Journal of Applied Fluid Mechanics, Vol. 5, No. 3, pp. 1-10, Available online at ISSN , EISSN

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1 Jornal of Applied Flid Mechanics, Vol. 5, o. 3, pp. 1-10, 01. Available online at ISS , EISS Finite Element Soltion of Heat and Mass Transfer in MHD Flow of a Viscos Flid past a Vertical Plate nder Oscillator Sction Velocit J. Anand Rao 1, R. Srinivasa Ra and S. Sivaiah 3 1 Department of Mathematics, Universit College of Science, Osmania Universit, Hderabad, , Andhra Pradesh, India. Department of Basic Science and Hmanities, Padmasri Dr. B. V. Ra Institte of Technolog, arsapr, Medak (Dt), 50313, Andhra Pradesh, India. 3 Department of Mathematics, Gitam Universit, Hderabad Camps, Hderabad, 5039, Andhra Pradesh, India. Corresponding Athor srivass999@gmail.com (Received September 11, 009; accepted Jl 13, 011) ABSTRACT The std of hdromagnetic heat and mass transfer in MHD flow of an incompressible, electricall condcting, viscos flid past an infinite vertical poros plate along with poros medim of time dependent permeabilit nder oscillator sction velocit normal to the plate has been made. It is considered that the inflence of the niform magnetic field acts normal to the flow and the permeabilit of the poros medim flctate with the time. The problem is solved, nmericall b Galerkin finite element method for velocit, temperatre, concentration field and the expressions for skin friction, sselt nmber and Sherwood nmber are also obtained. The reslts obtained are discssed for Grashof nmber (Gr > 0) corresponding to the cooling of the plate and (Gr < 0) corresponding to the heating of the plate with the help of graphs and tables to observe the effects of varios parameters. Kewords: Heat and mass transfer, MHD flow, Vertical plate, Sction velocit, Viscos flid, Galerkin finite element method. 1. ITRODUCTIO In indstries and natre, man transport processes exist in which heat and mass transfer takes place simltaneosl as a reslt of combined boanc effect of thermal diffsion and diffsion of chemical species. The phenomenon of heat and mass transfer is observed in boanc indced motions in the atmosphere, in bodies of water, qasi solid bodies, sch as earth and so on. Unstead oscillator free convective flows pla an important role in chemical engineering; trbo machiner and aerospace technolog sch flows arise de to either nstead motion of a bondar or bondar temperatre. Besides, nsteadiness ma also be de to oscillator free stream velocit and temperatre. In the past decades an intensive research effort has been devoted to problems on heat and mass transfer in view of their application to astrophsics, geo-phsics and engineering. In addition, the phenomenon of heat and mass transfer is also encontered in chemical process indstries sch as polmer prodction and food processing. Man researchers have stdied the problems on free convection and mass transfer flow of a viscos flid throgh poros medim. In these stdies, the permeabilit of the poros medim is assmed to be constant. However, a poros material containing the flid is a non-homogeneos medim and the porosit of the medim ma not necessaril be constant. Gebhart and Pera (1971) discssed the natre of vertical natral convection flows reslting from the combined boanc effects thermal and mass diffsion. Singh and Singh (1983) stdied mass transfer effects on nstead MHD free convective flow past an infinite vertical poros plate with variable sction. Raptis and Sondalgekar (1984) discssed the stead laminar free convection flow of an electricall condcting flid along a poros hot vertical plate in the presence of heat sorce/sink. Lai (1991) presented copled heat mass transfers b mixed convection form a vertical plate in a satrated poros medim. Jha and Prasad (199) discssed the effects of applied magnetic field on transient free convective flow in a vertical channel. Abdr Sattar (1994) intiated free convection and mass transfer flow throgh a poros medim past an infinite vertical poros plate with time dependent temperatre and concentration. Singh (1994) showed the effect of mass transfer on free convection in MHD flow of a viscos flid. Singh and Kmar (1995) stdied an integral treatment for combined heat and mass transfer b natral convection in a poros medim. Sondalgekar et al. (1995) discssed copled heat mass

