Corresponding Author (corresponding author) (Received October 7, 2014; accepted April 15, 2015) ABSTRACT

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1 Journal of Applied Fluid Mechanics Vol. 9 No. 3 pp Available online at.jafmonline.net ISSN EISSN DOI:.8869/acadpub.jafm Soret and Dufour Effects on MHD Mixed onvection Heat and Mass Transfer of a Stagnation Point Flo toards a Vertical Plate in a Porous Medium ith hemical Reaction Radiation and Heat Generation S. Karthikeyan M. Bhuvanesari S. Sivasankaran and S. Rajan Department of Mathematics Erode Arts and Science ollege Erode 6389 Tamilnadu India Institute of Mathematical Sciences University of Malaya Kuala Lumpur 563 Malaysia orresponding Author sd.siva@yahoo.com (corresponding author) (Received October 7 4; accepted April 5 5) ABSTRAT The objective of this paper is to analyze the effects of heat and mass transfer in the presence of thermal radiation internal heat generation and Dufour effect on an unsteady magneto-hydrodynamic mixed convection stagnation point flo toards a vertical plate embedded in a porous medium. The non-linear partial differential equations governing the flo are transformed into a set of ordinary differential equations using suitable similarity variables and then solved numerically using shooting method together ith Runge- Kutta algorithm. The effects of the various parameters on the velocity temperature and concentration profiles are depicted graphically and values of skin- friction coefficient Nusselt number and Sherood number for various values of physical parameters are tabulated and discussed. It is observed that the temperature increases for increasing values of the internal heat generation thermal radiation and the Dufour number and hence thermal boundary layer thickness increases. Keyords: Stagnation point flo; Mixed convection; Porous medium; Heat generation; Thermal radiation. NOMENLATURE B c p c s Df D m g Gr Gc k K K ' K K ~ KT M Nu Pr S strength of magnetic field concentration of the fluid specific heat concentration susceptibility Dufour number diffusion coefficient acceleration due to gravity Thermal Grashof number Solutal Grashof number thermal conductivity permeability parameter mean absorption coefficient chemical reaction permeability of the porous medium thermal diffusion ratio magnetic field parameter Nusselt number Prandtl number heat generation/absorption parameter Ri Solutal Richardson number Sc Schmidt number Sh Sherood number Sr Soret number T temperature of the fluid (u v) velocity components (x y) cartesian coordinates * e thermal diffusivity thermal expansion coefficient concentration expansion coefficient dimensionless variable dimensionless temperature viscosity kinematic viscosity density electrical conductivity Stefan-Boltzmann constant fluid electrical conductivity

2 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp Q heat generation/absorption coefficient q r radiative heat flux R radiation parameter Re x Reynolds number Ri T Thermal Richardson number. INTRODUTION Heat and mass transfer by natural convection in a fluid-saturated porous medium has alays been an active area of research due to the plenty of applications such as oil recovery geothermal reservoirs drying of porous solids and cooling of nuclear reactors. The flo in the neighborhood of a stagnation line has attracted many researchers during the last century. Stagnation-point flos may be produced by a solid all hile in other cases a free stagnation point or line exists in the fluid domain. Hiemenz (9) made the first contribution in this direction by examining the to-dimensional flo of a fluid near a stagnation point and obtained an exact similarity solution of the governing equations. Thermal effects in such a flo ere introduced by Eckert (94). The analysis by Devi et al. (99) presented the unsteady mixed convection in stagnation point flos for arbitrary distribution of surface temperature and concentration or surface heat and mass flux conditions. They found that dual solutions exist for a certain range of the buoyancy parameter hen the flo is opposing. The effect of radiation on MHD flo and heat transfer problems has become industrially more important. Many engineering processes occur at high temperatures and hence the knoledge of radiation heat transfer is essential for designing appropriate equipment. In designing some of