Heat and Mass Transfer Analysis of an Unsteady MHD Flow Past an Impulsively Started Vertical Plate in Presence of Thermal Radiation

Transcription

1 Internatinal Jurnal f Fluid Mechanics & Thermal Sciences 018; 4(): di: /j.ijfmts ISSN: (Print); ISSN: (Online) Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin Kamalesh Kumar Pandit, Sinam Ibtn Singh, Dipak Sarma Department f Mathematics, Gauhati University, Guwahati, India address: T cite this article: Kamalesh Kumar Pandit, Sinam Ibtn Singh, Dipak Sarma. Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin. Internatinal Jurnal f Fluid Mechanics & Thermal Sciences. Vl. 4, N., 018, pp di: /j.ijfmts Received: July 8, 018; Accepted: August 9, 018; Published: September 0, 018 Abstract: The present paper aims at investigating the bundary layer flw f an unsteady MHD free cnvectin heat and mass transfer flw f a viscus, incmpressible and electrically cnducting fluid ver an impulsively started infinite vertical plate in presence f thermal radiatin. The magnetic Reynlds number is cnsidered t be s small that the induced magnetic field can be neglected. Exact slutin f the gverning equatins is btained in clsed frm by Laplace transfrm technique. Expressin fr skin frictin, Nusselt number and Sherwd number are derived. The numerical values f fluid velcity, fluid temperature and species cncentratin are displayed graphically whereas the numerical values f skin frictin, the Nusselt number and the Sherwd number are presented in tabular frm fr varius values f pertinent flw parameters. It is being fund that the mass diffusin tends t reduce the fluid velcity whereas thermal radiatin has reverse effect n the fluid velcity thrughut the bundary layer regin. Mass diffusin tends t reduce the species cncentratin whereas time has reverse effect n the species cncentratin thrughut the bundary layer regin. The radiatin parameter and mass diffusin decreases the skin frictin whereas Prandtl number has reverse effect n it. The radiatin parameter reduces the rate f heat transfer whereas as time prgress the rate f heat transfer is getting accelerated. Keywrds: MHD, Thermal Radiatin, Heat and Mass Transfer, Impulsively Started Vertical Plate 1. Intrductin Study f MHD flw with heat and mass transfer plays an imprtant rle in bilgical Sciences. Effects f varius parameters n human bdy can be studied and apprpriate suggestins can be given t the persns wrking in hazardus areas having nticeable effects f magnetism and heat variatin. Study f MHD flws als has many ther imprtant technlgical and gethermal applicatins. Sme imprtant applicatins are cling f nuclear reactrs, liquid metals fluid, pwer generatin system and aer dynamics. The effects f radiatin n free cnvectin n the accelerated flw f a viscus incmpressible fluid past an infinite vertical prus plate with suctin has many imprtant technlgical applicatins in the astrphysical, gephysical and engineering prblem. The flw f an incmpressible viscus fluid past an impulsively started infinite plate, in its wn plane, was first studied by Stke s [1]. It is als knwn as Rayleigh s prblem in the literature. In many industrial applicatins, the flw past an infinite vertical plate, started impulsively frm rest, plays an imprtant rle. The flw f a viscus fluid past an infinite vertical plate mving impulsively in its wn plane was first studied by Sundalgekar [], where the effects f natural cnvectin current due t the cling r heating f the plate were discussed. The extensin f this prblem t hydrmagnetic was presented by Gergantpuls et al. [] when the magnetic field is fixed relative t the fluid. Further, Raptis and Singh [4] studied the effect f magnetic field n this prblem fr anther physical situatin when the magnetic lines f frce are fixed relative t the plate. Sundalgekar and Takhar [5] cnsidered MHD flw and heat transfer ver a semi-infinite plate under transverse magnetic field and Sundalgekar and Wavre [6] studied unsteady free

