Unsteady MHD Free Convective Flow Through Porous Media Past on Moving Vertical Plate with Variable Temperature and Viscous Dissipation

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1 ISS Avalable onlne at Internatonal ejornals Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp Unstead MHD Free Convectve Flow Throgh Poros Meda Past on Movng Vertcal Plate wth Varable Temperatre and Vscos Dsspaton E Rant Kmar, J Anand Rao, M Raa Shear 3 Department of Mathematcs, Raalaseema Unverst, Krnool, Andhra Pradesh, Inda Department of Mathematcs, Unverst College of Scence, Osmana Unverst, Hderabad, Telangana State, Inda 3 Department of Mathematcs, JTUH College of Engneerng Jagtal, Telangana State, Inda ABSTRACT The problem of an nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre n the presence of nclned magnetc feld and vscos dsspaton s stded. The governng eqatons are dscrtzed b Galern fnte-element method and are beng solved b Cran-colson method sng C-Program. mercal reslts for the veloct and temperatre are presented graphcall for varos vales of the parameter. ITRODUCTIO The std of heat and mass transfer problems are mportant n man reactve chemcals and chemcal formlatons. The analss of free convecton flows s great concern de to ts sefl applcatons n man branches of engneerng and scences eg. Geophscs, agrcltre and thermal nslaton. Elbashbesh [] nvestgated heat and mass transfer along a vertcal plate wth varable srface temperatre and concentraton n the presence of magnetc feld. Ja sngh [] stded vscos dsspaton and chemcal reacton effects on flow past a stretchng poros srface n a poros medm. Helm KA [3] stded the MHD nstead free convecton flow past a vertcal poros plate. Chamha AJ et.al [4] presented nstead MHD free convecton flow past an exponentall accelerated vertcal plate wth mass transfer, chemcal reacton and thermal radaton. Alao FI et.al [5] analzed the effects of thermal radaton, Soret and Dfor on an nstead heat and mass transfer flow of a chemcall

2 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear reactng fld past a sem nfnte vertcal plate wth vscos dsspaton. Free convecton heat and mass transfer MHD flow n a vertcal channel n the presence of chemcal reacton was presented b Bar R [6]. Tanvr Ahmed and Md. Mahmad [7] nvestgated a fnte dfference solton of MHD mxed convecton flow wth heat generaton and chemcal reacton. Mazmdar MK and Dea RK [8] have analzed MHD flow past an mplsve started nfnte vertcal plate n the presence of thermal radaton. Mohmed Abd E-Azz [9] presented nstead mxed convecton heat transfer along a vertcal stretchng srface wth varable vscost and vscos dsspaton. Rapts A et.al [] examned the effect of thermal radaton on MHD flow. The obectve of ths paper s to std an nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre n the presence of nclned magnetc feld and vscos dsspaton. FORMULATIO OF THE PROBLEM An nstead MHD flow of a vscos ncompressble fld b a vertcal plate and the plate mplsvel started movng wth veloct tang nto accont n the presence of vscos dsspaton. The x axs s taen along the vertcal plate n the pward drecton and the flow s assmed to be n ths drecton and - axs normal to the plate. A nform magnetc feld B s assmed to be appled on the plate wth angle. Intall the plate and the fld are at same temperatre T. At tme T w. The governng eqatons of flow feld are as nder follows: Momentm eqaton: sb sn a v = n + gb ( T - T ) + - t r K Energ eqaton: æ ö T T m = t rcp rc p ç è ø where, v are the veloct components n t, temperatre of the plate s ncreased to x, drectons, g s the gravtatonal acceleraton, b s the thermal expanson coeffcent respectvel, r s the fld denst,

