The Effects of Thermal Radiation, Hall Currents, Soret, and Dufour on MHD Flow by Mixed Convection over a Vertical Surface in Porous Media

Abstract

The study sought to investigate the influence of a magnetic field on heat and mass transfer by mixed convection from vertical surfaces in the presence of Hall, radiation, Soret (thermal-diffusion), and Dufour (diffusion-thermo) effects. The similarity solutions were obtained using suitable transformations. The similarity ordinary differential equations were then solved by MATLAB routine bvp4c. The numerical results for some special cases were compared with the exact solution and those obtained by Elgazery (2009) and were found to be in good agreement. A parametric study illustrating the influence of the magnetic strength, Hall current, Dufour, and Soret, Eckert number, thermal radiation, and permeability parameter on the velocity, temperature, and concentration was investigated.

1. Introduction

The range of free convective flows that occur in nature and in engineering practice is very large and has been extensively considered by many researchers (see, [1, 2], among others). When heat and mass transfer occur simultaneously between the fluxes, the driving potentials are of more intricate nature. An energy flux can be generated not only by temperature gradients but by composition gradients. The energy flux caused by a composition is called Dufour or diffusion-thermo effect. Temperature gradients can also create mass fluxes, and this is the Soret or thermal-diffusion effect. Generally, the thermal-diffusion and the diffusion-thermo effects are of smaller-order magnitude than the effects prescribed by Fourier's or Fick's laws and are often neglected in heat and mass transfer processes. However, there are exceptions. The thermal-diffusion effect, for instance, has been utilized for isotope separation and in mixture between gases with very light molecular weight (Hydrogen-Hellium) and of medium molecular weight (Nitrogen-air) the diffusion-thermo effect was found to be of a magnitude such that it cannot be neglected (see Kafoussias and Williams [3] and references therein). Kafoussias and Williams [3] considered the boundary layer-flows in the presence of Soret, and Dufour effects associated with the thermal diffusion and diffusion-thermo for the mixed forced natural convection.

In recent years, progress has been considerably made in the study of heat and mass transfer in magneto hydrodynamic flows due to its application in many devices, like the MHD power generator and Hall accelerator. The influence of magnetic field on the flow of an electrically conducting viscous fluid with mass transfer and radiation absorption is also useful in planetary atmosphere research. Kinyanjui et al. [4] presented simultaneous heat and mass transfer in unsteady free convection flow with radiation absorption past an impulsively started infinite vertical porous plate subjected to a strong magnetic field. Yih [5] numerically analyzed the effect of transpiration velocity on the heat and mass transfer characteristics of mixed convection about a permeable vertical plate embedded in a saturated porous medium under the coupled effects of thermal and mass diffusion. Elbashbeshy [6] studied the effect of surface mass flux on mixed convection along a vertical plate embedded in porous medium.

Chin et al. [7] obtained numerical results for the steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature. Pal and Talukdar [8] analyzed the combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium is analyzed. Mukhopadhyay [9] performed an analysis to investigate the effects of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium. Hayat et al. [10] analyzed a mathematical model in order to study the heat and mass transfer characteristics in mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid, by taking into account the diffusionthermo (Dufour) and thermal-diffusion (Soret) effects.

Postelnicu [11] studied simultaneous heat and mass transfer by natural convection from a vertical plate embedded in electrically conducting fluid saturated porous medium, using Darcy-Boussinesq model including Soret, and Dufour effects. Lyubimova et al. [12] dealt with the numerical investigation of the influence of static and vibrational acceleration on the measurement of diffusion and Soret coefficients in binary mixtures, in low gravity conditions. Abreu et al. [13] examined the boundary layer solutions for the cases of forced, natural, and mixed convection under a continuous set of similarity type variables determined by a combination of pertinent variables measuring the relative importance of buoyant force term in the momentum equation.

Alam et al. [14] studied numerically the Dufour and Soret effects on combined free-forced convection and mass transfer flow past a semi-infinite vertical plate, under the influence of transversely applied magnetic field. Alam and Rahman [15] studied numerically the Dufour and Soret effects on mixed convection flow past a vertical plate embedded in a porous medium. Li et al. [16] took an account of the thermal-diffusion and diffusion-thermo effects, to study the properties of the heat and mass transfer in a strongly endothermic chemical reaction system for a porous medium.

Gaikwad et al. [17] investigated the onset of double diffusive convection in a two-component couple of stress fluid layer with Soret, and Dufour effects using both linear and nonlinear stability analysis.

