R 2 C o 2 = 0 (40) A 2 R 2 ψ o 2 2 φ o 2 = 0 (41) l 2 θ so θ o = 0 (42) 2 C so 2 (43) With the corresponding boundary conditions: ψ o o o = 0, θ so = 0, C o = 0, C so = 0 at y = η ψ o o o = 1, θ so = 1, C o = 1, C so = 1 at y = η (44) 3 ψ o φ o 3 1 o = E 3 3 R 1 x x t 2 3 x η at y = ±η First order of δ 4 ψ θ 1 + Nl 1 P r 2 τ T 2 C A 1 1 R = S c1 C o 2 φ 1 2 ψ 1 = 3 ψ o + ψ o 2 2 y 2 θ s1 θ 1 = θ o + ψ o A 2 R 2 ψ φ 1 2 = R 3 φ o l 2 θ s1 θ 1 = θ so 2 C s1 2 = S c2 C so + 1 R 3 ψ o ψ o x y 2 x + ψ o θ o ψ o θ o x x C o ψ o C o x x + φ o 3 φ o φ o y 2 x y 2 x 1 φ o θ so + 1 φ o θ so R x R x C so 1 φ o C so x R x φ o 3 ψ o 3 (45) + 2Nl 1E c Rτ ν 303 φ o ψ 1 + ψ o φ 1 (46) (47) 3 φ o 3 (48) With the corresponding boundary conditions: ψ = 0, θ s1 = 0, C 1 = 0, C s1 = 0 at y = ±η (51) Solving equations (38-42) and (44-49) together with the boundary conditions (43) and (50), we get the stream functions, velocity, temperature and concentration of fluid and dust particles as: ψ = B 1 6 u = B 1 2 y 3 3η 2 y + δ B 1B 2 B y7 + B 2B y5 + B 4 6 y3 + B 5 y (52) y 2 η 2 + δ B 1B 2 B (49) (50) y 6 + B 2B 3 24 y4 + B 4 2 y2 + B 5 (53) θ = 1 2η y + η + δ D 5D 6 90 y10 + D 5D 7 56 y8 + D 5D 8 30 y6 + D 2D 4 20 y5 + D 5D 9 C = 1 y + η + δs F 2 2η c 20 y5 + F 1 6 y3 F 1η 2 + F 2η φ = B 1R 6 u s = B 1R 2 12 y4 + D 2D 3 6 y 3 + D 10 2 y2 + D 11 y + D 12 (54) y (55) y 3 3η 2 y + δ B 6 7 y7 + B 7 5 y5 + B 8 3 y3 + B 9 y (56) y 2 η 2 + δ B 6 y 6 + B 7 y 4 + B 8 y 2 + B 9 (57)

5 θ s = 1 y + η + δ D 5D 6 2η 1 l 2 D 4 y 3 + D 3 y 90 y10 + D 5D 7 56 y8 + D 5D 8 30 y6 + D 2D 4 20 y5 + D 5D 9 12 y4 + D 2D y 3 + D 10 2 y2 + D 11 y + D 12 C s = 1 y + η + δs F 2 2η c2 20 y5 + F 1 6 y3 F 1η 2 + F 2η 4 y (59) 6 20 Where B 1 B 9, D 1 D 12, F 1 and F 2 are defined in the appendix. 4. RESULTS AND DISCUSSION This study considered the effect of wall properties on peristaltic transport of a dusty fluid through porous channel with heat and mass transfer. The effects of the parameters entering the problem on the velocity, temperature and concentration of fluid and dust particles are shown graphically. In figures (2-4) the velocity of fluid u are plotted against y. It is observed from these figures that the velocity at fixed values of y increases by increasing the tension parameter E 1, the mass parameter E 2 and the damping parameter E 3. Also the effects of these parameters E 1, E 2 and E 3 on the velocity of dust particles are shown through figures (5-7). Its found that the velocity u s at fixed values of y increases by increasing E 1, E 2 and E 3. Figures (8-13) illustrate the effects of E 1, E 2, E 3, N, E c and P r on the temperature of the fluid. We can see that the temperature at fixed values of y decreases by increasing E 1, E 2 and E 3 also the temperature decreases by icreasing N, E c and P r. Figures (14-19) represents the effects of the above parameters on the temperature of dust particles when potted against y. It is clear that the temperature at fixed values of y decreases by increasing E 1, E 2 and E 3. Also, it is observed that the temperature decreases by increasing the number density of dust particles N and Eckert number E c. The tempereture also decrases by increasing the Prandtl number P r. Figures (20 and 21) are graphed to illustrate the effects of the tension parameter and the mass parameter on the concentration of the dusty fluid. It's found that, the concentration at fixed values of y decreases with an increase in E 1 and E 2, this occurs near the lower porous wall and the invers effect will occur near the uper porous wall. Figures (22 and 23) show the variation of the concentration of the fluid for different values of the damping parameter and the Schmidt number. It is found that the concentration at fixed values of y increases with an increase in E 3 and S c this occurs near the lower wall and the inverse effect occur near the upper wall. Figures (24-27) represent the effects of E 1, E 2, E 3 and S c on the concentration of dust particles when potted against y. It's found that, the concentration at fixed values of y decreases with an increases in the tension parameter E 1 and the mass parameter E 2, this occurs near the lower porous wall and the invers effect will occur near the upper porous wall, also the concentration increases with an increase in E 3 and S c this is occurs near the lower wall and the inverse effect occur near the upper wall.. Figures (28-33) represents the effects of wall parameters (E 1, E 2 and E 3 ) on the stream functions for fixed values of the other parameters. These figures shows that the number of the stream lines decreases with an increases in E 1, E 2 and E CONCLUSION In the present paper we investigated the effect of wall properties on peristaltic transport of a dusty fluid through porous channel with heat and mass transfer. The resulting equations which control the motion of a dusty fluid are solved analytically by using the perturbation technique. The stream functions, velocity, temperature and concentration of fluid and solid particles are obtained. The phenomenon of trapping and effects of the wall parameters (tension parameter E 1, mass paramerter E 2 and damping parameter E 3 ), the density number of dust particles N, Eckert number E c, Prandtl number P r and Schmidt number S c on these distributions are discussed and illustrated by a set of graphs. It is strongly believed that the results of this problem are of great interest for many scientific and engineering applications. Caption of figures Figure (2). The velocity of fluid particles u is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. Figure (3). The velocity of fluid particles u is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. Figure (4). The velocity of fluid particles u is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. Figure (5). The velocity of dust particles u s is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. Figure (6). The velocity of fluid particles u s is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. Figure (7). The velocity of fluid particles u s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1. (58)