the complex equipments such as Nuclear poer plants gas turbines and various propulsion devices for aircrafts the knoledge of radiation heat transfer becomes essential. In vie of these many authors have made contributions to the study of fluid flo ith thermal radiation. The problem of radiative natural convection heat transfer flo past an inclined surface embedded in a porous medium is investigated by Lee et al. (8). They observed that velocity increases and temperature decreases by increasing porosity parameter. Later this ork has been extended by Bhuvanesari et al. (). They have analyzed this problem ith the inclusion of internal heat generation and found that both the velocity and temperature increase significantly hen the value of the heat generation parameter increases. Karthikeyan et al. (3) have studied the influence of thermal radiation on magnetoconvection flo of an electrically conducting fluid past a semi-infinite vertical porous plate embedded in a porous medium ith time dependent suction. Radiation effects on boundary layer flo and heat transfer of a fluid ith variable viscosity along a symmetric edge is investigated by Mukhopadhyay (9). Babu et al. (4) have analyzed the steady MHD boundary layer flo due to an exponentially heat source parameter stream function Subscripts at all at free stream stretching sheet ith radiation in the presence of mass transfer and heat source or sink. onvective heat and mass transfer ith chemical reaction plays an important role in meteorological phenomena burning of haystacks spray drying of milk fluidized bed catalysis and cooling toers. Bhuvanesari et al. (9) examined the convective flo heat and mass transfer of an incompressible viscous fluid past a semi-infinite inclined surface ith first-order homogeneous chemical reaction by Lie group analysis. Soret and Dufour effects on similarity solution of hydro magnetic heat and mass transfer over a vertical plate ith a convective surface boundary condition and chemical reaction are studied by Gangadhar (3). Lie group analysis of natural convection over an inclined semi-infinite plate ith variable thermal conductivity is investigated by Bhuvanesari and Sivasankaran (4). They observed that the velocity and temperature for all angles increase hen the thermal conductivity parameter increases. An analysis of visco-elastic free convective MHD flo over a vertical porous plate through porous media in presence of radiation and chemical reaction is presented by houdhury and Kumar Das (4). An investigation of the effects of Hall current and rotation on unsteady hydro-magnetic natural convection flo ith heat and mass transfer of an electrically conducting viscous incompressible and time dependent heat absorbing fluid past an impulsively moving vertical plate in a porous medium taking thermal and mass diffusions into account is carried out by Seth et al. (5). Various aspects of stagnation-point flo and heat transfer have been studied by many researchers and thus a considerable literature have been generated on this domain. The problem of the flo near a stagnation point of a heated surface embedded in a fluid saturated porous medium has been subject of several numerical and analytical studies. Simultaneous heat and mass transfer by natural convection in a to dimensional stagnationpoint flo of a fluid saturated porous medium using the Darcy Boussinesq model including suction/bloing Soret and Dufour effects are studied by Postelnicu (). There has been many orks related to mixed convection flo near the stagnation point on a vertical surface/plate in a porous medium. The problem of unsteady mixed convection boundary layer flo near the region of a stagnation point on a heated vertical surface embedded in a fluid-saturated porous medium is considered by Nazar et al. (4). The stagnation-point flo due to a stretching sheet has received much attention because of its significance in industry such as extrusion of 448

