Hall Effects on Rayleigh-Stokes Problem for Heated Second Grade Fluid

Transcription

1 Proceedings of he Pakisan Academy of Sciences 49 (3): (1) Copyrigh Pakisan Academy of Sciences ISSN: Pakisan Academy of Sciences Original Aricle Hall Effecs on Rayleigh-Sokes Problem for Heaed Second Grade Fluid Masud Ahmad* and Muhammad Yaseen Deparmen of Mahemaics, Universiy of Sargodha, Sargodha, Pakisan Absrac: The aim of he presen paper is o discuss he influence of Hall curren on he flows of second grade fluid. Two illusraive examples have been considered (i) Sokes firs problem for heaed fla plae (ii) The Raleigh-Sokes problem for a heaed edge. Expressions for velociy and emperaure disribuions are obained. The resuls for hydrodynamic fluid can be obained as he limiing cases. PACS Code g Keywords: Heaed plae and edge, Hall effec, second grade fluid, Fourier Sine ransform 1. Inroducion Recenly, he Rayleigh Sokes problem for a fla plae and an edge has acquired a special saus. The soluion of Sokes firs problem for a Newonian fluid is obained employing similariy ransformaions in [1, ]. Bu for he same problem in second grade fluid such similariy ransformaions are no useful [3]. In general, he governing equaions of second grade fluid are one order higher han he Navier- Sokes equaion and o obain an analyic soluion is no so easy. Also for a unique soluion one needs an exra condiion. For he deail of his issue, I may refer he readers o he references [4-7]. In sudy [8] Bandelli e al. discussed he Sokes firs problem using Laplace ransformaion reamen. I is shown ha he resuling soluion does no saisfy he iniial condiion. Feecau and Zierep [9] removed his difficuly by using Fourier Sine ransform echnique. Chrisov and Chrisov [1] have given commens on [9] by showing ha soluion of [9] is incorrec and have given he correc soluion. Hea ransfer analysis on he unidirecional flows of a second grade fluid is examined by Bandelli [11]. In coninuaion Feecau and Feecau [1,13] analyzed he emperaure disribuion in second grade and Maxwell fluids for laminar flow on a heaed fla plae and in a heaed edge. The purpose of he presen invesigaion is o exend he analysis of reference [1] for Hall effecs. The corresponding resuls of Newonian fluid can be recovered by choosing α =. In absence of Hall effecs, he resuls can be obained by leing B =.. Basic Equaions For second grade fluid he Cauchy sress ensort is (1) where p is he scalar pressure, I is he ideniy ensor, µ is he coefficien of viscosiy, α i ( i = 1, ) are he maerial parameers of second grade fluid and Ai ( i = 1, ) are he firs wo Rivlin-Ericksen ensors defined hrough () (3) Received, November 11; Acceped, Sepember 1 *Corresponding auhor: Masud Ahmad, amasud11@yahoo.com

2 194 Masud Ahmad & Muhammad Yaseen in which is he ime. The issue regarding he signs of α 1 and α is conroversy. For deailed analysis relevan o his issue, one may refer he readers o he references [14, 15]. The equaions governing he MHD flow of heaed fluid are: Coninuiy equaion: Equaion of moion: Energy equaion: Maxwell equaions: Generalized Ohm s law: (4) (5) (6) (7) (8) In above equaions J is he curren densiy, B(=B b) is he oal magneic field, B is he applied magneic field, b is he induced magneic field, σ is he elecrical conduciviy of he fluid, E is he elecric field, µ m is he magneic permeabiliy, d is he maerial derivaive, e ( = cθ ) is he inernal d energy, ρ is he fluid densiy, c is he specific hea, θ is he emperaure, is he hea flux vecor, r is he radial heaing, we andτ e are he cycloron frequency and collision ime of elecron respecively. I is assumed ha E = and b =. Furher weτ e O(1) and wτ i i<< 1 (where w i andτ i are cycloron frequency and collision ime for ions respecively). Under he aforemenioned assumpions, Eq (5) becomes (9) where φ = weτe is he Hall parameer. 3. The firs problem of Sokes for a heaed fla plae wih Hall curren Le a second grade fluid, a res, fill he space above an infiniely exended plae in (y, z)-plane. When ime =, he plae sars suddenly o slide, in is own plane, wih velociy V. Le T() and f(x) denoe he emperaure of he plae for and he emperaure of he fluid a he momen =. The velociy and emperaure fields are θ = θ( x, ). (1) where is a uni vecor in he y-direcion. The coninuiy equaion (4) is idenically saisfied. Furhermore, Equaions ( 6) and (9) give vx (,) σb (1 iφ) vx (, ) ( υ α ) = x ρ(1 φ ) vx (,), x>, >, θ (,) x β g(,) x = x θ (,) x, x>, >, where α α ρ (11) (1) υ = µ is he kinemaic viscosiy, ρ = 1, β = k ρc υ vx (,) rx (,) g(,) x = ( ). c x ρc The relevan iniial and boundary condiions are vx (,) =, x>, v(, ) = V ( ), >, (13) θ( x,) = f( x), x> ; θ(, ) = T ( ),, (14) vx (,) vx (, ),, θ ( x, ), x θ (,) x as x. (15) x

