Visco-Elastic MHD Boundary Layer Flow with Heat and Mass Transfer over a Continuously Moving Inclined Surface in Presence of Energy Dissipation

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1 Visco-Elastic MHD Bondar Laer Flow with Heat and Mass Transfer over a Continosl Moving Inclined Srface in Presence of Energ Dissipation RITA CHOUDHURY, 2 PABAN DHAR & 3 DEBASISH DEY,2 Department of Mathematics, Gahati Universit, Gwahati -784, Assam, India rchodhr66@ahoo.in pabankmardhar@ahoo.com 2 Department of Mathematics, Dibrgarh Universit, Dibrgarh-7864, Assam, India debasish492@gmail.com Abstract: - The problem of two-dimensional free convective bondar laer flow of a visco-elastic flid over a continosl moving inclined srface has been investigated in presence of transverse magnetic field and energ dissipation. Effects of heat and mass transfer are also analzed in this paper. Heat is spplied from the plate to the flid at a niform rate. The sction at the plate is assmed to be constant and the plate is assmed to move continosl with a niform velocit U in the pward direction. The visco-elastic flid flow is characterized b Walters liqid (model B ). A niform magnetic field of strength B is applied in the direction perpendiclar to the plate. Highl non-linear momentm bondar laer eqation, thermal bondar laer eqation and concentration eqation are converted into non dimensional form and then solved analticall b sing reglar pertrbation techniqe. The analtical expressions for velocit profile, temperatre field, concentration field, shearing stress, rate of heat transfer and rate of mass transfer at the srface have been obtained. The effects of Grashof nmber (Gr), Hartmann nmber (M), Prandtl nmber (Pr) and Schmidt nmber (Sc) on velocit profiles and shearing stress at the srface have been illstrated graphicall for varios vales of visco-elastic parameter in combination with other flow parameters. Ke-Words: - Visco-elastic, Walters liqid (model B / ), MHD, heat and mass transfer, reglar pertrbation techniqe, Prandtl nmber, Schmidt nmber. Introdction One of the recent advances in the area of flid dnamics is the mechanism of non-newtonian flid theor. Researchers have shown their interest in this emerging and dnamic field for its applications in geo-phsics, chemical engineering (absorption, filtration), petrolem engineering, hdrolog, soil-phsics, bio-phsics, paper and plp technolog. In non-newtonian theor, the analsis of visco-elastic flid lies on the fact that the viscosit of the flid stdies the energ dissipation and its elasticit analses the energ stored dring the flow. The concept of simltaneos heat and mass transfer is sed in varios science and engineering problems. It is sed in food processing, wet-blb thermometer and polmer soltion and also in varios flids flow related engineering problems. In or dail life, the combined heat and mass transfer phenomenon is observed in the formation of fog. The convective flow associated with the combined heat and mass transfer has man applications in varios branches of science and engineering. Natre of vertical convection flows from the boanc effects of thermal and mass diffsion has been done b Gebhart and Pera []. Singh and Singh [2] have discssed the MHD free convection flow and mass transfer past a flat plate. Al-Qadat and Al-Azab [3] have stdied the inflence of chemical reaction on transient MHD free convective flow over a moving vertical plate. The std of combined heat and mass transfer in mixed convective MHD flow along a vertical plate in presence of heat sorce has been shown b Zeco and Ahmed [4]. Palani and Srikanth [5] have explained the mass transfer effects on MHD flow past a semi infinite vertical plate. Chadhar and Jain [6] have analsed the combined heat and mass diffsion in a MHD free convective flow past a srface embedded in a poros medim. Effects of chemical reaction on transient MHD free convective flow over a vertical plate in slip-flow regime are explained b Sahin [7]. The investigation E-ISSN: Isse 4, Volme 8, October 23