6 Figure (8). The temperature of fluid particles θ is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (9). The temperature of fluid particles θ is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (10). The temperature of fluid particles θ is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (11). The temperature of fluid particles θ is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (12). The temperature of fluid particles θ is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = 1, t = 1, P r =0.1, N=10, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (13). The temperature of fluid particles θ is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = 1, t = 1, E c =0.2, N=10, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (14). The temperature of dust particles θ s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = 1, t = 1, P r =0.1, N=10, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (15). temperature of dust particles θ s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = 1, t = 1, E c =0.2, N=10, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (16). temperature of dust particles θ s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, x = π, t = π, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (17). temperature of dust particles θ s is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (18). temperature of dust particles θ s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (19). temperature of dust particles θ s is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=1x = 1, t = 10,, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (20). The concentration of fluid C is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c1 =0.05. Figure (21). The concentration of fluid C is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c1 =0.05. Figure (22). The concentration of fluid C is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c1 =0.05. Figure (23). The concentration of fluid C is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (24). The concentration of dusty fluid C s is plotted against y, for R = 1, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c2 =0.05. Figure (25). The concentration of dusty fluid C s is plotted against y, for R = 1, E 1 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c2 =0.05. Figure (26). The concentration of dusty fluid C s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2, S c2 =0.05. Figure (27). The concentration of dusty fluid C s is plotted against y, for R = 1, E 1 =0.01, E 2 =0.01, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10,x = 1, t = 1, P r =0.1, E c =0.2, τ T =1, τ ν =1, l 1 =0.1, l 2 =2. Figure (28). The stream function is plotted for R = 1, E 1 =1, E 2 =1, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 1. Figure (29). The stream function is plotted for R = 1, E 1 =2, E 2 =1, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 0. Figure (30). The stream function is plotted for R = 1, E 1 =1, E 2 =2, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 0. Figure (31). The stream function is plotted for R = 1, E 1 =1, E 2 =3, E 3 =0.5, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 0. Figure (32). The stream function is plotted for R = 1, E 1 =1, E 2 =1, E 3 =1, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 0. Figure (33). The stream function is plotted for R = 1, E 1 =1, E 2 =1, E 3 =2, A 1 =0.2, A 2 =0.1, δ=0.01, N=10, t = 0. Appendix B 1 = ϵ((e 1 + E 2 )(2π) 3 Cos[2π(x t)] E 3 (2π) 2 Sin[2π(x t)]) B 2 = ϵ((e 1 + E 2 )(2π) 4 Sin[2π(x t)] + E 3 (2π) 3 Cos[2π(x t)]) B 3 = ((A 1 R/A 2 ) + 1) B 4 = (B 2 B 1 η 4 /12)(B 3 4) (η 2 /2)(B 2 B 3 ) 305

