*Author for Correspondence. Keywords: Radiation, Memory Fluid Flow, Suction, MHD, Viscous Dissipation, Heat Sink and Chemical Reaction

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1 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. RDITION, HET ND MSS TRNSFER EFFECTS ON MGNETOHYDRODYNMIC UNSTEDY FREE CONVECTIVE WLTER S MEMORY FLOW PST VERTICL PLTE WITH CHEMICL RECTION THROUGH POROUS MEDIUM Srinathuni Lavana, *Chenna Kesavaiah D. and Sudhakaraiah. Department of H&S, Naraana Engineering & Technical Campus, Haatnagar, R.R. Dist, T.S, India Department of H & BS, Visvesvaraa College of Engg & Tech, Ibrahimpatnam, R.R. Dist, T.S, India Department of Future Studies, Sri Venkateswara Universit, Tirupathi, ndhra Pradesh, India *uthor for Correspondence BSTRCT n analtical stud is performed to stud the influence of radiation and mass transfer on unstead hdromagnetic free convective memor flow of viscous, incompressible and electricall conducting fluids past an infinite vertical porous plate in the presence of constant suction and heat absorbing sink with chemical reaction taking into an account. pproximate solutions have been derived for the mean velocit, mean temperature and mean concentration using multi-parameter perturbation technique and these are presented in graphical form. The effects of different phsical parameters such as magnetic parameter, Grashof number, modified Grashof number, Prandtl number, Schmidt number, Eckert number, Radiation parameter; Chemical reaction parameter and heat sink strength parameter are discussed. Kewords: Radiation, Memor Fluid Flow, Suction, MHD, Viscous Dissipation, Heat Sink and Chemical Reaction INTRODUCTION The most common tpe of bod force, which acts on a fluid, is due to gravit, so that the bod force can be defined as in magnitude and direction b the acceleration due to gravit. Sometimes, electromagnetic effects are important. The electric and magnetic fields themselves must obe a set of phsical laws, which are expressed b Maxwell s equations. The solution of such problems requires the simultaneous solution of the equations of fluid mechanics and electromagnetism. One special case of this tpe of coupling is known as magnetohdrodnamic. Coupled heat and mass transfer phenomenon in porous media is gaining attention due to its interesting applications. The flow phenomenon in this case is relativel complex than that in pure thermal/solutal convection process. Processes involving heat and mass transfer in porous media are often encountered in the chemical industr and formation and dispersion of fog, distribution of temperature and moisture over agricultural fields and groves of fruit trees, crop damage due to freezing and environmental pollution, in reservoir engineering in connection with thermal recover process, in the stud of dnamics of hot and salt springs of a sea and designing of chemical processing equipment. Underground spreading of chemical waste and other pollutants, grain storage, evaporation cooling, and solidification are a few other application areas where combined thermosolutal convection in porous media are observed. For some industrial applications such as glass production and furnace design and in space technolog applications, such as cosmical flight aerodnamics rocket, propulsion sstems, plasma phsics and spacecraft re-entr aerothermodnamics which operate at higher temperatures, radiation effects can be significant. However, the exhaustive volume of work devoted to this area is ampl documented b the most recent books b Beard and Walters (964) Elastico-viscous boundar laer flows, two dimensional flows near a stagnation point. Trevisan and Bejan (985) have studied the problem of combined heat and mass transfer b free convection in a porous medium. The studied the natural convection phenomenon occurring inside a porous laer with both heat and mass transfer from the side and derived the natural circulation b a combination of buoanc effects due to both temperature and concentration variations. Kafoussias (99) Copright 4 Centre for Info Bio Technolog (CIBTech) 57