2 of bondar laer flow and mass transfer past a vertical plate in a poros medim with constant heat flx in presence of transverse magnetic field has been done b Makinde [8]. The theor of MHD is extensivel sed in man applications sch as extrsion of plastics in the manfactre of Raon and Nlon, prification of crde oil, textile indstr etc. The natral convection bondar laer flow of an electricall condcting flid p a hot vertical wall in the presence of strong magnetic field has been stdied b several athors sch as Sparrow and Cess [9], Rile [] and Kiken [] etc. Simltaneos occrrence of boanc and magnetic field forces in the flow of an electricall condcting flid p a hot vertical flat plate in the presence of a strong cross magnetic field was stdied b Singh and Cowling [2]. Crammer and Pai [3] have presented a similarit soltion for the above problem with niform heat flx b formlating it in terms of both a reglar and inverse series expansions of characterizing co-ordinate that provided a link between the similarit state closed to and far from the leading edge. Hossain et al. [4] have stdied the convection flow from an isothermal plate inclined at a small angle to the horizontal. The problem of bondar laer flow past a stretching sheet are extensivel sed in varios technological process like hot rolling, wire drawing, glass-fibre and paper prodction etc. Man researchers have done their work on the basis of the classical works done b Sakiadis [5, 6], et.al. [7] and Crane [8]. Ericksen et al. [9] have stdied the combined heat and mass transfer on a moving continos plate with sction or injection. Gpta and Gpta [2] have investigated the heat and mass transfer with sction and blowing. Vajravel [2, 22] has analsed hdromagnetic flow with heat transfer and hdromagnetic convection respectivel over a moving srface. A theoretical soltion for hdromagnetic convection over a continosl moving vertical srface with niform sction has been obtained b Kmar et al. [23]. The mechanisms of visco-elastic bondar laer flow are sed in varios manfactring processes sch as fabrication of adhesive tapes, extrsion of plastic sheets, coating laers into rigid srfaces etc. Varios blood flow problems are also explained b sing the visco-elastic bondar laer theor. The secondorder visco-elastic flid flow past a stretching sheet has been stdied b Rajagopal et al. [24]. The [25] also have analzed the non similar bondar laer flow on a stretching sheet of second-order viscoelastic flid with niform free stream. The investigation on visco-elastic flid flow with heat transfer over a stretching sheet has been stdied b Dhanpat and Gpta [26]. Walters liqid (Model B / ) is a tpe of visco-elastic flid which resists shear flow and strains linearl with time nder the application of an applied stress bt when the stress is removed it qickl retrns to its original position. Ahmed et al. [27] have stdied the visco-elastic bondar laer flow characterized b Walters liqid past a stretching plate in presence of heat transfer. The std of visco-elastic (Walters liqid, Model B / ) flow past a stretching plate with sction has been done b Siddappa and Abel [28]. The combined analsis of heat and mass transfer of a visco-elastic, electricall condcting flid past a continos stretching sheet has been discssed b Kell et al. [29]. Sbhas et al. [3] have stdied the MHD effects on visco-elastic flid flow over a stretching srface in presence of heat transfer. A mathematical analsis of heat and mass transfer phenomena in a visco-elastic flid over an accelerating stretching sheet in the presence of heat sorce/sink, viscos dissipation and sction/blowing has been analzed b Sonth et al. [3]. Abel et al. [32] have investigated the visco-elastic bondar laer flow behavior over a stretching sheet in presence of non-niform heat sorce and energ dissipation. The natre of nstead bondar laer flow over a stretching sheet has been stdied b Ahmad and Mishra [33]. Chodhr and Das [34] have examined the flow behavior of visco-elastic bondar laer flow on a continosl moving srface in presence of transverse magnetic field. Misra et al. [35] have investigated the nstead bondar laer flow over a stretching sheet with variable thermal condctivit. In this paper, we have stdied the two dimensional stead free convective hdromagnetic bondar laer flow of visco-elastic over a continosl moving inclined srface in presence of heat and mass transfer. The dissipation of energ is also considered. The visco-elastic flid flow is characterized b Walters liqid (Model B / ). The constittive eqation for Walters liqid (Model B / ) is, (.) Where is the stress tensor, is isotropic pressre, is the metric tensor of a fixed coordinate sstem is the velocit vector, the contravarient form of is given b E-ISSN: Isse 4, Volme 8, October 23

