Peristaltic Pumping of a Conducting Sisko Fluid through Porous Medium with Heat and Mass Transfer

Transcription

1 Amerian Journal of Computational Mathematis Published Online September 5 in SiRes Peristalti Pumping of a Conduting Sisko Fluid through Porous Medium with Heat and Mass Transfer Nabit Tawfiq Mohamed El-Dabe Ahmed Younis Ghaly Sallam Nagy Sallam Khaled Elagamy Yasmeen Mohamed Younis Department of Mathematis Faulty of Eduation Ain Shams University Cairo Egypt mastermath3@gmailom Reeived July 5; aepted 4 September 5; published 7 September 5 Copyright 5 by authors and Sientifi Researh Publishing In This work is liensed under the Creative Commons Attribution International Liense (CC BY) Abstrat The mathematial model is presented for the flow of peristalti pumping of a onduting non- Newtonian fluid obeying Sisko model through a porous medium under the effet of magneti field with heat and mass transfer The solutions of the system of equations whih represent this motion are obtained analytially using perturbation tehnique after onsidering the approximation of long wave length The formula of the veloity with temperature and onentration of the fluid is obtained as a funtion of the physial parameters of the problem The effets of these parameters on these solutions are disussed numerially and illustrated graphially through some graphs Keywords Peristalti Sisko Fluid MHD Heat and Mass Transfer Chemial Reation Introdution Peristalsis is a form of fluid transport indued by a progressive wave of area ontration or expansion along the walls of distensible dut ontaining a liquid or mixture A peristalti pump is a devie for pumping fluids generally from a region of lower to higher pressure by means of a ontration wave traveling along a tube like struture Shapiro et al [] explained the basi priniples and brought out learly the signifianes of the various parameters governing the flow The non-newtonian effets in peristalti motion were inluded in Kaimal [] Later several mathematial and experimental models have been developed to understand the fluid mehanial aspets of peristalti motion A large body of work already exists on mathematial and experimental models ontaining a Newtonian or non-newtonian fluid in a hannel How to ite this paper: El-Dabe NTM Ghaly AY Sallam SN Elagamy K and Younis YM (5) Peristalti Pumping of a Conduting Sisko Fluid through Porous Medium with Heat and Mass Transfer Amerian Journal of Computational Mathematis

2 Peristalsis also have industrial and biologial appliations like sanitary fluid transport blood pumps in heart lungs mahines and peristalti transport of toxi liquid is used in nulear industries Some reent investigations made to disuss the mehanism of peristalsis inlude the works Radhakrishnamaharya and Srinivasulu [3] studied the influene of wall properties on peristalti transport with heat transfer Mekheimer and Abd Elmaboud [4] analyzed the influene of heat transfer and magneti field on peristalti transport of Newtonian fluid in a vertial annulus Hayat et al [5] studied the effet of heat transfer on the peristalti flow of an eletrially onduting fluid in a porous spae Krishna Kumari et al [6] studied the peristalti pumping of a magnetohydrodynami Casson fluid in an inlined hannel Ravi Kumar et al [7] onsidered power-law fluid in the study of peristalti transport The effet of porous medium on the motion of the fluid has been studied by many authors Elshehawey et al [8] studied the effet of porous medium on peristalti motion of a Newtonian fluid Eldabe [9] studied magnetohydrodynami flow through a porous medium fluid at a rear stagnation point Eldabe et al [8] studied MHD flow and heat transfer in a visoelasti inompressible fluid onfined between a horizontal strething sheet and a parallel porous wall Elshehawey et al [9] studied the peristalti motion of a Generalized Newtonian fluid through a porous medium El-Dabe et al [] studied the magnetohydrodynami flow and heat transfer for a peristalti motion of Carreau fluid through a porous medium The study of the influene of mass and heat transfer on Newtonian and non-newtonian fluids has beome important in the last few years This importane is due to a number of industrial proesses Examples are food-proessing biohemial operations and transport in polymers Eldabe et al [] studied the heat and mass transfer in hydromagneti flow of the non-newtonian fluid with heat soure over an aelerating surfae through a porous medium Srinivas and Kothandapani [] dealt with peristalsis and heat transfer Hayat et al [3]-[5] analyzed the peristalti mehanism with heat transfer Eldabe et al [6] studied the effet of ouple stresses on the MHD of a non-newtonian unsteady flow between two parallel porous plates The main aim of this study is to investigate the problem of the peristalti flow of a onduting Sisko fluid in a porous hannel with heat and mass transfer; the system of non-linear partial differential equations whih desribe this motion with heat and mass transfer subjeted to the appropriate boundary onditions is solved analytially by using perturbation method The expressions of the veloity the temperature and the onentration are determined The effets of different parameters on these expressions are disussed through graphs Basi Equations The basi equations governing the flow of an inompressible fluid are expressed as follows: The ontinuity equation V = () The momentum equation dv µ ρ = divτ + µ e J B V () dt k where τ = PI+ S and ( ) T S = m+ η A A = L + L L = grad V = tr ( A ) (3) where m η and n are the material parameters of the fluid Note that for n< o the fluid desribes shear thinning for n > o the fluid desribes shear thiking for n = the Newtonian fluid is reovered and for m= o the generalized power-law model an be obtained The temperature equation T ρp + ( V ) T = κ T +Φ t (4) The onentration equation C Dκ T + ( V ) C = D C+ T AC ( C ) t (5) T m 35