2 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. transfer b natral convection from vertical srface in poros medim. Singh et al. (1996) explained free convection heat and mass transfer along a vertical srface in a poros medim. Singh (1996) stdied the mass transfer effects on the flow past a vertical poros plate. Srikanth et al. (1996) have analzed the effect of mass transfer on nstead free convection flow past infinite vertical poros plate. Singh et al. (1999) discssed the hdromagnetic free convective and mass transfer flow of a viscos stratified liqid. Achara et al. (000) have reported magnetic field effects on the free convection and mass transfer flow throgh poros medim with constant sction and constant heat flx. Singh (000) presented an oscillator hdromagnetic coette flow in a rotating sstem. Kinani et al. (001) presented magnetohdrodnamic free convection heat and mass transfer of a heat generating flid past an implsivel started infinite vertical poros plate with hall crrent and radiation absorption. Kmar et al. (00) stdied an nstead oscillator laminar free convection flow of an electricall condcting flid throgh a poros medim along a poros hot vertical plate with time dependent sction in the presence of heat sorce/sink. Takhar et al. (00) stdied MHD flow over a moving plate in a rotating flid with magnetic field, hall crrents and free stream velocit. Singh et al. (003) stdied the effects of permeabilit variation and oscillator sction velocit on free convection and mass transfer flow of a viscos flid past an infinite vertical poros plate to a poros medim when the plate is sbected to a time dependent sction velocit normal to the plate in the presence of niform transverse magnetic field. The permeabilit of the poros medim is considered to be ( ) i t Ko t Ko(1 e ) and the sction velocit is assmed to be i t v( t ) v (1 e ) where v 0 o > 0 and ε << 1 is a positive constant. Ganesh and Pilani (004) stdied Finite Difference analsis of nstead natral convection MHD flow past an inclined plate with variable srface heat and mass flx. Abdr Sattar and Abdl Maleqe (005) estimated the effects of variable properties and hall crrent on stead MHD laminar convective flid flow de to a poros rotating disk. Samad et al. (005) stdied MHD bondar laer flow over a heated stretching sheet with variable Viscosit. Venkateshwarl and Anand Rao (005) have given the nmerical soltion of heat and mass transfer in MHD flow of a viscos flid past a vertical plate nder oscillator sction velocit. Prasad et al. (006) stdied transient radiative hdromagnetic free convection flow past an implsivel started vertical plate with niform heat and mass flx. Ogl et al. (007) stdied the nstead MHD free Convective flow of compressible flid past a moving vertical plate in the presence of radiative heat transfer. Sharma et al. (007) stdied the hall effect on MHD mixed convective flow of a viscos incompressible flid past a vertical poros plate immersed in poros medim with heat sorce/sink. Prasad et al. (011) discssed finite difference analsis of radiative free convection flow past an implsivel started vertical plate with variable heat and mass flx. Sneetha et al. (011) discssed Radiation and Mass transfer effects on MHD free convective Dissipative flid in the presence of heat sorce/sink. Vas et al. (011) stdied the radiation and mass transfer effects on transient free convection flow of a dissipative flid past semi-infinite vertical plate with niform heat and mass flx. In all these stdies, the oscillator sction velocit in presence of time dependent viscosit along with the inflence of niform magnetic field are not stdied while sch flows are encontered in varios fields, sch as, astrophsics, geophsics, engineering, aerodnamics, and soil sciences. In the present paper, the same