INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE MHD COUETTE FLOW OF DUSTY FLUID WITH HEAT TRANSFER

Transcription

1 INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE MHD COUETTE FLOW OF DUSTY FLUID WITH HEAT TRANSFER HAZEMA.ATTIA Received December 5; Revised February 6; Acceped 9 May 6 This paper sudies he effec of variable viscosiy on he ransien Couee flow of dusy fluid wih hea ransfer beween parallel plaes. The fluid is aced upon by a consan pressure gradien and an exernal uniform magneic field is applied perpendicular o he plaes. The parallel plaes are assumed o be porous and subjeced o a uniform sucion from above and injecion from below. The upper plae is moving wih a uniform velociy while he lower is kep saionary. The governing nonlinear parial differenial equaions are solved numerically and some imporan effecs forhe variable viscosiy and he uniform magneic field on he ransien flow and hea ransfer of boh he fluid and dus paricles are indicaed. Copyrigh 6 Hazem A. Aia. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.. Inroducion The sudy of he flow of dusy fluids has imporan applicaions in he fields of fluidizaion, combusion, use of dus in gas cooling sysems, cenrifugal separaion of maer from fluid, peroleum indusry, purificaion of crude oil, elecrosaic precipiaion, polymer echnology, and fluid droples sprays. The hydrodynamic flow of dusy fluids was sudied by a number of auhors [6 8,, ]. Laer, he influence of he magneic field on he flow of elecrically conducing dusy fluids was sudied [, 5,,, 6]. Mos of hese sudies are based on consan physical properies.moreaccuraepredicionforheflowandhearansfercanbeachievedby aking ino accoun he variaion of hese properies, especially he variaion of he fluid viscosiy wih emperaure [9]. Klemp e al. [] havesudiedheeffec of emperauredependen viscosiy on he enrance flow in a channel in he hydrodynamic case. Aia and Kob [] sudied he seady MHD fully developed flow and hea ransfer beween wo parallel plaes wih emperaure-dependen viscosiy. Laer, Aia [] hasexended he problem o he ransien sae. Hindawi Publishing Corporaion Differenial Equaions and Nonlinear Mechanics Volume 6, Aricle ID 759, Pages DOI.55/DENM/6/759

2 MHD Couee flow of dusy fluid In he presen work, he effec of variable viscosiy on he unseady flow of an elecrically conducing, viscous, incompressible dusy fluid and hea ransfer beween parallel nonconducing porous plaes is sudied. The fluid is flowing beween wo elecrically insulaing infinie plaes mainained a wo consans bu differen emperaures. An exernal uniform magneic field is applied perpendicular o he plaes. The upper plae is moving wih a uniform velociy while he lower is kep saionary. The magneic Reynolds number is assumed small so ha he induced magneic field is negleced. The fluid is aced upon by a consan pressure gradien and is viscosiy is assumed o vary exponenially wih emperaure. The flow and emperaure disribuions of boh he fluid and dus paricles are governed by he coupled se of he momenum and energy equaions. The Joule and viscous dissipaion erms in he energy equaion are aken ino consideraion. The governing coupled nonlinear parial differenial equaions are solved numerically using he finie difference approximaions. The effecs of he exernal uniform magneic field and he emperaure-dependen viscosiy on he ime developmen of boh he velociy and emperaure disribuions are discussed.. Descripion of he problem The dusy fluid is assumed o be flowing beween wo infinie horizonal plaes locaed a he y =±h planes. The dusy paricles are assumed o be uniformly disribued hroughou he fluid. The wo plaes are assumed o be elecrically nonconducing and kep a wo consan emperaures, T forhelowerplaeandt for he upper plae wih T >T. The upper plae is moving wih a uniform velociy U o while he lower is kep saionary. A consan pressure gradien is applied in he x-direcion and he parallel plaes are assumed o be porous and subjeced o a uniform sucion from above and injecion from below. Thus he y-componen of he velociy is consan and denoed by v o.auniform magneic field B o is applied in he posiive y-direcion. By assuming a very small magneic Reynolds number he induced magneic field is negleced [7]. The fluid moion sars from res a =, and he no-slip condiion a he plaes implies ha he fluid and dus paricles velociies have neiher a z-nor an x-componen a y =±h. The iniial emperaures of he fluid and dus paricles are assumed o be equal o T and he fluid viscosiy is assumed o vary exponenially wih emperaure. Since he plaes are infinie in he x- and z-direcions, he physical variables are invarian in hese direcions. The flow of he fluid is governed by he Navier-Sokes equaion [7] ρ u + ρv u o y = dp dx + μ u ) σb y y ou KN ) u u p,.) where ρ is he densiy of clean fluid, μ is he viscosiy of clean fluid, u is he velociy of fluid, u p is he velociy of dus paricles, σ is he elecric conduciviy, p is he pressure acing on he fluid, N is he number of dus paricles per uni volume, and K is a consan. The firs hree erms in he righ-hand side are, respecively, he pressure gradien, viscosiy, and Lorenz force erms. The las erm represens he force erm due o he relaive moion beween fluid and dus paricles. I is assumed ha he Reynolds number of he relaive velociy is small. In such a case he force beween dus and fluid is proporional

