Peristaltic Pumping of a Conducting Jeffrey Fluid in a Vertical Porous Channel with Heat Transfer

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1 vailable online at dvances in pplied Science Researc,, (6): ISSN: CODEN (US): SRFC Peristaltic Pumping of a Conducting Jeffrey Fluid in a Vertical Porous Cannel wit Heat Transfer S. V. H. N. Krisna Kumari. P, Y. V. K. Ravi Kumar *, M. V. Ramana Murty 3, S. Sreenad 4 Department of Matematics, Boj Reddy Engineering College for Women, Hyderabad, India Department of Matematics, Stanley College of Engineering and Tecnology for Women, Hyderabad, India 3 Department of Computer Science, King bdul ziz University, Rabig, KS 4 Department of Matematics, Sri Venkateswara University, Tirupati, India _ BSTRCT Peristaltic pumping of a conducting Jeffrey fluid in a vertical porous cannel wit eat transfer is presented. Te perturbation metod is used to find te solution. Te expressions for temperature, velocity, pressure rise and volume flow rate are obtained. Te effect of various parameters on te temperature and te pumping caracteristics are discussed troug graps. Keywords: Peristalsis; Jeffrey fluid; eat transfer. _ INTRODUCTION Peristaltic motion in a cannel/tube is now known as an important type of flow occurring in several engineering and pysiological processes. Te peristalsis is well known to te pysiologists to be one of te major mecanisms of fluid transport in a biological system and appears in urine transport from kidney to bladder troug te ureter, movement of cyme in te gastrointestinal tract, te movement of spermatozoa in te ductus effeerentes of te male reproductive tract and te ovum in te female fallopian tube, te transport of lymp in te lympatic vessels and vasomotion of small blood vessels suc as arterioles, venules and capillaries. Suc mecanism as several applications in engineering and in biomedical systems including roller and finger pumps. Te need for peristaltic pumping may arise in circumstances were it is desirable to avoid using any internal moving part suc as pistons in pumping process. fter te experimental work of Latam [] on peristaltic transport, Sapiro et al. [] made a detailed investigation of peristaltic pumping of a Newtonian fluid in a flexible cannel and a circular tube. Sud et al. [3] analyzed 439

2 te pumping action of blood flow in te presence of a magnetic field. Even toug it is observed in living systems for many centuries, te matematical modeling of peristaltic transport began wit trend setting works by Sapiro et al.[4] using wave frame of reference and Fung and Yin[5] using laboratory frame of reference. Hayat et al. [6] studied te peristaltic flow of a micropolar fluid in a cannel wit different wave frames. Hayat and li [7] investigated te peristaltic motion of a Jeffrey fluid under te effect of a magnetic field. Vajravelu et al. [8] studied te peristaltic transport of a Casson fluid in contact wit a Newtonian fluid in a circular tube wit permeable wall. In pysiological peristalsis, te pumping fluid may be considered as a Newtonian or a non- Newtonian fluid. Kapur [9] made teoretical investigations of blood flows by considering blood as a Newtonian as well as non- Newtonian fluids. Radakrisnamacarya and Srinivasulu [] studied te influence of wall properties on peristaltic transport wit eat transfer. Mekeimer and bd Elmaboud [] analyzed te influence of eat transfer and magnetic field on peristaltic transport of Newtonian fluid in a vertical annulus. Hayat et al. [] studied te effect of eat transfer on te peristaltic flow of an electrically conducting fluid in a porous space. Krisna Kumari et.al[3] studied te peristaltic pumping of a magnetoydrodynamic casson fluid in an inclined cannel. Ravi Kumar et.al[4] considered power-law fluid in te study of peristaltic transport. In tis paper, peristaltic flow of a conducting Jeffrey fluid in a vertical porous cannel wit eat transfer is studied. Using te perturbation tecnique, te nonlinear governing equations are solved. Te expressions for velocity, temperature and te pressure rise per one wave lengt are determined. Te effects of different parameters on te temperature and te pumping caracteristics are discussed troug graps. MTHEMTICL FORMULTION We consider te motion of a MHD Jeffrey fluid in a two-dimensional vertical porous cannel induced by sinusoidal waves propagating wit constant speed c along te cannel walls. For simplicity, we restrict our discussion to te alf widt of te cannel. We assume tat a uniform magnetic field strengt B is applied as sown in Figure. and te induced magnetic field is assumed to be negligible. Te wall deformations are given by π Y = H ( x, t) = a + b cos ( x ct) (rigt wall) λ () π Y = H ( x, t) = a b cos ( x ct) (left wall) λ () were a is te widt of te cannel, b is amplitude of te waves and λ is te wave lengt. Te constitutive equations for an incompressible Jeffrey fluid are T = p I + s (3) 44

