Unsteady Electromagnetic Free Convection Micropolar Fluid Flow through a Porous Medium along a Vertical Porous Plate

Unsteady electromagnetic free convection flows of two-dimensional micropolar 
fluid through in a porous medium parallel to a vertical porous plate have been investigated 
numerically. Similarity analysis has been used to transform the governing equations 
into its non-dimensional form by using the explicit finite difference method to 
obtain numerical solutions. Estimated results have been gained for various values 
of Prandtl number, Grashof number, material parameters, micropolar parameter, electric 
conductivity, electric permeability, thermal relaxation time and the permeability 
of the porous medium. The effects of pertinent parameters on the velocity, electric induction, magnetic induction, 
microrotation and temperature distributions have been investigated briefly and illustrated graphically.


Introduction
Fluids with microstructure are micropolar fluids which are randomly oriented or composed of spherical particles that are rigid with their rotation and also ceased in a viscous medium. It has been known that Navier-Stokes equations are unable to explain the phenomena at micro and nanoscales; on the other hand, MFD can express the physical phenomena at micro and nanoscales owing to its additional degree of freedom for circulation. Physical examples of micropolar fluids may present in the non-Newtonian fluids, blood flows, polymer fluids and liquid crystals and all of them containing intrinsic polarities. The presence of dust or fumes in a gas can be especially modeled using micropolar fluid dynamics. The porous media heat transfer problems have various practical uses in engineering applications such as geothermal systems, crude oil extraction and groundwater pollution. Eringen first proposed [1] and [2] the general theory of micropolar fluids which illustrate certain microscopic effects arising from the microstructure and micro motions of the fluid flow. The interaction of natural convection with thermal radiation in laminar boundary layer flow over an isothermal, horizontal flat plate is studied by Ali et al. [3]. Harutha and Devasena [4] investigated the steady mixed convection flow of a viscous incompressible micropolar fluid through a porous medium towards a stagnation point over a vertical surface when the buoyancy forces assist. Hudimoto and Tokuoka [5] have devised the two-dimensional parallel shear flow of a linear micropolar fluid. They analyzed and compared it with the colloidal suspensions. Rees and Pop [6] expressed the steady micropolar free convection fluid flow from a vertical isothermal flat plate. Elbarbary [7] discussed a new Chebyshev finite difference method is proposed for solving the governing equations of the boundary layer flow.
Nandhini and Ramya [8] analyzed the heat and mass transfer of the free convection flow in a micropolar fluid past an inclined stretching sheet. Hassanien and Glora [9] analyzed the heat transfer on a non-isothermal stretching sheet to a micropolar fluid. Kartini Ahmad et al. [10] described a micropolar fluid flow and heat transfer past a non-linearly stretching plate. Khonsari and Brewe [11] investigated and compared the parameters of micropolar fluids with finite length lubricated that resulted significantly higher load carrying capacity than Newtonian fluids. Effects of free convection currents with one relaxation time on the flow of a viscoelastic conduction fluid through a porous medium, which is bounded by a vertical plane surface, have studied by Ezzatand Abd-Ellal [12].
Edlabe and Mohammed [13] determined the heat and mass transfer occurring in the hydromagnetic flow of the non-Newtonian fluid on a linearly accelerating surface with temperature dependent heat source subject to suction or blowing.
Edlabe and Ouaf [14] are obtained the heat and mass transfer in a hydro magnetic flow of a micropolar fluid past a stretching surface with Ohmic heating and viscous dissipation. Aydin and Pop [15] analyzed the two-dimensional steady laminar natural convective flow and heat transfer of micropolar fluids in a square enclosure. Muthu et al. [16] investigated the oscillatory flow of micropolar fluid in an annular region with constriction, provided by variation of the outer tube radius. Glora [17] presented an unsteady combined convection of a micropolar fluid among a vertical plate. Hsu and Wang [18] presented a numerical study of the laminar mixed convection of micropolar fluids in a square cavity with localized heat source Lok et al. [19]  In the present work, our aim is to study that the numerical investigation on unsteady two-dimensional electromagnetic free convection micropolar fluid flows through a porous medium along a vertical porous plate. The obtained equations are non-linear coupled partial differential equations, which are solved by using explicit finite difference method and the results are shown graphically and also discussed its behavior in detail for the velocity, induced magnetic field, induced electric field, micro rotation and temperature distribution with respect to its pertinent parameters.