2 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. discussed the effects of mass transfer on free convective flow of a viscous fluid past a vertical isothermal cone surface. He obtained the effects of the buoanc parameter and Schmidt number on the flow field. The problem of convective heat transfer in an electricall conducting fluid at a stretching surface with uniform free stream is investigated b Vajravelu and Hadjinicolaou (997). Bestman and djepong (998) analzed unstead hdromagnetic free convection flow with radiative heat transfer in a rotating fluid. Ingham and Pop (998, ), Vafai (), and Pop and Ingham () studied the problem of transient flow of a fluid past a moving semi-infinite vertical porous plate. However, man problem areas which are important in applications, as well as in theor still persist. bd et al., () carried out the finite difference method for the problem of radiation effects on MHD unstead free-convection flow over vertical plate with variable surface temperature. The problem of flow of a micropolar fluid past a moving semi infinite vertical porous plate with mixed radiative convection is studied b Kim and Fedorov (). bel and Mahesha (8) have investigated the effects of thermal conductivit, non-uniform heat source and viscous dissipation in the presence of thermal radiation on the flow and heat transfer in viscoelastic fluid over a stretching sheet, which is subjected to an external magnetic field. Numerical stud of transient free convective mass transfer in a Walters-B viscoelastic flow with wall suction was analzed b Chang et al., (). The stud of electricall conducting viscous fluid that flows through convergent or divergent channels under the influence of an external magnetic field not onl is fascinating theoreticall but also finds applications in mathematical modeling of several industrial and biological sstems. possible practical application of the theor we envisage is in the field of industrial metal casting, the control of molten metal flows. Moreover, the magnetohdrodnamic (MHD) rotating fluids in the presence of a magnetic field are encountered in man important problems in geophsics, astrophsics, and cosmical and geophsical fluid dnamics. It can provide explanations for the observed maintenance and secular variations of the geomagnetic field. It is also relevant in the solar phsics involved in the sunspot development, the solar ccle, and the structure of rotating magnetic stars. The effect of the Coriolis force due to the Earth s rotation is found to be significant as compared to the inertial and viscous forces in the equations of motion. The Coriolis and electromagnetic forces are of comparable magnitude, the former having a strong effect on the hdromagnetic flow in the Earth s liquid core, which plas an important role in the mean geomagnetic field. Several investigations are carried out on the problem of hdrodnamic flow of a viscous incompressible fluid in rotating medium considering various variations in the problem. lagoa et al., (999) studied radiative and free convection effects on MHD flow through porous medium between infinite parallel plates with time-dependent suction. Nield and Bejan (999) convection in porous media. Chowdhur and Islam () were studied the MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate. Seddeek () discussed the problem of thermal radiation and buoanc effects on MHD free convective heat generating flow over an accelerating permeable surface with temperature-dependent viscosit. bel et al., (8) investigated the effects of viscous and ohmic dissipation in MHD flow of viscoelastic boundar laer flow. Mustafa et al.,(8) obtained the analtical solution of unstead MHD memor flow with oscillator suction, variable free stream and heat source. Gireesh et al., (9) analzed the effects of the chemical reaction and mass transfer on MHD unstead free convection flow past an infinite vertical plate with constant suction and heat sink. Gireesh Kumar and Satanaraana () mass transfer effects on MHD unstead free convective Walter s memor flow with constant suction and heat sink. Kesavaiah et al., () investigated effects of the chemical reaction and radiation absorption on an unstead MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with heat source and suction. Srinathuni Lavana and Chenna Kesavaiah (4) Radiation and Soret effects to MHD flow in vertical surface with chemical reaction and heat generation through a porous medium. Viscoelastic flows arise in numerous processes in chemical engineering sstems. Such flows possess both viscous and elastic properties and can exhibit normal stresses and relaxation effects. n extensive range of mathematical models has been developed to simulate the diverse hdrodnamic behavior of these non- Newtonian fluids. Rivlin Ericksen second order model b Metzner and White (965). Kafousias and Copright 4 Centre for Info Bio Technolog (CIBTech) 58