3 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp polymers glass fiber the cooling of metallic plate and the aerodynamics. Further the inclusion of magnetic field in the study of stagnation point flo has many practical applications for example the cooling of turbine blades here the leading edge is a stagnation point or cooling the nose cone of the rocket during re-entry. Hayat et al. () analyzed the problem of steady MHD to-dimensional mixed convection boundary layer flo of a viscous and incompressible fluid near the stagnation-point on a vertical stretching surface embedded in a fluidsaturated porous medium and thermal radiation. A study on the effects of heat generation/absorption and chemical reaction on unsteady MHD flo heat and mass transfer near a stagnation point of a three dimensional porous body in the presence of a uniform magnetic field is presented by hamkha et al. (). Sharma and Singh (9) have investigated the effects of variable thermal conductivity heat source/sink and variable free stream on flo of a viscous incompressible electrically conducting fluid and heat transfer on a non-conducting stretching sheet in the presence of transverse magnetic field near a stagnation point. It has been found that the rate of heat transfer at the sheet increases due to increase in the Prandtl number hile it decreases due to increase in the Hartmann number. Sinha (4) presented a numerical solution on a MHD stagnation-point flo ith heat transfer over a shrinking sheet in the presence of magnetic field. The effects of slip velocity and thermal slip on an electrically conducting viscous incompressible fluid are investigated in this paper. Steady to-dimensional stagnation point flo and heat transfer of a nanofluid over a porous stretching sheet ith heat generation is investigated analytically by Malvandi et al. (4). Their results indicate that the reduced Nusselt number declines ith increasing in Leis number hereas the reduced Sherood number increases. Mahapatra and Gupta () investigated the steady to-dimensional stagnation-point flo of an incompressible viscous fluid toards a stretching surface. Mahapatra et al. (7) further analyzed the same problem for the MHD flo. The steady to-dimensional stagnation-point flo of a viscous and incompressible fluid over a stretching vertical sheet in its on plane as investigated theoretically by Ishak et al. (6). Unsteady boundary layer flo in the stagnation point region on a stretching flat sheet here the unsteadiness is caused by the impulsive motion of the free stream velocity and by the suddenly stretched surface has been analyzed by Nazar et al. (4). Singh et al. () considered the convective heat and mass transfer in the presence of the volumetric rate of heat generation/ absorption hich depends on local specie concentration. This ork has been extended by Makinde () to include hydro-magnetic mixed convection stagnation point flo ith thermal radiation past a vertical plate embedded in a porous medium. Kazem et al. () have analytically studied the problem of the todimensional stagnation-point flo in a porous medium of a viscous incompressible fluid impinging on a permeable stretching surface ith heat generation/absorption. Motivated by the above studies e investigate the behavior of MHD mixed convection stagnationpoint flo toards a vertical plate embedded in a highly porous medium ith heat and mass transfer in the presence of thermal radiation internal heat generation Soret and Dufour effects in this paper.. MATHEMATIAL ANALYSIS onsider the MHD steady to-dimensional stagnation-point flo of a viscous incompressible electrically conducting fluid near a stagnation point at a surface coinciding ith the plane y the flo being in a region y. The configuration and the coordinate system of this stagnation point flo are shon in the Fig.. To equal and opposing forces are applied along the x-axis so that the surface is stretched keeping the origin fixed. The potential flo that arrives from the y-axis and impinges on a flat all placed at y divides into to streams on the all and leaves in both the directions. The velocity distribution in the potential flo in the neighborhood of the stagnation point is given byu cx here c is a positive constant. A constant magnetic field B is applied in the y direction. Since the magnetic Reynolds number of the flo is taken to be very small the induced magnetic field is neglected. It is also assumed that the external electric field is zero and the electric field due to polarization of charges is negligible. The temperature and the concentration of the ambient fluid are T and and those at the stretching surface are T and respectively. Fig.. Physical model and coordinate system. It is also assumed that the viscous and electrical dissipation are neglected. The MHD equations for to dimensional stagnation-point flo of heat and mass transfer toards a heated vertical plate are u x v y () 449