The interaction of buoyancy with thermal radiation has increased greatly during the last decade due to its importance in many practical applications. The thermal radiation effect is important under many isothermal and nonisothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then thermal radiation could be important. The knowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with a sought characteristics. Abd El-Aziz [18] studied the thermal-diffusion and diffusion-thermo effects on the heat and mass transfer characteristics of free convection past a continuously stretching permeable surface in the presence of magnetic field, blowing/suction, and radiation.

Osalusi et al. [19] investigated thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady hydromagnetic convective and slip flow due to a rotating disk in the presence of viscous dissipation and Ohmic heating. Motsa [20] investigated the effect of both the Soret, and Dufour effects on the onset of double diffusive convection. Mansour et al. [21] investigated the effects of chemical reaction, thermal stratification, Soret number, and Dufour number on MHD-free convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid on a vertical stretching surface embedded in a saturated porous medium. Shateyi [22] investigated thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing.

Afify [23] carried out an analysis to study free convective heat and mass transfer of an incompressible, electrically conducting fluid over a stretching sheet in the presence of suction and injection with thermal-diffusion and diffusion-thermo effects. Hence, based on the mentioned investigations and applications, the present paper considers the effect of both the Soret, and Dufuor effects on MHD convective heat and mass transfer from vertical surfaces with Hall and radiation effects. Elgazery [24] analyzed numerically the problem of magneto-micropolar fluid flow, heat and mass transfer with suction and blowing through a porous medium under the effects of chemical reaction, Hall, ion-slip currents, variable viscosity, and variable thermal diffusivity.

Motivated by the above referenced work and the numerous possible industrial applications of the problem (like in isotope separation), it is of paramount interest in this study to investigate the effects of Hall currents, thermal radiation, Soret, and Dufour on boundary layer mixed convection MHD flow over a vertical surface in the presence of suction. None of the above investigations simultaneously studied the effects of Hall currents, thermal radiation, Soret, and Dufour on boundary layer mixed MHD flow over a vertical surface through a porous medium. Hence, the purpose of this paper is to extend Afify [23], to study the more general problem which includes thermal radiation, Soret, and Dufour effects on mixed convection MHD flow with heat and mass transfer past a vertical plate with suction through a porous medium in the presence of Hall currents.

The momentum, thermal, and solutal boundary layer equations are transformed into a set of ordinary differential equations and then solved using MATALAB bvp4c. The analysis of the results obtained in the present work shows that the flow field is appreciably influenced by Dufour and Soret numbers, Hall, and thermal radiation parameters and suction on the wall. To reveal the tendency of the solutions, selected results for the velocity components, temperature, and concentration are graphically depicted. The rest of the paper is structured as follows. In Section 2, we formulate the problem; in Section 3, we give the method of solution. Our results are presented and discussed in Section 4, and in Section 5, we present some brief conclusions.

2. Mathematical Formulation

We consider mixed free-forced convective and mass transfer flow of a viscous incompressible fluid over an isothermal semi-infinite vertical flat plate through a porous medium. The flow is steady, laminar, and three dimensional and the viscous fluid is an electrically conducting one. The flow configuration and coordinate system are as shown in Figure 1 with the 𝑥-axis along the vertical plate and the 𝑦-axis normal to it.

Figure 1 

The coordinate system for the physical model of the problem.