3 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. Raptis (98) have discussed the mass transfer and free convection effects on the flow past an accelerated vertical infinite plate with variable suction of injection. Ji et al., (99) studied the Von Karman Oldrod-B viscoelastic flow from a rotating disk using the Galerkin method with B-spline test functions. n eloquent exposition of viscoelastic fluid models has been presented b Joseph (99). Khan et al., () also investigated the effect of work done b deformation in Walter s liquid B but with uniform heat source. Haat et al., (8) also investigated the effects of work done b deformation in second grade fluid with partial slip condition, in this no account of heat source has been taken into consideration. The Oldrod model (95). The mixture of polmethl mehacrlate and pridine at 5 C containing.5g of polmer per liter behaves ver nearl as the Walter s liquid model-b, (96, 96). Siddappa and Khapate (975) studied the second order Rivlin Ericksen viscoelastic boundar laer flow along a stretching surface. Rochelle and Peddieson (98) used an implicit difference scheme to analze the stead boundar-laer flow of a nonlinear Maxwell viscoelastic fluid past a parabola and a paraboloid. Raptis and Tzianidis (98) have studied the flow of a Walter s liquid B model in the presence of constant heat flux between the fluid and the plate and taking into account the influence of the memor fluid on the energ equation. Rao and Finlason (99) used an adaptive finite element technique to analze viscoelastic flow of a Maxwell fluid. MHD free convection flow of an elasto viscous fluid past an infinite vertical plate was analzed b Samria et al., (99). Renard (997) the upper convicted Maxwell model and the Walters-B model. Both stead and unstead flows have been investigated at length in a diverse range of geometries using a wide spectrum of analtical and computational methods. Rao (999) Johnson Seagalman model, Thermo solutal instabilit of Walter s (model-b) visco-elastic rotating fluid permeated with suspended particles and variable gravit field in porous medium was studied b Sharma and Rana (). Sharma et al., () have analses the Raleigh-Talor instabilit of Walter B elastic-viscous fluid through porous medium. Sharma and Chaudhar () effect of variable suction on transient free convective viscous incompressible flow past a vertical plate with periodic temperature variations in slip flow regime. Ramanamurth et al., (7) have discussed the MHD unstead free convective Walter s memor flow with constant suction and heat sink. Effects of the chemical reaction and radiation absorption on free convection flow through porous medium with variable suction in the presence of uniform magnetic field were studied b Sudheer Babu and Satanaraana (9). Rajesh () Heat source and mass transfer effects on MHD flow of an elasto-viscous fluid through a porous medium. Nabil et al., () Numerical stud of viscous dissipation effect on free convection heat and mass transfer of MHD non-newtonian fluid flow through a porous medium. Rita and Sajal () free convective MHD Flow of a Non-Newtonian fluid past an infinite vertical plate with constant suction and heat sink. Rita and Paban (4) Effects of MHD visco-elastic fluid flow past a moving plate with Double Diffusive convection in presence of heat generation. Pillai et al., (4) investigated the effects of work done b deformation in viscoelastic fluid in porous media with uniform heat source. The present stud is to stud the radiation, heat and mass transfer effects on unstead hdromagnetic free convective memor flow of viscous, incompressible and electricall conducting fluid flow an infinite vertical plate in the presence of chemical reaction taking into an account. Our main interest is to observe how various parameters affect the flow past an infinite vertical porous plate. Therefore, the main idea of the present work is to make a mathematical modeling of this phenomenon and the out purpose is to find the relation between the different parameters and the external forces with the solutions of the problem. Formulation of the Problem Consider unstead hdromagnetic free convective flow of viscous, incompressible and electricall conducting and radiating fluid past an infinite vertical porous plate in the presence of constant suction and heat absorbing sink with chemical reaction. Consider the infinite vertical plate embedded an infinite mass of the fluid. Initiall the temperature and concentration of both being assumed at T andc. t time t, the plate temperature and concentration are raised to T and C, and a periodic temperature and/concentration are assumed to be superimposed on this mean constant temperature/ concentration of the Copright 4 Centre for Info Bio Technolog (CIBTech) 59

4 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. plate. Let the x axis be taken in the verticall upward direction along the infinite vertical plate and axis is normal to it. The magnetic field of uniform strength is applied and induced magnetic field is neglected. Boussineq s approximation, the problem is governed b the following set of equations. v () u u u u u B v g T T g C C B u t t T T T qr u v S T T t Cp Cp C C C v D Kr C C (4) t From () we have v v (5) On disregarding the Joulean heat dissipation, the boundar conditions of the problem are: it it u, v v, T Tw Tw T e, C Cw Cw Ce at (6) u, T T, C C as Introducing the non-dimensional quantities and parameters: u v tv T T C C K u,, t, T, C, K v 4 T T C C C w w p S v Kr B v Pr, Sc, S, Ec, Kr, Rm D v C T T v p w B g T T g C C 4I M Gr Gc R v v v C v w w,,, p where Gr is the thermal Grashof number, Gc is modified Grashof Number, Pr is Prandtl Number, M is the magnetic field, Sc is Schmdit number, Kr is Chemical Reaction, K is Porous Permeabilit, S is Heat source parameter, R is the radiation parameter respectivel. The radiative heat flux q is given b equation (5) in the spirit of Cogl et al., (968) q r where 4 T T I r eb I Kw d, Kw is the absorption coefficient at the wall and eb is Planck s function, T I is absorption coefficient The equations (), () and (4) reduce to following non-dimensional form: u u u u u GrT GcC R m M u 4 t t Pr T T T u Pr S RPrT Ec Pr 4 t Copright 4 Centre for Info Bio Technolog (CIBTech) 6 () () (7) (8) (9)

