International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 Mixed convective Heat and Mass transfer flow in a vertical channel with Soret effect and Radiation Absorption J. Deepthi Research scholar,Department of Mathematics Rayalaseema University Kurnool, Andhra Pradesh-INDIA D.R.V.Prasada Rao Professor(Rtd) ,Department of Mathematics S.K. University Anantapur, Anantapur(Dt), Andhra Pradesh-INDIA ABSTRACT In this paper we, investigate the combined influence of Radiation parameter(F),Thermodiffusion(Sr), Radiation absorption(Q1) and Forchheimer number(A) effects on non-Darcy mixed convective heat and mass transfer flow in a vertical channel with heat generating sources.By using Galerkin finite element analysis method the governing equations have been solved and the effect of various parameters on all the flow-characteristics have been investigated.The velocity, temperature, concentration, and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter. INTRODUCTION Non – Darcy effects on natural convection in porous media have received a great deal of attention in recent years because of the experiments conducted with several combinations of solids and fluids covering wide ranges of governing parameters which indicate that the experimental data for systems other than glass water at low Rayleigh numbers, do not agree with theoretical predictions based on the Darcy flow model. This divergence in the heat transfer results has been reviewed in detail in Cheng [5]and Prasad et al. [18] among others. Extensive effects are thus being made to include the inertia and viscous diffusion terms in the flow equations and to examine their effects in order to develop a reasonable accurate mathematical model for convective transport in porous media. The work of Vafai and Tien [23] was one of the early attempts to account for the boundary and inertia effects in the momentum equation for a porous medium.They found that the momentum boundary layer thickness is of order of . Vafai and Thiyagaraja [24] presented analytical solutions for the velocity and temperature fields for the interface region using the Brinkman Forchheimer –extended Darcy equation. Detailed accounts of the recent efforts on non-Darcy convection have been recently reported in Tien and Hong [21], Cheng [5], and Kalidas and Prasad [8].Poulikakos and Bejan [17] investigated the inertia effects through the inclusion of Forchheimer’s velocity squared term, and presented the boundary layer analysis for tall cavities.Prasad and Tuntomo [18] reported an extensive numerical work for a wide range of parameters, and demonstrated that effects of Prandtal number remain almost unaltered while the dependence on the modified Grashof number, changes significantly with an increase in the Forchheimernumber.A numerical study based on the Forchheimer-Brinkman- ©2017 RS Publication, [email protected] Page 1 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 Extended Darcy equation of motion has also been reported recently by Beckerman et al [4].Jha and Singh [9] studied the free convection and mass transfer flow in an infinite vertical plate moving impulsively in its own plane taking into account the Soret effect. Kafousias [10] studied the MHD free convection and mass transfer flow taking into account Soret effect. The analytical studies of Jha and Singh[9] and Kafousias [10] were based on Laplace transform technique. Abdul Sattar and Alam [1] have considered an unsteady convection and mass transfer flow of viscous incompressible and electrically conducting fluid past a moving infinite vertical porous plate taking into the thermal diffusion effects.Malasetty et al [15] have studied the effect of both the soret coefficient and Dufour coefficient on the double diffusive convective with compensating horizontal thermal and solutal gradients. Umadevi et al[22] have studied the chemical reaction effect on Non-Darcy convective heat and mass transfer flow through a porous medium in a vertical channel with heat sources. Deepthi et al [7] and Kamalakar et al[11] have discussed the numerical study of non-Darcy convective heat and mass transfer flow in a vertical channel with constant heat sources under different conditions.The effects of radiation on MHD flow and heat transfer problem have become more important industrially. Keeping these applications in view