2 19 Kamalesh Kumar Pandit et al.: Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin cnvectin flw past an infinite vertical plate with variable suctin and mass transfer. Gupta et al. [7] have analyzed free cnvectin effects n the flw past an accelerated vertical plate in an incmpressible dissipative fluid. Mass transfer effects n the flw past an expnentially accelerated vertical plate with cnstant heat flux was studied by Jha et al. [8]. Sundalgekar and Takhar [9] have cnsidered radiatin effects n free cnvectin flw past a semi-infinite vertical plate. Das et al. [10] have studied effects f mass transfer n flw past an impulsively started vertical infinite plate with cnstant heat flux and chemical reactin. Takhar et al. [11] have studied radiatin effects n MHD free cnvectin flw f a radiating gas past a semi-infinite vertical plate. Nn - Darcy cnvective bundary layer flw past a semi-infinite vertical plate in saturated prus media was studied by Takhar and Beg [1]. Radiatin effect n mixed cnvectin alng a vertical plate with unifrm surface temperature was studied by Hssain and Takhar [1]. Radiatin and free cnvectin flw past a mving plate was cnsidered by Raptis and Perdikis [14]. Muthucumaraswamy et al. [15] analyzed theretical slutin f flw past an impulsively started vertical plate with variable temperature and mass diffusin. Muthucumaraswamy and Ganesan [16] have studied heat transfer effects n flw past an impulsively started semi-infinite vertical plate with unifrm heat flux. Samad and Rahman [17] studied thermal radiatin interactin with unsteady MHD flw past a vertical prus plate immersed in a prus medium. Sme effects like radiatin and mass transfer n MHD flw were studied by Muthucumaraswamy and Janakiraman [18], Chaudhary and Jain [19], Prasad et al. [0] and Muthucumaraswamy et al. [1]. Muthucumaraswamy and Vijayalakshmi [] studied effects f heat and mass transfer n flw past an scillating vertical plate with variable temperature. Mhamed et al. [] have investigated cmbined radiatin and free cnvectin frm a vertical wavy surface embedded in prus media. Unsteady MHD flw past a vertical scillating plate with thermal radiatin and variable mass diffusin was studied by Deka and Neg [4]. On the ther hand Muthucumaraswamy et al. [5] studied unsteady flw past an accelerated infinite vertical plate with variable temperature and unifrm mass diffusin. Radiatin and mass transfer effects n MHD free cnvectin flw past an expnentially accelerated vertical plate with variable temperature was studied by Rajesh and Verma [6]. Sangapatnam et al. [7] studied radiatin and mass transfer effects n MHD free cnvectin flw past an impulsively started isthermal vertical plate with dissipatin. Suneetha et al. [8] investigated the effects f thermal radiatin n the natural cnvective heat and mass transfer f a viscus incmpressible gray absrbing-emitting fluid flwing past an impulsively started mving vertical plate with viscus dissipatin. Suneetha et al. [9] studied the effects f thermal radiatin n MHD free cnvectin flw past an impulsively started vertical plate with variable surface temperature and cncentratin. Mahmud [0] studied the effect f thermal radiatin n unsteady MHD free cnvectin flw past a vertical plate with temperature dependent viscsity. Seth et al. [1] studied the effects f thermal radiatin and rtatin n unsteady hydrmagnetic free cnvectin flw past an impulsively mving vertical plate with ramped temperature in a prus medium. Rajput and Kumar [] studied the effect f thermal radiatin n MHD flw past an impulsively started vertical plate with variable heat and mass transfer. The radiative effects have imprtant applicatins in physics and engineering prcesses. The radiatins due t heat transfer effects n different