3 ... µ s = γ + λγ (4) + λ were T and s are Caucy stress tensor and extra stress tensor respectively, p is te pressure, I is te identity tensor, λ is te ratio of relaxation to retardation times λ is te retardation time,. γ is sear rate and dots over te quantities indicate differentiation wit respect to time. Figure. Pysical model In laboratory frame, te continuity equation is U V + = X Y Te equations of motion are U U P S XX S XY µ ρ U + V = + + U + ρ gα( T T ) X Y X X Y k U U P S XY S YY µ ρ U V + = + + V X Y X X Y k (5) (6) (7) 44

4 Te equation of energy is T T T T U V V U µ ρ cp U + V = k ( + ) + µ µ U X Y X Y X Y X Y k Te boundary conditions on velocity and temperature fields are U = and T = T at Y = H X ( ) U T = and = at Y = Y Y (9) (8) were U, V are te velocity components in te laboratory frame ( X, Y ), ρ is density, µ is te coefficient of viscosity of te fluid, c p is te specific eat at constant pressure, α is te coefficient of linear termal expansion of te fluid, k is te termal conductivity, k is permeability and T is temperature of te fluid. We sall carry out tis investigation in a coordinate system moving wit te wave speed c, in wic te boundary sape is stationary. Te coordinates and velocities in te laboratory frame ( X, Y ) and te wave frame ( x, y) are related by x = X ct, y = Y, u = U c, v = V, p = P( x, t) were u, v are te velocity components and p, P are te pressures in wave and fixed frames. Equations (5)-(9) can be reduced into wave frame as follows u v + = x y u u p S xx S xy µ ρ ( u + c) + v = + + ( u + c) + ρ gα ( T T ) x y x x y k v v p S xy S yy µ ρ ( u c) v + + = + + v x y y x y k T T T T u v v u µ ρ cp u + c + v = k + + µ µ u c x x y x x k ( ) ( ) Boundary conditions in wave frame are u + c = and T = T at y = H ( x) u T = and = at y = y y () () () (3) (4) we introduce te following non dimensional quantities : 44

5 π x y u v π a π a p π ct H x =, y =, u =, v =, δ =, p =, t =, =, λ a c cδ λ µ cλ λ a 3 ( ) a b a µ α g T T a S = S, φ =, σ =, γ =, T = θ ( T T ) + T, Gr =, µ c a k ρ γ µ c p ρca ca Gr c Pr =, R = =, G =, Ec =, N = Ec Pr k µ γ R c T T ( ) p (5) were R is te Reynolds number, δ is te dimension less wave number, σ is te permeability parameter, Gr is te Grasof number, Pr is te Prandtl number, γ is te Kinematic viscosity of te fluid, Ec is te Ecet number and N is te perturbation parameter. Te basic equations ()-(3) can be expressed in te non-dimensional form as follows u v + = x u u p S ( ) xx S xy δ R u v δ + + ( σ M )( u ) Gθ x y = x x 3 v v p Sxy S yy δ R ( u + ) + v δ δ δ ( σ M ) v x y = x x θ θ θ θ u v δ Pr R ( u + ) + v δ δ N x y = x x v u N δ N σ M u ( ) ( + ) + x were δ δλc v u Sxx = + u + + λ a x δ x δλc v u v Sxy = + u + + δ + λ a x δ x δ δλc v u S yy = + u + + λ a x δ nd (6) (7) (8) (9) sxy u = y + λ δ Te non-dimensional boundary conditions are u = and θ = at y = 443

6 u θ = and = at y = () Using long wave lengt approximation and dropping terms of order δ and iger, It follows equations (7) to () are p u = + ( σ + M )( u + ) + Gθ x + λ p = y θ u = + N( ) + N( σ + M )( u + ) u = and θ = at y = u θ = and = at y = () () (3) (4) Te dimensional volume flow rate in te laboratory and wave frames are given by Q = ( x, t ) U ( X, Y, t) dy, q ( x ) = u ( x, y ) d y and now tese two are related by te equation Q = q + c (x) Te time averaged flow over a period T at a fixed position x is Q T = T Q d t SOLUTION OF THE PROBLEM Equations () and (3) are non-linear because tey contain two unknowns u and θ wic must be solved simultaneously to yield te desired velocity profiles. Due to teir nonlinearity tey are difficult to solve. However te fact N is small in most practical problems allows us to employ a perturbation tecnique to solve tese non-linear equations. We write u = u + Nu θ = θ + Nθ (5) Using te above relations, te equations (), (3) and (4) become d( p + Np ) ( u + Nu ) = + ( σ + M )( u + Nu + ) + G( θ + Nθ ) dx + λ (6) 444