3 , (.2) It is the convected derivative of the deformation rate tensor defined b (.3) Here is the limiting viscosit at the small rate of shear which is given b (.4) N( ) being the relaxation spectrm. This idealized model is a valid approximation of Walters liqid (Model B ) taking ver short memories into accont so that terms involving With these phsical considerations and b sing Bossinesq approximation, the eqations governing the flid flow are as follows:, (.5) have been neglected. 2 Mathematical Formlations: We consider a stead bondar laer free convective visco-elastic flow of an electricall condcting visco-elastic flid over a continosl moving inclined srface, issing from a slot and moving with a constant velocit U in a flid. Heat is spplied to the plate at the constant rate. A magnetic field of niform strength B is applied in the transverse direction. The magnetic Renolds nmber is assmed to be so small that the indced magnetic field is neglected. Let x / axis be taken along the plate in the pward direction and / axis be taken normal to the srface. The inclination of the angle is assmed to so small that. The phsical model of the problem is shown in Figre. v =constant U The initial and bondar conditions are 3 Method of Soltion: Introdcing the following non-dimensional qantities: x g B α slot Figre : Phsical Model of the problem where Gr is the Grashof nmber for heat transfer, Gm is the Grashof nmber for mass transfer, Pr is the Prandtl nmber, M is the Hartmann nmber, k is the dimensionless visco-elastic parameter, Sc is the Schmidt nmber and Ec is the Eckert nmber. We make se of the assmption that the velocit and temperatre fields are independent of E-ISSN: Isse 4, Volme 8, October 23

4 the distance parallel to the srface (as given in Schlichting 968). Eqations (2.2), (2.3) and (2.4) are redced to the following ordinar differential eqations. Sbstitting the eqations (3.4) into the eqations (3.8), (3.) and (3.) and after eqating the like powers of k, we get the following sets of ordinar differential eqations: (3.5) The corresponding initial and bondar conditions in non-dimensional forms are ScA4e Sc α2ae α2+pra8e Pr+ScA9e Sc Pr α2a2e α2+pra3e Pr+ScA4e Scα22A2 e α2 Pr2A3e Pr Sc2A4e Sc Solving (3.4) with conditions (3.5) we get Again, the phsical variables, θ are expanded in the power of Eckert nmber (Ec) as Ec is ver small as compared to nit for incompressible flid. Ths the expressions for velocit, temperatre are expressed as follows: (3.7) Using (3.7) in eqations (3.2) & (3.3) and eqating the co-efficient of like powers of Ec, we obtain the following set of differential eqations: Sbject to bondar conditions (3.8) (3.9) (3.) (3.) The soltions of the eqations (3.9), sbject to the bondar conditions given in (3.2) are as follows: Again, to solve the eqations (3.8), (3.) and (3.), we se the mlti-parameter pertrbation techniqe and the velocit and temperatre components are expanded in the power of viscoelastic parameter k as k<< for small shear rate. Ths the expressions for velocit and temperatre components are considered as The modified bondar conditions are The differential eqations (3.5) to (3.8) are solved sbject to the bondar condition (3.9). The soltions of the differential eqations are not presented here for the sake of brevit. 4 Reslts and Discssion The velocit profile is given b The non dimensional shearing stress at the plate is given b The object of the present paper is to investigate the effects of visco-elasticit on free convective MHD bondar laer flow past an inclined moving srface in presence of heat and mass transfer. The viscoelastic effect is exhibited throgh the nondimensional parameter k. The nonzero vales of the parameter k characterize the visco-elastic flid and k= represents the Newtonian flid flow phenomenon. The velocit profile and the shearing stress at the plate are analzed graphicall for varios vales of flow parameters involved in the soltion. Figres 2 to 7 demonstrate the pattern of velocit profile against the displacement for varios vales of visco-elastic parameter. These profiles conclde that speed diminishes as the flid moves far awa from the plate. Also it is visalized from these figres that the flid decelerates rapidl in the E-ISSN: Isse 4, Volme 8, October 23

