Peristaltic transport of a Maxwell fluid in a porous asymmetric channel through a porous medium

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1 Akram et al., Cogent Engineering 04, : BIOMEDICAL ENGINEERING RESEARCH ARTICLE Peristaltic transport of a Maxwell fluid in a porous asymmetric channel through a porous medium Safia Akram *, M. Hanif and S. Nadeem Received: 4 February 04 Accepted: 7 September 04 Published: 03 December 04 *Corresponding author: Safia Akram, Department of Basic Sciences, MCS, National University of Sciences and Technology, Islamabad, Pakistan drsafiaakram@gmail.com Reviewing editor: Zhongmin Jin, Xian Jiao Tong University, China; Leeds University, UK Additional information is available at the end of the article Abstract: The present study investigates the peristaltic flow of a Maxwell fluid in a porous asymmetric channel through a porous medium. An analytical solution has been found using regular perturbation method. The stream function and average mean velocity are obtained. The graphical results are presented to discuss the physical behavior of various parameters appearing in the problem. Keywords: peristaltic flow, Maxwell fluid, porous medium, porous boundaries, asymmetric channel. Introduction Since, the first investigation done by Latham 966 a large amount of literature is available on peristaltic motion of Newtonian and non-newtonian fluid different flow geometries Akram & Nadeem, 03; Akram, Nadeem, & Hanif, 03; Ali, Hayat, & Asghar, 009; Eldesoky, 0, 03; Eldesoky & Mousa, 00; El-Shehawey, El-Dabe, & El-Desoky, 006; Mittra & Prasad, 973; Nadeem & Akram, 00; Nadeem, Akram, & Akbar, 03; Shapiro, Jaffrin, & Weinberg, 969. Peristaltic mechanism is a fluid transport induced by a progressive wave of area contraction or expansion along the walls of a distensible tube containing fluid. This mechanism occurs in many practical applications involving physiological and biomechanical processes. The study of fluid flow in a porous plate is another area which has been investigated by many researchers Hayat & Hutter, 004; Hayat, Ellahi, S. Asghar, & Siddiqui, 004; Rajagopal & Gupta, 984; Wang & Hayat, 004 because of its applications. Few of applications of porous boundaries are transpiration cooling, gaseous diffusion, and process of dialysis of blood in an artificial kidney. Only a limited attention has been focused to the study of peristaltic motion of fluid suction and injunction. The idea of peristaltic motion of fluid in porous boundaries was first investigated by Lukashev 993. He has formulated a model of peristaltic transport of liquid motion caused by the auto-wave process of mass transport through a porous capillary wall. Later on, El-Shehawey and Husseny 000 discussed the effects of porous boundaries on peristaltic transport through porous medium. Recently, Haroun 000 discussed the effects of wall compliance on peristaltic transport of a Newtonian fluid in an asymmetric channel. ABOUT THE AUTHOR Safia Akram is an assistant professor at National University of Sciences and technology, MCS, Islamabad. Her field of research is applied mathematics and computational fluid mechanics. She is recipient of research productive award in year 00 0 and 0 0 by Pakistan Council for Science and Technology. Her research articles have been published in international journals. PUBLIC INTEREST STATEMENT The present paper has many further applications in the field Fluid Mechanics and Mathematics. Nowadays peristaltic flow has received a considerable attentions of many researchers due to its wide range of applications in engineering, industry, and physiology. Peristaltic flows basically occurs in swallowing food through the esophagus, urine transport from kidney to bladder, lymph transport in the lymphatic vessels, in vasomotion of small blood vessels, in cell separators, arthropumps, artificial blood pumps and heart lung machine, finger and roller pumps, toxic liquid in nuclear industry, etc. 04 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 3.0 license. Page of 4