3 Hazem A. Aia o he relaive velociy []. The moion of he dus paricles is governed by Newon s second law [] u p m p = KN ) u u p,.) where m p is he average mass of dus paricles. The iniial and boundary condiions on he velociy fields are, respecively, given by For >, he no-slip condiion a he plaes implies ha = :u = u p =..) y = h : u =, y = h : u = U o..) Hea ransfer akes place from he upper ho plae owards he lower cold plae by conducion hrough he fluid. Also, here is a hea generaion due o boh he Joule and viscous dissipaions. The dus paricles gain hea energy from he fluid by conducion hrough heir spherical surface. Two energy equaions are required which describe he emperaure disribuions for boh he fluid and dus paricles and are, respecively, given by [5] ρc T ) + ρcv T o y = k T u y + μ + σb y ou + ρ pc s Tp T ),.5) γ T T p = γ T Tp T ),.6) where T is he emperaure of he fluid, T p is he emperaure of he paricles, c is he specific hea capaciy of he fluid a consan pressure, C s is he specific hea capaciy of he paricles, k is he hermal conduciviy of he fluid, γ T is he emperaure relaxaion ime = Prγ p C s /c), γ p is he velociy relaxaion ime = ρ s D /9μ), ρ s is he maerial densiy of dus paricles = ρ p /πd N), and D is he average radius of dus paricles. The las hree erms in he righ-hand side of.5) represen he viscous dissipaion, he Joule dissipaion, and he hea conducion beween he fluid and dus paricles. The iniial and boundary condiions on he emperaure fields are given as :T = T p =, >, y = h : T = T,.7) >, y = h : T = T. The viscosiy of he fluid is assumed o depend on emperaure and is defined as μ = μ o f T). For pracical reasons which are shown o be suiable for mos kinds of fluids [, ], he viscosiy is assumed o vary exponenially wih emperaure. The funcion

