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The present paper, a theoretical analysis of steady fully developed flow and heat transfer of two immiscible magneto hydrodynamic and viscous fluid, partially filled with porous matrix and partially with clear fluid bounded by two vertical plates, has been discussed, when both the plates are moving in opposite directions. The plates are maintained at unequal temperatures. The Brink-man-extended Darcy model has described the momentum transfer in a porous medium. The effect of various parameters and Darcy number are discussed in the flow field and the temperature profiles numerically and are expressed by graphs. The non-dimensional governing momentum and energy equations are analytically solved by applying the homotopy perturbation technique and the method of ordinary differential equation. It is observed that magnetic parameter (M) has a retarding effect on the main flow velocity and is to enhance the temperature distribution, whereas the reversal phenomenon occurs for the Darcy dissipation parameter (Da). The skin-friction component has also been determined and is presented with the help of a table. The magnetic parameter (M) reduces the skin friction coefficient for clear fluid region and is to increase the skin friction coefficient for porous region. It is also evident from table that getting bigger the width of the clear fluid layer increases the skin friction. The skin friction coefficient on both the plates (comparing at y = 0 and at y = 1 for A or B) increases when those are heated.

The convective flow and heat transfer of electrically conducting fluid in the channel have been an important research topic for the last few decades under the influence of magnetic field because of its applications in magneto-hydrodynamics (MHD) power generators, solar technology, the dispersion of metals, application in fusion reactors, aerodynamic heating, petroleum industry, crude oil purification, fluid droplet sprays and many more. Many relevant, pertinent studies have been reported as Hartmann [

Heat transfer aspects associated with flow systems comprising multi layer flow in a region, part of which is occupied by porous matrix and part by clear fluid under the effect of transverse magnetic field, have a pivotal importance due to its wide range of applications in both geophysical and industrial environments, including such as thermal energy storage system, flow and heat transfer behavior of lubricants in a porous journal bearings and porous rollers, oil recovery, groundwater hydrology, petroleum reservoir engineering and many others in which a porous matrix is set up adjacent to clear fluid.

The comprehensive view of technological point in above fluid mechanics, some slip of the fluid over the fluid-porous boundary may occur so slip velocity ought to be found in empirical way. A series of work has been investigated on the problem of immiscible fluid. