Viscous Dissipation, Soret and Dufour Effects on MHD Free Convection Flow of Doubly Stratified Darcy Porous Medium with Internal Heat Generation

 

K. Govardhan¹, K. Kaladhar², Balasiddulu Malga¹, G. Nagaraju¹, S.Renuka³

¹Department of Engineering Mathematics, GITAM University, Telangana.

²Department of Mathematics, NIT Pududchery, Karaikal.

³Department of Mathematics, Osmania University, Hyderabad, Telangana.

 

ABSTRACT:

In this paper, the steady Magnetohydrodynamic free convection heat and mass transfer from a vertical surface in a doubly stratified Darcian porous medium has been studied. In addition viscous dissipation, internal heat generation, Soret and Dufour effects are presented numerically. The non–linear partial differential equations are transformed by a similarity transformation into a system of ordinary differential equations, which is solved numerically by using implicit Crank–Nicolson finite difference scheme. The effect of various parameters such as Lewis number, magnetic parameter, Buoyancy ratio, Soret number, Dufour number, thermal and solutal stratification and Heat generation parameters on velocity, wall temperature and concentration are analyzed through plots.

 

 

 

INTRODUCTION:

Coupled heat and mass transfer phenomenon in porous media is gaining much attention due to its interesting applications. The flow phenomenon in this case is relatively more complex than that in the pure thermal/solutal convection process. Processes involving heat and mass transfer in porous media are often encountered in the chemical industry, in reservoir engineering in connection with thermal recovery process and in the study of the dynamics of hot and salty springs of a sea. Underground spread of chemical waste and other pollutants, grain storage, evaporation cooling, and solidification are few other application areas where combined thermo solutal convection in porous media can be observed.

 

Due to the coupling of temperature and concentration, new parameters such as buoyancy ratio and Lewis number (diffusion ratio) arise, and they influence the convective transport to a greater extent. Several authors, Chamkha et al. [1] and Srinivasacharya et al. [2] to mention but a few, have studied the free convective transport over a different surface geometries in a fluid saturated non-Darcy porous media. A detailed review of convective heat and mass transfer in Darcian and non-Darcian porous medium can be found in the book by Nield and Bejan [3]. Many problems of Magneto-hydrodynamics (MHD) flows of porous media (Darcian and non- Darcian) saturated with Newtonian as well as non-Newtonian fluids (see Chamkha and Aly, [4]; Hamada et al., [5]) have been analyzed and reported in the literature due to its importance in the various fields. Most recently, Rashidi et al. [6] studied the mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation. In many problems of practical interest, natural/mixed convection flows arise in a thermally stratified environment. The input of thermal energy in enclosed fluid regions, due to the discharge of hot fluid or heat removal from hot bodies, often leads to the generation of a stable thermal stratification. Stratification of fluid arises due to temperature variations, concentration differences or the presence of different fluids. Several investigations have explored the importance of convective heat and mass transfer in doubly stratified porous media using Darcian and non-Darcian models. This phenomenon was first shown theoretically by Prandtl [7] for an infinite wall and later on by Jaluria and Himasekhar [8] for semi-infinite walls. Lakshmi Narayana and Murthy [9] reported that the temperature and concentration became negative in the boundary layer depending on the relative intensity of the thermal and solutal stratification. Most recently, Ram Reddy et al. [10] presented the nature of thermally stratified nanofluid flow in a saturated non-Darcy porous medium numerically.  When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate nature. The energy flux caused by concentration gradient is termed as diffusion-thermo (Dufour) effect. On the other hand, mass fluxes can also be created by temperature gradients and this embodies the thermal-diffusion (Soret) effect. In most of the studies related to heat and mass transfer process, Soret and Dufour effects are neglected on the basis that they are of smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. But these effects are considered as second order phenomena and may become significant in areas such as hydrology, petrology, geosciences, etc. The Soret effect, for instance, has been utilized for isotope separation and in mixture between gases with very light molecular weight and of medium molecular weight. The importance of the Soret effect at low Rayleigh number has been analyzed by Bergman and Srinivasan [11], while Hurle and Jakerman [12] considered the thermo solutal convection due to large temperature gradient. In view of combined applications, Srinivasacharya and Surender [13] presented the non-Darcy mixed convection in a doubly stratified porous medium with soret - dufour effects.

