**ARTICLES**

**On MHD mixed convection with soret and dufour effects past a vertical plate embedded in a porous medium**

**O. D. Makinde**

*Faculty of Engineering, Cape Peninsula University of Technology, P. O. Box 652, Cape Town 8000, South Africa. (makinded@cput.ac.za)*

** Abstract - **The paper studies the mixed convection flow of an incompressible Boussinesq fluid under the simultaneous action of buoyancy and transverse magnetic field with Soret and Dufour effects over a vertical porous plate with constant heat flux embedded in a porous medium. Under suitable nondimensionalization, the governing non-linear coupled differential equations are solved numerically using shooting quadrature. Tabular and graphical results are presented and discussed quantitatively. Results obtained which compare favourably well with published data show that the local skin friction is enhanced by the Sorets and Dufour effects.

** Keywords - **MHD Mixed Convection; Vertical Plate; Heat and Mass Transfer; Porous Medium; Soret and Dufour Effects.

**I. INTRODUCTION**

The study of mixed convection flow with simultaneous heat and mass transfer past a vertical plate under the influence of a magnetic field and chemical reaction has applications in many areas of science and engineering (Chen and Armaly, 1987). Heat and mass transfer occur in processes, such as drying, evaporation at the surface of a water body, and energy transfer in a wet cooling tower. In astrophysics and geophysics, it is applied to study the stellar and solar structures, interstellar matter and radio propagation through the ionosphere. In industries, it finds its application in cooling of nuclear reactors and magnetohydrodynamic (MHD) power generators, MHD pump, chemical vapour deposition on surfaces, formation and dispersion of fog, distribution of temperature and moisture over agriculture fields, etc. The problem of mixed convection under the influence of magnetic field has attracted numerous researchers in view of its wide applications. Soundalgekar* et al.* (1979) investigated the problem of free convection effects on stokes problem for a vertical plate with transverse applied magnetic field whereas Elbashbeshy (1997) studied MHD heat and mass transfer problem along a vertical plate under the combined buoyancy effects of thermal and species diffusion. Singh and Dikshit (1988) studied hydrodynamic flow past a continuously moving semi-infinite plate with large suction, while in Bestman (1990) the focus was on chemically reacting species. Abd El-Aziz (2007) studied the combined effects of MHD heat and mass transfer on a boundary layer flow of a viscous incompressible fluid having temperature dependent viscosity over a stretching surface. A comprehensive account of the boundary layers flow over a vertical flat plate embedded in a porous medium can be found in Pop and Ingham (2001), Geindreau and Auriault (2002), Makinde and Ogulu (2008).

*et al*. (1954). They explained the phenomena and derived the necessary formulae to calculate the thermal diffusion coefficient and the thermal-diffusion factor for monatomic gases or for polyatomic gas mixtures. Recently, Bég

*et al*. (2009) investigated numerically the free convection magnetohydrodynamic heat and mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects.

The present study extends the recent work of Makinde (2009) to include MHD mixed convection with Dufour and Soret effects past a vertical plate embedded in a porous medium. Solutions are presented in both graphical and tabular form and given in terms of the local skin friction, plate surface temperature and local mass transfer rate for various parametric values. It is hoped that the results obtained will not only provide useful information for applications, but also serve as a complement to the previous studies.

**II. MATHEMATICAL FORMULATION**

We consider steady, unidirectional flow of a laminar, incompressible, electrically conducting fluid past a semi-infinite porous vertical plate with constant heat flux embedded in a porous medium in the presence of a transversely imposed magnetic field (see Fig. 1). In addition, there is no applied electric field and all of the Hall effects are neglected. Since the magnetic Reynolds number is very small for most fluid used in industrial applications, we assume that the induced magnetic field is negligible. Let the *x*-axis be taken along the direction of plate and *y*-axis normal to it, then under the Boussinesq and boundary-layer approximations, the fluid equations for momentum, energy balance and concentration governing the problem under consideration can be written in dimensionless form as;

