International Journal of Mathematics Trends and Technology (IJMTT) Volume 54 Number 3- February 2018

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1 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 Exponentiall Varing Temperature and Concentration Boundar Laer Influence on Thermal Diffusiveheat Generating Fluid in Conducting Field K. Viaa Kumar and E. Keshava Redd Reseach Scholar, Department of Mathematics, JNTUA College of Engineering, Ananthapuramu, A.P., India. Department of Mathematics, JNTUA College of Engineering, Ananthapuramu, A.P., India. Abstract-Numerical analsis is carried out for the case of thermal diffusion effect on heat generating fluid in the presence of exponentiall varing temperature and concentration in conducting field. The presence of homogeneous chemical reaction is considered. A magnetic field of uniform strength is applied perpendicular to the plate. The set ofdimensionlessequations are solved numericall b using finite difference method. The relevant phsical flow parameters on the flow quantities are studied through graphs. Also the variations in skin friction, Nusselt number and Sherwood number are presented in tables. Kewords:Thermal diffusion, Porous medium, Conducting field, Thermal radiation, Chemical reaction, finite difference method. I. INTRODUCTION In recent times convection flows b means of simultaneous heat and mass transfer under the impact of a magnetic field and chemical reaction occurs in man engineering applications. Which plas an essential role in man industries namel,in the chemical industr, power and cooling industr for dring,cooling of nuclear reactors and magneto hdrodnamic (MHD) power generators. Man transport processes exist in industrial applications and in nature, in which the simultaneous heat and mass transfer occurs as a result of combined buoanc effects of the diffusion of chemical species. Some fields of interest in which collective heat and mass transfer pla a vital role in the design of chemical processes in gear, development and dispersion of fog, distribution of temperature and vapor over food production fields. Earlier Tripath et al. [] studied the effect of Chemical reaction effect on MHD free convective surface over a moving vertical plate through porous medium. Srinivasachara and Swam Redd [] analzed the effect of free convection in a non-newtonian power law Fluid saturated porous medium with Chemical reaction and radiation effects. Kandasam et al. [3] discussed chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effect. Mahd et al. [4] analzed the effects of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in a porous media with variable viscosit. Anali Devi and Kandaswami [5] presented effects of a chemical reaction heat and mass transfer on MHD flow past a semi-infinite plate. Khan and Rashad [6] studied combined effects of radiation and chemical reaction on heat and mass transfer b MHD stagnation point flow of a micro polar fluid towards a stretching surface. Barik [7] analzed the effect of Chemical Reaction and Radiation Effects of MHD Free Convective Flow past an Impulsivel Moving Vertical Plate with Ramped Wall Temperature and Concentration. Nehad Ali shah et al. [8] investigated the general solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction. Chamkha et al. [9] discussed MHD flow of uniforml stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Seth et al. [] presented MHD natural convection flow with radiative heat transfer past an impulsivel moving plate with ramped wall temperature. Sattar et al. [] presented an analtical solution of the free convection and mass transfer flow with thermal diffusion. Makinde et al. [] studied hdromagnetic heat and mass transfer over a vertical plate with a convective surface boundar condition. Pattnaik et al. [3] discussed effect of heat and mass transfer on MHD free convection flow past an impulsivel moving infinite vertical plate with ramped wall temperature. When heat and mass transfer arise simultaneousl in a fluid flow, the relationships between the driving potentials and fluxes are of more important. It has been detected that energ flux can be produced b temperature gradients ISSN: Page

