Unsteady Magnetohydrodynamic Boundary Layer Flow near the Stagnation Point towards a Shrinking Surface

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1 Journal of Applid Mathmatics and Physics, 15, 3, Publishd Onlin July 15 in SciRs. Unstady Magntohydrodynamic Boundary Layr Flo nar th Stagnation Point toards a Shrinking Surfac Santosh Chaudhary, Pradp Kumar Dpartmnt of Mathmatics, Malaviya National Institut of Tchnology, Jaipur, India d11.santosh@yahoo.com, pradp17matrix@gmail.com Rcivd 3 Jun 15; accptd 7 July 15; publishd 3 July 15 Copyright 15 by authors and Scintific Rsarch Publishing Inc. This ork is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY). Abstract Th unstady to-dimnsional, laminar flo of a viscous, incomprssibl, lctrically conducting fluid toards a shrinking surfac in th prsnc of a uniform transvrs magntic fild is studid. Taking suitabl similarity variabls, th govrning boundary layr quations ar transformd to ordinary diffrntial quations and solvd numrically by a prturbation tchniqu for a small magntic paramtr. Th ffcts of various paramtrs such as unstadinss paramtr, vlocity paramtr, magntic paramtr, Prandtl numbr and Eckrt numbr for vlocity and tmpratur distributions along ith local Skin friction cofficint and local Nusslt numbr hav bn discussd in dtail through numrical and graphical rprsntations. Kyords Unstady Flo, Magntohydrodynamic, Boundary Layr, Stagnation Point, Shrinking Surfac 1. Introduction Stagnation flo of an incomprssibl viscous fluid ovr a shrinking sht has many important practical applications in nginring and industrial procsss, such as th xtrusion of a polymr in a mlt-spinning procss, continuous casting of mtals, th arodynamic xtrusion of plastic shts, th cooling of mtallic shts or lctronic chips and many othrs. In all ths cass, a study of fluid flo and hat transfr has important significanc bcaus th quality of th final product dpnds on th rat of cooling and th procss of strtching. In rcnt yars, th boundary layr flo du to a shrinking sht has attraction of many rsarchrs bcaus of its usful applications. A vry intrsting xampl in hich th shrinking sht situation occurs is of a rising shrinking balloon. Shrinking film is also a common application of shrinking sht problms in nginring and Ho to cit this papr: Chaudhary, S. and Kumar, P. (15) Unstady Magntohydrodynamic Boundary Layr Flo nar th Stagnation Point toards a Shrinking Surfac. Journal of Applid Mathmatics and Physics, 3,

2 industris. Shrinking film is vry usful in packaging of bulk products bcaus it can b unrappd asily ith adquat hat. From th strtching cas, th flo of shrinking sht is diffrnt and th fluid is attractd toards a slot. Physically, th gnratd vlocity at shrinking sht has an unstady flo du to th application of inadquat suction and is not confind ithin th boundary layr. In vi of all ths applications, Sakiadis [1] initiatd th study of boundary layr flo ovr a continuous solid surfac moving ith constant spd. Latr Cran [] considrd th problm of th flo ovr a linarly strtching sht in an ambint fluid and gav a similarity solution in closd analytical form for th stady to-dimnsional problm. Gupta and Gupta [3], and Vlggaar [4] hav invstigatd th solution of strtching flo problms at th constant surfac tmpratur hil Soundalgkar and Ramana [5] and Grubka and Bobba [6] hav analysd th solution of strtching flo problms ith a variabl surfac tmpratur. Many rsarchrs such as Magyari and Kllr [7], Elbashbshy and Bazid [8], Jat and Chaudhary [9]-[11], Bachok t al. [1] and Zhng t al. [13] hav analyzd th strtching sht problms ith diffrnt aspcts of fluid, such as th hat transfr, th prmability of th surfac and th unstadinss flo. Mahapatra and Nandy [14] [15] studid th unstady stagnation-point flo and hat transfr ovr an unstady shrinking sht. Rcntly Aly and Vajravlu [16] and Chaudhary and Kumar [17] discussd numrical solutions of boundary layr flo problms ovr diffrnt surfacs in a porous mdium. Mor rcntly Nandy t al. [18] and Rosca and Pop [19] invstigatd th unstady boundary layr flo ovr a prmabl strtching or shrinking surfac. Ralizing th incrasing tchnical applications of th magntohydrodynamic ffcts, th aim of th prsnt ork is concrnd ith a stady, to-dimnsional unstady stagnation flo of an lctrically conducting fluid ovr a shrinking surfac in th prsnc of a uniform transvrs magntic fild.. Formulation of th Problm Considr an unstady to-dimnsional stady flo (,,) uv of a viscous incomprssibl lctrically conducting fluid nar a stagnation point ovr a continuously shrinking surfac placd in th plan y = of a Cartsian coordinat systm in th prsnc of tim dpndnt fr stram. Th x-axis is takn along th shrinking surfac in th dirction of motion and y-axis is prpndicular to it. A uniform magntic fild of constant strngth, H, is assumd to b applid normal to th shrinking surfac (Figur 1). Th surfac is assumd to b ( ) Figur 1. Coordinat systm for th shrinking surfac. 9