4 MHD Couee flow of dusy fluid f T) akes he form [, ], f T) = e bt T), where he parameer b has he dimension of [T] and such ha a T = T, f T ) = and hen μ = μ o. This means ha μ o is he viscosiy coefficien a T = T.Theparameera may ake posiive values for liquids such as waer, benzene, or crude oil. In some gases like air, helium, or mehane a i may be negaive, ha is, he coefficien viscosiy increases wih emperaure [, ]. The emperaure variaions wihin a convecive flow give rise o variaions in he properies of he fluid, in he densiy and viscosiy, for example. An analysis including he full effecs of hese is so complicaed ha some approximaions become essenial. The equaions are commonly used in a form known as he Boussinesq approximaion. In he Boussinesq approximaion, variaions of all fluid properies oher han he densiy are ignored compleely. Variaions of he densiy are ignored excep insofar as hey give rise o graviaional force [8]. Therefore, a buoyancy force erm may be included in he Navier- Sokes equaion which equals αρδt,where α is he coefficien of expansion of he fluid. Such a buoyancy erm may be negleced on he basis of eiher ΔT small, ha is, T T is small, or small α which is a reasonable approximaion for liquids and perfec gases [8]. The problem is simplified by wriing he equaions in he nondimensional form. The characerisic lengh is aken o be h and he characerisic velociy is U o. We define he following nondimensional quaniies: x, ŷ) = x, y) h, = U o h, P = P ρuo, λ = d p d x, ûp, v p ) = up,v p ) U o, T = T T T T, T p = T p T T T, u,v) û, v) =, U o.8) f T) = e bt T) T = e a T, a is he viscosiy variaion parameer, Ha = σb oh /μ o, Ha is he Harmann number, R = KNh /μ o is he paricle concenraion parameer, G = m p U o /hk) is he paricle mass parameer, S = v o /U o is he sucion parameer, Pr = μ o c/k is he Prandl number, Ec = U o /ct T )) is he Ecker number, L o = ρh /μ o γ T is he emperaure relaxaion ime parameer. In erms of he above nondimensional variables and parameers.) o.7) akehe form he has are dropped for convenience) u + S u y = λ + f T) u y + ft) y u y Ha u R ) u u p,.9) G u p = ) u u p,.) :u = u p =, >, y = :u =, >, y = :u =,.)

5 T + S T y = T +Ecf T) Pr y u y Hazem A. Aia 5 ) +EcHa u + R Tp T ),.) Pr T p = L o Tp T ),.) :T = T p =, >, y = :T =,.) >, y = :T =. Equaions.9),.),.), and.) represen a sysem of coupled and nonlinear parial differenial equaions which are solved numerically under he iniial and boundary condiions.) and.) using he finie difference approximaions. A linearizaion echnique is firs applied o replace he nonlinear erms a a linear sage, wih he correcions incorporaed in subsequen ieraive seps unil convergence is reached. Then he Crank-Nicolson implici mehod is used a wo successive ime levels []. An ieraive scheme is used o solve he linearized sysem of difference equaions. The soluion a a cerain ime sep is chosen as an iniial guess for nex ime sep and he ieraions are coninued ill convergence, wihin a prescribed accuracy. Finally, he resuling block ridiagonal sysem is solved using he generalized Thomas algorihm []. Finie difference equaions relaing he variables are obained by wriing he equaions a he midpoin of he compuaional cell and hen replacing he differen erms by heir second-order cenral difference approximaions in he y-direcion. The diffusion erms are replaced by he average of he cenral differences a wo successive ime levels. The compuaional domain is divided ino meshes each of dimension Δ and Δy in ime and space, respecively. We define he variables v = u/ y and H = θ/ y o reduce he second-order differenial equaions.9)and.)o firs-order differenialequaions which are ) ) ui+,j+ u i,j+ + u i+,j u i,j vi+,j+ + v i,j+ + v i+,j + v i,j + S Δ ) ) ) ) f = α + T) i,j+ + f T) i,j vi+,j+ + v i,j+ vi+,j + v i,j Δy ) ) f + T) i,j+ f T) i,j vi+,j+ + v i,j+ + v i+,j + v i,j Δy ) ) Ha ui+,j+ + u i,j+ + u i+,j + u i,j ui+,j+ + u i,j+ + u i+,j + u i,j R + R upi+,j+ + u pi,j+ + u pi+,j + u pi,j ),