4 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp u u u u v g( T T ) g*( ) x y y eb uuu K du dx r ( ) m T y sp y T T T q D K u v QTT x y k y cc () T T ( ) T T ( ) (9) here ( x y ) is the stream function defined by u y and v x so as to satisfy Eq.() identically. By substituting Eqs. (7) - (9) into Eqs. () - (5) e obtain the folloing nonlinear ordinary differential equations: DmKT T u v Dm K x y y Tm y The boundary conditions are u v T T at y (3) (4) u U cx T T as y (5) Using the Rosseland approximation the radiative heat flux in the y-direction is given by * 4 4 T qr 3K y (6) * here and K are the Stefan-Boltzmann constant and the mean absorption coefficient respectively. On the assumption that the temperature differences ithin the flo are 4 sufficiently small e can express T as a linear function of temperature T using a truncated Taylor series about the free stream temperature T as T 4T T 3T (7) We introduce the folloing non-dimensional variables 3 g T T x Gr Q S c * 3 T 4 R kk K Pr ck 3 g x Gc DmKT Df cscp T T Sc D m M e B (8) c KT T T K Sr U x r Re x Tm Gr T Re Gc Ri Ri x Rex No e introduce the folloing similarity transformations as c y ( x y) vcxf ( ) v f ff f RiTRi ( K M)( f ) () 4 Pr 3 R f S Df () ScSrScf rsc () here K is the porous medium permeability parameter R is the thermal radiation parameter Gr is the local thermal Grashof number Gc is the local solutal Grashof number Ri T is the thermal Richardson number Ri is the solutal Richardson number Pr is the Prandtl number Sr is the Soret number Df is the Dufour number Sc is the Schmidt number M is the magnetic field intensity parameter r is the chemical reaction parameter and S is the internal heat generation parameter. The corresponding boundary conditions (5) no becomes f f at f as. (3) The set of equations ()-() under the boundary conditions (3) have been solved numerically by applying the shooting iteration technique together ith Runge-Kutta fourth-order integration scheme. From the process of numerical computation theskin-friction coefficient the local Nusselt number and the local Sherood number are respectively given by f Sh here q U xq m Dm u. y y xq Nu k T * 4 T 4 T k y 3K y qm D. y y T y y (4) (5) Substituting (7) (8) and (3) into (5) e obtain the expressions for the skin-friction coefficient the 45

5 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp local Nusselt number and the local Sherood number as / Re x f f () (6) / Rex Nu 4 R / 3 / Re x Sh (). 3. RESULTS AND DISUSSION Equations ()-() constitute a highly non-linear coupled boundary value problem of third and second order for hich the closed-form solution cannot be obtained. Hence the problem has been solved numerically using shooting technique along ith fourth order Runge Kutta scheme. The basic idea of the shooting method for solving boundary value problem is to try to find appropriate initial condition for hich the computed solution hits the target so that the boundary conditions at the other points are satisfied. Furthermore the higher order non-linear differential equations are converted into simultaneous linear differential equations of first order and they are further transformed into initial valued problem. The iterative solution procedure as carried out until the error in the solution became less than a predefined tolerance level. We have computed the values of the skin-friction coefficient the local heat transfer rate and the local mass transfer rate and made a comparison ith the corresponding values obtained by Makinde () in Table. One can observe that the present results are in good agreement ith that of the ork by Makinde (). Table presents the variations in the values of the skin-friction coefficient f () the local heat transfer rate and the local mass transfer rate () for different values of the governing parameters K M Ri T Ri S R r Sr and Df hen Pr =. Sc =.5. We observe that the skinfriction at the plate surface increases ith increasing values of K M Ri Ri T S and Df and decreases ith increasing values of R r and Sr. It can thus be understood that the influence of magnetic field buoyancy forces permeability of porous medium thermal radiation and Dufour effect tend to increase skin-friction. The rise in the local Nusselt number is seen for increasing