2 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 and also concentration gradients. On the other hand, mass flux can also be affected b temperature gradients and this represents the thermal-diffusion (Soret) effect. In various studies belonging to Soret effects are neglected on the basis that the are of a smaller order of magnitude than the effects described b Fourier s and Fick s laws. But these effects ma become essential when the are considered as second order phenomena and in the areas of petrolog, geosciences, hdrolog etc. The Soret effect has been emploed for isotope separation and in mixture between gases with ver less molecular weight and of standard molecular weight. Chandra reddet al. [4] investigated Soret and Dufour effects on MHD free convection flow of Rivlin-Ericksen fluid past a semi-infinite vertical plate. Aat et al. [5] discussed heat and mass transfer for Soret and Dufour s effect on mixed convection boundar laer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Chandra redd et al. [6] studied thermal and solutal buoanc effect on MHD boundar laer flow of a visco-elastic fluid past a porous plate with varing suction and heat source in the presence of thermal diffusion. Ahammad et al. [7] analzed Numerical stud of MHD free convection flow and mass transfer over a stretching sheet considering Soret and Dufour effects in the presence of magnetic field. Siva redd et al. [8] considered Soret effects on Unstead MHD free convective flow past a semi-infinite vertical plate in the presence of viscous dissipation. Cheng [9] presented a stud on Soret and Dufour effects on heat and mass transfer b natural convection from a vertical truncated cone in a fluid saturated porous medium with variable wall temperature and concentration. Srinivasachara et al. [] discussed the effects of Soret and Dufour on mixed convection along a vertical wav surface in a porous medium with variable properties. Chamka et al. [] considered Double-Diffusion MHD free convective flow along a sphere in the presence of homogeneous-chemical reaction and soret and dufuor effects. Shankar Goud et al. [] analzed finite element stud of soret and radiation effects on mass transfer flow through a highl porous medium with heat generation and chemical reaction. Maleque et al. [3] studied Dufour and Soret effects on unstead MHD convective heat and mass transfer flow due to a rotating disk. Recentl, unstead magnetohdrodnamic free convective double-diffusive viscoelastic fluid flow past an inclined permeable plate in the presence of viscous dissipation and heat absorption was presented b Umamaheswar et al. [4]. Dash et al. [5] addressed about free convective MHD flow through porous media of a rotating Visco-elastic fluid past an infinite vertical porous plate with heat and mass transfer in the presence of chemical reaction. Vivek et al. [6] investigated on hdrodnamic and hdromagnetic tripl diffusive convection in a viscoelastic fluid through porous medium. Mukesh et al. [7] have considered MHD flow and heat transfer through non-darc porous medium bounded between two parallel plates with viscous and Joule dissipation. Choudhur et al. [8] studied effect of viscoelastic MHD boundar laer flow with heat and mass transfer over a continuousl moving inclined surface in presence of energ dissipation. Srinivasachara and Redd [9] investigated a numerical stud on free convection in a non-newtonian power law fluid in a saturated porous medium in the presence of chemical reaction and radiation. Recentl, Redd et al. [3] carried out unstead MHD free convection flow of a visco-elastic fluid past a vertical porous plate in the presence of thermal radiation, radiation absorption, heat generation/absorption and chemical reaction. Several authors [37-45] contributed to this kind of studies. Motivated b the above studies, in this manuscript, we have studied Soret effects on radiation absorption fluid past a linearl accelerated vertical porous plate in the presence of exponentiall varing temperature and concentration in conducting field, b extending the work of Muthucumaraswam and Velumurugan [36]. This is not the simple extension work, we have alsochanged the boundar conditions. The previous studies were confined to isothermal boundar laer and uniform mass transfer or the plate temperature as well as concentration levels near the plate are linear functions of time. This indicates that small changes in temperature and concentration levels were focused much but large variations in temperature and concentration levels were ignored. Hence in this stud we have considered exponentiall varing temperature and also exponentiall varing concentration. This kind of applications can be found in the process of cooling the nuclear reactors where large temperatures will be reduced to small temperatures or vice versa. The novelt of this stud is the consideration of radiation absorption effect along with Soret effects in the presence of chemicall reacting and radiating fluid in porous medium along with the exponentiall varing temperature and concentration boundar laers. II. MATHEMATICAL FORMULATION We consider a viscous incompressible, electricall conducting, heatgenerating and radiating fluid flow past an infinite vertical porous plate in the presence of thermo diffusion. A magnetic field of uniform strength B is applied perpendicular to the plate. Let x -axis is taken along the plate in the verticall upward direction and the - axis is taken perpendicular to the plate. At time t, the plate is maintained at the temperature higher than ambient temperature T and the fluid is at rest. At time t >, the plate is linearl accelerated with increasing time in its own ISSN: Page 3