The 𝑧-direction coincides with the leading edge of the plate. A normal magnetic field is assumed to be applied in the 𝑦-direction and the induced magnetic field is negligible in comparison with the applied one which corresponds to very small magnetic Reynolds number. The surface is maintained at uniform constant temperature and concentration. The flow has significant thermal radiation, Hall, Soret, and Dufour effects. Under the electromagnetic Boussinesq approximations the basic boundary-layer equations are given by𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝜕𝑦=𝜈²𝑢𝜕𝑦²+𝑔𝛽_𝑡(𝑇−𝑇_∞)+𝑔𝛽_𝑐(𝐶−𝐶_∞)−𝜎𝐵²₀𝜌(1+𝑚²)𝜇(𝑢+𝑚𝑤)−𝜌𝑘^∗𝑢𝑢,𝜕𝑤𝜕𝑥+𝑣𝜕𝑤𝜕𝜕𝑦=𝜈²𝑤𝜕𝑦²+𝜎𝐵²₀𝜌(1+𝑚²)𝜇(𝑚𝑢−𝑤)−𝜌𝑘^∗𝑢𝑤,𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝑘𝜕𝑦𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦²+𝐷𝑘_𝑡𝑐_𝑠𝑐_𝑝𝜕²𝐶𝜕𝑦²+𝜎𝐵²₀𝜌𝑐_𝑝(1+𝑚²)(𝑢²+𝑤²1)−𝜌𝑐_𝑝𝜕𝑞_𝑟𝑢𝜕𝑦,𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝜕𝑦=𝐷²𝐶𝜕𝑦²+𝐷𝑘_𝑡𝑇_𝑚𝜕²𝑇𝜕𝑦², (2.1)(2.2)(2.3)(2.4)(2.5) where 𝑢,𝑣, and 𝑤 are the fluid velocity components along the 𝑥-, 𝑦-, and 𝑧-axes, respectively. 𝑇 and 𝐶 are the fluid temperature and concentration, respectively. 𝜈 is the kinematic viscosity, 𝜇 is the dynamic viscosity, 𝑔 is the gravitational force due to acceleration, 𝜌 is the density, 𝛽_𝑡 is the coefficient of volume expansion, 𝛽_𝑐 is the volumetric coefficient of expansion with concentration, 𝑘 is the thermal conductivity of the fluid, 𝐵₀ is the magnetic field of constant strength, 𝐷 is the coefficient of mass diffusivity, 𝑐_𝑝 is the specific heat at constant pressure, 𝑇_𝑚 is the mean fluid temperature, 𝑘_𝑡 is the thermal diffusion ratio, 𝑘^∗ is the permeability, 𝑚 is the Hall parameter, and 𝑐_𝑠 is the concentration susceptibility.

The boundary conditions are𝑢(𝑥,0)=𝑈_𝑠=𝐴₀𝑥,𝑣(𝑥,0)=−𝑉_𝑤,𝑤(𝑥,0)=0,𝑇(𝑥,0)=𝑇_𝑤,𝐶(𝑥,0)=𝐶_𝑤,𝑢(𝑥,∞)=𝑤(𝑥,∞)=0,𝑇(𝑥,∞)=𝑇_∞,𝐶(𝑥,∞)=𝐶_∞. (2.6)𝑈_𝑠 is the surface velocity, 𝐴₀ is a constant with dimension (𝑡𝑖𝑚𝑒)⁻¹, and 𝑉_𝑤,𝑇_𝑤, and 𝐶_𝑤 are the suction (>0) or injection (<0) velocity, the fluid temperature, and concentration at the plate, respectively. 𝑇_𝑤 and 𝐶_𝑤 are the temperature and concentration of the plate. The radiative heat flux 𝑞_𝑟 is described by the Rosseland approximation such that𝑞_𝑟=−4𝜎^∗3𝐾𝜕𝑇⁴𝜕𝑦, (2.7) where 𝜎^∗ and 𝐾 are the Stefan-Boltzman constant and the mean absorption coefficient, respectively. Following Chamkha [25], we assume that the temperature differences within the flow are sufficiently small so that the 𝑇⁴ can be expressed as a linear function after using Taylor series to expand 𝑇⁴ about the free stream temperature 𝑇_∞ and neglecting higher-order terms. This results in the following approximation:𝑇⁴≈4𝑇³_∞𝑇−3𝑇⁴_∞. (2.8) Using (2.7) and (2.8) in (2.4), we obtain 𝜕𝑞_𝑟𝜕𝑦=−16𝜎^∗𝑇³_∞𝜕3𝐾²𝑇𝜕𝑦². (2.9) Following Elgazery [24] we nondimensionalize (2.1)–(2.5) using the following transformations:𝜂=𝐴₀𝜈𝑦,𝑢=𝐴₀𝑥𝑓^′(𝜂),𝑣=−𝐴₀𝜈𝑓(𝜂),𝑤=𝐴₀𝜈ℎ(𝜂),𝑇=𝑇_∞+(𝑇_𝑤−𝑇_∞)𝜃(𝜂),𝐶=𝐶_∞+(𝐶_𝑤−𝐶_∞)𝜙(𝜂), (2.10) where 𝑓(𝜂),ℎ(𝜂),𝜃(𝜂), and 𝜙(𝜂) are the dimensional stream, microrotation functions, temperature, and concentration distribution functions, respectively.