values of K and R. But it decreases for increasing values of Ri T S r Sr and Df. Further e notice that except for the thermal radiation parameter R the local Sherood number increases for increasing values of all other parameters. The effects of the permeability parameter K magnetic field parameter M thermal Richardson number Ri T solutal Richardson number Ri and the heat generation parameter S on the velocity field have been analyzed in Figures -6. Figures 7-9 depict the variations of temperature due to the influence of heat generation parameter S radiation parameter R and Dufour number Df hereas Figures and are plotted for variations of concentration profiles ith respect to Sr and r. 3. Velocity Profiles The effect of increasing values of K on the velocity profile is presented in Figure. It is clear that the velocity decreases gradually on increasing values of K. Fig. 3 shos the influence of magnetic field on the velocity. As M increases e observe that the velocity also increases and attains its peak value hen M = 8 before sliding don rapidly to reach the free stream velocity. f'() f'() K =.. K =.5 K = Fig.. Velocity profiles for different K values ith r=. Df=. Ri = T Ri =.5 M=. R=. S=. Sr= M = M = 4 M = Fig. 3. Velocity profiles for different M values ith r=. Df=. Ri T= Ri =.5 K= R=. S=. Sr=.. For the case of different values of thermal Richardson number Ri T the velocity profiles in the boundary layer are shon in Fig. 4. As expected it is observed that an increase in Ri T leads to an increase in the values of velocity due to the enhancement in buoyancy force. Fig. 5 provides the velocity profiles in the boundary layer for various values of the solutal Richardson number Ri. This figure indicates that the velocity increases due to the increase in the species buoyancy force. The velocity attains a distinctive maximum in the vicinity of the plate and then decelerates to approach the free stream velocity. 45

6 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp Table omputations shoing the comparison ith Makinde () for different values of S hen Ri T = Ri c =.5 Pr = Sc =.5 M = K = r = Rd = f Makinde Makinde Makinde S () Present () Present () Present Table Values of f () and () for different values of the governing parameters hen Pr =. Sc =.5 K M Gr Gc S R r Sr Df f () () f'() f'( ).5 Ri T = Ri T = Ri T = Fig. 4. Velocity profiles for different Gr values ith r=. Df=. Ri =.5 K= M=. R=. S=. Sr=...5 Ri = Ri = Ri = Fig. 5. Velocity profiles for different Gc values ith r=. Df=. Ri T = K= M=. R=. S=. Sr=.. 45

7 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp The effect of increasing the internal heat generation parameter S on velocity is illustrated in Fig. 6. It is observed that the velocity increases ith an increase in the internal heat generation and there is an overshoot in the fluid velocity toards the plate surface. This is due to the fact that the rise in S has the tendency to increase the fluid temperature hich in turn causes an increased fluid velocity along the plate due to buoyancy effect. '() R = R =.5 R =. f'() Fig. 8. Temperature profiles for different R values ith r=. Df=. Ri T= Ri =.5 K= M=. S=. Sr=...4 S = -.5. S = S = Fig. 6. Velocity profiles for different S values ith r=. Df=. Ri T= Ri =.5 K= M=. R=. Sr=.. '() Df = Df =.5 Df =. 3. Temperature Profiles The temperature of the flo suffers a substantial change ith the variations of heat generation parameter S radiation parameter R and Dufour number Df. The effect of heat generation parameter S is presented in Fig. 7. Here e notice that there is a steep rise in temperature for the increasing values of S. The thermal boundary layer thickness increases ith an increase in internal heat generation. An increase in thermal radiation R results in an increase in the fluid temperature ithin the boundary layer is shon in Fig. 8. As a consequence thermal boundary layer also increases. The effect of Dufour number Df on temperature field is as shon in Fig. 9. From this figure it is observe that an increase in the Dufour number Df increases the temperature inside the boundary layer. '() S = -.5 S = S = Fig. 7. Temperature profiles for different S values ith r=. Df=. Ri T= Ri =.5 K= M=. R=. Sr= Fig. 9. Temperature profiles for different Df values ith r=. Ri T= Ri =.5 K= M=. R=.S=. Sr= oncentration Profiles The variations in the concentration boundary layer corresponding to Soret number Sr and chemical reaction parameter r are depicted in Fig. and. Figure plots the concentration profiles for different values of Sr. As seen from this graph that concentration of species increases ith increasing values of the Soret number leading to an increase in thermal boundary value thickness. '() Sr = Sr =.5 Sr = Fig.. oncentration profiles for different Sr values ith r=. Df=. Ri T= Ri =.5 K= M=. R=. S=.. 453