investigation is obtained b sing Galerkin finite element method, which is more economical from comptation viewpoint.. MATHEMATICAL AALYSIS An nstead hdromagnetic flow of viscos, incompressible, electricall condcting flid past an infinite vertical poros plate in a poros medim of time dependent permeabilit and sction velocit is considered as shown in Fig. 1. Fig. 1. Phsical sketch and geometr of the problem In Cartesian co ordinate sstem, x axis is assmed to be along plate in the direction of the flow and - axis normal to it. A niform magnetic field is introdced normal to the direction of the flow. In the analsis, it is assmed that the magnetic Renolds nmber is mch less than nit so that the magnetic indced field, Frther, all the flid properties are assmed to be constant except that of the inflence of the densit variation of the temperatre. Therefore, the basic flow in the medim is entirel de to boanc force cased b temperatre difference between the wall and medim. Initiall t 0, the plate as well as flid is assmed to be at the same temperatre and the concentration of species is ver low so that the Soret and Dofor effect are neglected. When t > 0, the temperatre of the plate is instantaneosl raised (or lowered) to T w and the concentration of the species is raised (or lowered) to C w. Under the stated assmptions and taking the sal Bossinesq s approximation in to accont, the governing eqations for momentm, energ and concentration in dimensionless form are:

3 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. Continit Eqation: v 0 t Momentm Eqation: v g T T t g CC B 0 K o Energ Eqation: T T k T v t C p Concentration Eqation: C C v D C t () (1) (3) (4) where is the velocit along the x - axis, is the kinematic coefficient of viscosit, g is the acceleration de to gravit, β is the coefficient of volme expansion for the heat transfer, β * is the volmetric coefficient of expansion with species concentration, T is the flid temperatre, T is the flid temperatre at infinit, C is the species concentration, C is the species concentration at infinit, D is the chemical moleclar diffsivit, - porosit of the poros medim, k e - Mean absorption co-efficient, K o is the constant permeabilit of the medim, μ is the coefficient of viscosit, C p is the specific heat at constant pressre, η is the freqenc of oscillation, ρ is the densit of the flid, k r - the chemical reaction parameter and t is the time. The corresponding bondar conditions are t 0: 0, T T, C C for all i t 0, TTw Tw T e, i t t 0: CCw Cw C e at 0 0, T T, C C as From the continit eqation, it can be seen that v is either a constant or a fnction of time. So assming sction velocit to be oscillator abot a non zero constant mean, one can write i t v v (1 e ) 0 where v 0 is the mean sction velocit, η is freqenc of oscillation and v 0 > 0, ε << 1 is a positive constant. The negative sign indicates that the sction velocit is directed towards the plate. The permeabilit of the poros medim is considered to be i t Ko( t) Ko(1 e ). (5) The non dimensionless qantities introdced in these eqations are defined as: v v t o ; t o ; 4v 4v 4 n ; ; v o vo T T C C T ; C ; Tw T Cw C K v K o o o ( Permeabilit of the medim ); g * ( T T w ) Gr Grashof mber ; v 3 o Sc Schimidt mber; D C p Pr Pr andtl mber ; K t B o M Hartmann nmber v o g C w C Gm v 3 o Modified Grashof mber.; (6) The governing eqations for momentm, energ and concentration in dimensionless form are: 1 1 e int Gr T Gm C 4 t M K 1 e int o (7) 1 T int 1 (1 e ) T T (8) 4 t Pr 1 C 1 (1 int C C e ) (9) 4 t Sc The relevant bondar conditions in dimensionless form are 0, T 1 e int, C 1 e int at 0 (10) 0, T 0, C 0 as 3. METHOD OF SOLUTIO B appling Galerkin finite element method for Eq. (7) is: over the element (e), ( e) ( e) 1 k T 4 t d ( e ) ( e ) A R P k 0 (11) 3