5 neighborhood of the plate. This flow behavior is noticed in both Newtonian and visco-elastic flid. Another interesting characteristic is noticed dring the growth of visco-elasticit in the neighborhood of the inclined srface. An accelerated flow is observed dring the modification of visco-elasticit in comparison with the simple Newtonian flid. redces the speed for both Newtonian as well as visco-elastic flid..2 Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=9,k= Gr=9,k=. Gr=9,k=.2.2 M=3,k= M=3,k=. M=3,k=.2 M=4.5,k= M=4.5,k=. M=4.5,k= Figre 3: velocit profile against for M=3, Gm=, Pr=, Sc= Figre 2: velocit profile against for Gr=7, Gm=, Pr=, Sc= The effects of Hartmann nmber in the bondar laer of both Newtonian and visco-elastic flid are shown in figre 2. The application of transverse magnetic field generates a resistive force called Lorentz force which increases the viscosit of the flid flow and hence a retarding trend is observed in the speed of the flid flow. Both the visco-elastic flid and Newtonian flid experience a decreasing trend dring the rise of magnetic parameter. Grashof nmber stdies the behavior of free convection and it is defined as the ratio of boanc force to viscos force. It plas an important role in heat and mass transfer technolog. Gr characterizes the free convection parameter for heat transfer and Gm characterizes the free convection parameter for mass transfer. In or std, the reslts are discssed for the flow past an externall cooled plate (Gr > ) and flow past an externall heated plate (Gr < ). Figre 3 stdies the behavior of varios flid flows dring the positive vales of free convection parameter. The enlargement of Grashof nmber for heat transfer Pr=8,k= Pr=8,k=. Pr=8,k=.2 Pr=,k= Pr=,k=. Pr=,k= Figre 4: velocit profile against for M=3, Gm=, Gr=7, Sc= Prandtl nmber plas a significant role in heat transfer flow problems as it helps to std the E-ISSN: Isse 4, Volme 8, October 23

6 simltaneos effects of momentm and thermal diffsion in flid flow. The effects of Prandtl nmber on both visco-elastic flid and Newtonian flid are analzed in figre 4. It states that the ascending vale of Prandtl nmber raises the viscosit of the flid flow in case of cooling problems and the flid becomes thick. Ths a redction in speed is experienced for varios flid flow sstems Gm=,k= Gm=,k=. Gm=,k=.2 Gm=2,k= Gm=2,k=. Gm=2,k=.2 that the free stream concentration is less than the concentration at the bondar srface. The increasing vales of free convection parameter for mass transfer diminishes the viscosit of varios flid flow mechanisms and which in trn accelerates the flow of both simple Newtonian flid and complex visco-elastic flid flows. In mass transfer problems, the importance of Schmidt nmber cannot be neglected as it stdies the combined effect of momentm and mass diffsion. Figre 6 notifies the effect of Schmidt nmber in this paper. The rising natre of Schmidt nmber increases the viscosit of the flid flow and hence it will slow down the speed of both Newtonian as well as visco-elastic flid. The maximm effects of both Prandtl nmber and Schmidt nmber are detected in the neighborhood of the inclined srface..2 Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= Figre 5: velocit profile against for M=3, Pr=, Gr=7, Sc= Sc=,k= Sc=,k=. Sc=,k=.2 Sc=4,k= Sc=4,k=. Sc=4,k= Figre 7: velocit profile against for M=3, Pr=, Sc=, Gm= Figre 6: velocit profile against for M=3, Pr=, Gr=7, Gm= Figre 5 represents the effect of Gm on both viscoelastic flid and Newtonian flid. Gm > indicates The difference in the natre of varios flids for the flow past a heated plate and for the flow past a cooled plate is shown in figre 7. Dring visco-elastic flid flows, maximm discrepanc is noticed in comparison with the Newtonian flid flow. Also, a decelerating trend is observed in case of flow past an externall heated plate for both Newtonian and non-newtonian flid. Knowing the velocit field, it is important from a practical point of view to know the effect of viscoelastic parameter on shearing stress or viscos drag. Figres 8 to 2 depict the shearing stress at the plate for the visco-elastic flid in comparison with the E-ISSN: Isse 4, Volme 8, October 23