5 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. Sc C C C Sc KrSc C () 4 t (fter dropping the asterisks) The corresponding boundar conditions are it it u, T e, C e at () u, T, C as Solution of the Problem To solve equations (8), (9) and (), we assume to be ver small and the velocit, temperature and concentration in the neighborhood of the plate as it u u e u it nt T T e T C C e C Where u, T, and C are mean velocit, mean temperature and mean concentration respectivel. Using () in equations (8), (9) and (), equating harmonic and non-harmonic terms for mean velocit, mean temperature and mean concentration, after neglecting coefficient of, we get R u u u Mu Gr T GcC () m T PrT S R PrT Pr Ecu (4) C Sc C Sc Kr C (5) The equation () is third order differential equation due to presence of elasticit. Therefore u is expanded using (Beard and Walters rule, 964; Chowdhar and Islam, ) u u R u (6) m Zero-Order of R m u u Mu Gr T GcC (7) First-Order of R m u u Mu u (8) Using multi parameter perturbation technique and assuming Ec, we write u u Ec u u u Ec u T T EcT C C Ec C Using equations (9) in equations (4), (5), (7) and (8) and equating the coefficient of Ec and Ec, we get the following sets of differential equations Zero order of Ec u u Mu Gr T GcC () u u Mu u () T PrT S R PrT () C Sc C Sc Kr C () Copright 4 Centre for Info Bio Technolog (CIBTech) 6 () (9)

6 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. First order of Ec u u Mu Gr T GcC (4) u u Mu u (5) T PrT S R PrT Pr Ecu (6) C Sc C Sc Kr C (7) Here primes denote differentiation with respect to The respective boundar conditions are u u u u, T, T, C, C (8) u u u u, T T, C C Solving these differential equations from () (7) using boundar conditions (8), then making use of equations (9) and finall with the help of (6), we obtain mean velocit u, mean temperature T and mean concentration C as follows. u e e e Ec e e e e e m6 m m8 m m8 m6 m 8 9 e e e Rm e e e e m6m8 m m6 m m8 m4 m8 m6 m m Ec e e e e e e e m 4 m m8 m6 m m6 m8 m m mm8 m6 e e 4 m6 m8 m6 m m6 m8 m m6 m m8 m T e Ec B e B e B e B e B e B e B e C e m RESULTS ND DISCUSSION 5 Sc=.65,M=.,S=-.5,Kr=.5,R=. Ec=., Pr=.5, Gc=5., Rm=. u 5 Gr=5.,.,5.,. 5 4 Figure (): : Mean velocit profiles for for different values values of Gr of Gr Copright 4 Centre for Info Bio Technolog (CIBTech) 6

7 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. 5 Sc=.65,M=.,S=-.5,Kr=.5,R=. Ec=., Pr=.5, Gr=5., Rm=. u 5 Gc=5.,.,5.,. 5 4 Figure (): Mean velocit profiles for different values of Gc Figure : Mean velocit profiles for different values of Gc 5 Sc=.65, S=-.5, Kr=.5, Ec=. Pr=.5,Gr=5.,Gc=-5.,R=.,Rm=. u 5 M=.,.,., Figure (): Mean velocit profiles for different values of M Figure : Mean velocit profiles for different values of M 5 Sc=.65, M=., Kr=.5, Ec=. Pr=.5,Gr=5.,Gc=5.,R=.,Rm=. u 5 5 S=.5,.,.5, Figure (4): Mean velocit porifles for different values of S Figure 4: Mean velocit profiles for different values of S The problem of radiation and mass transfer on unstead hdromagnetic free convective memor flow of incompressible and electricall conducting fluids past an infinite vertical porous plate in the presence of constant suction and heat absorbing sink with chemical reaction has been formulated, analsed and solved b using multi-parameter perturbation technique. pproximate solutions have been derived for the mean velocit, mean temperature and mean concentration. The effects of the flow parameters such as Hartmann number (M), suction parameter (S), thermal Grashof number for Gr and Solutal Grashof number Gc, Schmidt number (Sc), Prandtl number (Pr) and Eckert number Ec. n insight into the effects of these parameters of the flow field can be obtained b the stud of the mean velocit components, mean temperature and mean concentration distributions. The components of the velocit u mean temperature T and mean concentration C sets of the values of the parameters. have been plotted against the dimension for several Copright 4 Centre for Info Bio Technolog (CIBTech) 6