7 ( θ + Nθ) ( u + Nu ) = + N + N( σ + M )( u + Nu + ) u + Nu = and θ + Nθ = at y = ( u + Nu) ( θ + Nθ) = and = at y = (7) (8) Zerot order solution By comparing constant terms on bot sides of te above equations we get te zerot order equations as below dp u = + ( σ + M )( u + ) + Gθ (9) dx + λ θ = y u = and θ = at y = u θ = and = at y = Solving te equations (9) and (3) wit te boundary conditions (3), we obtain dp dp G cos G dx β + λ y + β u dx = β cos β + λ β θ = (33) were = Using te relation (7.5) we obtain zerot order dimensionless mean flow in te laboratory and in te wave frame =F + dp sin β + λ Gsi nσ + λ ( β G) = 3 3 dx β λ cos β λ β + + β + λ cos β + λ β Te pressure gradient is given by G si n β + λ ( β G ) F d p β + λ cos β + λ β = dx sin β + λ 3 β + λ co s β + λ β (3) (3) (3) (34) 445

8 = Q G si n β + λ ( β G ) β + λ co s β + λ β sin β + λ 3 β + λ co s β + λ β (35) Te non-dimensional zerot order pressure rise is given by dp p = dx (36) dx Time mean flow (time averaged flow rate) = =F + (37) First order solution From equations (7.6), (7.7) and (7.8) we obtain te first order equations dp u = + ( σ + M ) u + Gθ dx + λ θ u = + + ( σ + M )( u + ) u = and θ = at y = u θ = and = at y = Solving te equations (38) and (39) wit te use of boundary conditions (4) we obtain dp cos β + λ y G ( ) u = cos + β + λ y + dx β cos β λ β cos β λ G y cos β + λ y + + λ ysin β + λ y β 3β β y θ = cos β + λ y + cos β + λ y + D 3 4 dp G dx were λ = = β, cos β + λ + λ 3 = 4 = 8 β ( + λ )cos β + λ β ( + λ, )cos β + λ D = ( ) 5 = 3 cos β + λ 6 =, 7 =,, β ( + λ ) 3 = cos 8 β λ β 3β = + λ sin β + λ, β (38) (39) (4) (4) (4) 446

9 G = + D β β ( + λ ) Using te relation (7.5) we obtain first order dimensionless mean flow in te laboratory and in te wave frame = F u dy = dp F = + G ( ) + (43) dx Te pressure gradient is given by dp F + G ( ) = dx ( ) ( Q + G ( ) ) = (44) were S in β + λ =, 3 β + λ C os β + λ β G ( + ) Si n β + λ C os β λ + β + λ 8 9 = Cos β + λ S in β + λ = + λ β β λ ( β + λ ) S in β + λ =, β 6β + λ Te non-dimensional first order pressure rise is given by dp p = dx (45) dx Te expression for te velocity is given by u = u + N u (46) were u and u are given by te equations (3) and (4) Te expression for te temperature is obtained as θ = θ + N θ (47) were θ and θ are given by te equations (33) and (4) Te expression for te pressure rise is 447