2 Akram et al., Cogent Engineering 04, : Motivated from the above discussion, the aim of the present paper is to discuss the peristaltic motion of a Maxwell fluid in a porous asymmetric channel through a porous medium. To the best of authors knowledge, no attempt has been made to discuss the peristaltic motion of non-newtonian fluid in a porous channel. The porosity of the channel means that there will be both suction and injection phenomena. The governing highly non-linear partial differential equations are solved analytically by employing perturbation method. The expressions for stream function and average mean velocity have been computed. Numerical illustrations that show the physical effects and the pertinent features are investigated at the end of the paper.. Mathematical formulation Let us consider the peristaltic flow of an incompressible, non-newtonian fluid linear Maxwell fluid in a two dimensional channel having width d and d. The walls of the channel are porous, flexible, and also there are imposed traveling sinusoidal waves of small amplitude. The equations which governs the flow are defined as u x + u y = 0 +τ t u t u u +u +v = +τ p x y t ρ x +ν u x + u ν y W u +τ v v v +u +v = +τ p t t x y t ρ y +ν v x + v ν y W v 3 where u and v are the velocity components along x and y-axis respectively, p is the pressure, τ is the relaxation times, W is the permeability parameter, and ρ is the fluid density. Making use of stream function Ψ reduce to u = Ψ, v y = Ψ x and eliminating pressure, Equations and 3 +τ t t Ψ+Ψ y Ψ x Ψ x Ψ y = ν 4 Ψ ν W Ψ 4 where = x + y Let the vertical displacement for upper wall is η and lower wall is η. Further, η and η are assumed to be in the form of a sinusoidal wave of different amplitudes and phases. Thus, η = a cos π π x ct, η λ = a cos x ct+θ λ where a and a are the amplitude, λ is the wave length, c is the speed of light, and θ is the phase difference which varies in the range 0 θ π in which θ = 0 corresponds to symmetric channel wave out of phase and θ = π, the waves are in phase, further, a, a, d, d, and θ satisfies the condition 5 a +a +a a cos θ d +d Page of 4

3 Akram et al., Cogent Engineering 04, : The horizontal displacement is assumed to be zero. The fluid is entering the flow region through one plate at the same rate as it is leaving through the other plate velocity V in the positive direction of the y-axis. Therefore, the boundary conditions are Ψ y = 0, Ψ x = V η t at y = d +η 6 Ψ y = 0, Ψ x = V + η t at y = d η 7 Defining x = x d, ȳ = y d, ū = u c, v = v c, p = p a = α α ρc, V = V c, η = η d, η = η d, W = W d t = ct, ε = α, h = d, d d d, Ψ= Ψ cd, τ = τc, R = cd, α = πd d ν λ 8 Using the above non-dimensional quantities into Equations 4 7, the resulting equations after dropping the bars can be written as +τ t t Ψ+Ψ y Ψ x Ψ x Ψ y = 4 Ψ R W Ψ 9 η = ε cos αx t, η = aε cos αx t+θ 0 Ψ y = 0, Ψ x = V αε sin αx t at y = h+η Ψ y = 0, Ψ x = V +aαε sinαx t+θ at y =+η where ε is the amplitude ratio, α is the wave number and R is the Reynolds number. 3. Solution of the problem To find the solution of Equation 9, we expand Ψ and p x in the form Ψ=Ψ 0 +εψ +ε Ψ + p p p p 4 x = +ε +ε + x 0 x x p The first term corresponds to an imposed pressure gradient which is considered as a constant. x 0 The higher order terms correspond to peristaltic motion and are considered as function of x, y, and t. Making use of Equation 3 in Equations 9 and collecting the like powers of ε, we obtain three sets of coupled linear differential equations their corresponding boundary conditions in Ψ 0, Ψ and Ψ. To avoid the lengthy calculations, the solution for Ψ 0, Ψ, and Ψ are directly defined as follows Solution for Ψ 0 The solution for Ψ 0 is obtained by adopting the similar producer as discussed by El-Shehawey and Husseny 000 subject to suction and injection is written as Ψ 0 x, y = kw e y λ h e λ e λ e e λ e λ h λ h e λ e λ + eλ e λ h e λ e λ e λ h + e λ e λh y eλ λ λ λ e λ y λ Vx e λ 5 Page 3 of 4