3 Hall Effecs on Rayleigh-Sokes Problem 195 By Fourier Sine ransform, he soluion for v is υξ (1 φ ) { B (1 i ) (1 )} { σ B (1 iφ ) υξ (1 φ )} vx (,) = π σ φ υξ φ V ( ) V ()exp sin ξxdξ (1 φ )(1 αξ ) { (1 ) B (1 i )} { B i } { B (1 i ) (1 )} ξ υ φ ασ φ (1 αξ ) σ (1 φ) υξ (1 φ ) π σ φ υξ φ V ( τ )exp (1 φ )(1 ) αξ ( τ) dτ sin ξxdξ. (16) If he plae moves wih consan velociy V, hen and he above equaion simplifies o: (17) A res, he emperaure disribuion is he same in presence of Hall currens as for a second grade fluid and for a Newonian one. Furher if he radian heaing rx (,) is negligible quaniy he relaion () akes he same form as in [1], i.e., θ (,) x = T 1 Erf ( ) β ()1 τ Erf ( ) dτ β( τ) sin( xξ ) fs ( ξ)exp( βξ d ) ξ. π (1) From he above resuls we see ha, if, hen θ ( x, ) T( ). Moreover, if he iniial emperaure of he fluid is zero and he plae is kep o he consan emperaure T, Eq. (1) gives For B =, we ge he resuls of reference [1] as υξ sinξ vx (, ) = V1 exp dξ. π 1 αξ ξ (18) θ (,) x = T 1 Erf ( ), β and θ (,) x T as. () When α he above equaion yields x v(,) x = V 1 Erf, υ (19) where Erf (x) is he error funcion of Guass. Employing he same mehodology as for v we obain θ (,) x = T 1 Erf ( ) β ()1 τ Erf ( ) dτ β( τ) 4. The Rayleigh-Sokes problem for heaed edge wih Hall curren Consider a second grade fluid a res occupying he space of he firs dial of recangular edge (x, - < y <, z ). For =, he exended edge is impulsively brough o he consan speed V. The walls of he edge have emperaure T (). The velociy and emperaure fields are V = vxz (,,) ˆ j, (3) θ = θ( xz,, ). Equaions (6) and (9) here are of he following forms: T = lim T ( ) () ν α ( ) vxz (,,) x z σb (1 iφ) vxz (,,) = vxz (,,), ρ(1 φ ) x >, z >, >, (4)

4 196 Masud Ahmad & Muhammad Yaseen β ( ) θ(, xz,) x z gxz (,,) = θ (, xz,), x >, z >, >, where vxz (,,) ν x rxz (,,) g(,) x = ( ). c vxz (,,) ρc z (5) The corresponding iniial and boundary condiions are vxz (,,) =, x>, z>, sinξxsin ηzdξdη. (8) When plae has consan velociy V, and he above equaion reduces o 4 vxz (,,) = π µξ ( η )(1 φ ) ηξ σ B (1 iφ ) µ ( ξ η )(1 ϕ ) { } { σb (1 iφ ) µξ ( η )(1 ϕ )} ρ(1 φ) { 1 αξ ( η) } hen V ( ) Vexp sinξxsin ηzdξdη. (9) For B = above equaion reduces o he resul of [1]. v(, z, ) = vx (,, ) = V, >, (6) θ ( xz,,) = f( xz, ), x> ; z> ; θ(, z, ) = θ( x,, ) = T ( ),, (7) where he funcion f( xz, ) represens he emperaure disribuion of he fluid a he momen =. Moreover, vxz (,,), θ (, xz,) and heir parial derivaives wih respec o x and z have o end o zero as x z. Following he same mehod of soluion as in secion 3, we obain 4 υξ ( η )(1 φ ) vxz (,,) = π ηξ σ φ υ ξ η ϕ { B (1 i ) ( )(1 )} { σb (1 iφ) υξ ( η )(1 ϕ )} (1 φ) { 1 αξ ( η) } V ( ) V()exp 4 ( ξ η ) sinξxsinηzdξdη π ηξ 1 α ( ξ η ) { } ασ B (1 iφ ) µ (1 ϕ ) σb (1 iφ ) µξ ( η )(1 ϕ ) { σb (1 iφ) υξ ( η )(1 ϕ ) } ( )exp ( ) V τ τ dτ (1 ){ 1 ( )} φ αξ η The expression of emperaure is (3) x z θ ( x, z, ) = T 1 Erf ( ) Erf ( ) β β x z ( τ) 1 Erf ( ) Erf ( ) dτ β( τ) β( τ) exp { βξ ( η) } sin( xξ)sin( zη) π fs( ξη, ) gs( ξητ,, ) exp { βξ ( η ) τ} dτdξdη. (31) in which fs( ξη, ) and gs( ξη,, ) are he double Fourier sine ransforms of he funcions f ( x, z) and g( x, z, ) wih respec o he variables x and z. Whenα, relaions (3) and (31) reduce again o hose resuling from he Navier- Sokes fluids. Thus, we recover he universal profile of velociy []. x z v(, x z,) = V 1 Erf Erf, ν ν (3)