(1) |

**Fig. 1**: Flow configuration and coordinate system.

(2) | |

(3) | |

(4) | |

(5) |

It is important to note that Eqs. (1)-(5) are made dimensionless by introducing the following variables and quantities into the governing conservation equations:

(6) |

The set of Eqs. (1)-(3) under the boundary conditions (4)-(5) have been solved numerically by applying the Nachtsheim and Swigert (1965) shooting iteration technique together with Runge-Kutta sixth-order integration scheme. From the process of numerical computation, the surface temperature, the local skin-friction coefficient and the local Sherwood number, which are respectively proportional to *θ*(0), and are worked out and their numerical values are presented in a tabular form, where the prime symbol denotes differentiation with respect to *y.*

**III. RESULTS AND DISCUSSION**

*Pr*= 0.71, which corresponds to air and the values of Schmidt

**Table 1**: Computations showing comparison with earlier results for *Du=Sr=Gr=Gm*=0.1, *Pr* =1, *K*=1, *U* =0.5.

**Table 2**: Computations showing values of *'*(0), *u'*(0) and *θ*(0) for *Pr* =0.71, *U* = 0.3, *Sr = Du = Gr = Gm *= 0.1, *K*=1.

**Table 3**: Computations showing values of *'*(0), *u'*(0) and *θ*(0) for *Pr* =0.71, *U* = 0.3, *Ec = M = Gr = Gm* = 0.1, *Sc*=0.24

**Fig. 2**. Velocity profiles. *Du*=*Sr*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71;*K*=10; __*M*= 0; ooo*M *= 0.5; ++*M*= 3; ..*M*=10

**Fig. 3.** Velocity profiles. *Du*=*Sr*=*M*=*Gr*=*Gm*=*Ec*=0.1; *U*=0.3; *Pr*=0.71; *K*=10; __*Sc*= 0.24; oo*Sc *= 0.6;++*Sc*= 0.78; ..*Sc*=2.62

**Fig. 4.** Velocity profiles. *Du*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; *K*=10; __*Sr*= 0; oo*Sr *= 1; ++*Sr*= 3;……*Sr*=5

**Fig. 5**. Velocity profiles.

*Sr*=

*M*=

*Gr*=

*Gm*=

*Ec*=0.1;

*Sc*=0.6;

*U*=0.3;

*Pr*=0.71;

*K*=10;__

*Du*= 0; oo

*Du*= 1; ++

*Du*= 3; …

*Du*=5

**Fig. 6**. Velocity profiles. *Sr*=*Du*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; ___*K*= 0.1; oo*K *= 1; +++*K*= 3; …..*K*=10

**Fig. 7**. Velocity profiles.* Sr*=*M*=*Gr*=*Gm*=*Du*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; *K*=10; __*Ec*= 0; oo*Ec *= 1;++*Ec*= 3; …*Ec*=5

**Fig. 8**. Velocity profiles. *Sr*=*M*=*Ec*=*Gm*=*Du*=0.1; *Sc*=0.6; *U*=0.3;*Pr*=0.71;*K*=10;__*Gr*=0.1;oo*Gr*=0.5;++*Gr*=1; …*Gr*=1.2

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**Fig. 9**. Velocity profiles. *Sr*=*M*=*Ec*=*Gr*=*Du*=0.1; *Sc*=0.6; *U*= 0.3;*Pr*=0.71;*K*=10;__*Gm*=0.1;oo*Gm*=0.5;++*Gm*=1;…*Gm*=2.