Upon substituting (2.10) into (2.1)–(2.5) we get the following similarity equations:𝑓^′′′+𝑓𝑓^′′−(𝑓^′)²𝑀+Gr𝜃+Gm𝜙−1+𝑚²(𝑓^′+𝑚√Reℎ)−Ω𝑓^′ℎ=0,_′′+𝑓ℎ^′+𝑀1+𝑚²(𝑚√Re𝑓^′(1−ℎ)−Ωℎ=0,Pr+𝑅)𝜃^′′+𝑓𝜃^′+Du𝜙^′′+𝑀Ec1+𝑚²((𝑓^′)²+ℎ²1Re)=0,𝜙Sc^′′+𝑓𝜙^′+Sr𝜃^′′=0, (2.11) where the primes denote differentiation with respect to 𝜂. Where 𝑀=𝜎𝐵²₀/𝜌𝐴₀ is the magnetic parameter, Pr=𝜈/𝛼 is the Prandtl number, Sc=𝜈/𝐷 is the Schmidt number, Sr=𝐷𝑘_𝑡(𝑇_𝑤−𝑇_∞)/𝜈𝑇_𝑚(𝐶_𝑤−𝐶_∞) is the Soret number, Du=𝐷𝑘_𝑡(𝐶_𝑤−𝐶_∞)/𝜈𝑇_𝑚(𝑇_𝑤−𝑇_∞) is the Dufour number, Gr=𝑔𝛽_𝑡(𝑇_𝑤−𝑇_∞)/𝑈_𝑠𝐴₀ is the local Grashof number, Gm=𝑔𝛽_𝑐(𝐶_𝑤−𝐶_∞)/𝑈_𝑠𝐴₀ is the local modified Grashof number, Ec=𝑈²_𝑠/𝑐_𝑝(𝑇_𝑤−𝑇_∞) is the Eckert number, Ω is the permeability parameter, Re=𝑥𝑈_𝑠/𝜈 is the Reynolds number, Sc=𝜈/𝐷 is the Schmidt number, and 𝑅=4𝜎𝑇³_∞/𝑘𝐾 is the dimensionless thermal radiation coefficient. In view of the similarity transformations, the boundary conditions transform into:𝑓(0)=𝑓_𝑤,𝑓^′𝑓(0)=1,ℎ(0)=0,𝜃(0)=1,𝜙(0)=1,_′(∞)=0,ℎ(∞)=0,𝑇(∞)=0,𝐶(∞)=0, (2.12) where 𝑓_𝑤=𝑉_𝑤/𝐴₀𝜈 is the mass transfer coefficient such that 𝑓_𝑤>0 indicates suction and 𝑓_𝑤<0 indicates blowing at the surface.

3. Method of Solution

The governing nonlinear similarity equations (in the case 𝑚=Gr=Gm=0), together with the boundary conditions 𝑓(0)=𝑓_𝑤, 𝑓^′(0)=1, and 𝑓^′(∞)=0, has an exact solution in the following form:𝑓(𝜂)=𝑓_𝑤+1𝑎(1−𝑒^−𝑎𝜂𝑓),𝑎=^𝑤+𝑓²_𝑤+4(𝑀+Ω+1)2. (3.1) The full set of (2.11) were reduced to a system of first-order differential equations and solved using a MATLAB boundary value problem solver called bvp4c. This program solves boundary value problems for ordinary differential equations of the form 𝑦^′=𝑓(𝑥,𝑦,𝐩),𝑎≤𝑥≤𝑏, by implementing a collocation method subject to general nonlinear, two-point boundary conditions 𝑔(𝑦(𝑎),𝑦(𝑏),𝐩)=0. Here p is a vector of unknown parameters. Boundary value problems (BVPs) arise in most diverse forms. Just about any BVP can be formulated for solution with bvp4c. The first step is to write the 𝑂𝐷𝐸𝑠 as a system of first-order ordinary differential equations. The details of the solution method are presented in Shampine and Kierzenka [26] and references therein. The numerical results were compared with the exact solution for the skin friction −𝑓^′′(0) for various values of 𝑘_𝑝=1/Ω. Table 1 gives a comparison between the exact solution for the skin friction, the present numerical scheme, and results obtained by Elgazery [24] who used a Chebyshev pseudospectral method of solution. The table shows an excellent agreement between our numerical results and the exact solution. The results in Table 1 were generated using 𝑀=1,𝑓_𝑤=−0.7,and𝑚=Gr=Gm=0

Table 1 

Values of the skin friction, −𝑓^′′(0), the exact solution, the present method, and results of Elgazery [24].