8 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. Sc=.65, M=., S=-.5, Ec=. Pr=.5,Gr=5.,Gc=5.,R=.,Rm=. 5 Sc=.65, M=., S=-.5, Ec=. Pr=.5,Gr=5.,Gc=5.,Kr=.5,Rm=. 8 u 6 Kr=.,.,.,.4 u R=.5,.,.5,. 4 Figure (5): Mean velocit profiles for different values of Kr Figure 5: Mean velocit profiles for different values of Kr Figure (6): Mean velocit porfiles for different values of R Figure 6: Mean velocit profiles for different values of R M=., S=-.5, Kr=.5, Ec=. Pr=.5,Gr=5.,Gc=5.,R=.,Rm=..8 Sc=.65,M=.,S=.5,Kr=. Pr=., Gr=5., Gc=5.,R=. u 8 6 Sc=.6,.7,.8,.9 T.6.4 Ec=.,.,., Figure (7): Mean velocit profiles for different values of Sc Figure 7: Mean velocit profiles for different values of Sc. 4 5 Figure (8): Mean temperature for different values of Ec Figure 8: Mean velocit profiles for different values of Ec.8 Sc=.65, M=., S=.5,Kr=. Gr=5.,Gc=5.,R=.,Ec=..8 Sc=.65, M=., S=.5,Kr=. Ec=.,Pr=.,Gc=5.,R=..6.6 Gr=5.,.,5.,. T.4 Pr=.7,.8,.9,. T Figure (9): Mean temperature profiles for different values of Pr Figure 9: Mean velocit profiles for different values of Pr 5 5 Figure (): Mean temperature profiles for different values of Gr Figure : Mean velocit profiles for different values of Gr Copright 4 Centre for Info Bio Technolog (CIBTech) 64

9 C C International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al..8 Sc=.65, M=., S=.5,Kr=. Ec=.,Pr=.,Gr=5.,R=..8 Sc=.65, M=., Pr=.,Kr=. Gr=5.,Gc=5.,R=.,Ec=. T.6.4 Gc=5.,.,5.,. T.6.4 S=.,.,., Figure (): Mean temperature profiles for different values of Gc Figure : Mean velocit profiles for different values of Gc 4 5 Figure (): Mean Temperature profiles for different values of S Figure : Mean velocit profiles for different values of S.8 Sc=.6.8 Kr= Kr=.,.,.,4..4 Sc=.6,.,.4, Figure (): Concentration profiles for different values of Kr Figure : Mean velocit profiles for different values of Kr 4 5 Figure (4): Concentration for different values of Sc Figure 4: Mean velocit profiles for different values of Sc Mean velocit profiles shows from figures () (7). Figures () - (4) represent the mean velocit profiles due to variations in thermal Grashof number Gr, Solutal Grashof number Gc, Magnetic parameter M and Sink strength parameter S. It is observed that the mean velocit increases with increase of thermal Grashof number and solutal Grashof number. It also observed that mean velocit decrease with increase in magnetic parameter and sink strength parameter, this is an indication that the force which tends to oppose the fluid flow increases with increase in the magnetic field parameter. Figures (5) - (7) reveals the mean velocit profiles due to variations in chemical reaction parameter Kr, radiation parameter R and Schmidt number Sc. It is noticed that whenever radiation and Schmidt number increases the mean velocit decrease. lso, from the figures, it can be concluded that the Newtonian fluid shows a rising trend as compared to visco-elastic fluid for both kind of surface sstems. Further, slightl awa from the plate the dispersion in the velocit profiles is considerable as compared to the initial stage. The reverse effect observed in chemical reaction parameter Kr in mean velocit. Mean temperature profiles shows from figures (8) - (). These figures reveals the mean temperature profiles due to variations in Eckert number Ec, Prandtl number Pr, thermal Grashof number Gr, Solutal Grashof number Gc and Sink strength parameter S. It is noticed that whenever Prandtl number, sink strength parameter and Eckert number increases the mean temperature decrease. It is also Copright 4 Centre for Info Bio Technolog (CIBTech) 65