6 6 MHD Couee flow of dusy fluid ) upi+,j+ u pi,j+ + u pi+,j u pi,j G Δ = ui+,j+ + u i,j+ + u i+,j + u i,j u p i+,j+ + u pi,j+ + u pi+,j + u pi,j ) ) Ti+,j+ T i,j+ + T i+,j T i,j Hi+,j+ + H i,j+ + H i+,j + H i,j + S Δ Pr ) ) ) ) f = T) i,j+ + f T) i,j Hi+,j+ + H i,j+ Hi+,j + H i,j Pr Δy +Ec ) ) f T) i,j+ + f T) i,j vi+,j+ + v i,j+ + v i+,j + v i,j ) ) vi+,j+ + v i,j+ + v i+,j + v i,j +EcHa ui+,j+ + u i,j+ + u i+,j + u i,j ) ui+,j+ + u i,j+ + u i+,j + u i,j + R Tpi+,j+ + T pi,j+ + T pi+,j + T pi,j T ) i+,j+ + T i,j+ + T i+,j + T i,j, Pr Tpi+,j+ T pi,j+ + T pi+,j T pi,j ) ), Δ = L o Tpi+,j+ + T pi,j+ + T pi+,j + T pi,j T ) i+,j+ + T i,j+ + T i+,j + T i,j..5) The variables wih bars are given iniial guesses from he previous ime seps and an ieraive scheme is used a every ime o solve he linearized sysem of difference equaions. Compuaions have been made for R =.5, G =.8, λ = 5, Pr =, Ec =., and L o =.7. Grid-independence sudies show ha he compuaional domain << and <y< can be divided ino inervals wih sep sizes Δ =. and Δy =.5 for ime and space, respecively. Smaller sep sizes do no show any significan change in he resuls. Convergence of he scheme is assumed when all of he unknowns u, v, u p, T, H, and T p for he las wo approximaions differ from uniy by less han 6 for all values of y in <y< a every ime sep. Less han 7 approximaions are required o saisfy hese convergence crieria for all ranges of he parameers sudied here.. Resuls and discussions The exponenial dependence of he viscosiy on emperaure resuls in decomposing he viscous force erm in he momenum equaion ino wo erms. The variaions of hese

7 Hazem A. Aia 7 u 5.5 Figure.. The evoluion of u for differen values of a H, S = ). up 5.5 Figure.. The evoluion of u p for differen values of a H, S = ). resuling erms wih he viscosiy variaion parameer a and heir relaive magniudes have an imporan effec on he flow and emperaure fields in he absence or presence of he applied uniform magneic field. Figures. and. indicae he variaions of he velociies u and u p a he cener of he channel y = ) wih ime for differen values of he viscosiy variaion parameer a and for HandS =. The figures show ha increasing a increases he velociy and he ime required o approach he seady-sae. The effec of he parameer a on he seadysae ime is more pronounced for posiive values of a han for negaive values. Noice ha u reaches he seady sae faser han u p. This is because he fluid velociy is he source for he dus paricles velociy. Figure. shows also ha he influence of a on u p is negligible for some ime and hen increases as he ime develops. Figures. and. presen he variaions of he emperaures T and T p a he cener of he channel y = ) wih ime for differen values of he viscosiy variaion parameer a for HandS =. The figures show ha increasing a increases he emperaures and he seady-sae imes. Increasing he posiive values of a decreases he emperaure for some ime and hen he emperaure increases wih he incremen in a as he ime develops. Thus, increasing a increases he seady-sae value of he emperaure wih he appearance of he cross-over of he emperaure curves corresponding o differen values of a. The ime a which he curves inersec increases wih he incremen in a and is longer

