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  and  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. . Harutha and Devasena  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  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  expressed the steady micropolar free convection fluid flow from a vertical isothermal flat plate. Elbarbary  discussed a new Chebyshev finite difference method is proposed for solving the governing equations of the boundary layer flow. Nandhini and Ramya  analyzed the heat and mass transfer of the free convection flow in a micropolar fluid past an inclined stretching sheet. Hassanien and Glora  analyzed the heat transfer on a non-isothermal stretching sheet to a micropolar fluid. Kartini Ahmad et al.  described a micropolar fluid flow and heat transfer past a non-linearly stretching plate. Khonsari and Brewe  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 . Edlabe and Mohammed  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  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  analyzed the two-dimensional steady laminar natural convective flow and heat transfer of micropolar fluids in a square enclosure. Muthu et al.  investigated the oscillatory flow of micropolar fluid in an annular region with constriction, provided by variation of the outer tube radius. Glora  presented an unsteady combined convection of a micropolar fluid among a vertical plate. Hsu and Wang  presented a numerical study of the laminar mixed convection of micropolar fluids in a square cavity with localized heat source Lok et al.  studied a microplar mixed convection boundary layer fluid flow near the region of the stagnation point of on a double-infinite vertical flat plate. Zakaria  investigated the influence of a transverse magnetic field on the motion of an electrically conducting micropolar fluid through a porous medium in one-dimensional and used the Laplace transformation with -algorithm technique to find its solution in the Laplace transformation domain numerically.
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.
2. 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 to , where and 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 Figure 1.
The flow is governed by the equation of continuity, the momentum equation, induction magnetic field equation, the electric field equation, angular momentum equation and the energy equation are as follows:
Figure 1. Physical Model of the system.
with boundary conditions are as follows:
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:
where is the Grashof number, is the Prandtl number, is the micropolar parameter, is the Permeability parameter. Also , , and are the dimensionless material parameter.
3. 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.
4. Results and Discussion
The behavior of the velocity (u), induced magnetic field (H), induced electric field (E), microrotation (N) and temperature ( ) distributions have been analyzed for the different values of Prandtl number ( ), Grashof number ( ), permeability of porous medium (K), micropolar parameter (R) and thermal relaxation time ( ) with the values of time . The flow characteristics have been shown graphically from Figures 2-25.
Figure 2. Time sensitivity on velocity u.
Figure 3. Mesh sensitivity on velocity u.
Figure 4. (a) Velocity distribution u for different values of Gr [M. Zakaria ; (b) Velocity distribution u for different values of Gr [Our Results].
Figure 5. Velocity distribution u for different values of K.
Figure 6. Velocity distribution u for different values of Pr.
Figure 7. Velocity distribution K for different values of R.
Figure 8. Velocity distribution u for different values of t0.
Figure 9. Magnetic field distribution H for different values of Gr.
Figure 10. Induced Magnetic field distribution H for different values of K.
Figure 11. Induced Magnetic field distribution R for different values of Pr.
Figure 12. Induced Magnetic field distribution H for different values of R.
Figure 13. Induced electric field distribution K for different values of Gr.
Figure 14. Induced electric field distribution E for different values of K.
Figure 15. Induced electric field distribution E for different values of Pr.
Figure 16. Microrotation distribution N for different values of K.
Figure 17. Microrotation distribution N for different values of R.
Figure 18. Temperature distribution θ for different values of Pr.
Figure 19. (a) Illustration of local shear stress τxL for different values of Gr; (b) Illustration of Average shear Stress τxA for different values of Gr.
Figure 20. (a) Illustration of local shear Stress τxL for different values of K; (b). Illustration of average shear Stress τxA for different values of K.
Figure 21. (a) Illustration of local shear stress τxL for different values of Pr; (b). Illustration of local average shear stress τxA for different values of Pr.
Figure 22. (a) Illustration of local shear stress τxL for different values of R; (b). Illustration of average shear stress τxA for different values of R.