4 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. int Where A = 1 e, R = 1 M, Ko A P = ( Gr) T ( Gm ) C Integrating the first term in Eq. (11) b parts one obtains () e () e T () e 1 () e k () e T k 4 t d 0 () e () e T A () e R P (1) eglecting the first term in Eq. (1), one gets: ( e) ( e) T ( e ) T k d 0 ( e) ( e) 1 ( e) A R P 4 t ( e) ( e) ( e) Let be the finite element approximation soltion over the element k T where ( e) ( e), k k and k, are the basis fnctions. k k k One obtains: k k d k k k k 1 k k d 4 k k k k A k k d () e l k k k k k R k d 6 k k k k k P d k Simplifing we get () e l k k A 1 1 R 1 () e l k k P 1 1 Where prime and dot denotes differentiation w.r.to and time t respectivel. Assembling the element eqations for two consective elements i1 i and i i1 following is obtained: i 1 1 ( ) e i l i 1 i i i i 1 A ( e ) i l i i 1 R i (13) 0 1 i 1 1 P 1 ow pt row corresponding to the node i to zero, ( e ) from Eq. (13) the difference schemes with l h is: i 1 i i 1 i i i 1 h 4 (14) A R 4 P h i1 i1 6 i1 i i1 Appling the trapezoidal rle, following sstem of eqations in Crank-icholson method are obtained: n1 n1 n1 A 1 i1 A i A 3 i1 n n n A 4 i1 A 5 i A 6 i1 1Phk (15) ow from Eqs. (7) and (8), following eqations are obtained: 4

5 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. BT n1 B T n1 BT n1 1 i1 i 3 i1 BT n BT n n BT (16) i i 6 i1 CC n C 1 C n C n 3 C i i i1 C C n C n n C C C i i 6 i 1 (17) where A 1 = h + Rk 1rh + 6Ak; A = 4h + 4rh +8Rkh; A 3 = h + Rk 1rh - 6Ak; A 4 = h - Rk + 1rh - 6Ak; A 5 = 4h - 4rh - 8Rkh; A 6 = h - Rk + 1rh + 6Ak; P 4h( Gr ) kt i 4h( Gm) kc i ; B 1 = h(pr) + 6Ak(Pr) -1rh; B = 4h(Pr) + 4rh; B 3 = h(pr) - 6Ak(Pr) -1rh; B 4 = h(pr) - 6Ak(Pr) +1rh; B 5 = 4h(Pr) - 4rh; B 6 = h(pr) + 6Ak(Pr) +1rh; C 1 = h(sc) + 6Ak(Sc) -1rh; C = 4h(Sc) + 4rh; C 3 = h(sc) - 6Ak(Sc) -1rh; C 4 = h(sc) - 6Ak(Sc) +1rh; C 5 = 4h(Sc) - 4rh; C 6 =h(sc) + 6Ak(Sc) + 1rh; k Here r = and h, k are mesh sizes along h direction and time-direction respectivel. Index i refers to space and refers to the time. In the Eqs. (15), (16) and (17), taking i = 1(1) n and sing bondar conditions (10), then the following sstem of eqations are obtained: A i X i B i i 1(1)3 (18) Where Ai s are matrices of order n and X i, Bi s are colmn matrices having n-components. The soltions of above sstem of eqations are obtained b sing Thomas algorithm for velocit, temperatre and concentration. Also, nmerical soltions for these eqations are obtained b C programme. In order to prove the convergence and stabilit of Galerkin finite element method, the same C programme was rn with smaller vales of h and k and no significant change was observed in the vales of, T and C. Hence the Galerkin finite element method is stable and convergent. 4. SKI-FRICTIO, RATE OF HEAT AD MASS TRASFER Skin Friction coefficient (τ) at the plate is 0 Heat transfer coefficient ( ) at the plate is T 0 Mass transfer coefficient (S b ) at the plate is S b C 0 5. RESULTS AD DISCUSSIO Some nmerical calclations have been carried ot for the non-dimensional velocit (), temperatre (T), concentration (C), skin friction coefficient ( ) and heat and mass transfer coefficients in terms of sselt nmber ( ) and Sherwood nmber (S b ) respectivel. The effects of material parameters sch as Prandtl nmber (Pr), Schmidt nmber (Sc), Hartmann nmber (M), permeabilit parameter (K o ), Grashof nmber (Gr) and modified Grashof nmber (Gm) have been observed. The nmerical calclations of these reslts are presented graphicall in Figs. to 15. Fig.. Effect of Grashof nmber Gr on velocit field for cooling of the plate when Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/. Fig. 3. Effect of modified Grashof nmber Gm on velocit field for cooling of the plate when Gr = 10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/. Dring the corse of nmerical calclations of the velocit, temperatre and concentration, the vales of the Prandtl nmber are chosen for air (Pr = 0.71), electroltic soltion (Pr = 1.0), water (Pr = 7.0) and water at 4 o C (Pr = 11.40). To focs ot attention on nmerical vales of the reslts obtained in the std the vales of Sc are chosen for the gases representing diffsing chemical species of most common interest in air namel Hdrogen (Sc = 0.), Water vapor (Sc = 0.60), Oxgen (Sc = 0.66), Ammonia (Sc = 0.78), Methanol (Sc = 1.00) and Propl-benzene (Sc =.6) at 0 o C and one atmospheric pressre. 5