3 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 plane and the temperature and as well as concentration are assumed to var exponentiall with respect to time. It is assumed that the effect of viscous dissipation and the induced magnetic field produced b fluid motion are neglected in comparison to applied magnetic field. This is ustified because magnetic Renolds number is ver small for liquid metals and partiall ionized fluids which are commonl used in various industrial processes (Cramer and Pai [35], Seth et al. []. B usual Boussineq s and boundar laer approximations followed b Mahd [4] Sattar and Alam [] and Umamaheswar [4] and based on the above considerations the unstead flow is governed b the following equations: u u Bu g T T g C C u t k T T Cp Q T T t q C C Dk T m T D k ( C C ) Tm t The corresponding initial and boundar conditions are 3 r u, T T, C C for all, t 3 At At t : u At,T T ( Tw T ) e, C C ( Cw C ) e at u u, T T, C C as u where A () () (3) (4) The local radiant absorption for the case of an opticall thin gra gas, followed b Ahmed et al. [33] and Raptis and q 4 4 Perdikis [3] is expressed as r 4 a ( T T ) (5) Where σ and a are the Stefan- Boltzmann constant and the mean absorption coefficient, respectivel. It is assumed 4 that the temperature difference with in the flow are so small so that T can be expressed as linear function of T after using Talor s series to expand 4 T about the free stream temperature T and neglecting higher order terms This approximation results as follows: T 4T T 3 T. (6) T T 3 C p Q T T 6 a T ( T T ) t The non-dimensional quantities are as follows: u 3 3 3, u u, T T C C Cp U u t t Y,, C, Pr, T T C w w C 3 B 3 ( Dmk Tw T ) Kr k, M, Sc, S T, u D ( u Tm Cw C ) (7) ISSN: Page 4

4 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 B 3 g TwT g CwC M, Gr, Gc, u 3 3 u u u , u k Q at K Q, Ra u. (8) B introducing the above non-dimensional quantities, the equations (), (3) and (6) reduce to following forms. U U Gr GcC M U U t Y K Pr Q Ra t Y C C KrC S t Sc Y Y (9) () () B introducing the non-dimensional quantities, the boundar conditions reduce to the following forms. u,, C for all Y, t t t t : U t, e, C e at Y () u,, C as Y III. SOLUTION OF THE PROBLEM Equations (9), () and () are coupled partial differential equations for which the exact solutions are not possible. Hence these equations along with the set of boundar conditions () are solved b semi-implicit finite difference method. The corresponding finite difference schemes of equations (9) - () are as follows: U i, U i, U i, U i, U i, Gr i, GcC i, M U i, U t K i, Y i, i, i, i, i, Pr Q i, Ra t i, Y C i, C i, C i, C i, C i, i, i, i, KrC i, S t Sc Y Y Equations (9) () can be written as follows: U i, U i, U i, U i, U i, t Gr i, t GcC i, t M U i, t U i, t Y K t i, i, i, Q Ra i, i, Pr Pr i, t Pr i, t Y (3) (4) (5) (6) (7) ISSN: Page 5

5 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 t C i, C i, C i, i, i, i, C i, C i, KrC i, S t Sc Y Y (8) Here, the suffix i relates to and to time. The mesh sstem is divided b taking =.. From the initial condition in (8), we have the following equivalent: U( i,), ( i,), C( i,) for all i (9) The boundar conditions from (8) are expressed in finite-difference form as follows U(, ) t, (, ) e t, C(, ) e t U( imax, ), ( imax, ), C( imax, ) (Here i max was taken as ) The velocit at the end of time step viz., u (i, +)(i=,) is computed from (6) in terms of velocit, temperature and concentration at points on the earlier time-step. After that θ (i, +) is computed from (7) and then C (i, +) is computed from (8). The procedure is repeated until t =.5 (i.e. = 5). During computation t was chosen as.. Skin-friction: The skin-friction in non-dimensional form is given b the relation U, where Y Y Rate of heat transfer: U The dimensionless rate of heat transfer in terms of Nusselt number is given b Nu Y Y Rate of mass transfer:. The dimensionless rate of mass transfer in terms of Sherwood number is given b C Sh Y Y IV. RESULTS AND DISCUSSION To get a clear idea of the phsics of the flow field, a numerical stud has been carried out through the finite difference scheme and the influence of various phsical parameters on concentration, temperature, velocit are discussed with the help of Figs. -. Apart, variations in skin friction coefficient, rate of heat transfer in the form of Nusselt number and the rate of mass transfer in the form of Sherwood number are also presented in tables -3. Figs. -3 show the variations in the fluid concentration under the influence of Soret number, Schmidt number and chemical reaction parameter. Fig. exhibits that the concentration of the fluid decreases for an increase in chemical reaction parameter values. () ISSN: Page 6