3 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear s the coeffcent of vscost, T s the thermal temperatre nsde the thermal bondar laer s the thermal condctvt, K s the permeablt of the medm C p s the specfc heat at constant pressre, s s the electrc condctvt, T s the temperatre of the fld far awa from the plate, t - s tme. The bondar condtons are: = = " üï ï ïïï t :, T T t ( w ) t > : =, T = T + T - T at = ý v ïïïï T T as ï ïþ 3 where p s the veloct of the fld, tme. Introdcng the followng non- dmensonal qanttes, Tw s the temperatre of the wall respectvel, n T - T vr Cp K üï =, =, n=, q=,pr =, K =, ï n n Tw - T v ï g v ýï b ( Tw - T ) t s B l U ï Gr =, t =, Ha =, Ec = 3 v m cp( Tw - T ) ï ïþ 4 t s the where Gr, Pr, Ha, Ec, and Pr are the thermal Grashof nmber, Prandtl nmber, Hartmann nmber, Ecert nmber, respectvel. Wth the help of non- dmensonal qanttes, eqatons () and () becomes t K Gr M... 5 Pr Pr Ec t 6 The correspondng dmensons less bondar condton are 3

4 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear t : =, q = " üï t > : =, q = t at = ï ý ïïï q as ïþ 7 METHOD OF SOLUTIO B applng Galern fnte element method for eqaton (5) over the element (e), s: ( et ) e P d t... 8 Where P ( ) Gr, M K Integratng the frst term n eqaton (8) b parts we obtan T T T T Pd t 9 eglectng the frst term n eqaton (9), we get: e Pd t T ( et ) Let = be the lnear pecewse approxmaton solton over the element (e) ( ) where, T and, are the bass fnctons. We obtan d d B d P d ' ' ' ' ' ' ' ' ' ' ' ' On smplfng we get 4

5 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear ( ) ( ) ( ) e e e l l Pl 6 6 l where prme and dot denote dfferentaton wth respect to and tme t respectvel. Assemblng the element eqatons for two consectve elements followng s obtaned: P 4 4 l 6 6 ow pt row correspondng to the node to zero, from eqaton () the dfference schemes wth ( l e ) h s: 4 4 P 6 h 6 Applng Cran cholson method to the above eqaton, we get A + A + A = A + A + A + P and where A 6r A 6r 4 A 8 r 4 A 8 r 4 5 A 6r A 6r 3 6 P P Gr ow from eqaton (6) followng eqaton s obtaned: B B B B B B P Where 5

6 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear B Pr 3r B Pr 3r 4 B 4Pr 6r B 4Pr 6r 5 B Pr 3r B Pr 3r Pr P P Ec Here, r and h, are mesh sze along the drecton and the tme drecton respectvel. h Index refers to the space, and refers to the tme. In the eqatons () and (), tang =,..., n and sng bondar condtons (7), the followng sstem of eqatons are obtaned: A X = B, =,..., n 3 where A s are matrx of order n and X, B s colmn matrces havng n components. The soltons of above sstems of eqatons are obtaned b sng the Thomas algorthm for veloct, temperatre and concentraton. Also the nmercal soltons are obtaned b exectng the C-program wth the smaller vales of h and. o sgnfcant change was observed n, and θ then the Galern fnte element method s stable and convergent. RESULTS AD DISCUSSIOS The nvestgaton of an nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre n the presence of nclned magnetc feld and vscos dsspaton has been carred ot n the prevos secton. The nmercal vales were compted wth respect to the phscal parameters, le angle of nclnaton of magnetc feld α, Hartmann nmber Ha, thermal Grashof nmber Gr, permeablt K, Prandtl nmber Pr, Ecert nmber Ec and tme t s shown n fgres. Fgre represents the veloct profles for dfferent vales of angle of nclnaton of magnetc feld α. It s observed that an ncrease n α leads to a decrease n the vale of veloct. The veloct profle decreases wth an ncrease n Hartmann nmber as shown n fgre. Fgre 3 and 4 llstrate the behavor of veloct profle for dfferent vales of Thermal Grashof nmber Gr and permeablt parameter K. The nmercal reslt shows that ncreasng n the vales of Gr and K reslts veloct ncreases. 6