6 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. vale. It is noticed that the velocit increases with increasing vales of the Soltal Grashof nmber. Fig. 4. Effect of Magnetic nmber M on velocit field for cooling of the plate when Gr = 10.0, Gm = 10.0, Pr = 0.71, Sc = 0., K o = 10.0, ε = and nt = π/. For the phsical significance, onl the real part of complex qantit is invoked for the nmerical discssion in the problem and at t = 1.0, stable vales for velocit, temperatre and concentration fields are obtained. To examine the effect of parameters related to the problem on the velocit field and skin-friction nmerical comptations are carried ot at (Pr = 0.71) which corresponds to air at 5 o C and one atmospheric pressre. The vales of Grashof nmber Gr and modified Grashof nmber Gm are taken to be positive and negative as the respectivel represent smmetric cooling of the plate when Gr > 0 and smmetric heating of the plate when Gr < 0. Since the flow is continos flow which is tends to infinit. For finding soltion of this problem we have placed infinite vertical plate in a finite length in the flow and hence we solved the entire problem in a finite bondar. However, in the graph vales var from 0 to 4, velocit, temperatre and concentration tends to zero as tends to 4. This is tre for an vale of, ths we have considered finite length. The temperatre and the species concentration are copled to the velocit via Grashof nmber Gr and modified Grashof nmber Gm as seen in Eq. (7). Figres -13 displa the effects of material parameters sch as Gr, Gm, M, Sc, Pr and K o on the velocit field for both externall cooling (Gr > 0) and heating (Gr < 0) of the plate. It is observed that an increase in the Grashof nmber or modified Grashof nmber leads to increase in the velocit field in both the presence of cooling and heating of the plate. For varios vales of Grashof nmber and modified Grashof nmber, the velocit profiles are plotted in Figs. and 3. The Grashof nmber Gr signifies the relative effect of the thermal boanc force to the viscos hdrodnamic force in the bondar laer. As expected, it is observed that there is a rise in the velocit de to the enhancement of thermal boanc force. Here, the positive vales of Gr correspond to cooling of the plate. Also, as Gr increases, the peak vales of the velocit increases rapidl near the poros plate and then decas smoothl to the free stream velocit. The modified Grashof nmber Gm defines the ratio of the species boanc force to the viscos hdrodnamic force. As expected, the flid velocit increases and the peak vale is more distinctive de to increase in the species boanc force. The velocit distribtion attains a distinctive maximm vale in the vicinit of the plate and then decreases properl to approach the free stream The effect of Hartmann nmber M is shown in the Fig. 4 in case of cooling of the plate. It is observed that the velocit of the flid decreases with the increase of Hartmann nmber vales. As expected, the velocit decreases with an increase in the Hartmann nmber. It is becase that the application of transverse magnetic field will reslt in a resistive tpe force (Lorentz force) similar to drag force which tends to resist the flid flow and ths redcing its velocit. Also, the bondar laer thickness decreases with an increase in the Hartmann nmber. We also see that velocit profiles decrease with the increase of magnetic effect indicating that magnetic field tends to retard the motion of the flid. Magnetic field ma control the flow characteristics. Fig. 5. Effect of Schmidt nmber Sc on velocit field for cooling of the plate when Gr = 10.0, Gm = 10.0, Pr = 0.71, M = 0.5, K o = 10.0, ε = and nt = π/. Fig. 6. Effect of Prandtl nmber Pr on velocit field for cooling of the plate when Gr = 10.0, Gm = 10.0, M = 0.5,Sc = 0., K o = 10.0, ε = and nt = π/. Fig. 7. Effect of Permeabilit parameter K o on velocit field for cooling of the plate when Gr = 10.0, Gm = 10.0, M =0.5,Sc = 0., Pr = 0.71, ε = and nt = π/. 6