6 C C International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8. Sc=. So=..8 Kr=,,3, Fig. : Effect of Kr on Concentration profiles.. Kr=. So=..8 Sc=.,.6,.78, Fig. : Effect of Sc on Concentration profiles. ISSN: Page 7

7 C International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8. Sc=.66 Kr= So=.5,,.5, Fig. 3: Effect of S on Concentration profiles. Fig. illustrates that the concentration of the fluid decreases for ascending values of Schmidt number. Phsicall it is true since the increase in Sc means decrease of molecular diffusivit and therefore decrease in concentration boundar laer. Hence the concentration of species is higher for smaller values of Sc and lower for larger values. When heat and mass transfer occur simultaneousl in a moving fluid, the mass flux generated b temperature gradients is termed as thermal-diffusion (Soret effect). The velocit of the fluid as well as species concentration increases for ascending values of Soret number. This is noticed from Fig.3 that the concentration of the fluid increases with an increase in Soret number. The variations in temperature field are shown in Figs It is evident from Fig.4 that temperature of the fluid falls down with an increase in the values of Prandtl number. Phsicall this is true because thermal conductivit of the fluid decreases with increasing values of Prandtl number, resulting decrease in thermal boundar laer thickness. ISSN: Page 8

8 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar Pr=.7,,3,7 Sc=.66 Q=. Ra=.5 So=.5 Kr= Fig. 4: Effect of Pr on Temperature Sc=.66 Pr=.7 Ra=.5 So=.5 Kr=.5.4. Q=,,3, Fig. 5: Effect of Q on Temperature. ISSN: Page 9

9 U International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8.4. Pr=.7 Sc=.66 Q=.5 So=.5 Kr= Ra=,,3, Fig. 6: Effect of Ra on Temperature M=3 Pr=.7 Sc=. Q=.5 Ra=.5 So=.5 Gr=Gm=5 Kr= K=.,.4,.7, ISSN: Page

10 U U International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 Fig.7: Effect of K on Velocit M=,,3,4 K= Pr=.7 Sc=. Q=.3 Ra=.5 So=.5 Gr=Gm=5 Kr= Fig.8: Effect of M on Velocit Pr=.7,,3,7 M=3 K= Sc=. Q=. Ra=. So=.5 Gr=Gc=5 Kr= ISSN: Page