4 Akram et al., Cogent Engineering 04, : where λ = RV + RV +4 W, λ = RV RV +4 W The expression for velocity is obtained as follows u 0 = kw + eλ e λ h e λy e λ e λ h e e λ h e λ e λ e λ h e λ e λ λ y 6 If lim lim u V 0 W 0 = k y which leads to the classical Poisseuille flow in the absence of V and /W. 3.. Solution for Ψ and Ψ The solution for Ψ and Ψ can be calculated by using the following expressions according to their boundary conditions. Ψ = φ ye iαx t +φ ye iαx t 7 Ψ = φ 0 y+φ ye iαx t +φ ye iαx t 8 in which asterisk denotes the complex conjugate. Substituting Equations 7 and 8 into the differential equations and their corresponding boundary conditions in Ψ and Ψ leads to the following differential equations d α +iαr iαrψ +RV iατ VR d dy 0 W dy +Rα τ d α φ dy +Rτα φ Ψ 0 Ψ 0 Rτα φ α φ +iαrψ φ = 0 0 φ { } { } { } = ae iθ Ψ 0 { } { } φ = ae iθ 9 0 φ 0 RVφ 0 W φ = 0 iαr[ φ φ φ ] φ φ 0 { } { ± { } = a } a cos θ { } Ψ 0 φ { } { +φ } 3 4α d 4α +iαδ dy dy R φ W VRδ d 4α φ dy = iαδ RΨ d 0 4α φ dy + iαrδ φ φ φ φ iαδ RΨ φ 0 d 4 Page 4 of 4

5 Akram et al., Cogent Engineering 04, : φ { φ { } = { 4 a e iθ } ± { 4 ae iθ } { Ψ 0 } { φ } = 0 } { ae iθ } { φ } 5 6 Using the similar procedure as discussed in El-Shehawey and Husseny 000, the solution of Equations 9 is straight forward written as φ = A cos h αy +A sin h αy +A 3 e ω y +A 4 e ω y 7 φ 0 y=fy+ D e λ h+λ y +e λ h+λ y +D e λ +λ y e λ +λ y e λ λ h e λ λ h + Fe λ h+λ y e λ h+λ y e λ h+λ y +e λ h+λ y λ +λ +Fe λ +λ y e λ +λ y e λ λ h e λ λ h e λ +λ y e λ h+λ y e λ +λ y +e λ h+λ y +C + e λ λ h e λ λ h where ω = RVδ + RVδ +4β, ω = RVδ RVδ +4β 8 β = α iαrδ + W, δ = iατ z = αw w +e +hw +w αe w +hw +e w +hw w w cos h +hα +e w +hw +e w +hw w w +α sin h +hα z = α w w +ae iθ w e w +hw w e w +hw cos h α α w e w +hw +aw w e +hw +w +iθ w e w +hw w w e w +hw e w +hw ae iθ sin h α+sin h α cos h α z 3 = ae iθ w w e w +hw e w +hw cos h α+e w +hw e w +hw w w cos h α +α w w +ae iθ w e w +hw w ew +hw sin h α +α w e w +hw +aw w e +hw +w +iθ w e w +hw sin h α z 4 = αw e hw e w +hw ae iθ +w e hw +ae w +hw +iθ cos h +hα αe hw +aew +hw +iθ sin h +hα z 5 = αw e hw e w +hw ae iθ +w e hw +ae w +hw +iθ cos h +hα αe hw +aew +hw +iθ sin h +hα A = z z, A = z 3 z, A 3 = z 4 z, A 4 = z 5 z Page 5 of 4

6 Akram et al., Cogent Engineering 04, : { Fy = iαr A3 b ω α e ω y + A b 4 3 ω α e ω y A b b b cos h αy 3 b +A sin h αy 4 A3 b ω α e ω y + A b 4 4 ω α e ω y A b b b sin h αy 3 b +A cos h αy 4 A 3 b α ω e ω y A 4 b 3 α ω e ω y + + A cos h αy +A sin h αy b b b 3 b 4 A 3 b α ω e ω y A 4 b 4 α ω e ω y + b b b A sin h αy +A cos h αy 3 b 4 A A 3 3 ω ω e ω + ω y A A 4 3 ω ω e ω +ω y + ω +ω + RV ω +ω W ω +ω RV ω +ω W A A 3 4 ω ω e ω +ω y A A 4 4 ω ω e ω +ω y + ω +ω + RV ω +ω W ω +ω RV ω +ω W b = α +ω RVω W, b 3 = α +ω RVω W, b = αω RV b 4 = αω RV D = φ += 0 α [ A +A cos h α +A +A sin h α] +ω A 3 eω +ω A 4 eω +ω A 3 eω +ω A 4 eω D =φ = 0 a cos θα [A +A cos h αh A +A sin h αh]+ωa 3 e ω h +ω A 4 e ω h +ω A h 3 e ω +ω A h 4 e ω In above problem, C is arbitrary constant, we can choose C as C p R = x The main time average velocities becomes ū = ε φ = ε 0 Fy+ D e λ h+λ y +e λ h+λ y +D e λ +λ y e λ +λ y e λ λ h e λ λ h Fe λ h+λ y e λ h+λ y e λ h+λ y +e λ h+λ y λ +λ +Fe λ +λ y e λ +λ y e λ λ h e λ λ h p e λ +λ y e λ h+λ y e λ +λ y +e λ h+λ y +WR + x e λ λ h e λ λ h 4. Graphical results and discussion In this section, the graphical results are displayed. Figures 8 are prepared to see the behavior of D and D values of porosity V permeability parameter W, relaxation time τ, and amplitude of wave a. From Figures and, it is observed that D increases the increase in V and decreases the increase in W. From Figures 3 and 4, it is shown that D Page 6 of 4