4. Results and Discussion

The similarity equations (2.11) are nonlinear, coupled ordinary differential equations which possess no closed-form solution. Thus, we solve these equations numerically subject to the boundary conditions given by (2.12). Graphical representations of the numerical results are illustrated in Figure 2 through Figure 23 to show the influences of different parameters on the boundary layer flow.

Figure 2 

The variation of the tangential velocity distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

In this study, we investigate the influence of the Dufour and Soret effects separately in order to clearly observe their respective effects on the velocity, temperature, and concentration profiles of the flow. The variation of tangential velocity distribution with 𝜂 for different values of the Dufour variable Du is shown in Figure 2. It can be clearly seen in this figure that as the Dufour effects increase, the tangential velocity increases.

The variation of lateral velocity distribution with 𝜂 for different values of the Soret variable Sr is shown in Figure 3. It can be clearly seen that the velocity distribution in the boundary layer increases with the Soret parameter. In Figure 4 we observe that as the Eckert number value increases, the tangential velocity increases.

Figure 3 

The variation of the tangential velocity distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 4 

The variation of the tangential velocity distribution with increasing Eckert parameter with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Du}=0.$

The effect of the magnetic field parameter (𝑀) is shown in Figure 5. It is observed that the tangential velocity of the fluid decreases with the increase of the magnetic field parameter values. The decrease in the tangential velocity as the Hartman number (𝑀) increases is because the presence of a 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 study. This resistive force slows down the fluid velocity component as shown in Figure 5. Figure 6 depicts the tangential velocity profiles as the Hall parameter 𝑚 increases. We see that 𝑓^′ increases as 𝑚 increases. It can also be observed that 𝑓^′-profiles approach their classical values when the Hall parameter 𝑚 becomes large (𝑚>5). In Figure 7 we observe that the tangential velocity decreases as the values of the permeability parameter are increased as more fluid is taken away from the boundary layer.

Figure 5 

The variation of the tangential velocity distribution with increasing Hartman number parameter $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 6 

The variation of the tangential velocity distribution with increasing Hall parameter with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 7 

The variation of the tangential velocity distribution with increasing surface permeability parameter with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

The influence of thermal radiation on the tangential velocity is shown on Figure 8. Increasing the thermal radiation parameter produces an increase in the tangential velocity of the flow. This is because large values of 𝑅 correspond to an increased dominance of conduction over absorption radiation thereby increasing buoyancy force and thickness of the momentum boundary layer (thus velocity).

Figure 8 

The variation of the tangential velocity distribution with increasing thermal radiation parameter with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

The effects of Dufour parameter (Du) are depicted in Figure 9. It is observed in this figure that the lateral velocity component increases with the increase in diffusion thermal effects. For each value of Dufour there exists a local maximum value for the lateral velocity profile. Figure 10 shows the effect of Soret number on the lateral velocity distribution. From the figure it can be seen that this velocity component increases with the increase in Soret parameter Sr. It can also be seen that at each value of Sr there exist local maximum values in the lateral velocity profile in the boundary region.

Figure 9 

The variation of the lateral velocity distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 10 

The variation of the lateral velocity distribution with increasing Soret number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

In Figure 11 we have the influence of the magnetic field parameter on the lateral velocity. It can be seen that as the values of this parameter increase, the lateral velocity increases.

Figure 11 

The variation of the lateral velocity distribution with increasing magnetic field with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

In Figure 12, we see that ℎ-profiles increase for 𝑚≤1 and decrease for 𝑚>1. In Figure 13, we see the influence of the Eckert number on lateral velocity of the flow. It can be seen that as the Eckert number increases, this velocity component increases as well. Figure 14 shows that the increase of permeability of the surface reduces the lateral distribution of the fluid. The effects of thermal radiation on lateral velocity are shown in Figure 15. We observe that the lateral velocity increases as the value of the radiation parameter 𝑅 increases.

Figure 12 

The variation of the lateral velocity distribution with increasing Hall current with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 13 

The variation of the lateral velocity distribution with increasing Eckert number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Du}=0.$

Figure 14 

The variation of the lateral velocity distribution with increasing permeability number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 15 

The variation of the lateral velocity distribution with increasing thermal radiation with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=0,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figures 16 and 17 depict the behaviour of Du and Sr, on 𝜙, respectively. In Figure 16 we see the effects of the Dufour number on the concentration profiles. The Dufour effects reduce the concentration boundary layer in the fluid.