11 U International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8.8 Fig.9: Effect of Pr on Velocit M=3 K= Sc=. Q=. Ra=.5 Pr=.7 Gr=Gm=5 Kr=..3. So=,,3, Fig.: Effect of Soret number on Velocit. The influence of heat absorption parameter on temperature is conferred in Fig.5. It is examined that the temperature increases for increasing values of heat absorption parameter. This is due to the fact that the heat absorption causes an increase in kinetic energ as well as thermal energ of the fluid. As a result, the temperature of the thermal boundar laer increases in the case of heat generating fluids. Fig.6 depicts the variations in temperature in the presence of radiation parameter. Decrease in temperature is detected under the influence of radiation parameter. This is in conformit with the results of Raptis and Perdikis [3]. Figs. 7- shows the variations in the velocit distribution under the influence of several parameters involved in this stud. From Fig.7 velocit profiles are displaed with the variations in permeabilit of the porous medium parameter K. From this figure, it is noticed that the velocit of the fluid increases from the moving plate to free surface with the increase in the values of the permeabilit of the porous medium. Phsicall, an increase in the permeabilit of porous medium leads the rise in the flow of fluid through it. When the holes of the porous medium become large, the resistance of the medium ma be neglected. So that velocit at the plate is observed to be maximum and step b step it reaches the free surface. This result is in good concordance the result of Rau et al. [34]. Fig.8 represents the effect of magnetic parameter on fluid velocit. The velocit decreases with increasing values of magnetic parameter. The central reason behind this effect is that the application of transverse magnetic field perpendicular to the flow causes a flow-resistive force called the Lorentz force which deeds in the opposite direction of the fluid flow. This force results in slowing the motion of the fluid. Fig.9 illustrates the effect of Prandtl number on velocit of the fluid. It is identified that the velocit of the fluid decreases as the values of Prandtl number increases. The fluid of low Prandtl number has high thermal diffusivit and hence it gains higher temperature in stead state, which in turn means more buoanc force i.e. high fluid velocit with respect to reasonabl high Prandtl fluid. The variation of the velocit boundar-laer under the effect of Soret number is shown in Fig.. An increase in velocit boundar laer is examined with ascending values of Soret number. Fig. depict the variations in velocit with the effect of Grashof number and modified Grashof number. It is shown that ISSN: Page

12 U International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 the velocit increases under the influence of both the numbers. It can be identified from table that the Nusselt number increases with rising value of Prandtl number. Increasing values of Schmidt number (Sc), heat absorption parameter (Q), heat generation parameter results in declining the Nusselt number. Table displas the variations in Sherwood number. Sherwood number value increases for increasing values of Schmidt number (Sc), chemical reaction parameter (Kr) and it decreases for ascending value of Soret number (S ). Table 3 displas the variations in skin friction. The skin friction increases with an increase in magnetic parameter (M) and Prandtl number (Pr) whereas it declines under the effect of porosit parameter (K), Grashof number (Gr) and modified Grashof number (Gc), heat absorption parameter (Q), Soret number (S ). Table.: Variations in Nusselt Number Pr Q Sc Nu Table. : Variations in Sherwood Number Sc Kr S Sh M= K= Pr=.7 Sc=. Q=. Ra=. So=.5 Kr=. Gr=5 Gr=6 Gr=8 Gc=7 Gc=9 Gc= Fig.: Effect of Gr and Gc on Velocit Table 3: Variations in Skin-friction S M K Q Pr Gr Gc ISSN: Page 3

13 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar NOMENCLATURE A, a Constants Cp Specific heat at constant pressure [J.kg - K - ] D m Thermal diffusivit Gr Thermal Grashof number Gcmodified Grashof number gacceleration due to gravit [m.s - ] Pr Prandtl number K porosit parameter Q heat absorption parameter Ra radiation parameter M magnetic parameter Kr chemical reaction parameter S Soret number Sc Schmidt number Nu Nusselt number Sh Sherwood number өtemperature [K] ttime [s] u Velocit of the plate [m.s - ] C Concentration [g/ml] Greek smbols βvolumetric coefficient of thermal expansion [K - ] β Volumetric coefficient of expansion for mass transfer kthermal conductivit [W.m -.K - ] μcoefficient of viscosit [Pa.s] ᴦ Heat generation parameter. νkinematic viscosit [m.s - ] ρdensit of the fluid [kg.m -3 ] τ skin friction σ Electrical conductivit [ohm - s - ] Subscripts s surface of the plate Conditions in the free stream Superscript Dimensional Coordinate axis normal to the plate [m] V. CONCLUSIONS ISSN: Page 4