flws are very imprtant in space technlgy and high temperature prcesses. But very little is knwn abut the effects f radiatin n the bundary layer. Thermal radiatin effects may play an imprtant rle in cntrlling heat transfer in plymer prcessing industry where the quality f the final prduct depends, t sme extent t the heat cntrlling factrs (Mukhpadhyay et al. []). High temperature plasmas, cling f nuclear reactrs, liquid metal fluids, and pwer generatin systems are sme imprtant applicatins f radiative heat transfer frm a vertical wall t cnductive gray fluids. The purpse f present study is t analyze the effect f thermal radiatin and heat absrptin parameter n an unsteady MHD free cnvectin flw f a viscus, incmpressible, electrically cnducting fluid past an impulsively started vertical plate. The gverning equatins are first transfrmed int a set f nrmalized equatins and then slved analytically by using Laplace transfrm technique and a general slutin is btained. The effects f different invlved parameters such as magnetic field parameter, Schmidt number, Prandtl number, Grashf number fr heat transfer and mass transfer and thermal radiatin n the fluid velcity, temperature and cncentratin distributins are pltted and discussed.. Mathematical Frmulatin We cnsider the unsteady MHD free cnvective heat and mass transfer flw f a viscus, incmpressible, electrically cnducting fluid alng an impulsively started vertical plate. Crdinate system is chsen in such a way that x -axis is cnsidered alng the plate in upward directin and y -axis nrmal t plane f the plate in the fluid. Initially i.e., at time t 0, bth the fluid and plate are at rest and are maintained at a unifrm temperature T. Als species cncentratin at the surface f the plate as well as at every pint within the fluid is maintained at unifrm cncentratin C. At time t > 0, the plate is given an impulsive mtin in the vertical directin with cnstant velcity u 0, and at the plate cnstant heat and mass flux are impsed. Since plate is f infinite extent in x and z directins and is electrically nncnducting, all physical quantities except pressure depend n y and t nly. Als n applied r plarized vltages exist s the effect f plarizatin f fluid is negligible. This crrespnds t the case where n energy is added r extracted frm the fluid by electrical means (197). It is assumed that the induced magnetic field generated by fluid

3 Internatinal Jurnal f Fluid Mechanics & Thermal Sciences 018; 4(): mtin is negligible in cmparisn t the applied ne. This assumptin is justified because magnetic Reynlds number is very small fr liquid metals and partially inized fluids which are cmmnly used in industrial applicatins [4]. In view f the abve assumptins under usual Bussinesq s apprximatin, the unsteady flw past an infinite vertical plate is gverned by the fllwing equatins: Cnservatin f mmentum: u υ gβ ( T T ) ( ) gβ C C (1) u = + + t Cnservatin f energy: 1 q = r T k T t ρcp ρcp Cnservatin f species cncentratin: C C = D t Initial and bundary cnditins fr the fluid flw prblem are given belw: () () u = 0, T = T, C = C fr all y and t 0 (4) 0, T q, C j u = u = = k D at y = 0 fr t > 0 (5) u 0, T T, C C as y fr t > 0 (6) Fr an ptically thick fluid, in additin t emissin there is als self absrptin and usually the absrptin c-efficient is wavelength dependent and large s we can adpt the Rsseland apprximatin fr radiative heat flux vectr q r. Thus q r is given by 1 4 4σ1 T qr = k We assume that the temperature differences within the flw is sufficiently small, then equatin (7) can be linearized by expanding T 4 int Taylr s series abut the free stream temperature T and neglecting secnd and higher rder terms in ( ) T T. This results f the fllwing apprximatins: Frm (7) and (8) we have (7) T T T T (8) q r 4σ 16 = = y k k 4 1 T σ1t T 1 1 Thus the energy equatin () reduces t (9) 16σ1T y 1ρ p T k T = + T t ρc k C p (10) In rder t reduce the gverning equatins (1), () and (10), int nn-dimensinal frm, the fllwing dimensinless variables and parameters are intrduced. yu u tu T T y =, u =, t =, T =, υ u υ υ ( j υ Du ) ( q Ku ) ( q Ku ) 16 T C C gβυ υ σ C =, Gr =, N = u Kk ( ) 1 gβ υ j υ Du µ Cp υ Gm =,Pr =, Sc =, u k D Equatin (1), () and (10) reduces t u u = + GrT + GmC t T Pr = 1+ t ( N ) T C 1 C = t Sc 1 (11) (1) (1) The crrespnding initial and bundary cnditins in nndimensinal frm becme: u = 0, T = 0, C = 0 fr all y andt 0 (14) T C u = 1, = 1, = 1 at = 0 y fr t > 0 (15) u 0, T 0, C 0 as y fr t > 0 (16). Slutin f the Prblem y 4t y te u ( y, t) = erfc + ( A1 + A ) t π ( 6 ) 1 y y t + y erfc 6 t y Preff 4 te t t y eff A1 π ( 4 + Pr ) ( 4t + y ) 1 y Preff y Preff ( 6t + y Preff ) erfc 6 t

4 1 Kamalesh Kumar Pandit et al.: Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin te A y Sc 4 t ( 4 t + y Sc ) 1 y Sc y Sc ( 6t + y Sc) erfc 6 t π y Preff Pr 4t t y eff T ( y, t ) = e yerfc. π Preff t y Sc 4 t y Sc t C ( y, t) = e yerfc. πsc t (17) (18) (19) Nte that the slutin given by equatin (15) are fr Preff 1 and Sc 1. The slutin fr Preff = 1 and Sc = 1, can be easily btained by substituting Preff = 1and Sc = 1 int equatins (1) and (1) and fllw the same prcedure as discussed abve..1. Skin Frictin The expressin fr the skin frictin at the plate, is defined as.. Nusselt Number = u τ y y =0 ( A1 A A1 Pr ) 1 = + + eff + A Sc t πt (0) The Nusselt number Nu, which measures the rate f heat transfer at the plate is defined as 1 T 1 Nu = = T y T t ( 0) ( 0, ) y = 0 fr the purpse f analyzing the effect f thermal Grashf number (Gr), slutal Grashf number (Gm), Prandtl number (Pr), thermal radiatin parameter (N), Schmidt number (Sc) and time (t) n the flw field, numerical values f the fluid velcity, fluid temperature and species cncentratin in the bundary layer regin were cmputed and are displayed graphically versus bundary layer c-rdinate y in Figures The numerical values f skin frictin, heat transfer cefficient in terms f Nusselt number (Nu) and mass transfer c-efficient in terms f Sherwd number (Sh) are depicted in Tables 1-6. During the curse f numerical calculatins f the fluid velcity, the temperature and the species cncentratin, the values f the Prandtl number are chsen fr air at 5 C and ne atmspheric pressure (Pr=0.71), Mercury (Pr=0.05), electrlytic slutin (Pr=1.0) and water (Pr=7.0). T fcus ur attentin n numerical values f the results btained in the study, the values f Sc are chsen fr the gases representing diffusing chemical species f mst cmmn interest in air, namely, hydrgen (Sc=0.), water-vapur (Sc=0.60) and ammnia (Sc=0.78). T examine the effect f parameters related t the prblem n the velcity field, the skin frictin numerical cmputatin are carried ut at Pr=0.71 and Sc=0.. Figures 1 and depict the influence f thermal and cncentratin buyancy frces n fluid velcity. It is perceived frm Figures 1 and that the fluid velcity increases n increasing values f Gr and Gm thrughut the bundary layer regin. Gr represents the relative strength f thermal buyancy frce t viscus frce and Gm represents the relative strength f cncentratin buyancy frce t viscus frce. Therefre, Gr increases n increasing the strengths f thermal buyancy frce whereas Gm increases n increasing the strength f cncentratin buyancy frce. In this prblem, natural cnvectin flw induced due t thermal and cncentratin buyancy frces; therefre, thermal and cncentratin buyancy frce tends t accelerate the fluid velcity thrughut the bundary layer regin which is clearly evident frm Figures 1 and. 1 π.preff = t (1).. Sherwd Number The Sherwd number Sh, which measures the rate f mass transfer at the plate, is given by 1 C 1 Sh = = C y C t ( 0) ( 0, ) y = 0 1 π. Sc = t () 4. Results and Discussins Figure 1. Fluid velcity u against y fr different values f Gr. In rder t get the physical understand f the prblem and