Problem Formulation
Considered unsteady MHD micropolar fluid flow embedded in a porous medium along a vertical porous plate. The velocity at the wall is zero and also outside of the boundary layer is zero. The temperature of the plate is raised from w T to T ∞ , where w T and T ∞ is the temperature at the plate and outside of the boundary layer respectively. The magnetic Reynolds number is taken large enough so that the induced magnetic field equation is considerable for our assumption. The Physical model of the system is shown in the following with boundary conditions are as follows:

Similarity Analysis
Now introducing the following non-dimensional quantities as Using these quantities into the above Equations (1)- (7), we obtain the following dimensionless form of the given equations: with the corresponding boundary conditions: is the micropolar parameter,

Method of Solution
The explicit finite difference method has been used to solve the governing non-linear coupled dimensionless partial differential Equations (8) to (13) together with its boundary conditions. The finite difference schemes with respect to t, x and y are as follows: Here, the subscript i and j refer to x and y and the superscript k refers to time t.
Finite difference Schemes for the other variables have been written in the same way. The graphical representations of this problem have been illustrated by using Compaq visual FORTRAN 6.6 a tools.

Results and Discussion
The

Time and Mesh Sensitivity Test
To get the steady-state solution, the computations are carried out for different time 20, 25, 29 t = and 30 with time increment 0.001 t ∆ = for the velocity distribution, which have shown in Figure 2. It is found that after

Comparison with Previous Results
Zakaria [20] investigated the influence of the Grashof number on the velocity u, which is shown in Figure 4(a). Here the velocity decreases with the increase of Grashof number G r . But in Figure 4(b), it is found that the velocity increases with the same increasing values of Grashof number. In this case the maximum time has taken 1 t = . Figure 5 depicts that the velocity u is increased with the increase of K. In Figure  6, it is observed that the velocity u is decreased with the increasing values of P r . Figure 7, showed a cross-flow for the velocity, here velocity distribution is decreased within the interval 0 6 y < < (approx.) and thereafter it has very minor increasing effect with the increase of R. Figure 8 represented that the velocity has an increasing effect with the increase of t 0 . Figure 9 and Figure 10 illustrate that the induced magnetic field distribution H has a cross-flow for the different values of G r . It is obvious that near the plate, H has a minor increasing effect and thereafter found a large decreasing effect for increasing values of G r and K. Figure 11 represents that H has a very minor increasing effect near the plate and thereafter a decreasing effect with the increase of P r . But from Figure 12, it is observed that H has an increasing effect with the rising values of R.

Induced Electric Field Distributions
Profiles in Figure 13 and Figure 14, represented that the induced electric field E is decreased with the increase of G r and K respectively. But E has an increasing effect with the rising values of P r which is shown in Figure 15.

Microrotation Distributions
The microrotation N has a cross-flow depicts in Figure 16. It has a decreasing effect within 0 2.9 y < < (approx.) and thereafter it has an increasing effect with the increase of K. But Figure 17 noticed that the increasing values of R, the micropolar rotation N has a decreasing effect within 0 2.2 y < < (approx.) and then it has an increasing effect. As shown, temperature is decreasing with the increasing of P r .

Skin-Friction, Current Density and Rate of Heat Transfer
The effects of various parameters on local and average shear stress from the velocity profile have been investigated. The non-dimensional form of the local shear stress and average shear stress in x-direction is given by the relations  Figure 19 and Figure

Conclusions
In the present study, the influence of various values of Prandtl number, Grashof number, permeability parameter, micropolar parameter, electric conductivity, electric permeability and thermal relaxation time has been investigated. The non-linear coupled governing equations have been solved numerically and the main findings can be summarized as follows: 1) The velocity u increases with the increase of G r , K and 0 τ , while it decreases with the increase of P r and R.
2) Induced magnetic field H has cross-flow near the plate. But in major space, it has been increasing effect with the increase of P r and R, while it decreases with the increase of G r and K. Open Journal of Applied Sciences 3) Induced electric field E increases with the increase of P r , while it decreases with the increase of G r and K. 4) Microrotation N has cross-flow for all the different values of all the parameters. First portion near the plate N has increasing effect with the increase of R. Thereafter, it has a decreasing effect. But for G r and K, it has reverse effect. 5) Temperature θ decreases with the increase of P r . 6) Local (or average) Shear stress increases of G r and K, while it decreases with the increase of P r and R. 7) Local (or average) Current density decreases with the increase of G r and K. 8