several authors have studied the effect of radiation on flow in different configurations under varriedconditions.Bharathi [3] has studied thermo-diffusion effect on unsteady convective Heat and Mass transfer flow of a viscous fluid through a porous medium in vertical channel. Radiative flow plays a vital role in many industrial and environmental process e.g. heating and cooling chambers , fossil fuel combustion energy process, evaporation form larger open water reservoirs, astrophysical flows, solar power technology and space vehicle re-entry. Taneja et al [20] studied the effects of magnetic field on free convective flow through porous medium with radiation and variable permeability in the slip flow regime. Kumar et al [13] studied the effect of MHD free convection flow of viscous fluid past a porous vertical plate through non homogeneous porous medium with radiations and temperature gradient dependent heat source in slip flow regime. The effect of free convection flow with thermal radiation and mass transfer past a moving vertical porous plate was studied by Makinde [14]. Ayani et al [2] studied the effect of radiation on the laminar natural convection induced by a line source. Raptis [19] have discussed the effect of radiation and free convection flow through porous medium. MHD oscillating flow on free convection radiation through porous medium with constant suction velocity was investigated by El.Hakiem[8]. Muralidhar[16] has analysed the thermo-diffusion effect of non-Darcy convective heat and mass transfer flow in a vertical channel. Recently, Das et al [6] have studied the mixed convective magnetohydrodynamic flow in a vertical channel filled with nanofluid. Recently,Kristaiah [12] investigated the combined influence of thermal radiation ,chemical reaction and thermo-dffusion effects on convective heat and mass transfer flow of an electrically conducting fluid in vertical channel. In this paper, to investigate the combined influence of Radiation parameter(F),Thermodiffusion(Sr), Radiation absorption(Q1) and Forchheimer number(A) effects on non-Darcy mixed convective heat and mass transfer flow in a vertical channel with heat generating sources.By using Galerkin finite element analysis method the governing equations have been solved and the effect of various parameters on all the flow-characteristics have been investigated.The velocity, temperature, concentration, and rate of Heat and Mass transfer are evaluated numerically for different variations of parameter. FORMULATION OF THE PROBLEM We consider a fully developed laminar convective heat and mass transfer flow of a viscous, electrically conducting fluid through a porous medium confined in a vertical channel bounded by flat walls. We choose a Cartesian co-ordinate system O(x,y,z) with x- axis in the vertical direction and y-axis normal to the walls and the walls are taken at y= L. The walls are maintained at constant temperature and concentration. The temperature gradient in ©2017 RS Publication, [email protected] Page 2 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 the flow field is sufficient to cause natural convection in the flow field .A constant axial pressure gradient is also imposed so that this resultant flow is a mixed convection flow.The porous medium is assumed to be isotropic and homogeneous with constant porosity and effective thermal diffusivity. The thermo physical properties of porous matrix are also assumed to be constant and Boussinesq- approximation is invoked by confining the density variation to the buoyancy term. In the absence of any extraneous force flow is unidirectional along the x-axis which is assumed to be infinite. x Tw=T0+Ax Tw=T0+Ax C w= C0+Bx C w= C0+Bx y y=-L g y=+L Fig.1 : Configuration of the problem The equations governing the non-darcy flow heat and mass transfer taking thermal radiation into account are p 2u 2 H 2 F 2 ( ) 2 ( e o )u u g 0 (1) x y o k 16 2Te3 2T T 2T k f 2 Q(T To ) Q1' (C Co ) x y 3 r y 2 C 2C 2T u D1 2 k1C k11 2 x y y 0C p u The relevant boundary conditions are T=Tw , C=Cw at y=L u 0, (2) (3) (4) Following Tao and Das et.al [29],we assume that the temperature and concentration of the both walls is Tw T0 Ax , Cw C0 Bx where A and B are the vertical temperature and concentration gradients which are positive for buoyancy –aided flow and negative for buoyancy – opposed