5 Internatinal Jurnal f Fluid Mechanics & Thermal Sciences 018; 4(): 18-6 Figure. Fluid velcity u against y fr different values f Gm. The influence f Schmidt number (Sc) n the fluid velcity and cncentratin prfiles are depicted in Figures and 4 respectively. It is nticed frm Figures and 4 that, fluid velcity and cncentratin prfiles decreases n increasing the values f Sc. The Schmidt number embdies the rati f the mmentum t the mass diffusivity. The Schmidt number therefre quantifies the relative effectiveness f mmentum t mass transprt by diffusin in the hydrdynamic (velcity) and cncentratin (species) bundary layers. As the Schmidt number increases the cncentratin decreases. This cause the cncentratin buyancy effects t decrease yielding a reductin in the fluid velcity. The reductins in the velcity and cncentratin prfiles are accmpanied by simultaneus reductins in the velcity and cncentratin bundary layers. These behaviurs are clear frm Figures and 4. Figure 4. Species cncentratin C against y fr different values f Sc. Figure 5. Fluid velcity u against y fr different values f N. Figure. Fluid velcity u against y fr different values f Sc. Figure 6. Fluid temperature T against y fr different values f N. Figures 5 and 6 demnstrate the influence f thermal radiatin parameter (N) n fluid velcity and fluid temperature respectively. It is evident frm Figures 5 and 6

6 Kamalesh Kumar Pandit et al.: Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin that, the thermal radiatin has accelerating effect n fluid velcity and fluid temperature. The rate f radiative heat transfer t the fluid decreases and cnsequently, the fluid velcity increases with increasing values f thermal radiatin parameter fr ther fixed parameters. The influence f Prandtl number (Pr) n the fluid velcity and fluid temperature are depicted in Figures 7 and 8 respectively. It is evident frm Figure 7 and 8 that, fluid u y t and fluid temperature T decreases n increasing Pr. An increase in Prandtl number reduces the thermal bundary layer thickness. Prandtl number signifies the rati f mmentum diffusivity t thermal diffusivity. It can be nticed that as Pr decreases, the thickness f the thermal bundary layer becmes greater than the thickness f the velcity bundary layer accrding t the well-knwn velcity (, ) relatin δt δ 1 Pr where δ T the thickness f the thermal bundary layer and δ the thickness f the velcity bundary layer, s the thickness f the thermal bundary layer increases as Prandtl number decreases and hence temperature prfile decreases with increase in Prandtl number. In heat transfer prblems, the Prandtl number cntrls the relative thickening f mmentum and thermal bundary layers. When Prandtl number is small, it means that heat diffuses quickly cmpared t the velcity (mmentum), which means that fr liquid metals, the thickness f the thermal bundary layer is much bigger than the mmentum bundary layer. Hence Prandtl number can be used t increase the rate f cling in cnducting flws. Figures 9, 10 and 11 illustrate the influence f time n fluid velcity, fluid temperature and species cncentratin respectively. It is evident frm Figures 9, 10 and 11 that, the fluid velcity, fluid temperature and species cncentratin are getting accelerated with the prgress f time thrughut the bundary layer regin. Als it may be nted that, unabated mass diffusin int the fluid stream, the mlar cncentratin f the mixture rises with increasing time and s there is an enhancement in species cncentratin with the prgress f time thrughut the bundary layer regin. Figure 8. Fluid temperature T against y fr different values f Pr. Figure 9. Fluid velcity u against y fr different time t. Figure 10. Fluid temperature T against y fr different time t. Figure 7. Fluid velcity u against y fr different values f Pr.