7 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear Fgre 5 represent the veloct profle for dfferent vales of Ecert nmbers. It observed that veloct ncreases wth an ncrease n Ec vales. The effect of the Prandtl nmber (Pr) on the veloct and temperatre are shown n fgres 6 and 7, the veloct and temperatre decreases as there s an ncrease n the Pr vales. Fgres 8 and 9 llstrate the behavor of the veloct and temperatre profles for dfferent vales of t. As the tme ncreases, the veloct and temperatre ncreases. COCLUSIOS In ths paper, an nstead MHD free convecton flow of a vscos ncompressble fld b a movng vertcal plate wth varable temperatre n the presence of nclned magnetc feld and vscos dsspaton has been stded nmercall. From the present std we see that the momentm bondar laer decreases wth an ncrease n angle of nclnaton of magnetc feld, Hartmann nmber and Prandtl nmber, whle there s an ncreases wth an ncrease n thermal Grashof nmber, permeablt, Ecert nmber and tme. The thermal bondar laer thcness decreases wth an ncrease n Prandtl nmber whereas t ncreases wth an ncrease n tme.. Pr =.7, Gr =, K =., Ha = 4, α = 5, 3, Fgre : Effect of Ha on veloct profle Pr =.7, Gr =, K =., t =., α = 3, Ec=. Ha =, 4, Fgre.: Effect of on veloct profle 7

8 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear Pr =.7, Ha = 4, K =., t =., α = 3, Ec=. Gr =, Fgre 3: Effect of Gr on Veloct profle Pr =.7, Gr =, Ha =, t =., α = 3, Ec =. K =.,.4, Fgre 4: Effect of K on veloct profle Pr =.7, Gr =, Ha = 4, K =., α = 3, t =. Ec =.,.4, Fgre 5: Effect of Ec on veloct profle 8

9 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear Gr =, Ha = 4, K =., t =., Pr =.7, Fgre 6: Effect of Pr on veloct profle Gr =, Ha = 4, K =., t =., α = 3, Ec =. Pr =.7, Fgre 7: Effect of Pr on temperatre profle. Pr =.7, Gr =, Ha = 4, K =., t =.,.5, Fgre 8: Effect of t on veloct profle 9

10 Internatonal ejornal of Mathematcs and Engneerng (7) Vol. 8, Isse, pp E.Rant Kmar, J.Anand Rao and M..Raa Shear Pr =.7, Gr =, Ha = 4, K =., α = 3, Ec =. t =.,.5, Fgre 9: Effect of t on temperatre profle REFERECES. Elbashbesh EMA (997): Heat and mass transfer along a vertcal plate wth varable srface temperatre and concentraton n the presence of magnetc feld, Int. J. Eng. Sc., 34, pp Ja sngh () : Vscos dsspaton and chemcal reacton effects on flow past a stretchng poros srface n a poros medm, Adv. Theor. Appl. Mech., 5(8), pp Helm.K.A (998) : MHD nstead free convecton flow past a vertcal poros plate, ZAMM Z. Angew. Math. Mech., 78(4), pp Chamha AJ, Ra MC, Sdhaar Redd T and Varma SVK () : Unstead MHD free convecton flow past an exponentall accelerated vertcal plate wth mass transfer, chemcal reacton and thermal radaton, Internatonal Jornal of Mcroscale and anoscale Thermal, 5(), pp Alao FI et. al. (6) : Effects of thermal radaton, Soret and Dfor on an nstead heat and mass transfer flow of a chemcall reactng fld past a sem nfnte vertcal plate wth vscos dsspaton, J. geran Mathematcal Socet, 35(), pp Bar R (3) : Free convecton heat and mass transfer MHD flow n a vertcal channel n the presence of chemcal reacton, Int. J. of Analss and Applcaton, 3(), pp Tanvr Ahmed and Md.Mahmad (3) : Fnte dfference solton of MHD mxed convecton flow wth heat generaton and chemcal reacton, Proceda Engneerng, 56, pp Mazmdar M.K and Dea R.K (7): MHD flow past an mplsve started nfnte vertcal plate n the presence of thermal radaton, Rom. Jornal Phs., 5(5-7), pp Mohmed Abd E-Azz (4): Unstead mxed convecton heat transfer along a vertcal stretchng srface wth varable vscost and vscos dsspaton, Jornal of the Egptan Mathematcal Socet,, pp Rapts A, Perd C, and Tahar HS (4): Effect of thermal radaton on MHD flow, Appled Mathematcs and Comptaton, 53, pp