flow, respectively, T0 and C 0 are the upstream reference wall temperature and concentration respectively. For the fully developed laminar flow in the presences of transverse magnetic field, the velocity depend only on the normal coordinate and all the other physical variables except temperature, concentration and pressure are functions of y and x, x being the vertical co-ordinate. The temperature and concentration inside the fluid can be written as T T ( y ) Ax , C C ( y ) Bx We define the following non-dimensional variables as ©2017 RS Publication, [email protected] Page 3 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm u Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 u p , ( x, y) ( x, y ) / L , p ( / L) ( 2 / L2 ) (5) T -T0 C -C0 (y) , , C, ALP1 BLP1 Introducing these non-dimensional variables the governing equations in the dimensionless form reduce to (on dropping the dashes) d 2u 1 ( M12 )u G ( NC ) 2 u 2 (6) 2 dy d 2 4 F d 2 (Pr)u Q1C dy 2 3 dy 2 d 2C ScSo d 2 C ( Sc ) u N dy 2 dy 2 (7) (8) where gAL3 FD ( Inertia or Fochhemeir parameter), G (Grashof Number) 2 e2 H o2 L2 M2 (Hartmann Number), Sc (Schmidt number) 2 D1 Cp B N (Buoyancy ratio), Pr (Prandtl Number) kf A 1 / 2 k1 L2 QL2 (Heat source parameter), (Chemical reaction parameter) D1 kf F k T 4 Te3 (Radiation parameter), S 0 11 (Soret parameter) k f R T 3F 3F Q1' BL2 , 1 (Radiation absorption parameter), P1 Pr 3 4F 3 4F kf A dp (Constant pressure gradient) dx Q1 The corresponding boundary conditions are u 0 , 0 , C 0 on y 1 THE METHOD OF SOLUTION (9) The Galerkinfinite element method has been implemented to obtain numerical solutions of coupled non-linear equations [6] to [8] of third-order inf and second order in h,, under boundary conditions [9]. This technique is extremely efficient and allows robust solutions of complex coupled, nonlinear multiple degree differential equation systems. The fundamental steps comprising the method are 1]Discretization of the domain into elements 2] Derivation of element equations 3] Assembly of Element Equations 4] Imposition of boundary conditions 5] Solution of assembled equations ©2017 RS Publication, [email protected] Page 4 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 COMPARISON In the absence of radiation absorption parameter (Q1=0) and Forchheimernumber(A=0) the results are in good agreement with Kristahiah (12). Sr 0.5 1.0 1.5 0.5 0.5 Table 1 : Comparison of Nu and Sh at η=±1 with Kristaiah(12) with A=0, Q1=0 Kristaiah[12]Results Present Results(A=0, Q1=0) Pr Nu(+1) Nu(-1) Sh(+1) Sh(-1) Nu(+1) Nu(-1) Sh(+1) Sh(-1) 0.71 0.71 0.71 1.71 7.0 -0.2607 -0.2598 -0.2599 -0.2617 -0.4595 -4.0053 -3.9895 -3.9885 -4.1086 -5.9582 0.2534 0.2516 0.2508 0.2686 0.4595 14.7935 14.7929 14.7915 14.8592 15.1084 -0.2606 -0.2596 -0.2598 -0.2616 -0.4593 -4.0054 -3.9896 -3.9884 -4.1084 -5.9599 0.2536 0.2518 0.2507 0.2685 0.4591 14.8014 14.7932 14.7916 14.8591 15.1079 DISCUSSION OF THE NUMERICAL RESULTS In order to get physical insight into the problem we have carried out numerical calculations for non-dimensional velocity, temperature and concentration, Nusselt number andSherwood number by assigning some specific values to the parameters entering into the problem.We have investigated the influence of F, Sr,Q1andAby fixing the other parameters as G=2,M=2,D-1=0.2,Sc=1.3,N=1,Pr=0.71,=0.5 unless otherwise stated. Figs.1a-1c represents the velocity components,temperature and concentration withRadiation parameter(F). From Fig.1a it can be seen from the profiles that an increase in F increases the magnitude of the velocity thus higher the radiative heat flux larger the magnitude of u in the flow region.Fig.1b represents with radiation parameter F.It can be seen from the profiles that increase in F leads to an enhancement in the actual temperature in the flow region, this may be attributed to the fact that increasing F results in the enhancement of thickness of the thermal boundary layer.Fig.1c represents the concentration C with radiation parameter(F).It can be seen from the profiles that higher the radiative heat flux larger the actual concentration in the flow field. Figs. 2a-2c shows the variation of the velocity components,temperature and concentration with Soret parameter(Sr).Fig.2a shows the variation of u with Soret parameter Sr , and it can be seen that higher the thermo-diffusion