7 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. From Figs. 5 and 6 it is observed that an increase in Sc or Pr decreases the velocit field. A comparison of velocit distribtion crves de to cooling of the plate show that in the vicinit of the plate the velocit falls ver rapidl and thereafter steadil indicating that the crves rise gradall after attaining minimm vale near the plate. Figre 7 shows the effect of the permeabilit of the poros medim parameter K o on the velocit distribtion. As shown, the velocit is increasing with the increasing dimensionless poros medim parameter. The effect of the dimensionless poros medim K o becomes smaller as K o increase. Phsicall, this reslt can be achieved when the holes of the poros medim ma be neglected. In the Figs. 8 13, on velocit field mentioned above, compare to the case of cooling of the plate opposite effects are observed in the case of heating of the plate. Fig. 8. Effect of Grashof nmber Gr on velocit field for heating of the plate when Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/. Fig. 9. Effect of modified Grashof nmber Gm on velocit field for heating of the plate when Gr = -10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/. Fig. 10. Effect of Magnetic nmber M on velocit field for heating of the plate when Gr = -10.0, Gm = 10.0, Pr = 0.71, Sc = 0., K o = 10.0, ε = and nt = π/. Fig. 11. Effect of Schmidt nmber Sc on velocit field for heating of the plate when Gr = -10.0, Gm = 10.0, Pr = 0.71, M = 0.5, K o = 10.0, ε = and nt = π/. Fig. 1. Effect of Prandtl nmber Pr on velocit field for heating of the plate when Gr=-10.0, Gm=10.0, M=0.5,Sc=0., K o =10.0, ε=0.005 and nt = π/. Fig. 13. Effect of Permeabilit parameter K o on velocit field for heating of the plate when Gr = ,Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, ε = and nt = π/. An increase in Prandtl nmber decreases the Temperatre field (Fig. 14). Also, Temperatre field falls more rapidl for Water in comparison to Air and the Temperatre field crve is exactl linear for Mercr, which is more sensible towards change in Temperatre. From this observation it is conclded that Mercr is most effective for maintaining Temperatre differences can be sed efficientl in the laborator. Air can replace Mercr, the effectiveness of maintaining the Temperatre changes are mch less than Mercr. If Temperatres are maintained, Air can be better and cheap replacement for indstrial prposes. 7

8 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. Table 1 Skin Friction coefficient of (τ) for cooling of the plate. Gr Gm M Sc Pr K o τ Table Skin Friction coefficient of (τ) for heating of the plate. Gr Gm M Sc Pr K o τ Fig. 14. Effect of Prandtl nmber Pr on temperatre field T when Gr =10.0, Gm =10.0, M = 0.5, Sc = 0., K o =10.0, ε = and nt = π/. Water vapor can be sed for maintaining normal Concentration field and Hdrogen can be sed for maintaining effective Concentration field. In order to ascertain the accrac of the nmerical reslts, the present reslts are compared with the previos reslts of Venkateshwarl and Anand Rao (005) for Gr = 10.0, Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/ in Fig. 16. The are fond to be in an excellent agreement. Figre 16. Effect of Permeabilit parameter K o on velocit field for cooling of the plate when Gr = 10.0, Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, ε = and nt = π/. Fig. 15. Effect of Schmidt nmber Sc on concentration field C when Gr =10.0,Gm =10.0, M = 0.5, Sc = 0., K o =10.0, ε = and nt = π/. From Fig. 15, shows that an increase in Schmidt nmber decreases the concentration field. Also Concentration field falls slowl and steadil for Hdrogen and Helim bt falls ver rapidl for Oxgen and Ammonia in comparison to Water vapor. Ths Table 1 represents the nmerical vales of skin-friction coefficient (τ) for variations in Gr, Gm, M, Sc, Pr, and K o respectivel, corresponding to cooling of the plate. An increase in Gr or Gm or K o leads to an increase in the vale of skin friction coefficient while in increase in M or Sc or Pr leads to a decrease in the vale of skin friction coefficient. Table represents the nmerical vales of skin-friction coefficient (τ) for variations in Gr, Gm, M, Sc, Pr, and K o respectivel, corresponding to heating of the plate. An increase in Gr or Gm or Pr leads to an increase in 8