8 8 MHD Couee flow of dusy fluid.5 T.5.5 Figure.. The evoluion of T for differen values of a H, S = ). Tp Figure.. The evoluion of T p for differen values of a H, S = ). for T han for T p,ast p always follows T. I is noiced ha he seady-sae values of T p coincide wih he corresponding seady-sae values of T, and he ime required for T p o reach he seady sae, which depends on a, is longer han ha for T. The applicaion of he uniform magneic field adds one resisive erm o he momenum equaion and he Joule dissipaion erm o he energy equaion. Figures.5 and.6 presen he influence of he viscosiy variaion parameer a on he evoluion of boh he velociies u and u p a he cener of he channel, respecively, for Ha = ands =. The magneic field resuls in a reducion in he velociies and he seady-sae ime for all values of a due o is damping effec. Figures.7 and.8 presen he influence of he viscosiy variaion parameer a on he evoluion of he emperaures T and T p a he cener of he channel, respecively, for Ha = ands =. Increasing he magneic field increases he emperaures for all posiive values of a excep for very small ime. This is because he magneic field has a resisive effec which becomes more pronounced as ime develops especially wih he case of negaive a which has he same resisive effec. Figures.9 and. indicae he variaions of he velociies u and u p a he cener of he channel y = ) wih ime for differen values of he viscosiy variaion parameer a and for HandS =. I is clear ha he sucion velociy decreases boh u and u p and

9 Hazem A. Aia 9 u.5 Figure.5. The evoluion of u for differen values of a Ha =, S = ). up.5 Figure.6. The evoluion of u p for differen values of a Ha =, S = )..5 T.5.5 Figure.7. The evoluion of T for differen values of a Ha =, S = ). heir seady-sae imes as a resul of pumping he fluid from he slower lower half region o he cener of he channel. The influence of sucion on u and u p is more pronounced for higher values of he parameer a. Figures. and. presen he influence of he viscosiy variaion parameer a on he evoluion of he emperaures T and T p a he cener of he channel, respecively, for

10 MHD Couee flow of dusy fluid.5 Tp.5.5 Figure.8. The evoluion of T p for differen values of a Ha =, S = ). u.5 Figure.9. The evoluion of u for differen values of a H, S = ). up.5 Figure.. The evoluion of u p for differen values of a H, S = ). HandS =. I is shown ha increasing sucion velociy decreases boh T and T p and heir seady-sae imes. This resuls from pumping he fluid from colder lower half region o he cener of he channel. The effec of sucion on T and T p is more apparen for higher values of a.

11 Hazem A. Aia T Figure.. The evoluion of T for differen values of a H, S = ). Tp Figure.. The evoluion of T p for differen values of a H, S = ). u.5.5 y.5 Figure.. Seady-sae profile of u for various values of a H.5, S =.5). Figures. and. presen he influence of he viscosiy variaion parameer a on he seady-sae profile of he velociies u and u p, respecively, for H.5 ands =.5. I is clear ha increasing a increases u and u p for all values of y due o he increase in viscosiy. I is clear also ha he seady-sae velociy aains more han hree imes he wall velociy due o he effec of he applied pressure gradien. Figures.5 and.6

12 MHD Couee flow of dusy fluid up.5.5 y.5 Figure.. Seady-sae profile of u p for various values of a H.5, S =.5)..5 T y.5 Figure.5. Seady-sae profile of T for various values of a H.5, S =.5). Tp y.5 Figure.6. Seady-sae profile of T p for various values of a H.5, S =.5). presen he influence of he viscosiy variaion parameer a on he seady-sae profile of he emperaures T and T p, respecively, for H.5andS =.5. Increasing a increases boh T and T p as a resul of increasing he velociies and heir gradiens which increase he viscous and Joule dissipaions. Also, i is shown ha he emperaures exceed uniy