 

The effects of Soret and Dufour on free convection heat and mass transfer from a vertical surface in a doubly stratified Darcy porous medium has been presented by Lakshmi Narayana and Murthy [14].  But they have not considered the effects of MHD, Viscous dissipation and internal heat generation. Also, the effect of Soret and Dufour with viscous dissipation and magnetic field on free convection along a vertical plate in a doubly stratified Darcy porous medium saturated with internal heat generation has not been reported in the literature. There is a more common practice situation, where heat and mass transfer occurs at the boundary surface to or from a fluid flowing on the surface at a known temperature and concentration, e.g. in heat exchangers, condensers and reboilers. B.S Malga and Govardhan[15] studied finite element analysis of fully developed free convection flow heat and mass transfer of a MHD / Micropolar Fluid over a Vertical Channel. In view of the above said applications, the present study investigates the combined effects of magnetic, thermal and solutal stratification on free convection along a vertical plate saturated with MHD and viscous dissipation in Darcy porous medium. The implicit Crank - Nicolson finite difference scheme is used to solve the nonlinear system of the present system of equations with internal heat generation. The effects of the emerging parameters on velocity, temperature and concentration are presented through plots.

 

FORMULATION OF THE PROBLEM:

Free convection heat and mass transfer from an impermeable vertical flat wall in a stable and doubly stratified fluid in saturated porous medium with thermal diffusion (Soret) and diffusion-thermo (Dufour) effects is considered along with MHD and viscous dissipation. The schematic drawing is shown in Fig. 1. The wall is maintained at constant heat and mass flux conditions q_w and q_m respectively. The temperature and the mass concentration of the ambient medium are assumed to be in the form,   where T_∞,0 and C_∞,0 are the temperature and concentration at any reference point inside the boundary layers respectively. Also assume that the flow is slow such that the Darcy law is valid. Viscous resistance due to the solid boundary is neglected under the assumption that the medium has low permeability. The non-uniform transverse magnetic field B₀ is imposed along the y-axis.

 

Then the governing equations for the boundary layer flow, heat and mass transfer from the wall y=0 into the fluid saturated stratified porous medium x ≥0 and y>0 (by Boussinesq approximation) are given by (see Nield and Bejan [3])

                                                                                                                                                                (1)

                                                                                                           (2)

                                                                                          (3)

                                                                                                                              (4)

with the boundary conditions

                                                                                        (5)

 

Here x and y are the Cartesian coordinates, u and v are the averaged velocity components in the x - and y - directions respectively, T and C are the dimensional temperature and concentration respectively, β_T and β_C are the coefficient of thermal expansion and coefficient  of   solutal  expansion  respectively,  is the kinematic viscosity of the fluid, K is the permeability, g is the acceleration due to gravity, α and D are the thermal and solutal diffusivities of the medium respectively, k is the thermal diffusion ratio, C_P is the specific heat at constant pressure and C_S is the concentration susceptibility. Variables q_w and q_m are constant heat and mass flux at the wall, respectively. A and B are constants, varied to alter the intensity of stratification in the medium. The subscripts w, (¥,0) and ¥ indicate the conditions at the wall, at some reference point in the medium and at the outer edge of the boundary layer respectively.

 

Making use of the following similarity transformation

                                                                                                                           (6)

The governing Equations (1) - (4) become

                                                                                                                                           (7)

                                                                                                (8)

                           

                                                                                                                 (9)

and the boundary conditions (5) transform into

                                                                                                          (10)

 

where  is the Darcy - Rayleigh number, is the diffusivity ratio and  is the buoyancy ratio. () indicates the aiding buoyancy at which both the thermal and solutal buoyancies are in the same directions and  indicates opposing buoyancy when the solutal buoyancy is in the opposite direction to the thermal buoyancy).  is the Eckert number, and are stratification parameters. When , ,  and  will be independent of  x and allows the similarity solution.  is the heat generation parameter,  is the magnetic parameter,  is the Dufour parameter and  is the Soret parameter.

 

RESULTS AND DISCUSSION:

The system of non-linear ordinary differential Equations (7- 9) with boundary conditions (10) have been solved numerically by using Crack-Nickelson implicit finite difference scheme. In order to see the effects of step size, the code has been ran with three different step sizes as Dh =0.001, Dh=0.01 and Dh=0.05 and in each case we found very good agreement between them on different profiles with almost 10^-5. After some trials we imposed a maximal value of η at ∞ of 10 and a grid size of η as 0.01. Then the effects of D_f, S_r, M, N, ε₁, ε₂, E_C and Q on velocity, temperature and concentrations are presented through plots.

 

Extensive calculations have been performed to obtain the flow, temperature and concentration fields for the following range of parameters 0≤ε_1, ε₂≤ 0.5, 0≤Le≤5, -1≤N≤1 and 0≤S_r, D_ƒ<0.5 (to avoid change of sign of temperature concentration value in the boundary layer).

 

The effects of Lewis number (Le), magnetic parameter (M) on velocity f’ with an aiding buoyancy (N=1) is shown in Figs. 2-3 for fixed ε₁=0, ε₂=0, D_f = 0.09, S_r=0.01. It can be seen from Figure 2 that the non dimensional velocity f’ decreases with an increase in the Lewis parameter Le. It is also observed form Figure 3 that as the magnetic parameter M increases, the non dimensional velocity f’ decreases.