3 The dissipation funtion Φ V Φ=τ ij X i j (6) Maxwell s equations B = µ J (7) e Ohm s equation B E = t B = ( ) J = σ E+ V B (8) d where V is the veloity vetor ρ is the density dt is the material time derivative k is the permea- bility J is the urrent density σ is the eletrial ondutivity µ e is the magneti permeability E = is the eletri field and τ is the Cauhy stress T and C are temperature and onentration of the fluid κ is the thermal ondutivity p is the speifi heat apaity at onstant pressure D is the oeffiient of mass diffusivity κ T is the thermal diffusion ratio T m is the mean fluid temperature and A is the reation rate onstant 3 Mathematial Formulation Consider the two-dimensional motion of an inompressible Sisko fluid in an infinite hannel of width a see Figure In the upper wall we assume an infinite sinusoidal wave train moving ahead with onstant veloity along it while the lower plate is fixed at Y = The wavy surfae is defined as π Y = H ( X t ) = a + bsin ( X t ) λ (9) where b is the wave amplitude λ is the wave length and t is the time Now Equations ()-(6) an be written in two-dimensional ( ) U X V Y + = X Y as follows: S S ρ t X Y X X Y k P X X X Y µ + U + V U = + + U σbu o () () Figure Sketh of the problem 36

4 S S ρ t X Y Y X Y k P X Y Y Y µ + U + V V = + + V T T T U V T T S U S V ρ U S V t X Y X Y X X Y Y p + + = κ X X X Y + + Y Y C C C C C Dκ T T t X Y X Y T X Y T + U + V = D + AC + + ( C ) m () (3) (4) where S X X U U V V U = m+ η X Y X Y X (5) S X Y U U V V U V = m+ η X Y X Y Y X (6) S Y Y U U V V V = m+ η X Y X Y Y U V U V Φ= S + S + + S X X Y Y X X X Y Y Y Subjeted to the following appropriate boundary onditions: Choose the wave frame ( ) In whih ( ) U = T = T C = C at Y = Y π U = T = T C = C at Y = h = a + bsin X t λ ( ) x y moving with speed where ( ) ( ) (7) (8) (9) x = X t y = Y u = U v = V p x y = P X Y t () u v are omponents of the veloity in the moving oordinates system Then the Equations ()-(9) an be written as: u x v + = () ρ u v u u Bu x x x k p x x x y µ + σ = + + o ρ u v v v x x k p x y y y µ + = + + T T T T u v u v ρ u v κ s s s x x x x p + = x x x y y y () (3) (4) 37