**Fig. 10**. Temperature profiles *Du*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3;*Pr*=0.71;*K*=10;__*Sr*=0;oo*Sr*=1;++*Sr*= 3;...*Sr*=5

**Fig. 11**. Temperature profiles *Sr*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; *K*=10;__*Du*= 0; oo*Du *= 1;++*Du*= 3; …*Du*=5

**Fig. 12**. Temperature profiles *Sr*=*M*=*Gr*=*Gm*=*Du*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; *K*=10; __*Ec*= 0; oo*Ec *= 1;++*Ec*= 3; …*Ec*=5

]]>
**Fig. 13.** Temperature profiles *Sr*=*M*=*Ec*=*Gm*=*Du*=0.1; *Sc*=0.6; *U*=0.3; *Pr*=0.71; *K*=10;__*Gr*=0.1;oo*Gr*=0.5;++*Gr*=1;...*Gr*=1.2

**Fig. 14.** Temp. profiles *Sr*=*M*=*Ec*=*Gr*=*Du*=0.1; *Sc*=0.6;*U*=0.3; *Pr*=0.71;*K*=10;__*Gm*=0.1;oo*Gm*=0.5;++*Gm*=1;…*Gm*=2

**Fig. 15**. Concentration profiles *Du*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*=0.6; *U*=0.3;*Pr*=0.71;*K*=10;__*Sr*=0;oo*Sr*=1;++*Sr*=3;…*Sr*=5

**Fig. 16**. Concentration profiles. *Sr*=*M*=*Gr*=*Gm*=*Ec*=0.1; *Sc*= 0.6;*U*=0.3;*Pr*=0.71;*K*=10;__*Du*=0;oo*Du*=1;++*Du*=3;…*Du*=5

**Fig. 17**. Concentr. profiles

*Du*=

*Sr*=

*M*=

*Gr*=

*Gm*=

*Ec*=0.1;

*U*=0.3;

*Pr*=0.71;

*K*=10;__

*Sc*=0.24;oo

*Sc*=0.6;++

*Sc*=0.78;…

*Sc*=2.62

number (*Sc*) were chosen to be Sc = 0.24, 0.6, 0.78, 2.62, representing diffusing chemical species of most common interest in air like H_{2}, H_{2}O, NH_{3}, and Propyl Benzene respectively. For positive values of the buoyancy parameters i.e. *Gr* > 0 and *Gm* > 0, it is seen from tables (2)- (3) that the local skin friction on the plate surface increases with increasing parameter values of *Ec*, *M*, *Sr, Du*, *K* and decreases with increasing values of Sc. Further, it is found that the local Sherwood number at the plate surface (*Sh*) increases with increasing values of *Ec, Sc, M, Du, K* and decreases with increasing values of *Sr*.

**A. Velocity Profiles **

The influence of the various thermophysical parameters on the velocity profiles can be analyzed from Figs. 2-9. From these figures it is seen that the velocity starts from minimum value of zero at the plate surface and increases until it attains the peak value within the boundary layer and then starts decreasing until it reaches the free stream prescribed value satisfying the far field boundary condition. It is interesting to note that the effect of magnetic field (see Fig. 2) is to decrease the value of the velocity profile throughout the boundary layer. The effect of magnetic field is more prominent at the point of peak value i.e. the peak value drastically decreases with increasing intensity of the magnetic field, because the presence of magnetic field in an electrically conducting fluid introduces a force called the Lorentz force, which acts against the flow if the magnetic field is applied in the normal direction, as in the present problem. This type of resisting force slows down the fluid velocity as shown in this figure. Figure 3 illustrates the effect of increasing Schmidt number on the velocity profiles. The trend of the velocity profiles in this figure is same as those shown in Fig. 2. A decrease in the peak value of velocity with the boundary layer is observed with increasing Schmidt number due to low molecular diffusivity of the chemical species. From Figs. 4-5, it is noteworthy that the effects of increasing diffusion-thermo (Dufour) number (*Du*) and thermal-diffusion (Soret) number (*Sr*) is to increase the value of the velocity profile throughout the boundary layer. Soret and Dufour effects are more prominent at the point of peak value i.e. the peak value of velocity increases with increase in the values of Soret and Dufour numbers (*Sr* and *Du)*. Similar trend is observed in Figs. 6-9. The peak value of fluid velocity within the boundary layer increases with an increase in porous medium permeability (*K*), viscosity dissipation (*Ec*) and buoyancy parameters (*Gr*, *Gm*).