 

The effect of Lewis number Le and magnetic parameter M on temperature is exposed in figs. 4-5 in case of aiding buoyancy (N=1). It is observed that the temperature q increases with diffusion rate (Le), where as temperature decreases with an increase in magnetic parameter M. Figure 6 shows that the temperature profile in case of aiding buoyancy (N=1)  when ε₁< ε₂and ε₁>ε₂ for different values of soret number S_r. It is observed that as Soret number S_r increases, temperature decreases when ε₁< ε₂, where as the temperature increases as Soret number increases when ε₁>ε₂.

 

Figure 7 presents the temperature profile at different values of Lewis number Le in case of opposing buoyancy (N=-0.5). It shows that as Lewis number Le increases, the temperature q increases. Figure 8 displays the temperature profile for opposing buoyancy (N=-0.5) at different values of Le and S_r with fixed ε₁=0.4, ε₂=0.1 and D_f =0.09. It is observed that the temperature is increases as an increase in the parameters Le and S_r. Figure 9 illustrates the effect of soret number in case of ε₁< ε₂ and ε₁>ε₂. It is observed that the temperature profile increases with the increase of S_r when ε₁< ε₂, ε₁>ε₂ and D_ƒ = 0.09. It is observed from Fig 10 that the temperature decreases with N increases and also decreases as increase in Dufour parameter D_f in both aiding and opposing buoyancies.

 

The variation of temperature profile with Eckert number E_c (viscous dissipation) in case of opposing buoyancy is shown in Fig 11. We see that the dimensionless temperature increase with an increase in parameter E_c. Figure 12 and 13 present the effect of heat generation parameter on temperature in both (aiding and opposing) buoyancies. It can be noted that the dimensionless temperature increases as the internal heat generation increases.

 

The effect of magnetic parameter M on concentration profile is offered in Fig 14 in case of aided buoyancy.  It can be seen from Fig 14 that the non dimensional concentration increases as the magnetic parameter increases. It can be depict from Fig 15 that as an increase in Le, concentration decreases in presence of aided buoyancy.

 

The effect of Soret parameter on the concentration field is shown in Figs. 16 and 17. As Soret number increases, concentration profiles increases when ε₁< ε₂ and ε₁>ε₂ and ε₁= ε₂. Figures 18 and 19 illustrates the influence of internal heat generation in both cases (aided and buoyancy). It is noticed that the non dimensional concentration decreases with an increase in internal heat generation in case of both aiding and opposing buoyancies.

 

Figure 1: Schematic drawing of the problem

         

Fig. 2: Lewis number Le effect on velocity  profile when N=1           Fig. 3: Velocity profiles for different values of   M at N=1

     

Fig 4: Temperature profile for different values of Le at N=1            Fig 5: Magnetic parameter effect on Temperature under aiding

                                                                                                                  buoyancy

      

Fig. 6. Temperature profile at different ε₁, ε_2, S_r when N=1                   Fig. 7 . Lewis number effect on temperature under opposing

                                                                                                                                Buoyancy

         

Fig. 8. Effect of Le, S_r on temperature profile under opposing              Fig. 9. Temperature profile at different ε1, ε₂, S_r when N= -0.5

      Buoyancy

             

Fig. 10. Effect of Dufour number on Temperature both in                              Fig. 11. Variation of q for different when N=1

aiding and opposing buoyancies

           

Fig. 12. Variation of q for different values of when N=1                                Fig. 13. Variation of q for different values of when N= -0.5

              

Fig. 14. Concentration profiles for different values of M                                    Fig. 15. Effect of Le on Concentration at N=1

                    when N=1

             

Fig. 16. Concentration profiles for different   values of ε₁, ε₂ and                   Fig. 17. Effect of S_r on f  when N=-0.5 and N=1

S_r at N= - 0.5          

 

 Fig. 18. Effect of Q on f in case of aiding   Buoyancy                                        Fig. 19. Effect of Q on f in case of opposing Buoyancy

 

CONCLUSIONS:

In this study, the steady Magnetohydrodynamic free convection heat and mass transfer from a vertical surface in a doubly stratified Darcian porous medium has been studied. In addition viscous dissipation, internal heat generation, Soret and Dufour effects are presented numerically. Using the similarity variables, the governing equations are transformed into a set of non-similar parabolic equations and the numerical solution has been presented for a wide range of parameters. Results clearly show that the emerging parameters of the present study influences very effectively on velocity, temperature and concentration.

The main findings are summarized as follows:

·        In case of aiding buoyancy, the non dimensional velocity decreases as an increase in Lewis parameter and magnetic parameter.