10 p = p + N p (48) were p and p are given by te equations (36) and (45) RESULTS ND DISCUSSION Temperature is calculated from te equation (47) to study te effects of various parameters suc as permeability parameter, material parameter (Jeffrey parameter), Grasof number Gr, Reynolds number R and perturbation parameter N on it. Figure. is drawn to study te effect of Jeffrey parameter on te temperature wit fixed values of te remaining parameters. It is observed tat te temperature increases wit increasing λ. Te curve λ corresponds to Newtonian fluid. = Te effect of permeability parameter σ on te temperature is studied from figure. 3. It is observed tat te temperature decreases wit increasing. σ From figure.4 it is noticed tat te temperature increases wit increasing Grasof number Gr wit fixed a =.5, b =.5, σ =., x =, M = 5. It is observed from figure.5, tat te temperature decreases wit increasing values of Reynolds number R. Te effect of perturbation parameter N on te temperature is sown in figure.6. It is noticed tat te temperature increases wit increasing N. Figure.7 is plotted to study te effect of magnetic parameter M on te temperature. It is observed tat te temperature decreases wit increasing magnetic parameter M. Using equation (47) we ave calculated te variation of time averaged flux wit For different values of Jeffry parameter.8,.,.,.,. as sown in Figure.8. It is observed tat te pressure rise decreases wen increases. lso it is noticed tat for a given mean flow, increases wit increasing. Figure.9 sows te variation of pressure rise p wit time averaged flux for different values of N wit Ø =.8, σ =., =, Gr =., R =. and M=. It is observed tat te pressure rise p decreases wen increases. lso for a given, p increases wit increasing N. For a fixed p te mean flow increases wit increase in N. Te variation of pressure rise p wit time mean flow rate for different values of permeability parameter σ wit Ø =.8, N=., =, Gr =., R =. and M= and is sown in figure.. It is sown tat te pressure rise decreases wit te increase in te mean flow rate. lso for a fixed pressure rise p decreases wen σ increases. It is observed tat for a fixed pressure rise p, decreases wit te increase in σ. Te variation of pressure rise p wit time averaged volume flow rate for different values of Magnetic parameter wit Ø =.8, N=., =, Gr =., R =. and σ = and is sown in figure..it is observed tat for a given p, increases as te Magnetic Parameter M increases. lso it 448

11 is observed tat an increase in Magnetic Parameter M, increases te peristaltic pumping rate, pressure rise in pumping region. Figure. Temperature profiles for different values of Jeffrey parameter wit fixed.,.5,.,.,., 5,, Figure.3 Temperature profiles for different values of permeability parameter wit fixed...5,,.,.,., 5,, 449

12 Figure.4 Temperature profiles for different values of Grasoff number wit fixed.,.5,,.,.,., 5,, Figure.5 Temperature profiles for different values of Reynolds number wit fixed.,.5,,.,., 5,, 45

13 Figure.6 Temperature profiles for different values of Perturbation parameter wit fixed.,.5,,.,., 5,, Figure.7 Temperature profiles for different values of magnetic parameter wit fixed.,.5,,.,.,, 45

14 S. V. H. N. Krisna Kumari. P et al dv. ppl. Sci. Res.,, (6): Figure.8.Te variation of p wit Q for different values of λ wit φ =. 8, σ =., N =., Gr =., R =., M =. Figure.9. Te variation of p wit Q for different values of N N wit λ =, Gr =., R =., M =. cknowledgements One of te autors Dr. S. Sreeand, tanks DST,New Deli for providing financial support troug a major researc project. 45

15 REFERENCES [] Latam, T.W.,966, Fluid motion in a peristaltic pump. M.SC, Tesis. Cambridge, Mass: MIT-Press. [] Sapiro,.H., Jaffrin, M.Y and Weinberg, S.L., J. Fluid Mec,969,. 37, [3] Sud, V.K., Sekon, G.S. and Misra, R.K., Bull. Mat. Biol,977,. 39, no. 3, [4] Sapiro,.H.,,Proceedings of te Worksop in Ureteral Reflux in Cildren, 967,9-6. [5] Yin, F. and Fung, Y.C.,Trans. SME J. ppl. Mec.,969, 36, [6] Hayat,T.,Kan,M.,Siddiqui,.M.,and sgar.s.,communications in Nonlinear Science and Numerical Simulation, 7,,9-99. [7] Hayat, T and li, N.,Pysica : Statistical Mecanics and its pplications,6, 37,5-39. [8] Vajravelu,K.,Hemadri Reddy.R., Murugesan, M.,Int. Jr. of Fluid Mecanics Res.,9,36, Issue 3, [9] Kapur,J.N.,Matematical Models in Biology and Medicine,985,ffiliated East west press Pvt.Lts,New York. [] Radakrisnamacarya, G. and Srinivasulu,C., C.R. Mec.,7,335, [] Mekeimer, K.S. and bd Elmaboud, Y.,Pys. Lett.,8, 37, [] Hayat.T, Umar Quresi,M., Q.Hussain.Q.,pplied Matematical Modelling 9,33,pp [3] S.V.H.N.Krisna Kumari,P.,Ramana Murty,Cenna Krisna Reddy,M., Ravi Kumar Y.V.K.,dvances in pplied Sci.Res.,,(), [4] Y.V.K.Ravi Kumar,S.V.H.N.Krisna Kumari,P., M.V.Ramana Murty,S.Sreenad, dvances in pplied Sciences,,(3),