7 B 5 = B 2 B 3 ((η 4 /24) (B 1 η 6 /360)) (B 4 η 2 /2) B 6 = B 1 B 2 B 3 R/60 B 7 = (B 1 B 2 R 2 /12A 2 ) + (B 3 B 2 R/24) B 8 = (B 2 R 2 /2A 2 ) + (B 4 R/2) B 9 = B 6 η 6 B 7 η 4 B 8 η 2 D 1 = πϵcos[2π(x t)]/η 2, D 2 = P r + np r L 1 /τ T L 2, D 3 = D 1 + (B 1 D 1 η 2 /2) B 2 η/4 D 4 = (B 2 /12η) B 1 D 1 /2, D 5 = (2L 1 ne C P r /τ ν R), D 6 = (B 1 B 6 /2) (B 1 2 B 2 B 3 R/720) D 7 = (B 1 B 2 B 3 /48) + (B 1 B 7 /2) (B 1 B 6 η 2 /2) + (B 1 2 B 2 B 3 Rη 2 /720) D 8 = (B 1 B 4 R/4) + (B 1 B 8 /2) (B 1 B 2 B 3 Rη 2 /48) (B 1 B 7 η 2 /2) D 9 = (B 1 B 5 R/2) + (B 1 B 9 /2) (B 1 B 4 η 2 R/4) (B 1 B 8 η 2 /2) D 10 = (B 1 B 5 Rη 2 /2) + (B 1 B 9 η 2 /2) D 11 = (D 2 D 4 η 4 /20) (D 2 D 3 η 2 /6) D 12 = (D 5 D 6 η 10 /90) (D 5 D 7 η 8 /56) (D 5 D 8 η 6 /30) (D 5 D 9 η 4 /12) (D 10 η 2 /2) F 1 = D 1 + (B 1 D 1 η 2 /2) B 2 η/4 F 2 = (B 2 /12η) (B 1 D 1 /2) 306

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11 Figure 28 Figure 29 Figure 31 Figure

12 Figure 32 Figure 33 REFERENCES [1]. D. Srinivasacharya, G. Radhakrishnamacharya, C. Srinivasulu, "The effects of wall properties on peristaltic transport of a dusty fluid ", Turkish J. Eng. Env. Sci. 32, (2008), [2]. M. Suryanarayana, G. Sankar, M. Subba, K. Jayalakshmi, "Peristaltic MHD flow of a Bingham fluid through a porous medium in a channel ", African Journal of Scientific Research, 1, (2011), [3]. C. Vasudev, U. Rajeswara, G. Prabhakara, M. Subba, "Peristaltic flow of a Newtonian fluid through a porous medium in a vertical tube under the effect of magnetic field ", Int. J. Cur. Sci. Res., 3, (2011), [4]. M. Narahari, S. Sreenadh, " Peristaltic transport of a Bingam fluid in contact with a Newtonian fluid ", Int. J. of Appl. Math and Mach., 11, (2010), [5]. S. Mahmoud, N. Afifi, H. AL-Isede, " The effect of porous medium and magnetic field on peristaltic transport of a Jeffrey fluid ", Int. J. of Math Analysis, 21, (2011), [6]. S. Krishna, M. Ramana, Y. Ravi, S. Sreenadh, " peristaltic pumping of a Jeffrey fluid in a porous tube ", Arpn. J. of Engineering and Applied Sciences, 3, (2011), [7]. A. Medhavi, " The peristaltic pumping of a non Newtonian fluid ", Int. J. of Applications and Applied Mathematics, 1, (2008), [8]. M. Suryanarayana, G. Sankar, " Non linear peristaltic pumping of Johnson Segalman fluid in an asymmetric channel under effect of magnetic field ", European J. of Scientific Research, 1, (2010), [9]. C. Casudeu, U. Rajeswara, M. Subba, G. Prabhakara, "Peristaltic pumping of Williamson fluid through a porous medium in a horizontal channel with heat transfer ", American J. of scientific and Industrial Research, 3, (2010), [10]. T. Hayat, F. Abbasi, A. Hendi, "Heat transfer analysis for peristaltic mechanism in variable viscosity fluid ", Chin. Phys. Lett., 4, (2011), 1-3. [11]. C. Vasudev, U. Rajeswara, M. Subba, G. Prabhakara, "Effect of heat transfer on peristaltic transport of a Newtonian fluid through a porous medium in an asymmetric vertical channel ", European J. of Scientific Research, 1, (2010), [12]. N. El-Dabe, G. Saddeek, A. EL-Sayed, " Heat and mass transfer of MHD unsteady Maxwell fluid flow through porous medium past a porous flat plate", J. Egypt. Math. Soc., 13, (2005), [13]. N. El-Dabe, S. Elmohandis, A. Khodier,'' Dusty non-newtonian flow between two coaxial circular cylinders", Ain- Shams Science Bulletin, 32, (1994), (3-17). 311