effect larger u in the flow region and for higher Sr1.5 the velocity enhances u in the entire flow region.Fig.2b shows the variation of with respect to Soret parameter(Sr) we find that the higher the thermo-diffusion effect(Sr1.0) larger the actual temperature and for higher thermo-diffusion effects(Sr1.5) lesser the actual temperature in the entire flow region except in a narrow region adjacent to the left wall y=-1.From fig.2c we find that higher the thermo-diffusion(Sr)effects, smaller the actual concentration in the flow-region. Due to the fact that increase in Sr reduces the thickness of the solutary boundary layer,also higher the thermo-diffusion effects larger the actual concentration in the flow. Figs.3a-3c represents the velocity components,temperature and concentration with Radiation absorption parameter(Q1). Fig.3a represents the variation of u with radiation absorption parameter(Q1),it can be observed from the figure that |u| reduces with increase in Q 1.This can be attributed to the fact that the thickness of the momentum boundary layer decreases with increase in Q1.From fig.3b we find that the actual temperature decreases with increase in radiation absorption parameter(Q1≤1) and enhances with higher Q1≥1.5.Due to the fact that the thickness of the boundary layer increases with increase inQ1. Fig.3c represents the concentration with radiation ©2017 RS Publication, [email protected] Page 5 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 absorption parameter(Q1). An increase inQ1 reduces the thickness of the solutary boundary layer, which in turns leads to a depreciation in the actual concentration. u y 0 -1 -0.5 0 0.5 1 0.0010 F=0.5,1.5,3.5,5 -0.1 0.0008 -0.2 0.0006 -0.3 F=0.5,1.5,3.5,5 0.0004 -0.4 0.0002 -0.5 y 1.0 Fig. 1a: Variation of u with F G=2,M=2,D-1=0.2,Sc=1.3,N=1, Pr=0.71, A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5 0.5 0.5 1.0 Fig. 1b: Variation of with F G=2,M=2,D-1=0.2,Sc=1.3,N=1, ,Pr=0.71, A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5 Figs.4a-4c represents the velocity components, temperature and concentration with Forchheimer number(A).Fig.4a represents the velocity u with Forchheimer parameter A. From the figures we find that an increase in ‘A’ reduces |u| in the flow region ,the inclusion of nondarcy effect results in a reduction(or)depreciation in the velocity.Fig.4b depeicts a temperature with Forchheimer number(A).It can be seen from the profiles that an increase in A leads to a depreciation in the actual temperature in the flow region.Thus the inclusion of intertia and boundary effects reduces the actual temperature.Fig12c depicts the concentration with Forchheimernumber(A). From the fig we notice an enhancement in the actual concentration with increase in A. Thus the inclusion of the inertia and boundary effects results in an enhancement in the actual concentration in the flow region. ©2017 RS Publication, [email protected] Page 6 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 C 0.045 uf 0.04 1.0 0.035 0.5 0.5 1.0 y 0.03 0.1 Sr=1.0,1.5,0.5,2.0 0.025 0.02 0.2 0.015 0.01 F=0.5,1.5,3.5,5 0.3 0.005 0 -1 -0.5 y 0 0.5 0.4 1 Fig. 2a : Variation of u with Sr G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5 Fig. 1c : Variation of C with F G=2,M=2,D-1=0.2,Sc=1.3,N=1, Pr=0.71, A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5 C0. 0.008 0.035 0.030 0.006 0.025 0.020 0.004 0.015 Sr=1.0,1.5,0.5,2.0 0.010 0.002 0.005 Sr=0.5,1,1.5,2.0 y 1.0 0.5 0.5 1.0 1.0 Fig. 2b : Variation of with Sr G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5 0.5 0.5 u -0.5 y Fig. 2c : Variation of C with Sr G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5 0 -1 1.0 0.16 y 0 0.5 1 -0.05 0.14 Q=-0.5,-1.5 -0.1 0.12 -0.15 0.1 -0.2 0.08 -0.25 0.06 -0.3 0.04 -0.35 0.02 -0.4 Q1=0.5,1,1.5,2.0 Q1=0.5,1,1.5,2.0 0 -0.45 -1 Fig. 3a : Variation of u with Q1 G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5 ©2017 RS Publication, [email protected] -0.5 y 0 0.5 1 Fig. 3b : Variation of with Q1 G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5 Page 7 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 C 0.25 u y 0 -1 -0.5 0 0.5 1 -0.05 0.2 -0.1 0.15 A= 0.05,0.07 -0.15 -0.2 0.1 -0.25 -0.3 0.05 Q1=0.5,1,1.5,2.0 0 -1 -0.5 -0.4 y 0 0.5 A= 0.01,0.03 -0.35 1 -0.45 Fig. 3c : Variation of C with Q1 G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5 Fig. 4a : Variation of u with A G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5 0.09 C0. 