7 Newtonian flid for varios vales of flow parameters involved in the soltion. The graphical illstration has been given for both flow past an externall heated plate (Gr < ) and externall cooled plate (Gr > ). These graphs conclde that the growth of visco-elasticit raise the viscos drag at the srface. The shearing stress experienced b the visco-elastic flids past a heated plate is lesser in magnitde in compared to the flow past a cooled plate. -.5 τ - Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= τ Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= M Figre 8: shearing stress against M for Pr=, Gr=7,-7, Sc=, Gm= The effects of Lorentz force on viscos drag are investigated in figre 8. The shearing stress experienced b the Newtonian flid (k=) experiences a declined trend. In non-newtonian flid flow mechanism, the visco-elastic flid (k=.) feels a redction in shearing stress dring the growth of Hartmann nmber (M), bt when the elasticit factor k.2, an enhancement is noticed in shearing stress at the inclined srface. Figre 9, illstrates the natre of shearing stress of both tpes of flids against Prandtl nmber (Pr). Dring the flow past a cooled srface, the shearing stress prodced at the srface will sbde its magnitde bt a reverse behavior is observed for the flow past a heated plate Pr Figre 9: shearing stress against Pr for M=3, Gr=7,- 7, Sc=, Gm= τ Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= Gm Figre : shearing stress against Gm for M=3, Gr=7, -7, Sc=, Pr= On the other end, the shearing stress experiences a rising trend dring the growth of Grashof nmber for mass transfer (figre ). Schmidt nmber characterizes the behavior of simltaneos momentm and concentration diffsion. Figre exhibits that the increasing vale of Sc declines the shearing stress at the plate for both tpes of flids along with the increasing vales of visco-elastic parameter (k). E-ISSN: Isse 4, Volme 8, October 23

8 τ Fig : shearing stress against Sc for M=3, Gr=7,-7, Gm=, Pr= τ Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= Sc Gr=7,k= Gr=7,k=. Gr=7,k=.2 Gr=-7,k= Gr=-7,k=. Gr=-7,k= Ec Fig 2: shearing stress against Ec for M=3, Gr=7,-7, Sc=, Pr=, Gm= Eckert nmber (Ec) characterizes the dissipation of mechanical energ into thermal energ de to the presence of viscosit. Figre 2, reveals the behavior of shearing stress against Eckert nmber for varios vales of visco-elastic parameter. The viscos drag formed b both Newtonian and non- Newtonian flid experience a diminishing trend dring the growth of Eckert nmber. The rate of heat transfer and the rate of mass transfer do not differ significantl affected b the visco-elastic parameter. 4 Conclsion The behavior of free convective MHD bondar flow of a visco-elastic flid past an inclined moving srface in presence of heat and mass transfer has been investigated in this paper. Some of the important points are conclded as below:. The deceleration of the flid flow is sperior in the neighborhood of the plate in comparison with flid flowing at some distance from the plate. 2. The flid flow is accelerated dring the enhancement of visco-elastic parameter. 3. Both Newtonian and non-newtonian flids experience a decreasing trend nder the amplified magnitde of Hartmann nmber. 4. A decelerating trend is observed in case of flow past an externall heated plate for both Newtonian and non-newtonian flid in comparison with the flow past an externall cooled plate. 5. The growth of visco-elasticit raises the viscos drag at the srface. 6. Dring the flow past a cooled srface, the shearing stress prodced at the srface will sbde its magnitde. 7. The viscos drag formed b both Newtonian and non-newtonian flid experience a diminishing trend dring the growth of Eckert nmber. References: [] B. Gebhart and L. Pera. The natre of vertical convection flows reslting from the combined boanc effects of thermal and mass diffsion, Int. J. Heat Mass transfer, 4, 97, [2] N. P. Singh and A. K. Singh, MHD free convection and mass transfer flow past a flat plate, The Arabian J. Sci. & Engg., 32, A, 27, [3] M. Q. Al-Qadat and T. A. Al-Azab. Inflence of chemical reaction on transient MHD free convective flow over a moving vertical plate, Emirates J. Engg. Research, 2(3), 27, 5-2. [4] J. Zeco and S. Ahmed. Combined heat and mass transfer b mixed convection MHD E-ISSN: Isse 4, Volme 8, October 23