3 highly lastic and is shrinking in th x-dirction ith a vlocity is u and surfac tmpratur T hil th vlocity of th flo, xtrnal to th boundary layr is u and tmpratur T. Thrfor, undr th usual boundary layr and Boussinsq approximations, th systms of boundary layr quations ar givn by u v + = (1) x y u u u u u u σ µ H u + u + v = + u + υ t x y t x y ρ () T T T υ T µ u σ µ H u + u + v = + + t x y Pr y ρcp y ρcp hr υ is th cofficint of kinmatic viscosity, σ th lctrical conductivity, µ th magntic prma- µ C p bility, ρ th dnsity, Pr th Prandtl numbr =, µ th cofficint of viscosity, C p th spcific hat κ at constant prssur and κ th thrmal conductivity. Th othr symbols hav thir usual manings. Th boundary conditions ar cx y = : u = u =, v = ; T = T 1+ γ t ax y = : u = u = ; T = T 1+ γ t hr c is a constant, γ is th shrinking rat and a is th strngth of th stagnation point flo. (3) (4) 3. Analysis Th continuity Equation (1) is idntically satisfid by stram function ψ ( xyt,, ) ψ ψ u =, v = y x, dfind as (5) For th solution of th momntum and th nrgy Equations () and (3), th folloing dimnsionlss variabls ar dfind: ψ aυ 1+ γ t ( x, y, t) = xf ( η) (6) η = υ a ( 1+ γt) y (7) T T θ( η) = T T Equations (5) to (8), transform Equations () and (3) into 1 f + f + ηβ f + ( β Rm f ) f + 1 β = θ + f + ηβ θ + Ec f + Ec f = Pr Pr Pr R m (8) (9) (1) hr a prim (') dnots diffrntiation ith rspct to η, γ β = is th unstadinss paramtr, a c = is a 93

4 σ x th vlocity paramtr, Rm = µ H is th Magntic paramtr and ρu numbr. Th corrsponding boundary conditions ar η= : f =, f = ; θ = 1 η= : f = 1; θ = Ec = u ( ) Cp T T is th Eckrt For numrical solution of th Equations (9) and (1), through a prturbation tchniqu, by assuming th folloing por sris in a small magntic paramtr R m as f i ( η) ( R ) f ( η) i= m i (11) = (1) j ( ) ( R ) ( ) θ η = θ η (13) j= Substituting Equations (1) and (13) and its drivativs in Equations (9) and (1) and thn quating th cofficints of lik pors of R m, gt th folloing st of quations ith th boundary conditions m ( ) 1 f + f + ηβ f + β f f = β θ + Pr f + ηβ θ = Pr Ec f ( ) 1 f 1+ f + ηβ f 1+ β f f 1+ f f1= f 1 1 θ 1+ Pr f + ηβ θ 1= Pr f1θ Pr Ec f f 1+ f j ( ) ( ) ( ) 1 f + f+ ηβ f + β f f + f f= ff 1 1+ f 1+ f θ + Pr f + ηβ θ = Pr f1θ 1+ fθ Pr Ec f f + f 1 + f f 1 ( ) ( ) η = : f =, f =, f = ; θ = 1, θ = i j j η = : f = 1, f = ; θ = i, j > j Th Equation (14) is obtaind by Mahapatra and Nandy [14] for th non-magntic cas and th rmaining quations ar ordinary linar diffrntial quations and hav bn solvd numrically by Rung-Kutta mthod of fourth ordr. Th vlocity and tmpratur distributions for various valus of paramtrs ar shon in Figur to Figur 6 rspctivly. 4. Skin Friction and Surfac Hat Transfr Th physical quantitis of intrst, th local skin friction cofficint surfac hat transfr ar givn by: ρu i (14) (15) (16) (17) (18) (19) () C f and th local Nusslt numbr Nu i.. u µ τ y y= C f = = (1) ρu 94

5 ƒ'( ) Figur. Vlocity distribution against η for various valus of β and R m ith = θ( ) Figur 3. Tmpratur distribution against η for various valus of R m and Pr ith β =.1, = 1. and Ec =.. 95