7 Akram et al., Cogent Engineering 04, : Figure. Variation of D values of V at R = 0, τ =, a = 0.5, W = 0.0, h = 0.5, θ = π 3. Figure. Variation of D values of W at R = 0, τ =, a = 0.5, V = 0.05, h = 0.5, θ = π 3. Figure 3. Variation of D values of τ at R = 0, V = 0.05, a = 0.5, W = 0.0, h = 0.5, θ = π 3. Page 7 of 4

8 Akram et al., Cogent Engineering 04, : Figure 4. Variation of D values of a at R = 0, V = 0.05, τ =, W = 0.0, h = 0.5, θ = π 3. Figure 5. Variation of D values of V at R = 0, τ =, a = 0.5, W = 0.0, h = 0.5, θ = π 3. Figure 6. Variation of D values of W at R = 0, τ =, a = 0.5, V = 0.05, h = 0.5, θ = π 3. Page 8 of 4

9 Akram et al., Cogent Engineering 04, : Figure 7. Variation of D values of τ at R = 0, V = 0.05, a = 0.5, W = 0.0, h = 0.5, θ = π 3. Figure 8. Variation of D values of a at R = 0, V = 0.05, τ =, W = 0.0, h = 0.5, θ = π 3. Figure 9. Variation of mean different values of a at R = 30, τ =, V = 0.08, α = 0.5, θ = π 3, h = 0.5, W =.5, dp/dx = Page 9 of 4

10 Akram et al., Cogent Engineering 04, : Figure 0. Variation of mean different values of W at R = 30, V = 0.08, α = 0.5, τ =, a = 0.5, θ = π 3, h = 0.5, dp/dx = -.0. Figure. Variation of mean different values of W and a at R = 30, τ =, V = 0.08, α = 0.5, θ = π 3, h = 0.5, dp/dx = -.0. Figure. Variation of mean different values of V at R = 0, τ =, W = 3.5, α = 0.5, a = 0.5, θ = π 3, h = 0.5, dp/dx = -.0. Page 0 of 4

11 Akram et al., Cogent Engineering 04, : Figure 3. Variation of mean different values of τ at R = 30, V = 0.05, α = 0.5, a = 0.5, θ = π 3, h = 0.5, dp/dx = -.0. Figure 4. Variation of mean different values of a at R = 30, τ =, V = 0.08, α = 0.5, θ = π 3, h = 0.5, W =.5, dp/dx =.0. Figure 5. Variation of mean different values of W at R = 30, V = 0.08, α = 0.5, τ =, a = 0.5, θ = π 3, h = 0.5, dp/dx =.0. Page of 4

12 Akram et al., Cogent Engineering 04, : Figure 6. Variation of mean different values of W and a at R = 30, τ =, V = 0.08, α = 0.5, θ = π 3, h = 0.5, dp/dx =.0. Figure 7. Variation of mean different values of V at R = 0, τ =, W = 3.5, α = 0.5, a = 0.5, θ = π 3, h = 0.5, dp/dx =.0. Figure 8. Variation of mean different values of τ at R = 30, V = 0.05, α = 0.5, a = 0.5, θ = π 3, h = 0.5, dp/dx =.0. Page of 4