9 flow along a poros plate with chemical reaction in presence of heat sorce, Appl. Math. Mech. - Engl. Ed., 3(), 2, [5] G. Palani and U. Srikanth. MHD flow past a semi-infinite vertical plate with mass transfer, Non-linear analsis Modelling and Control, 4(3), 29, [6] R. C. Chadhar, And A. Jain. MHD heat and mass diffsion flow b natral convection past a srface embedded in a poros medim, Theoret. Appl. Mech., 36 (), 29, -27. [7] A. Sahin. Inflence of chemical reaction on transient MHD free convective flow over a vertical plate in slip-flow regime, Emirates J. Engg. Research, 5(), 2, [8] O. D. Makinde. On MHD bondar laer flow and mass transfer past a vertical plate in a poros medim with constant heat flx, Int. J. Nm. Meth. Heat & Flid Flow, 9(3/4), 29, [9] E. M. Sparrow and R. D. Cess. The effect of a magnetic field on free convection heat transfer, Int. J. Heat Mass Transfer, 3(4), 96, [] N. Rile. Magneto-hdrodnamic free convection, J. Flid Mech., 8(4), 964, [] H. K. Kiken, Magneto-hdrodnamic free convection in a strong cross field, J. Flid Mech., 4, 97, [2] K. R. Singh and T. G. Cowling. Effect of magnetic field on free convective flow of electricall condcting flids past a semi infinite flat plate, Qart. J. Mech. Appl. Math., 6, 963, -5. [3] E. M. Crammer and S. I. Pai. Magnetoflid Dnamics for Engineering and Applied Phsicists, McGraw-Hill, New York. 974 [4] M. A. Hossain, I. Pop and M. Ahmed. MHD free convection flow from an isothermal plate inclined at a small angle to the horizontal, J. Theo. Appl. Flid Mech.,, 996, [5] B.C. Sakiadis. Bondar laer behavior on continos solid srfaces: I. The bondar laer eqations for two dimensional and axi-smmetric flow. American Institte of Chemical Engineers, 7(), 96, [6] B. C. Sakiadis, Bondar Laer Behavior on Continos Solid Srfaces: II Bondar Laer on a Continos Flat Srface, AIChE Jornal, 7(2), 96, [7] F. K. Tso, E. M. Sparrow and R. J. Goldstein, Flow and Heat Transfer in the Bondar Laer on a Continos Moving Srfaces, International Jornal of Heat and Mass Transfer,, (2), 967, [8] L. J. Crane, Flow past a Stretching Plate, Zeitschrift für Angewandte Mathematik nd Phsik, 2(4), 97, pp [9] L.E. Erickson, L.T. Fan and V.G. Fox. Heat and mass transfer on a moving continos flat plate with sction or injection. Indstrial & Engineering Chemistr Fndamentals, 5(), 966, [2] P.S. Gpta and A.S. Gpta. Heat and mass transfer with sction and blowing. The Canadian Jornal of Chemical Engineering, 55(6), 977, [2] K. Vajravel. Hdromagnetic flow and heat transfer over a continos, moving, poros flat srface. Acta Mechanica, 64(3-4), 986, [22] K. Vajravel. Hdromagnetic convection at a continos moving srface. Acta Mechanica, 72(3-4), 988, [23] B.R. Kmar, D.R.S. Raghraman and R. Mthcmaraswam, Hdromagnetic flow and heat transfer on a continosl moving vertical srface. Acta Mechanica, 53(3-4), 22, [24] K. R. Rajagopal, T. Y. Na and A. S. Gpta, Flow of a Visco-Elastic Flid over a Stretching Sheet, Rheologica Acta, 23(2), 984, [25] K. R. Rajagopal, T. Y. Na and A. S. Gpta, A Non Similar Bondar Laer on a Stretching Sheet in a Non-Newtonian Flid with Uniform Free Stream, Jornal of Mathematical Phsics, 2, (2), 987, [26] B. S. Dhanpat and A. S. Gpta, Flow and Heat Transfer in a Visco-Elastic Flid over a Stretching Sheet, Inter-national Jornal of Non-Linear Mechanics, 24, (3), 989, [27] N. Ahmad, G. S. Patel and B. Siddappa, Visco-Elastic Bondar Laer Flow past a Stretching Plate and Heat Transfer, Zeitschrift für Angewandte Mathematik nd Phsik, 4(2), 99, [28] B. Siddappa and M. S. Abel, Visco-Elastic Bondar Laer Flow past a Stretching E-ISSN: Isse 4, Volme 8, October 23

10 Plate with Sction and Heat Transfer, Rheologica Acta, 25(3), 986, [29] D. Kell, K. Vjravel and L. Andrews, Analsis of Heat Mass Transfer of a Visco- Elastic, Electricall Condcting Flid past a Continos Stretching Sheet, Nonlinear Analsis: Theor, Methods & Applications, 36(6), 999, [3] M. Sbhas, A. Joshi and R. M. Sonth, Heat Transfer in MHD Visco-Elastic Flid Flow over a Stretching Srface, Zeitschrift für Angewandte Mathematik nd Me- chanik, 8(), 2, [3] R.M. Sonth, S. K. Khan, A. M. Sbhas and K. V. Prasad, Heat and Mass Transfer in a Visco-Elastic Flid Flow over an Accelerating Srface with Heat Sorce/Sink and Viscos Dissipation, Heat and Mass Transfer, 38(3), 22, [32] M. S. Abel, P. G. Siddheshwar and M. M. Nandep-panavar, Heat Transfer in a Visco- Elastic Bondar Laer Flow over a Stretching Sheet with Viscos Dissipation and Non-Uniform Heat Sorce, International Jornal of Mass and Heat Transfer, 5, (5-6), 27, [33] N. Ahmad and M. Mishra, Unstead Bondar Laer Flow and Heat Transfer over a Stretching Sheet, Pro-ceedings of 28th UIT Heat Transfer Congress, Brescia, 2-23 Jne 2, [34] R. Chodhr and U.J. Das Hdromagnetic flow and heat transfer on a visco-elastic flid on a continosl moving vertical srface, International Jornal of Applied Mathematics and Mechanics, 6(3), 2, -. [35] M. Misra, N. Ahmad and Z. U. Siddiqi, Unstead Bondar Laer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Condctivit, World Jornal of Mechanics, 2, 22, E-ISSN: Isse 4, Volme 8, October 23