10 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. observed that the increases in Eckert number, Prandtl number and Sink strength parameter causes the decrease in mean temperature; while the mean temperature profile due to variations in Thermal Grashof number Gr, Solutal Grashof number Gc. It is noticed that whenever thermal Grashof number and solutal Grashof number increases the mean temperature also increase. Mean concentration profiles shown in figure () and (). From these figures it is observed that the increases in chemical reaction parameter Kr and Schmidt number Sc causes the decrease in mean concentration. Conclusion The results indicate that as the radiation and magnetic parameters increase, the value of the velocit decreases. This conclusion meets the logic of the magnetic field exerting a retarding force on the free convection flow. Moreover, it is noted that there is a fall in the temperature due to the heat created b the viscous dissipation, free convection and heat source. n increase in the Grashof number, leads to a rise in the magnitude of fluid velocit due to enhancement in buoanc force. The peak value of the velocit an increase rapidl near the porous plate as buoanc force for heat transfer increases and then decas the free stream velocit. n increase in the chemical reaction parameter tends to increase the velocit and decrease the species concentration. The hdrodnamic and the concentration boundar laer become thin as the reaction parameter increases. n increase in Prandtl number leads to decrease in the thermal boundar laer and in general lower average temperature within the boundar laer region being the smaller values of Pr are equivalent to increase in the thermal conductivit of the fluid and therefore heat is able to diffuse awa from the heated surface more rapidl for higher values of Prandtl number. Hence for smaller Prandtl number, the rate of heat transfer is reduced. This problem has man scientific and engineering applications such as: Flow of blood through the arteries. Soil mechanics, water purification, and powder metallurg. Stud of the interaction of the geomagnetic field with in the geothermal region. The petroleum engineer concerned with the movement of oil, gas and water through the reservoir of an oil or gas field. It is hoped that the present work will serve as a vehicle for understanding more complex problems involving the various phsical effects investigated in the present problem. REFERENCES bd El-Nab M, Elbarbar EME and bdelazem NY (). Finite difference solution of radiation effects on MHD unstead free convection flow over vertical plate with variable surface temperature, Journal of pplied Mathematics 65. bel MS and Mahesha N (8): Heat transfer in MHD viscoelastic fluid over a stretching sheet with variable thermal conductivit, non-uniform heat source and radiation, pplied Mathematical Modelling () bel MS, Sanjaanand Emmanuel and Nandeppanavar Mahantesh M (8). Viscoelastic MHD flow and heat transfer over a stretching sheet with viscous and ohmic dissipations, Communication Nonlinear Science and Numerical Simulation (9) lagoa KD, Ta G and bbe TM (999). Radiative and free convection effects of a MHD flow through porous medium between infinite parallel plates with time dependent suction, strophsics and Space Science Beard DM and Walters K (964). Elastico-viscous boundar laer flows, two dimensional flows near a stagnation point. Proceedings of the Cambridge Philosophical Societ Copright 4 Centre for Info Bio Technolog (CIBTech) 66

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13 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. Vajravelu K and Hadjinicolaou (997). Convective heat transfer in an electricall conducting fluid at a stretching surface with uniform free stream, International Journal of Engineering Science 5() 7. Walter K (96). The motion of an elastic-viscous liquid contained between concentric spheres, Quarterl Journal of Mechanics and pplied Mathematics () 5-. Walter K (96). The motion of an elastic-viscous liquid contained between coaxial clinders (II), Quarterl Journal of Mechanics and pplied Mathematics (4) Walters K (96). Non-Newtonian effects in some elastico-viscous liquids whose behavior at small rates of shear is characterized b a general linear equation of state, Quarterl Journal of Mechanics and pplied Mathematics 5() PPENDIX Sc Sc 4Kr Sc Pr Pr 4S RPr m m 4, m 6 m 4M m8 m m4 m6 Gr Gc,,, m m M m m M m 8 4 m m M m 6 m 5, 6, m6 m6 M m m M GrB7 GrB GrB 8, 9, m m M 4m8 m8m 4m6 m6 M GrB GrB4 GrB5,, 4m m M m m m m M m m m m M 4 GrB m m m m M , m m m M m, m m M 8 7 8m, m m M m6 m m m8 m m m m M m m m m M 8 8 Pr Ecm8 B, 4m Pr m S R Pr B m m m M Copright 4 Centre for Info Bio Technolog (CIBTech) 69 8m, m m M 4 m m6, m m m m M Pr Ecm6 4 Pr Pr m m S R Pr Ecm B 4m Pr m S R Pr 4 m m Pr m m S R B Pr Ecm m Pr

14 International Journal of Phsics and Mathematical Sciences ISSN: 77- (Online) 4 Vol. 4 () Jul- September, pp. 57-7/Lavana et al. B B 6 m m Pr m m S R 5 Pr Ecm m m m Pr m m S R 6 Pr Ecm m 8 8 Pr B B B B B B B, Pr Copright 4 Centre for Info Bio Technolog (CIBTech) 7