7 Internatinal Jurnal f Fluid Mechanics & Thermal Sciences 018; 4(): rate f heat transfer at the plate whereas Prandtl number and thermal radiatin has the tendency t reduce the rate f heat transfer at the plate. The numerical values f mass transfer cefficient in terms f Sherwd number (Sh), cmputed frm the analytical expressin (), and are presented in tabular frm fr varius values f Sc and t in Tables 6. It is revealed frm Table 6 that, the rate f mass transfer increases n increasing t whereas it decreases n increasing Schmidt number. This implies that, time tends t enhance rate f mass transfer at the plate whereas mass diffusin has the tendency t reduce the rate f mass transfer at the plate. Table 4. Nusselt Number when N=1. Figure 11. Species cncentratin C against y fr different time t. Table 1. Skin Frictin when Pr=0.71, Sc=0., N=1, t=0.5. Gm τ Gr Table. Skin Frictin when Gr=5, Gm=5, Pr=0.71, Sc=0.. t τ N Table. Skin Frictin when Gr=5, Gm=5, N=1, t=0.5. Sc τ Pr The numerical values f the skin frictin τ, cmputed frm the analytical expressin (0), is presented in tabular frm fr varius values f Gr, Gm, Pr, Sc, N and t in Tables 1-. It is evident frm Tables 1- that, the skin frictin increases n increasing Pr whereas it decreases n increasing Gr, Gm, N, Sc and t. This implies that, thermal diffusin has the tendency t enhance the skin frictin cefficient whereas thermal buyancy frce, cncentratin buyancy frce, thermal radiatin, mass diffusin and time has the tendency t reduce the skin frictin cefficient at the plate. The numerical values f heat transfer cefficient in terms f Nusselt number (Nu), cmputed frm the analytical expressin (1), and are presented in tabular frm fr varius values f Pr, N and t in Tables 4-5. It is nticed frm Tables 4 and 5 that, Nusselt number increases n increasing time whereas it decrease n increasing Prandtl number and thermal radiatin. This implies that, time tends t enhance t Nu Pr Table 5. Nusselt Number when Pr=0.71. t Nu N Table 6. Sherwd Number fr different values f Sc. t Sh Sc Cnclusin The unsteady MHD natural cnvectin flw with heat and mass transfer f a viscus, incmpressible, electrically cnducting and thermal radiating fluid ver an impulsively started vertical plate.. Exact slutins f the gverning equatins were btained using Laplace transfrm technique. A cmprehensive set f graphical fr the fluid velcity, fluid temperature and species cncentratin is presented and their dependence n sme physical parameters is discussed. Significant finding are as fllws: 1. Mass diffusin tends t reduce the fluid velcity whereas thermal radiatin has reverse effect n the fluid velcity thrughut the bundary layer regin.. The fluid velcity decreases with the increase f Prandtl number thrughut the bundary layer regin.. Thermal radiatin and time has the tendency t enhance the fluid temperature whereas Prandtl number has the reverse effect n the fluid temperature thrughut the bundary layer regin. 4. Mass diffusin tends t reduce the species cncentratin whereas time has reverse effect n the species cncentratin thrughut the bundary layer regin.

8 5 Kamalesh Kumar Pandit et al.: Heat and Mass Transfer Analysis f an Unsteady MHD Flw Past an Impulsively Started Vertical Plate in Presence f Thermal Radiatin 5. Thermal radiatin and mass diffusin tends t reduce the skin frictin whereas Prandtl number has reverse effect n it. 6. Thermal radiatin and Prandtl number tends t reduce the rate f heat transfer whereas as time prgress the rate f heat transfer is getting accelerated. 