8 S. Karthikeyan et al. / JAFM Vol. 9 No. 3 pp It is observed from Fig. that increasing the value of chemical reaction parameter r decreases the concentration of species in the boundary layer and hence the solutal boundary layer thickness becomes thinner. '() r = -. r = r = Fig.. oncentration profiles for different r values ith Df=. Ri T= Ri =.5 K= M=. R=. S=. Sr=.. 4. ONLUSIONS In this study the behavior of MHD mixed convection stagnation-point flo toards a vertical plate embedded in a highly porous medium ith heat and mass transfer in the presence of thermal radiation internal heat generation Soret and Dufour effects is investigated. The transformed system of non-linear ordinary differential equations is solved numerically by shooting method together ith sixth-order Runge-Kutta scheme. From this analysis the folloing conclusions are dran. (i) (ii) The skin-friction coefficient and the local Sherood number increase ith increasing values of magnetic parameter (M). For increasing values of internal heat generation (S) and Dufour number (Df) the skin-friction coefficient and the local Sherood number increase hereas the local Nusselt number decreases. When the radiation parameter increases the local Nusselt number and the local Sherood number increase hereas the skin-friction coefficient decreases. The velocity distribution across the boundary layer are increased ith an increase in magnetic parameter thermal Richardson number solutal Richardson number and then internal heat generation hile they sho opposite trends ith increasing values of the permeability parameter. (iii) For increasing values of the internal heat generation thermal radiation and the Dufour number the temperature increases and hence thermal boundary layer thickness increases. (iv) The concentration of species increases ith increasing Soret number hereas increasing values of chemical reaction parameter decreases the concentration of species in the boundary layer and hence the solutal boundary layer thickness becomes thinner. AKNOWLEDGEMENT One of the authors (S. K) acknoledges the University Grants ommission India for financial support to carry out this research ork under the minor research project No.F MRP-49/4 (SERO/UG). One of the authors (S. S) acknoledges the University of Malaya Malaysia for financial support through the grants RG6- AFR and RPB-3AFR. REFERENES Babu P. R. J. A. Rao and S. Sheri (4). Radiation effect on MHD heat and mass transfer flo over a shrinking sheet ith mass suction. Journal of Applied Fluid Mechanics 7(4) Bhuvanesari M. and S. Sivasankaran (4). Free convection flo in an inclined plate ith variable thermal conductivity by scaling group transformations. AIP onf. Proc Bhuvanesari M. S. Sivasankaran and M. Ferdos (9). Lie group analysis of natural convection heat and mass transfer in an inclined surface ith chemical reaction. Nonlinear Analysis: Hybrid Systems Bhuvanesari M. S. Sivasankaran and Y. J. Kim ().Lie group analysis of radiation natural convection flo over an inclined surface in a porous medium ith internal heat generation. Journal of Porous Media 5() hamkha A. J. and S. E. Ahmed (). Similarity solution for unsteady MHD flo near stagnation point of a three-dimensional porous body ith heat and mass transfer heat generation/absorption and chemical reaction. Journal of Applied Fluid Mechanics 4() houdhury R. and S. Kumar Das (4). Viscoelastic MHD free convective flo through porous media in presence of radiation and chemical reaction ith heat and mass transfer. Journal of Applied Fluid Mechanics 7(4) Devi. D. S. H. S. Takhar and G. Nath (99). Unsteady mixed convection flo in stagnation region adjacent to a vertical surface. W armeund Stoff ubertragung 6() Eckert E. R. G. (94). Die Berechnung des Wärmeubergangs in der laminaren Grenzschicht umstromter Körper. VDI Forschungsheft Gangadhar K. (3). Soret and Dufour effects on 454

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