5 where s u C v C C D C Dκ T T AC C x y T + = ( ) x x Tm x s s x x u u v v u = m+ η x x x u u v v u v = m+ η x x x y y The boundary onditions are: u u v v v = m+ η x x u = T = T C = C y = at π u = T = T C= C at y = h = a+ bsin x λ In order to simplify the governing equations of the motion we may introdue the following dimensionless transformations: π π y u v a a h C C T x = x y = u = v = s = s p = p h = φ = θ = T λ a µ λµ a C C T T where T and C are the temperature and the onentration of the fluid at the lower wall of the hannel T and C are the temperature and the onentration of the fluid at the upper wall of the hannel Substituting (3) into Equations ()-(9) we obtain the following non-dimensional equations after dropping the star mark: u v δ + = x p xx xy Re δu + v u = + δ + u Mu x x x Da δ δ p δ δ xy yy Re u + v v = + + v x x Da θ θ θ θ u v u v Re δu + v = δ + E + δsxx + sxy δ + syy x Pr x x x φ φ φ φ θ θ Re δu + v = δ + S + r δ + R φ x S x x δ (5) (6) (7) (8) (9) (3) (3) (3) (33) (34) (35) s xx u u v v u η δ δ = x x x (36) 38

6 The boundary onditions are: s xy u u v v η δ δ u v δ = x x x s yy u u v v v η δ δ = x x u = θ = φ = at y = u = θ = φ = at y = h where the dimensionless parameters are defined by: πa ρa δ = is the Wave number R e = is the Reynolds number λ µ H a e M = σµ is the Magneti field parameter µ is the Prandtle number S r m ( ) ( ) Dκ T T T = T ν C C C E = T T p ( ) is the Soret number and R l η = a m is the Ekert number ( C) AC By using the following definition of stream funtion ψ as: (37) (38) (39) k K = is the Dary number a is the Non-Newtonian parameter S p P νρ r = κ ν = is the Shmidt number D = is the hemial reation parameter ν u = v = δ x The system of Equations (33)-(338) an be written as: p s s xx xy R ψ ψ ψ δ e = + δ + M ψ x x x x K p δ δ δ x x x x K x 3 xy yy δ Re + = θ θ θ θ Reδ = δ + E + δsxx + sxy δ δs yy x x Pr x xy x xy s xy φ φ φ φ θ θ Reδ = δ + S + r δ + R φ x x S x x s xx = δ + η δ + δ δ + xy x xy xy = + η δ + δ δ δ + xy x xy x (4) (4) (4) (43) (44) (45) (46) 39

7 The boundary onditions for the dimensionless stream funtion in the moving frame are given by: ψ ψ = = θ = φ = at y = = ψ = F θ = φ = at y = h The dimensionless form of the surfae of the peristalti wall an be written as: where h = + χsinx b χ = is the amplitude ratio or the olusion and a h h F= uy= y= h d d ψ ( ) ψ ( ) is the total flux number (48) 4 Solution of the Problem Aording to long wavelength approximation ( ) gradient beome: (47) δ [] Equations (4)-(46) after eliminating the pressure ψ s xy L = θ + ES xy = Pr φ θ S r + Rφ = S y y s xy S xx (49) (5) (5) = S = (5) = + η s xy yy ψ ψ = + η n In order to solve the Equations (49)-(5) subjeted to the boundary onditions we suppose the following perturbation for small non-newtonian parameter η : ψ = ψ + ηψ + o o o θ = θ + ηθ + φ = φ + ηφ + Substituting (55) into (49)-(54) and omparing the oeffiient of zero and first order of η we get 4 Zero Order System of η (53) (54) (55) 4 L = 4 θ + E = Pr (56) (57) 3