14 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 The dimensionless governing equationsare solved with the help of semi implicit finite difference method. Influence of several parameters on velocit, temperature and concentration are discussed with the help of graphs. Also the effect of some parameters on skin friction, Nusselt number and Sherwood number is also discussed thoroughl. The main conclusions of the present stud are stated below:. Fluid velocit increases for increasing values of Thermal Grashof number, modified Grashof number, Porosit parameter, Soret number and but it shows reverse trend in the case of magnetic parameter, chemical reaction parameter and Prandtl number.. Fluidtemperaturereduces for increasing values of Prandtl number, whereas it enhances in the case of heat absorption parameter and radiation parameter. 3. Cconcentration of the fluid increases when soret number increases but it falls down under the influence of Schmidt number and chemical reaction parameter. 4. Skin friction increases when magnetic parameter and Prandtl number increase whereas it decreases for increasing values of Thermal Grashof number, modified Grashof number, Porosit parameter, heat absorption parameter, Soret number. 5. Nusselt number increases with increasing values of Prandtl number, but an opposite behavior is noticed in the case ofheat absorption parameter and Schmidt number. 6. Sherwood number is enhanced for increasing values of Schmidt number and chemical reaction parameter but it shows reverse trend in the presence of Soret number. REFERENCES. R.S.Tripath, G.C. Dash, S.R. Mishra, S. Baag, Chemical reaction effect on MHD free convective surface over a moving vertical plate through porous medium, Alexandria Engineering Journal. 54(5) D. Srinivasacharaand G. Swam Redd,Free convection in a non-newtonian power law fluid saturated porous medium with Chemical reaction and radiation effects, Special Topics & Reviews in Porous Media - An International Journal, 4(3) (3) R. Kandasam,K. Periasamand K.K.S. Prabhu, Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. Int. J. Heat and Mass Transfer,48(-) (5) A. Mahd, Effect of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in a porous media with variable viscosit,international Communications in Heat and Mass Transfer,37 () S.P. Anali Devi, R. Kandasam,Effects of a chemical reaction heat and mass transfer on MHD flow past a semi-infinite plate,z. Angew. Math. Mech. 8 () W. A. Khan and A.M. 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Nandkeolar, MHD natural convection flow with radiative heat transfer past an impulsivel moving plate with ramped wall temperature, Heat and Mass Transfer,47(5) () M.A. Sattar and M.M. Alam, Analtical solution of the free convection and mass transfer flow with thermal diffusion, Dhaka Universit Journal of Science,49 () O.D. Makinde, Similarit solution of hdro magnetic heat and mass transfer over a vertical plate with a convective surface boundar condition. Int.J.Ph.Sci.vol.5(6) () J.R. Pattnaik, G.C. Dash and S. Singh, Effect of heat and mass transfer on MHD free convection flow past an impulsivel moving infinite vertical plate with ramped wall temperature, International Journal of Fluid Mechanics,4() () P. Chandra Redd, M.C. Rau, G.S.S. Rau, Soret and Dufour effects on MHD free convection flow of Rivlin-Ericksen fluid past a semi infinite vertical plate, Advances and Applications in Fluid Mechanics, 9 (6) doi:.7654/fm T. Aat, M. Mustafa, and I. 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15 International Journal of Mathematics Trends and Technolog (IJMTT) Volume 54 Number 3- Februar 8 8. Siva Redd Sheri and R.Srinivasa Rau,Soret effects on Unstead MHD free convective flow past a semi-infinite vertical plate in the presence of viscous dissipation, International ournal for computational methods in engineering science and mechanics, 6 (5) C.Y. Cheng, Soret and Dufour effects on heat and mass transfer b natural convection from a vertical truncated cone in a fluid saturated porous medium with variable wall temperature and concentration, International Communications in Heat and Mass Transfer, 37(8) () D. Srinivasachara, B. Mallikaruna and R. Bhuvanaviaa, Soret and Dufour effects on mixed convection along a vertical wav surface in a porous medium with variable properties, Ain Shams Engineering Journal6 (5), doi:.6/.ase A.J. Chamka, A.M. Ali and Z.A.S. 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Sailakumari "Heat and Mass Transfer Characteristics of Mhd Free Convective Rivlin-Ericksen Fluid Flow Past a Porous Plate", International Journal of Mathematics Trends and Technolog (IJMTT). V53(6):44-45 Januar 8. ISSN: Page 6