Figure 16 

The variation of the concentration distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 17 

The variation of the concentration distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 17 shows the influence of the Soret parameter, Sr on the concentration profiles. It can be seen from this figure that the concentration 𝜙(𝜂) increases with increasing Sr values. From this figure we observe that the concentration profiles increase significantly with increase of the Soret number values.

Figure 18 depicts the effects of the Dufour parameter on the fluid temperature. It can be clearly seen from this figure that diffusion thermal effects greatly affect the fluid temperature. As the values of the Dufour parameter increase, the fluid temperature also increases. The influence of thermal-diffusion effects is shown in Figure 19. We observe that as Sr increases, there is a decrease in the temperature of the fluid though the changes are not very significant. As expected the effect of Soret number Sr on the temperature is quite opposite to that of Du. It is also noticed that the behaviour of Du and Sr on the concentration and temperature distributions is opposite.

Figure 18 

The variation of the temperature distribution with increasing Dufour number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

Figure 19 

The variation of the temperature distribution with increasing Soret number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=0,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

In Figure 20 we see that the increase in the Eckert number values greatly affects the temperature of the fluid. The temperature of the fluid, increases as the Eckert number increases.

Figure 20 

The variation of the concentration distribution with increasing Eckert number with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Du}=0.$

Figure 21 shows that the temperature boundary layer becomes thick by increasing the magnetic parameter. The effects of a transverse magnetic field give rise to a resistive-type force called the Lorentz force. This force has the tendency to slow down the motion of the fluid and to increase its thermal boundary layer hence increasing the temperature of the flow.

Figure 21 

The variation of the temperature distribution with increasing magnetic field with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1.$

In Figure 22 we see that the temperature profiles approach their classical values when the Hall parameter 𝑚 becomes large. Temperature profiles 𝜃 decrease with increasing 𝑚. The effect of thermal radiation 𝑅 on the temperature profiles in the boundary layer is illustrated in Figure 23. It is obvious that an increase in the radiation parameter results in increasing the temperature profiles within the boundary layer as well as an increase in the thermal boundary layer thickness.

Figure 22 

The variation of the temperature distribution with increasing Hall current with $A_0=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}R=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=0,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{Du}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Ec}=1$.

Figure 23 

The variation of the temperature distribution with increasing thermal radiation with $A_0=1,\hspace{0.17em}\hspace{0.17em}\text{Gr}=\text{Gm}=1,\hspace{0.17em}\hspace{0.17em}m=1,\hspace{0.17em}\hspace{0.17em}\text{Pr}=0.72,\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega }=1,\hspace{0.17em}\hspace{0.17em}\mathrm{Re}=1,\hspace{0.17em}\hspace{0.17em}\text{Ec}=1,\hspace{0.17em}\hspace{0.17em}M=1,\hspace{0.17em}\hspace{0.17em}\text{Sc}=1,\hspace{0.17em}\hspace{0.17em}\text{Sr}=0,\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Du}=0.$

5. Conclusion

This work investigated the effects of diffusion-thermo and thermal-diffusion on MHD natural convection heat and mass transfer over a permeable vertical plate in the presence of radiation and hall current. The governing equations are approximated to a system of nonlinear ordinary differential equations by using suitable similarity transformations. Numerical calculations are carried out for various values of the dimensionless parameters of the problem using an efficient and easy to use MATLAB routine bvp4c. The results are presented graphically and we can conclude that the flow field and the quantities of physical interest are significantly influenced by these parameters. The velocity increases as the Soret, Dufour effects, Hall, Eckert parameter, and thermal radiation increase. However, the tangential velocity was found to decrease as the Hartman parameter increases and the lateral velocity component was increased by the increased values for the Hartman number. Both velocity components distributions were reduced by the increased values of the permeability of the plate.

The fluid temperature was found to increase as the Dufour parameter, Eckert number, magnetic strength, thermal radiation, and surface permeability increase and to decrease as the Hall current and Soret effects increase. The concentration decreases as the Dufour number, Eckert, Hall parameter, and radiation parameters increase and increases as the Soret effect, magnetic strength, and surface permeability increase.

In addition, the present analysis has shown that the Soret and Dufour numbers have significant effects on the distributions of the velocity, temperature, and concentration. We therefore conclude that thermal-diffusion and diffusion-thermo effects have to be considered in the fluid, heat, and mass transfer.