8 φ θ + S r R φ = S With the respetive boundary onditions ψ ψ = = θ = φ = at y = ψ = F = θ = φ = at y = h = + φsin x (58) (59) 4 First Order System of η We shall onsider the ase of Dielatent fluids when n > and we hoose (n = 3) then we have the following system of first order of η with the respetive boundary onditions 4 3 ψ ψ ψ L = 4 4 θ + E + = Pr φ θ + S r R φ = S ψ ψ = = θ = φ = at y = 43 Solution for System of Zero Order ψ = = θ = φ = at y = h = + χsin x ( ) (6) (6) (6) (63) ψ = y + G SinhLy ylcosh Lh (64) φ 4 PEG r L y y Ly y Lh h θ = Cosh Cosh + 4 L L h L L h PE S SG L PEG L PE S SG L PEG L Sinh e + Cosh ( 4L S ) ( ) rs L SrR r r r SR y r r r = A SR y + Ly + (65) (66) 44 Solution for System of First Order L ψ = GL ( CoshLy LCoshLh) G L Cosh3Lh + LCoshLh 6 (67) L + Sinh Cosh3 + Cosh + Sinh L h Lh Ly L Ly L y Ly CoshLy 3 4 3yL θ = Ay + B PrE GL A y G L SinhLy + L CoshLy Cosh4 Ly G L y (68) 3

9 RS Cosh y L Ly φ = ASinh RS y PEGLSSZ r r e + PEGLSS r r A + 4L RS RS A ( SinhLy CoshLy) 3LySinhLy CoshLy Sinh4Ly G L (69) 5 Results and Disussions The problem of the peristalti flow of a Sisko fluid through a porous medium with heat and mass transfer has been disussed The effets of non-newtonian dissipation and hemial reation on the fluid flow have been onsidered We obtained the solutions of the momentum heat and mass equations analytially by using the perturbation tehnique for small non-newtonian parameter η after onsidering the long wave approximations Figure illustrates the relation between the value of stream funtion ψ and the magneti parameter M it is lear that ψ inreases with inreasing the magneti parameter while it dereases with inreasing the permeability parameter K whih seen through Figure 3 In Figure 4 the relation between the value of stream funtion ψ and the amplitude ratio χ has been illustrated it is lear that ψ dereases with the derease of χ Figure 5 illustrates the relation between the value of temperature θ and the magneti parameter M it is lear that inreases with inreasing the magneti parameter In Figure 6 we an observe that the value of temperature inreases with inreasing the Prandtle number parameter P r while in Figure 7 the relation between the value of temperature θ and the Ekert number E has been illustrated It is lear that θ inreases with the inrease of E Figure Stream funtion profiles ψ ( y) for some values of M Figure 3 Stream funtion profiles ψ ( y) for some values of K 3

10 Figure 4 Stream funtion profiles ψ ( y) for some values of χ Figure 5 Temperature profiles θ ( y) for some values of M Figure 6 Temperature profiles θ ( y) for some values of P r Figure 7 Temperature profiles θ ( y) for some values of E 33

11 In Figure 8 the relation between the value of onentration φ and the magneti parameter M it is lear that φ dereases with inreasing the magneti parameter But we an observe from Figure 9 that φ dereases with inreasing the Shmidt parameter S also in Figure the relation between the value of onentration funtion φ and the Soret value S r has been illustrated It is lear that φ inreases with the inrease of R as in Figure 6 Conlusion and Appliations In this work we study the peristalti motion of magneto-hydrodynamis flow with heat and mass transfer for inompressible non-newtonian fluid through a porous medium The governing partial differential equations of this problem subjeted to the boundary onditions are solved analytially by using perturbation tehnique The Figure 8 Conentration profiles φ ( y) for some values of M Figure 9 Conentration profiles φ ( y) for some values of S Figure Conentration profiles φ ( y) for some values of S r 34