**B. Temperature Profiles**

Figures. 10-14 represent graph of temperature distribution with span-wise coordinate *y* for different values of thermophysical parameters. From these figures we note that, temperature is highest at the porous plate surface and asymptotically decreases to zero far away from the plate satisfying the boundary condition. It is interesting to note that increase in the thermal-diffusion (Soret) parameter (*Sr*) decreases the temperature distribution in the thermal boundary layer leading to a decrease in the thickness of the thermal boundary layer as illustrated in Fig. (10). In Figs. 11-14, it can easily be seen that the fluid temperature increases with increasing effects of diffusion-thermo (Dufour), viscous dissipation (*Ec*) and buoyancy parameters (*Gr*, *Gm*), leading to an increase in thermal boundary layer thickness.

**C. Concentration Profiles **

The chemical species concentration profiles against spanwise coordinate *y* for varying values of physical parameters in the boundary layer are demonstrated in Figs. 15-17. The thermal-diffusion (Soret) effect on the concentration profiles is illustrated in Fig. 15. It is clearly observed from this figure that the concentration of species value of 1 at the start of the plate surface, increases to its peak value within the boundary layer and decreases slowly till it attains the minimum value of zero far away from the plate surface with increasing value of Soret parameter *Sr*. In Fig. 16, an increase in the diffusion-thermo (Dufour) effects causes a slight increase in the solutal boundary layer thickness. The effect of Schmidt number Sc on concentration profiles is depicted in Fig. 17. As can be expected for all other pa-

rameter fixed, an increase in the Schmidt number *Sc* decreases the concentration boundary layer thickness which is associated with the reduction in the concentration profiles. Physically, the increase of *Sc* means decrease of molecular diffusion *D*. Hence, the concentration of the species is higher for small values of *Sc* and lower for larger values of *Sc*.

**IV. CONCLUSIONS**

In this study, a numerical analysis is presented to investigate the influence of magnetic field on the steady combined heat and mass flow of an electrically conducting fluid by mixed convection along a semi-infinite vertical porous plate embedded in a saturated porous medium with a constant heat flux taking into Soret and Dufour effects. Velocity, temperature and concentration profiles are presented graphically and analyzed. It is found that, the local skin friction on the plate surface increases with increasing parameter values of *Ec*, *M*, *Sr, Du*, *K* and decreases with increasing values of Sc while the local mass transfer rate at the plate surface increases with increasing values of *Ec, Sc, M, Du, K* and decreases with increasing values of *Sr*.

**NOMENCLATURE**

Cartesian Coordinates | |

constant wall heat flux | |

c_{w} | concentration at plate surface |

c_{s} | concentration susceptibility |

c | chemical species concentration |

Du | Dufour number |

Ec | Eckert number |

fluid temperature | |

free stream temperature | |

free stream concentration | |

g | gravitational acceleration |

M | magnetic field parameter |

B_{0} | magnetic field intensity |

D | mass diffusivity |

Pr | Prandtl number |

Sc | Schmidt number |

Sr | Soret number |

Gm | solutal Grashof number |

C_{p} | specific heat at constant pressure |

Gr | thermal Grashof number |

K_{T} | thermal diffusion ratio |

velocity components | |

V_{0} | wall suction velocity |

θ | dimensionless temperature |

dimensionless concentration | |

σ | electrical conductivity |

υ | fluid kinematic viscosity |

ρ | fluid density |

λ | thermal conductivity |

Β_{T} | thermal expansion coefficient |

Β_{c} | solutal expansion coefficient |

**ACKNOWLEDGEMENTS**

The author would like to thank the National Research Foundation of South Africa Thuthuka programme for financial support.

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**Received: December 8, 2009. Accepted: March 20, 2010. Recommended by Subject Editor Walter Ambrosini.**