6 θ( ) Figur 4. Tmpratur distribution against η for various valus of β and R m ith = 1., Pr = 1. and Ec = θ( ) Figur 5. Tmpratur distribution against η for various valus of and R m ith β =.1, Pr =.5 and Ec =.. 96

7 θ( ) Figur 6. Tmpratur distribution against η for various valus of R m and Ec ith β =.1, = 1. and Pr = 1.. and T x y Nu = y= ( T T ) hich, in th prsnt cas can b xprssd in th folloing forms and Nu = Cf = f 3 R = 3 R i= R θ R = ( ) i ( Rm) f i( ) ( ) j= ( j Rm ) θ j( ) () (3) (4) ux hr R = is th local Rynolds numbr. υ Numrical valus of th functions f ( ) and θ ( ), hich ar proportional to local skin friction and local hat transfr rat at th surfac rspctivly for various valus of th paramtr ar prsntd in Tabl 1 and Tabl. 97

8 Tabl 1. Numrical valus of f ( ) for various valus of th paramtrs β, & R m. f ( ) β =. β =.1 β =. R m =. R m =.5 R m =.7 R m =. R m =.5 R m =.7 R m =. R m =.5 R m = Tabl. Numrical valus of θ ( ) for various valus of th paramtrs β,, R m, Pr & Ec. Pr θ ( ) β =. Ec =. Ec =. Ec =.5 R m =. R m =.5 R m =.7 R m =. R m =.5 R m =.7 R m =. R m =.5 R m = β =.1 Pr Ec =. Ec =. Ec =.5 R m =. R m =.5 R m =.7 R m =. R m =.5 R m =.7 R m =. R m =.5 R m = β =. Pr Ec =. Ec =. Ec =.5 R m =. R m =.5 R m =.7 R m =. R m =.5 R m =.7 R m =. R m =.5 R m =

9 5. Rsults and Discussion Figur shos th variation of vlocity distribution against η for various valus of th unstadinss paramtr β, th vlocity paramtr and th magntic paramtr R m. It may b obsrvd that, for th fixd valu of th vlocity paramtr vlocity distribution incrass ith th dcrasing valu of th unstadinss paramtr β, and opposit phnomnon occur for th magntic paramtr R m, for a fixd η. Figur 3 to Figur 6 sho th variation of tmpratur distribution against η for th various valus of th paramtrs such as th unstadinss paramtr β, th vlocity paramtr, th magntic paramtr R m, th Prandtl numbr Pr and th Eckrt numbr Ec. From ths figurs it may b obsrvd that th tmpratur distribution dcrass ith incrasing valus of th unstadinss paramtr β, th vlocity paramtr, th magntic paramtr R m, th Prandtl numbr Pr and th Eckrt numbr Ec. In Tabl 1, th numrical valus of th function f ( ) for various valus of th unstadinss paramtr β, th vlocity paramtr and th magntic paramtr R m ar givn. It may b obsrvd from th tabl that th boundary valus f ( ) for th non-magntic flo ar th sam as thos obtaind by Mahapatra and Nandy [14]. Th valu of th function f ( ) dcrass ith th incrasing valus of th unstadinss paramtr β and th magntic paramtr R m rspctivly taking othr paramtrs constant and rvrs phnomnon occurs for th vlocity paramtr. In Tabl, th numrical valus of th function θ ( ) for th diffrnt valus of th unstadinss paramtr β, th vlocity paramtr, th magntic paramtr R m, th Prandtl numbr Pr and th Eckrt numbr Ec ar givn. It may b obsrvd from th tabl that th boundary valus θ ( ) for th non-magntic flo ar sam as thos obtaind by Mahapatra and Nandy [14]. Th valu of th function θ ( ) incrass ith th incrasing valu of th unstadinss paramtr β, considring othr paramtrs constant and sam phnomnon occurs for th vlocity paramtr, th magntic paramtr R m, th Prandtl numbr Pr <.5 and th Eckrt numbr Ec. It is furthr obsrvd that th function θ ( ) dcrass ith an incrasing valu of th Prandtl numbr Pr >.5 for fixd othr paramtrs. 6. Conclusions Th prsnt ork xtnds th to-dimnsional unstady stagnation flo of an lctrically conducting fluid, ovr shrinking surfac in th prsnc of magntic fild. Undr som spcial conditions, th problm ill rduc th rsults obtaind by prvious rsarchrs. Th ffcts of diffrnt paramtrs such as th unstadinss paramtr, th vlocity paramtr, th magntic paramtr, th Prandtl numbr and th Eckrt numbr ar studid in dtail. Th vlocity as ll as thrmal boundary layr thicknss dcrass ith th incrasing valus of th unstadinss paramtr, th vlocity paramtr, th magntic paramtr, th Prandtl numbr and th Eckrt numbr hras in th vlocity rvrs phnomnon occurs for th magntic paramtr. From th rsults it can b concludd that skin friction and Nusslt numbr vary according to th vlocity and thrmal boundary layrs thicknss rspctivly ith diffrnt paramtrs. Rfrncs [1] Sakiadis, B.C. (1961) Boundary-Layr Bhaviour on Continuous Solid Surfacs: I. Boundary-Layr Equations for To Dimnsional and Axisymmtric Flo. Journal of Amrican Institut of Chmical Enginrs (AIChE), 7, [] Cran, L.J. (197) Flo past a Strtching Plat. Zitschrift für Angandt Mathmatik und Physik, 1, [3] Gupta, P.S. and Gupta, A.S. (1977) Hat and Mass Transfr on a Strtching Sht ith Suction or Bloing. Th Canadian Journal of Chmical Enginring, 55, [4] Vlggaar, J. (1977) Laminar Boundary-Layr Bhaviour on Continuous, Acclrating Surfacs. Chmical Enginring Scinc, 3, [5] Soundalgkar, V.M. and Ramana Murty, T.V. (198) Hat Transfr in Flo past a Continuous Moving Plat ith Variabl Tmpratur. Warm-und Stoffubrtragung, 14, [6] Grubka, L.J. and Bobba, K.M. (1985) Hat Transfr Charactristics of a Continuous, Strtching Surfac ith Variabl Tmpratur. Journal of Hat Transfr, 17, [7] Magyari, E. and Kllr, B. (1999) Hat and Mass Transfr in th Boundary Layrs on an Exponntially Strtching 99