13 Akram et al., Cogent Engineering 04, : decreases the decrease in τ and a. Also it is observed that D decreases the increase in both V and W see Figures 5 and 6. While D increases the decrease in τ and a see Figures 7 and 8. This means that fluid entering through the lower plate acts as injection and the fluid leaving through upper plate acts as suction. The mean velocity distribution and reversal flow are displayed in Figures 9 8. It is observed from Figure 9 that the decrease in a, the mean velocity distribution increases. It is also observed from Figures 0 and that the decrease in W and a, the mean velocity distribution decreases. It is depicted from Figure that the decrease in V the mean velocity distribution increases in the lower half of the channel. It is seen from Figure 3 that the mean velocity distribution decreases in the upper half of the channel the decrease in τ. It is also observed from Figures 4 to 8 that the behavior of the reversal flow is quite similar as compared to the mean velocity distribution. Funding The authors received no direct funding for this research. Author details Safia Akram drsafiaakram@gmail.com M. Hanif hanif_muhammad@yahoo.com S. Nadeem snqau@hotmail.com Department of Basic Sciences, MCS, National University of Sciences and Technology, Islamabad, Pakistan. Department of Mathematics, Quaid-i-Azam University, Islamabad 4530, Pakistan. Citation information Cite this article as: Peristaltic transport of a Maxwell fluid in a porous asymmetric channel through a porous medium, S. Akram, M. Hanif & S. Nadeem, Cogent Engineering 04, : References Akram, S., & Nadeem, S. 03. Influence of induced magnetic field and heat transfer on the peristaltic motion of a Jeffrey fluid in an asymmetric channel: Closed form solutions. Journal of Magnetism and Magnetic Materials, 38, 0. Akram, S., Nadeem, S., & Hanif, M. 03. Numerical and analytical treatment on peristaltic flow of Williamson fluid in the occurrence of induced magnetic field. Journal of Magnetism and Magnetic Materials, 346, Ali, N., Hayat, T., & Asghar, S Peristaltic flow of a Maxwell fluid in a channel compliant walls. Chaos, Solitons & Fractals, 39, Eldesoky, I. M. 0. Influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube. International Journal of Applied Mathematics and Mechanics, 8, Eldesoky, I. M. 03. Effect of relaxation time on MHD pulsatile flow of blood through porous medium in an artery under the effect of periodic body acceleration. Journal of Biological System,, Eldesoky, I. M., & Mousa, A. A. 00. Peristaltic flow of a compressible non-newtonian Maxwellian fluid through porous medium in a tube. International Journal of Biomathematics, 3, El-Shehawy, E. F., El-Dabe, N. T., & El-Desoky, I. M Slip effects on the peristaltic flow of a non-newtonian Maxwellian fluid. Acta Mechanica, 86, El-Shehawey, E. F., & Husseny, S. Z. A Effects of porous boundaries on peristaltic transport through a porous medium. Acta Mechanica, 43, Haroun, M. H Effect of wall compliance on peristaltic transport of a Newtonian fluid in an asymmetric channel. Mathematical Problems in Engineering, 43, Hayat, T., Ellahi, R., Asghar, S., & Siddiqui, A. M Flow induced by non-coaxial rotation of a porous disk executing non-torsional oscillations and a second grade fluid rotating at infinity. Applied Mathematical Modelling, 8, Hayat, T., & Hutter, K Rotating flow of a second order fluid on a porous plate. International Journal of Non- Linear Mechanics, 39, Latham, T. W Fluid motion in a peristaltic pump MSc Thesis, Massachusetts Institute of Technology, Cambridge. Lukashev, A. E Mathematical model of the peristaltic transport of liquid initiated by the auto-wave process of mass transport through the porous capillary wall. Journal of Kaledney, 55, Mittra, T. K., & Prasad, S. N On the influence of wall properties and Poiseuille flow in peristalsis. Journal of Biomechanics, 6, Nadeem, S., & Akram, S. 00. Slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of induced magnetic field. International Journal for Numerical Methods in fluids, 63, Nadeem, S., Akram, S., & Akbar, N. S. 03. Simulation of heat and chemical reactions on peristaltic flow of a Williamson fluid in an inclined asymmetric channel. Iranian Journal of Chemistry and Chemical Engineering, 3, Rajagopal, K. R., & Gupta, A. S An exact solution for the flow of a non-newtonian fluid past an infinite porous plate. Meccanica, 9, BF Shapiro, A. H., Jaffrin, M. Y., & Weinberg, S. L Peristaltic pumping long wavelengths at low Reynolds number. Journal of Fluid Mechanics, 37, Wang, Y., & Hayat, T Hydromagnetic rotating flow of a fourth order fluid past a porous plate. Mathematical Methods in the Applied Sciences, 7, Page 3 of 4

14 Akram et al., Cogent Engineering 04, : The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 3.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Engineering ISSN: is published by Cogent OA, part of Taylor & Francis Group. Publishing Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page 4 of 4