·        Temperature increases as Lewis parameter increases where as there is a decrement in temperature with the increase of magnetic parameter in case of aiding buoyancy.

·        At N =1 (aiding buoyancy), increase in Soret number S_r leads to decrease in temperature when ε₁< ε₂, where as the temperature increases as Soret number increases at ε₁>ε₂.

·        In case of opposing buoyancy (N= - 0.5), temperature increases when soret and Lewis number increases.

·        Temperature increases with the increase of S_r  in both the cases ε₁< ε₂, ε₁>ε₂when N = -0.5.

·        Temperature decreases as an increase in Dufour parameter D_f in both aiding and opposing buoyancies.

·        As dissipation rate increases, temperature increases at N=-0.5.

·        When there is an increase in heat generation, temperature increases in both aided and opposing buoyancies.

·        In case of aiding buoyancy, the dimensionless concentration increases as magnetic parameter increases, where as concentration decreases as an increase in Lewis number.

·        Concentration increases as Soret parameter increases when ε₁< ε₂and ε₁>ε₂ in case of opposing buoyancy.

·        It is noticed that the presence of soret number, concentration increases in both opposing and aiding buoyancies.

·        It is observed that, increase in heat generation reduces the concentration when N=1 and N=-0.5.

 

REFERENCES:

1.       Chamkha A.J,  Gorla R.S.R,  and Ghodeswar, K.  Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Transp Porous Media. 2011;86:13–22.

2.       Srinivasacharya D,  Pranitha J,  and RamReddy Ch. Magnetic and double dispersion effects on free convection in a non-Darcy porous medium saturated with power-law fluid. International Journal for Computational Methods in Engineering Science and Mechanics. 2012;13: 210-218.

3.       D.A. Nield,  and A. Bejan. Convection in Porous Media. New York, Springer, (2013).

4.       A.J. Chamkha, and A.M. Aly. MHD free convection flow of a nanofluid past a vertical plate in the presence of heat generation or absorption effects. Chemical Engineering Communications. 198 (2010)  425–441.

5.       M.A.A. Hamada, I. Pop, and A.I. Md-Ismail. Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Analysis: Real World Applications. 12 (2011) 1338–1346.

6.       M. M. Rashidi, M. Ali, N. Freidoonimehr, B. Rostami,  and  M. Anwar Hossain. Mixed Convective Heat Transfer for MHD Viscoelastic Fluid Flow over a Porous Wedge with Thermal Radiation. Advances in Mechanical Engineering. 2014, Article ID 735939, 10 pages.

7.       L. Prandtl. Essentials of Fluid Dynamics, Blackie, London, (1952).

8.       Y. Jaluria, and K. Himasekhar. Buoyancy Induced Two Dimensional Vertical Flows in a Thermally Stratified Environment. Comput. Fluids. 11 (1983) 39–49.

9.       P. A. Lakshmi Narayana, and P. V. S. N. Murthy. Free Convective Heat and Mass Transfer in a Doubly Stratified Non-Darcy Porous Medium. J. Heat Transf, 128 (2006) 1204–1212.

10.     Ch. RamReddy, P. V. S. N. Murthy, A. M. Rashad,  and A. J. Chamkha. Numerical study of thermally stratified nano fluid flow in a saturated non-Darcy porous medium. The European Physical Journal Plus. 129 (2014) 25-35.

11.     T.L. Bergman, and R. Srinivasan. Numerical Solution of Soret Induced Double Diffusion in an Initially Uniform Concentration Binary Liquid. Int. J.Heat Mass Transfer. 32 (1989) 679-687.

12.     D.T. Hurle, and E. Jakerman. Soret Driven Thermo Solutal Convection, J. Fluid Mech.  447 (1989) 667-687.

13.     D. Srinivasacharya, and O. Surender. Non-Darcy Mixed Convection in a Doubly Stratified Porous Medium with Soret-Dufour Effects. International Journal of Engineering Mathematics. 2014, Article ID 126218, 9 pages.

14.     P. A. Lakshmi Narayana, and P. V. S. N. Murthy. Soret and Dufour Effects on Free Convection Heat and Mass Transfer in a Doubly Stratified Darcy Porous Medium. J. Porous Media. 10 (2007) 613–623.

15.     B.S Malga, and Govardhan, finite element analysis of fully developed free convection flow heat and mass transfer of a mhd / micropolar fluid over a vertical Channel. Jordan Journal of Mechanical and Industrial Engineering. Vol. 8(4), P:219 – 232.

16.     R. Nicole, “Title of paper with only first word capitalized,” J. Name Stand. Abbrev., in press.

17.     K. Elissa, “Title of paper if known,” unpublished.

 

 

 

 

Accepted on 07.12.2017      ©A&V Publications All right reserved