10 Continuous Surfac. Journal of Physics D: Applid Physics, 3, [8] Elbashbshy, E.M.A. and Bazid, M.A.A. (3) Hat Transfr ovr an Unstady Strtching Surfac ith Intrnal Hat Gnration. Applid Mathmatics and Computation, 138, [9] Jat, R.N. and Chaudhary, S. (8) Magntohydrodynamic Boundary Layr Flo nar th Stagnation Point of a Strtching Sht. Il Nuovo Cimnto, 13B, [1] Jat, R.N. and Chaudhary, S. (9) Unstady Magntohydrodynamic Boundary Layr Flo ovr a Strtching Surfac ith Viscous Dissipation and Joul Hating. Il Nuovo Cimnto, 14B, [11] Jat, R.N. and Chaudhary, S. (1) Radiation Effcts on th MHD Flo nar th Stagnation Point of a Strtching Sht. Zitschrift für Angandt Mathmatik und Physik, 61, [1] Bachok, N., Ishak, A. and Nazar, R. (11) Flo and Hat Transfr ovr an Unstady Strtching Sht in a Micropolar Fluid. Mccanica, 46, [13] Zhng, L., Wang, L. and Zhang, X. (11) Analytic Solutions of Unstady Boundary Flo and Hat Transfr on a Prmabl Strtching Sht ith Non-Uniform Hat Sourc/Sink. Communications in Nonlinar Scinc and Numrical Simulation, 16, [14] Mahapatra, T.R. and Nandy, S.K. (11) Unstady Stagnation-Point Flo and Hat Transfr ovr an Unstady Shrinking Sht. Intrnational Journal of Applid Math and Mchanics, 7, [15] Mahapatra, T.R. and Nandy, S.K. (13) Slip Effcts on Unstady Stagnation-Point Flo and Hat Transfr ovr a Shrinking Sht. Mccanica, 48, [16] Aly, E.H. and Vajravlu, K. (14) Exact and Numrical Solutions of MHD Nano Boundary Layr Flos ovr Strtching Surfacs in a Porous Mdium. Applid Mathmatics and Computation, 3, [17] Chaudhary, S. and Kumar, P. (14) MHD Forcd Convction Boundary Layr Flo ith a Flat Plat and Porous Substrat. Mccanica, 49, [18] Nandy, S.K., Sidui, S. and Mahapatra, T.R. (14) Unstady MHD Boundary Layr Flo and Hat Transfr of Nanofluid ovr a Prmabl Shrinking Sht in th Prsnc of Thrmal Radiation. Alxandria Enginring Journal, 53, [19] Rosca, N.C. and Pop, I. (15) Unstady Boundary Layr Flo ovr a Prmabl Curvd Strtching/Shrinking Surfac. Europan Journal of Mchanics B/Fluids, 51,