# MHD MIXED CONVECTION BOUNDARY LAYER FLOW THROUGH POROUS MEDIUM DUE TO POROUS NON-LINEARLY SHRINKING SURFACE.

Abstract

This study intends to investigate the steady, incompressible flow and heat transfer for viscous electrically conducting fluid. The fluid flows through porous medium due to a non- linearly porous shrinking surface. A magnetic field is applied normal to the surface. The coupled non linear partial differential equations used for the mathematical model of the problem have been transformed to the ordinary differential form. The resulting equations are then solved numerically by employing Mathematica software. The results for velocity and temperature distribution reveal the flow kinematics under the effects of key physical parameters of the problem namely non linear shrinking parameter s, magnetic parameter M, porosity parameter K, mixed convection parameter and Prandtl number Pr . The results are presented in graphical form.

Key Words: MHD Mixed Convection, Flow through porous surface, Non-linearly shrinking surface.

1. INTRODUCTION

The convective heat transfer in fluids flows through porous medium has numerous applications. The earth mantle is an example of porous medium to yield the geothermal energy through convection process. The well known Darcy's law describes the flow of fluid through porous media. This law states that fluid flow through porous media experiences a resistance. Several industrial problems such as heat drums to generate underground chemicals, solar power collectors, heat exchangers with porous materials, the techniques for thermal insulation and petroleum reservoir modeling involve the fluid flow with convective heat transfer through porous media. This research area has gained the attraction of researchers. Ojjela and Kumar  considered the unsteady two-dimensional incompressible MHD flow and heat transfer of a micropolar fluid in a porous medium between parallel plates with chemical reaction, Hall and ion slip effects.

Hsu and Cheng  reported mixed convection flow in porous medium about a vertical plate. Tien and Vafai  investigated the free convection boundary layer flow in a porous medium, owing to combined heat and mass transfer. Combined heat and mass transfer in fluid-saturated porous media finds applications in a variety of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, geothermal and geophysical engineering, moisture migration in a fibrous insulation and nuclear waste disposal, and others. Bejan and Khair  investigated the free convection boundary layer flow in a porous medium, owing to combined heat and mass transfer. Takhar et al  analyzed mixed convention of an incompressible viscous fluid in a porous medium past a vertical plate. Several authors [6-11] have presented various aspects of the convective heat flow problems through porous media.

El-Hakiem  discussed the effect of magnetic field and double dispersion on mixed convection heat and mass transfer in non-Darcy porous medium.

The fluid flow due to shrinking surface behaves differently as compared to the forward stretching flow, this was first observed by Wang . Physically, the vorticity generated due to the shrinking of the sheet is not confined within the boundary layer, and the steady flow exists only when adequate suction on the boundary is imposed as reported by Miklavcic and Wang . Goldstein  remarked that the shrinking flow is a backward flow. Recently, Sajjad considered MHD forced convective boundary layer flow of micropolar fluids past a shrinking porous sheet prescribed with variable heat flux and heat source. Ahmad and Hussain  studied convective heat transfer for MHD micropolar fluids flow through porous media over a stretching surface. Rosca  investigated the forced convection boundary flow past a permeable shrinking surface. Bagh et al  examined the magnetohydrodynamic fluid flow through porous media over an exponentially shrinking porous sheet with convective heat transfer.

This work examines the MHD mixed convection boundary layer flow through porous media due to porous non-linearly shrinking surface. The numerical solution of the problem describes the effects of physical parameters of the study on fluid velocity and heat function.

2. MATHEMATICAL ANALYSIS

The fluid flow is steady, incompressible and two-dimensional. The fluid is viscous and electrically conducting. The fluid flows over a porous shrinking surface. The surface coincides with the plane y = 0, and the flow is bounded in the region y greater than 0. A magnetic field of variable magnetic strength is applied perpendicular to the surface. Under the above assumptions the equations of motion become as:

(Equations)

Where (Eq.) is velocity and T is fluid temperature, (Eq.) and (Eq.) are kinematic viscosity and thermal diffusivity respectively, and the relation for electromagnetic body force as given by Rossow  is

(Equation)

Where (Eq.) is a magnetic field strength, (Eq.) is the current density, (Eq.) is electric field intensity and (Eq.) is the electrical conductivity, g is acceleration due to gravity, (Eq.) is coefficient of thermal expansion.

The boundary conditions are:

(Equations)

where the primes denote differentiation with respect to (Eq.).

The equation of continuity (1) is identically satisfied.

Substituting (5) in to equations (2) and (3), we have

(Equations)

3. RESULTS AND DISCUSION

The set of coupled non- linear equations (6) and (7) together the boundary conditions (8) do not lend themselves to analytical solution. Mathematica software has been employed to obtain numerical solutions for parametric study of the problem. Several computations have been made for representative values of the parameters that influence the nature of flow and heat transfer.

Fig.1 shows that the magnitude of horizontal velocity f' increases with increase in magnetic field strength, the situation for flow due to shrinking sheet is reverse as that for stretching sheet where usually the Lorentz force opposes the fluid flow. Also, the boundary layer decreases with an increase in the values of M. Fig.2 presents the effect of suction parameter (Eq.) on horizontal velocity (Eq.).

It is noticed that velocity increases with increase in suction, the result corresponds to the physical phenomenon and because of increase in the speed the boundary layer decreases.

Fig.3 shows that velocity (Eq.) increases with increase in the porosity parameter K. The non-linearity parameter s causes to reduce the flow and the boundary layer thickness increases as shown in fig.4. The fig.5 demonstrates the effect of convection parameter (Eq.), the velocity increases with an increase in the values of (Eq.).

Fig.6 and fig.7 respectively presents the curves for temperature distribution under the effects of suction parameter (Eq.) and Prandtle number (Eq.) . It is noticed that temperature decreases with an increase in the values of these parameters. The situation has relevance to the physical nature of these parameters. Fig.8 shows that temperature decreases with an increase in the values of parameter m. The magnetic field has no significant effect on temperature function.

4. CONCLUSION

From the above given results and discussuion, we conclude that:

- The magnitude of horizontal velocity f' increases with increase in magnetic field strength M.

- The boundary layer decreases with increase in the values of M.

- The velocity f' increases with increase in the porosity parameter K.

- The non-linearity parameter s causes to reduce the flow and makes the boundary layer thickness increase.

- The velocity f' increases with increase in the values of convection parameter (Eq.).

- Under the effects of suction parameter s and Prandtle number Pr, it is noticed that temperature decreases with increase in the values of these parameters.

- The temperature decreases with increase in the values of non-linearity parameter m.

- The magnetic field has no significant effect on temperature function.

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