13 Hazem A. Aia for some locaions i.e., he emperaure of he upper plae) due o he heaing effec of he dissipaions.. Conclusions In his paper he effec of a emperaure-dependen viscosiy, sucion and injecion velociy, and an exernal uniform magneic field on he unseady flow and emperaure disribuions of an elecrically conducing viscous incompressible dusy fluid beween wo parallel porous plaes has been sudied. The viscosiy was assumed o vary exponenially wih emperaure and he Joule and viscous dissipaions were aken ino consideraion. The mos ineresing resul was he cross-over of he emperaure curves due o he variaion of he parameer a and he influence of he magneic field in he suppression of such cross-over. On he oher hand, changing he magneic field resuls in he appearance of cross-over in he emperaure curves for a given negaive value of a. Also, changing he viscosiy variaion parameer a leads o asymmeric velociy profiles abou he cenral plane of he channel y = ) which is similaro he effec of variable percolaion perpendicular o he plaes. References [] A. L. Aboul-Hassan, H. Sharaf El-Din, and A. A. Megahed, Temperaure disribuion in a dusy conducing fluid flowing hrough wo parallel infinie plaes due o he moion of one of hem,proceedings of s Inernaional Conference of Engineering Mahemaics and Physics, Cairo, 99, pp [] W. F. Ames, Numerical Soluions of Parial Differenial Equaions, nd ed., Academic Press, New York, 977. [] H. A. Aia, Transien MHD flow and hea ransfer beween wo parallel plaes wih emperaure dependen viscosiy, Mechanics Research Communicaions 6 999), no., 5. [] H.A.AiaandN.A.Kob,MHD flow beween wo parallel plaes wih hea ransfer, AcaMechanica 7 996), no., 5. [5] A.K.BorkakoiandA.Bharali,Hydromagneic flow and hea ransfer beween wo horizonal plaes, he lower plae being a sreching shee, Quarerly of Applied Mahemaics 98), no., [6] L. A. Dixi, Unseady flow of a dusy viscous fluid hrough recangular ducs, Indian Journal of Theoreical Physics 8 98), no., 9. [7] A.K.Ghosh and D.K.Mira,Flow of a dusy fluid hrough horizonal pipes, Revue Roumaine de Physique 9 98), [8] R.K.GupaandS.C.Gupa,Flow of a dusy gas hrough a channel wih arbirary ime varying pressure gradien, Journal of Applied Mahemaics and Physics 7 976), 9. [9] H. Herwig and G. Wicken, The effec of variable properies on laminar boundary layer flow, Warme-und Soffuberragung 986), [] K. Klemp, H. Herwig, and M. Selmann, Enrance flow in channel wih emperaure dependen viscosiy including viscous dissipaion effecs, Proceedings of he rd Inernaional Congress of Fluid Mechanics, vol., Cairo, 99, pp [] A. A. Megahed, A. L. Aboul-Hassan, and H. Sharaf El-Din, Effec of Joule and viscous dissipaion on emperaure disribuions hrough elecrically conducing dusy fluid, Proceedings of 5h Miami Inernaional Symposium on Muli-Phase Transpor and Pariculae Phenomena, vol., Florida, 988, p..

14 MHD Couee flow of dusy fluid [] P. Mira and P. Bhaacharyya, Unseady hydromagneic laminar flow of a conducing dusy fluid beween wo parallel plaes sared impulsively from res, Aca Mechanica 9 98), no. -, 7 8. [] V.R.PrasadandN.C.P.Ramacharyulu,Unseady flow of a dusy incompressible fluid beween wo parallel plaes under an impulsive pressure gradien, Defense Science Journal 979), 5. [] P. G. Saffman, On he sabiliy of laminar flow of a dusy gas, Journal of Fluid Mechanics 96), 8. [5] H. Schliching, Boundary Layer Theory, McGraw-Hill, New York, 968. [6] K. K. Singh, Unseady flow of a conducing dusy fluid hrough a recangular channel wih ime dependen pressure gradien, Indian Journal of Pure and Applied Mahemaics 8 976), no. 9,. [7] G. W. Suon and A. Sherman, Engineering Magneohydrodynamics, McGraw-Hill, New York, 965. [8] D. J. Trion, Physical Fluid Dynamics, ELBS & Van Nosrand Reinhold, London, 979. Hazem A. Aia: Deparmen of Mahemaics, College of Science, Al-Qasseem Universiy, P.O. Box 7, Buraidah 8999, Saudi Arabia address: ah@yahoo.com