7. Mass diffusin tends t reduce the rate f mass transfer whereas as time prgress the rate f mass transfer is getting accelerated. Acknwledgements The authrs wuld like t thank the Gauhati University, Dept. f Mathematics fr their supprt in the develpment f the paper. Nmenclature C species cncentratin Cp specific heat at cnstant pressure D the cefficient f mass diffusivity g acceleratin due t gravity Gm the slutal Grashf number Gr the thermal Grashf number j mass flux per unit area at the plate k thermal cnductivity k 1 Rsseland mean absrptin c-efficient N thermal radiatin parameter Pr Prandtl number q heat flux per unit area at the plate q r radiative heat flux vectr Sc Schmidt number T the temperature f the fluid u the fluid velcity in the x -directin u 0 velcity f the plate β the vlumetric cefficient f thermal expansin β the vlumetric cefficient f expansin fr cncentratin ρ the fluid density σ electrical cnductivity σ 1 Stefan-Bltzmann cnstant υ the kinematic viscsity References [1] Stkes G. G. (1851). On the effect f internal frictin f fluids n the mtin f pendulums. Camb. Phil. Trans. IX, [] Sundalgekar V. M. (1977). Free cnvectin effects n the Stkes prblem fr an infinite vertical plate. ASME J. Heat Transfer, 99, [] Gergantpuls G. A., Dusks C. N., Kafusias N. G. and Gudas C. L. (1978). Lett. in Heat and Mass Transfer, 6, 79. [4] Raptis A. and Singh A. K. (198). MHD free cnvectin flw past an accelerated vertical plate. Int. Cmm. Heat Mass Transfer, 10, 1. [5] Sundalgekar V. M. and Takhar H. S. (1977). On MHD flw and heat transfer ver a semi-infinite plate under transverse magnetic field. Nuclear Engineering and Design, 4, -6. [6] Sundalgekar V. M. and Wavre P. D. (1977). Unsteady free cnvectin flw past an infinite vertical plate with variable suctin and mass transfer. Int. J. Heat Mass Transfer, 0, [7] Gupta A. S., Pp I. and Sundalgekar V. M. (1979). Free cnvectin effects n the flw past an accelerated vertical plate in an incmpressible dissipative fluid. Rev. Rum. Sci. Techn.-Mec. Apl., 4, [8] Jha B. K., Prasad R. and Rai S. (1991). Mass transfer effects n the flw past an expnentially accelerated vertical plate with cnstant heat flux. Astrphysics and Space Science, 181, [9] Sundalgekar V. M. and Takhar H. S. (199). Radiatin effects n free cnvectin flw past a semi-infinite vertical plate. Mdeling, Measurement and Cntrl B, 51, [10] Das U. N., Deka R. K. and Sundalgekar V. M. (1994). Effects f mass transfer n flw past an impulsively started vertical infinite vertical plate with cnstant heat flux and chemical reactin. Frschung in Ingenieurwesen, 60, [11] Takhar H. S., Grla R. S. and Sundalgekar V. M. (1996). Radiatin effects n MHD free cnvectin flw f a radiating gas past a semi-infinite vertical plate. Int. J. Numerical Methds Heat Fluid Flw, 6, [1] Takhar H. S. and Beg O. A. (1996). Nn-Darcy cnvective bundary layer flw past a semi-infinite vertical plate in saturated prus media. Heat Mass Transfer,, -44. [1] Hssain M. A. and Takhar H. S. (1996). Radiatin effect n mixed cnvectin alng a vertical plate with unifrm surface temperature. Heat and Mass Transfer, 1, [14] Raptis A. and Perdikis C. (1999). Radiatin and free cnvectin flw past a mving plate. Int. J. f App. Mech. and Eng., 4, [15] Muthucumaraswamy R., Ganesan P. and Sundalgekar V. M. (000). Theretical slutin f flw past an impulsively started vertical plate with variable temperature and mass diffusin. Frschung im Ingenieurwesen, 66, [16] Muthucumaraswamy R. and Ganesan P. (00). Heat transfer effects n flw past an impulsively started semi-infinite vertical plate with unifrm heat flux. Nuclear Engineering and Design, 15, [17] Samad Md. A. and Rahman M. M. (006). Thermal radiatin interactin with unsteady MHD flw past a vertical prus plate immersed in a prus medium. Jurnal f Naval Architecture and Marine Engineering,, [18] Muthucumaraswamy R. and Janakiraman (006). MHD and Radiatin effects n mving isthermal vertical plate with variable mass diffusin. Ther. Appl. Mech., (1), [19] Chaudhary and Jain A. (007). Cmbined heat and mass transfer effects n MHD free cnvectin flw past an scillating plate embedded in prus medium. Rm. Jurnal Phys., 5 (5-7), [0] Prasad V. R., Reddy N. B. and Muthucumaraswamy R. (007). Radiatin and mass transfer effects n twdimensinal flw past an impulsively started infinite vertical plate. Int. J. Thermal Sci., 46 (1),

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