12 Figure Conentration profiles φ ( y) for some values of R analytial forms for the stream distribution ψ the temperature θ and the onentration φ are obtained The effets of various physial parameters of the problem on these formulas are disussed and have been shown graphially It is seen that the ψ inreases with inreasing the magneti parameter M while it dereases with inreasing the permeability parameter K Also it is lear that θ inreases with the inrease of E The study of this phenomenon is very important beause the study of flow through porous medium has many appliations It has an important role in agriultural extrating pure petrol from rude oil and hemial engineering There are examples of natural porous media suh as wood filter paper otton leather and plastis As a good biologial example on the porous medium the human lung galls bladder and the walls of vessels The peristalti motion has been found to involve in many biologial organs suh as esophagus small and large intestine stomah the human ureter lymphati vessels and small blood vessels Also peristalti transport ours in many pratial appliations involving biomehanial systems suh as finger pumps Referenes [] Shapiro AH Jafferin MY and Weinberg SL (969) Peristalti Pumping with Long Wave Lengths at Low Reynolds Number Journal Fluid Mehanis [] Kaimal MR (978) Peristalti Pumping of Non-Newtonian Fluid at Low Reynolds Number under Long Wave Approximation Journal of Applied Mehanis [3] Radhakrishnamaharya G and Srinivasulu C (7) Influene of Wall Properties on Peristalti Transport with Heat Transfer Comptes Rendus Meanique [4] Hayat T Qureshi MU and Hussain Q (9) Effet of Heat Transfer on the Peristalti Flow of an Eletrially Conduting Fluid in a Porous Spae Applied Mathematial Modelling [5] Abd Elmaboud Y and Mekheimer KhS (8) Peristalti Flow of a Couple Stress Fluid in an Annulus: Appliation of an Endosope Physia A: Statistial Mehanis and Its Appliations [6] Krishna KumariP SVHN Ramana Murthy MV Chenna Krishna Reddy M and Ravi Kumar YVK () Peristalti Pumping of a Magnetohydrodynami Casson Fluid in an Inlined Channel Advanes in Applied Siene Researh [7] Ravi Kumar YVK Krishna KumariP SVHN Ramana Murthy MV and Sreenadh S () Peristalti Transport of a Power-Law Fluid in an Assymmetri Channel Bounded by Permeable Walls Advanes in Applied Sienes [8] Eldabe NT and Sallam SN (998) Magenetohydrodynami Flow and Heat Transfer in a Visoelasti Inompressible Fluid Confined between a Horizontal Strething Sheet and a Parallel Porous Wall Canadian Journal of Physis [9] El Shehawey EF and El Sebaei W () Peristalti Transport in a Cylindrial Tube through a Porous Medium International Journal of Mathematis and Mathematial Sienes [] El-Dabe NTM Fouad A and Hussein MM () Magnetohydrodynami Flow and Heat Transfer for a Peristalti Motion of Carreau Fluid through a Porous Medium Punjab University Journal of Mathematis

13 [] Eldabe NT and Mohamed MAA () Heat and Mass Transfer in Hydromagneti Flow of the Non-Newtonian Fluid with Heat Soure over an Aelerating Surfae through a Porous Medium Chaos Solitons and Fratals [] Srinivas S and Kothandapani M (8) Peristalti Transport in an Asymmetri Channel with Heat Transfer A Note International Communiations in Heat and Mass Transfer [3] Hayat T Abbasi FM and Alsaedi A (4a) Soret and Dufour Effets on Peristalti Flow in an Asymmetri Channel Arabian Journal for Siene and Engineering [4] Hayat T Hina S Hendi AA and Asghar S () Effet of Wall Properties on the Peristalti Flow of a Third Grade Fluid in a Curved Channel with Heat and Mass Transfer International Journal of Heat and Mass Transfer [5] Hayat T Noreen S Alhothuali MS Asghar S and Alhomaidan A () Peristalti Flow under the Effets of an Indued Magneti Field and Heat and Mass Transfer International Journal of Heat and Mass Transfer [6] Eldabe NT Hassan AA and Mohamed MAA (3) Effet of Couple Stresses on the MHD of a Non-Newtonian Unsteady Flow between Two Parallel Porous Plates Zeitshrift für Naturforshung Appendix L = + M K F + h G = SinhLh hlcoshlh hl G L G = Lh + L Lh + L Lh 6 SinhLh hlcoshlh Cosh3 Cosh Sinh PrE GL B = G GL 4 8 PrE GL GL CoshLh A = + GL h h 4 6 L 3hL B G L SinhLh CoshLh Cosh4 Lh G L h h GL 3 L 9G L 3 Z= + GL + GL L RS RS 3 4L RS 6 ( ) 36