Figure 23. (a) Illustration of Current density JwL for different values of Gr; (b). Illustration of average Current Density JwA for different values of Gr.
Figure 24. (a) Illustration of Current density at the wall JwL for different values of K; (b). Illustration of average Current density JwA for different values of K.
Figure 25. (a) Illustration of Nusselt number NuL for different values of Pr; (b): Illustration of average Nusselt Number NuA for different values of Pr.
4.1. Time and Mesh Sensitivity Test
To get the steady-state solution, the computations are carried out for different time and 30 with time increment for the velocity distribution, which have shown in Figure 2. It is found that after , there are very negligible changes. On the other hand, to choose the appropriate mesh, a solutions find out for different pairs of meshes like as ; and on the velocity distributions, which have shown in Figure 3. For those three chosen different values of meshes, the profiles are likely unchanged. There are same situations for the other distributions. Therefore our estimated steady-state time is at with time increment and mesh pair is with the fixed values of ; ; ; ; ; ; ; ; and .
4.2. Comparison with Previous Results
Zakaria  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 Gr. 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 .
4.3. Primary Velocity Distributions
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 Pr. But in Figure 7, showed a cross-flow for the velocity, here velocity distribution is decreased within the interval (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 t0.
4.4. Induced Magnetic Field Distributions
Figure 9 and Figure 10 illustrate that the induced magnetic field distribution H has a cross-flow for the different values of Gr. It is obvious that near the plate, H has a minor increasing effect and thereafter found a large decreasing effect for increasing values of Gr 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 Pr. But from Figure 12, it is observed that H has an increasing effect with the rising values of R.
4.5. Induced Electric Field Distributions
Profiles in Figure 13 and Figure 14, represented that the induced electric field E is decreased with the increase of Gr and K respectively. But E has an increasing effect with the rising values of Pr which is shown in Figure 15.
4.6. Microrotation Distributions
The microrotation N has a cross-flow depicts in Figure 16. It has a decreasing effect within (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 (approx.) and then it has an increasing effect.
4.7. Temperature Distributions
Figure 18 displays the effect of the Prandtl number Pr on the temperature ( ). As shown, temperature is decreasing with the increasing of Pr.
4.8. 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 and respectively. From the temperature profile, the effects of various parameters on local and average Nusselt numbers have been calculated. The local Nusselt number and the average Nusselt number are given by and respectively. Similarly, analyze the effects of various parameters on the local and average Sherwood numbers from the concentration field. The rate of mass transfer at the plate is defined as the Sherwood number; the local Sherwood number and the average Sherwood number is defined by and respectively.
Figures (a) and (b) of Figures 19-22 are displayed the variations of the local shear stress and average shear stress respectively. It is obtained from Figure 19 and Figure 20 that the local (or average) shear stress increases with the increase of Gr and K. The same effects of velocity are also represented. But from Figure 21 and Figure 22, show that both of the local and average shear stress decreases with the increase of Pr and R respectively.
Again, figures (a) and (b) of Figure 23, Figure 24 showed the variations of the local and average current density respectively. It is presented that JwL (or Jw) has both decreasing effect with the increasing value of Gr and K. But Figure 25(a), Figure 25(b) have shown the variation of local (or average) Nusselt number for different values of the Prandtl number (Pr), it depicts from the figure that Nusselt number increases with the increase of Pr.
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 Gr, K and , while it decreases with the increase of Pr 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 Pr and R, while it decreases with the increase of Gr and K.
3) Induced electric field E increases with the increase of Pr, while it decreases with the increase of Gr 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 Gr and K, it has reverse effect.
5) Temperature decreases with the increase of Pr.
6) Local (or average) Shear stress increases of Gr and K, while it decreases with the increase of Pr and R.
7) Local (or average) Current density decreases with the increase of Gr and K.
8) Local (or average) Nusselt number increases with the increase Pr.
The accuracy of this work is qualitatively good in case of all the flow parameters.
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