0.08 A= 0.01,0.03,0.05,0.07 0.20 A= 0.01,0.03,0.05,0.07 0.07 0.06 0.15 0.05 0.04 0.10 0.03 0.02 0.01 0.05 y 0 -1 -0.5 0 0.5 Fig. 4b : Variation of with A G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5 1 1.0 0.5 0.5 1.0 y Fig. 4c : Variation of C with A G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5, Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5 Table.1 Values of Nusselt number, Sherwood number at ƞ=1 ©2017 RS Publication, [email protected] Page 8 Parameter Nu(1) Nu(-1) Sh(1) Sh(-1) 0.5 0.0182438 -0.0182438 0.0585079 -0.0585079 F International Journal of Emerging Trends in Engineering and Development Issue 7, Vol. 2 (March 2017) 1.5 0.0181263 -0.0181263 0.0570322 -0.0570322 Available online on http://www.rspublication.com/ijeted/ijeted_index.htm ISSN 2249-6149 3.5 0.018034 -0.018034 0.0570511 -0.0570511 5.0 0.0179654 -0.0179654 0.0570652 -0.0570652 0.5 0.0182442 -0.0182442 0.0603368 -0.0603368 Sr 1.0 0.0181724 -0.0181724 0.0551637 -0.0551637 1.5 0.018172 -0.018172 0.0533047 -0.0533047 2.0 0.0181715 -0.0181715 0.0514458 -0.0514458 0.5 0.0182438 -0.0182438 0.0585079 -0.0585079 Q1 1.0 0.018208 -0.018208 0.0570156 -0.0570156 1.5 0.0182431 -0.0182431 0.0570085 -0.0570085 2.0 0.0182783 -0.0182783 0.0570014 -0.0570014 0.01 0.0182438 -0.0182438 0.0585079 -0.0585079 A 0.03 0.0181729 -0.0181729 0.0570227 -0.0570227 0.05 0.0181728 -0.0181728 0.0570221 -0.0570221 0.07 0.0181726 -0.0181726 0.0570212 -0.0570212 The rate of heat transfer (Nusselt number) Nu enhances on y=1, with increase in the strength of the heat generating source while an increase in the strength of the heat absorption source enhances Nu on y=+1 and reduces on y=-1. An increase in Forchheimer number A reduces |Nu| on y=+1,thus the inclusion of inertia and boundary effect reduces the rate of heat transfer on y=±1. The rate of heat transfer reduces with increase in Q1 ≤1.0 and enhances with higher Q1≥1.5 on y=±1. An increase in F≤3.5 enhances |Nu| and reduces with higher F≥5. Higher the thermo diffusion (Sr) lesser the rate of heat transfer on the walls y=1. The rate of mass transfer (Sherwood number) enhances with increase in F≤3.5 and reduces with higher F≥5 on y=±1. An increase in the radiation absorption Q1 (or) Forchheimer number A results in depreciation |Sh| on both the walls. Higher the thermo diffusion (Sr) lesser the rate of mass transfer on the walls y=1. CONCLUSIONS The coupled equations governing the flow,heat and mass transfer have been solved by using Galerkin finite element technique.The important conclusions of this analysis are 1) Higher the thermo-diffusion(Sr1.5) larger the velocity, temperature and concentration and for higher Sr1.5 ,the velocity and temperature reduces and the concentration enhances in the flow region. The rate of heat and mass transfer reduces on the walls with increase in Sr.It is found that an increase in soret parameter S r≤1.0 increases the velocity,temperature and reduces the concentration for higher Sr ≥1.5,we notice a depreciation in them. 2) Higher the thermo-diffusion(Sr1.5) larger the velocity, temperature and concentration and for higher Sr1.5 ,the velocity and temperature reduces and the concentration enhances in the flow region. The rate of heat and mass transfer reduces on the walls with increase in Sr.It is found that an increase in soret parameter S r≤1.0 increases the velocity,temperature and reduces the concentration for higher S r ≥1.5,we notice a depreciation in them. 3) The velocity and temperature decreases while the concentration increases with increase inForchheimer number A. The rate of heat and mass transfer reduces with increase in A. 4) The velocity, temperature and concentration reduces with increase in radiation absorption Q1>1 and for higherQ1≥1.5 the velocity and concentration reduces while the temperature enhances in the flow region .The Sherwood number reduces and Nusselt number increases with increase in Q1. REFERENCES ©2017 RS Publication, [email protected] Page 9 International Journal of Emerging Trends in Engineering and Development Available online on http://www.rspublication.com/ijeted/ijeted_index.htm Issue 7, Vol. 2 (March 2017) ISSN 2249-6149 [1]Md.AbdulSattar,Md.Alam. 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