9 J. Anand Rao et al. / JAFM, Vol. 5, o. 3, pp. 1-10, 01. the vale of skin friction coefficient while in increase in M or Sc or K o leads to a decrease in the vale of skin friction coefficient. Table 3 Heat transfer coefficient in terms of sselt nmber. increase of Permeabilit parameter K o for heating of the plate (Gr < 0). 4) The velocit increases with the increasing of Grashof nmber Gr and Modified Grashof nmber Gm. Pr Table 4 Mass transfer coefficient in terms of Sherwood nmber. S b Sc Table 3 represents the nmerical vales of heat transfer coefficient ( ) for different vales of Prandtl nmber Pr. An increase in Pr leads to an increase in heat transfer coefficient. Also the vale of is least for Mercr and highest for Water at 4 o C. Table 4 represents the nmerical vales of mass transfer coefficient (S b ) for different vales of Schmidt nmber Sc. An increase in Sc leads to an increase in mass transfer coefficient. Also, the vale of S b is least for Hdrogen and highest for Propl benzene. 6. COCLUSIO The problem Finite element soltion of heat and mass transfer in MHD flow of a viscos flid past a vertical plate nder oscillator sction velocit is stdied. The dimensionless eqations are solved b sing Galerkin finite element method. The effects of velocit, temperatre and concentration for different parameters like Gr, Gm, M. Sc, Pr and K o are stdied. The std concldes the following reslts: 1) The velocit decreases with the increasing of Hartmann nmber M. ) The velocit decreases with the increase of Prandtl nmber Pr and Schmidt nmber Sc for cooling of the plate (Gr > 0) and the velocit increases with the increase of Prandtl nmber Pr and Schmidt nmber Sc for heating of the plate (Gr < 0). 3) The velocit increases with the increase of Permeabilit parameter K o for cooling of the plate (Gr > 0) and the velocit decreases with the 5) The temperatre and Concentration decreases with increasing of Prandtl nmber Pr and Schmidt nmber Sc respectivel. 6) In order to ascertain the accrac of the nmerical reslts, the present reslts are compared with the previos reslts of Venkateshwarl and Anand Rao (005) for Gr = 10.0, Gm = 10.0, M = 0.5, Sc = 0., Pr = 0.71, K o = 10.0, ε = and nt = π/ in Fig. 16. The are fond to be in an excellent agreement. REFERECES Abdr Sattar and M.D. Kh. Abdl Maleqe (005). The effects of variable properties and hall crrent on stead MHD laminar convective flid flow de to a poros rotating disk, Int. ornal of heat and mass transfer 48, Abdsattar, M.D. (1994). Free convection and mass transfer flow throgh a poros medim past an infinite vertical poros plate with time dependent temperatre and concentration. Ind ornal of pre Application Math 5, Achara, M., G.C. Dash and L.P. Singh (000). Magnetic field effects on the free convection and mass transfer flow throgh poros medim with constant sction and constant heat flx. Ind Jornal of Pre Application Math 31, Ganesh, P. and G. Pilani (004). Finite difference analsis of nstead natral convection MHD flow past an inclined plate with variable srface heat and mass flx. Int. Jornal of Heat and Mass Transfer 47, Gebhart, B. and L. Pera (1971). The natre of vertical natral convection flows reslting from the combined boanc effects thermal and mass diffsion. International Jornal of Heat Mass Transfer 14, Jain, M.K. (1984). merical soltions of differential eqations. nd edition, Wile Eastern Limited, ew Delhi. Jha, B.K. and R. Prasad (199). Effects of applied magnetic field on transient free convective flow in a vertical channel. Jornal of Math Ph. Science 6, 1 8. Kinani, M., J.K. Kwanza and S.M. Uppal (001). Magnetohdrodnamic free convection heat and mass transfer of a heat generating flid past an implsivel started infinite vertical poros plate with Hall crrent and 9

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