5 Hall Effecs on Rayleigh-Sokes Problem 197 in which only similariy variables x / v and z / v occur. For z, vxz (,,) goes o vx (,) given by (19). The expression for θ (, xz,) is x z θ ( x, z, ) = T 1 Erf ( ) Erf ( ) β β x z ( τ) 1 Erf ( ) Erf ( ) dτ, β( τ) β( τ) which for z goes o θ (, x z,) = T 1 Erf ( ) β ()1 τ Erf ( ) dτ. β( τ) (33) (34) If he edge is mainained o he consan emperaure T, Eq. (33) akes he form x z θ ( x, z, ) = T 1 Erf ( ) Erf ( ), β β (35) and θ (, x z,) T as. 5. Conclusions In his paper, he exac soluions for laminar flow of an elecrically conducing non-newonian fluid are obained. The velociy field and he emperaure disribuion in a second grade fluid on heaed fla plae and on a heaed edge in he presence of Hall curren are deermined. These soluions are obained using simple and double Fourier sine ransforms and presened as a sum of seady sae and ransien soluions. For large values of ime, when ransien disappear, hese soluions reduce o seady-sae soluions. Direc calculaions show ha θ (,) x and θ (, xz,) of Eqs. () and (31) as well vx (,) and vxz (,,) of Eqs. (16) and (3) saisfy he corresponding parial differenial equaions ogeher wih he iniial and boundary condiions. Puing B = in Eqs. (17) and (9), we obain he resuls of [1]. If we pu α in (17), (), (3) and (31), we find corresponding soluions for he Navier-Sokes fluid. In a fluid a res he emperaure disribuion is he same wheher i is second grade or no. If he radian heaing is negligible θ (,) x and θ (, xz,) become T ( ) as or T if he plae and he edge are mainained o he consan emperaure. The corresponding resuls in absence of Hall curren can be obained by choosing B =. 6. REFERENCES 1 Zierep, J. Similariy Laws and Modeling. Marcel Dekker, New York (1971). Zierep, J. Das Rayleigh-Sokes problem fur die Ecke. Aca Mechanica 34: (1979). 3 Tiepel, I. The impulsive moion of a fla plae in a viscoelasic fluid. Aca Mechanica 39: (1981). 4 Rajagopal, K.R. & A.S. Gupa. An exac soluion for he flow of a non-newonian fluid pas an infinie porous plae. Mechanica 19: (1984). 5 Rajagopal, K. R. On he creeping flow of he second-order fluid. Journal of Non-Newonian. Fluid Mechanics 15: (1984). 6 Rajagopal, K.R. & N.P. Kaloni. Some Remarks on Boundary Condiions for Flows of Fluids of he Differenial Type. Proceedings of Canadian Applied Mahemaics Sociey, Barnaby, Canada (1989). 7 Rajagopal, K.R. On boundary condiions for fluids of differenial ype, In: Navier-Sokes equaions and relaed non-linear problems, A. Sequiera (Ed.). Plenum Press, New York (1995). 8 Bandelli, R., K.R. Rajagopal, & G.P. Galdi. On some unseady moions of fluids of second grade. Archive Mechanics 47 (4): (1995). 9 Feecau, C., & J. Zierep. On a class of exac soluions of he equaions of moion of a second grade fluid. Aca Mechanica 15: (1). 1 Chrisov, I.C. & C.I. Chrisov. Commens on On a class of exac soluions of he equaions of moion of a second grade fluid. Aca Mechanica 15: 5-8 (1). 11 Bandelli, R. Unseady unidirecional flows of second grade fluids in domains wih heaed boundaries. Inernaional Journal of Non-Linear Mechanics 3: (1995). 1 Feecau, C., & C. Feecau. The Rayleigh-Sokes problem for heaed second grade fluids. Inernaional Journal of Non- Linear Mechanics 37: (). 13 Feecau, C., & C. Feecau. The Rayleigh-Sokes- Problem for a fluid of Maxwellian ype. Inernaional Journal of Non-Linear Mechanics 38: (3).

6 198 Masud Ahmad & Muhammad Yaseen 14 Dunn, J.E. Rajagopal, K.R. Fluids of differenial ype: criical review and hermodynamic analysis. Inernaional Journal of Engineering Sciences 33: (1995). 15 Fosdick, R.L. & K.R. Rajagopal. Anomalous feaures in he model of second order fluids. Archive for Raional Mechanics and Analysis 7: (198). 16 Sneddon, I.N. Funcional Analysis, In: Encyclopedia of Physics, Vol. II. Springer, Berlin, (1955).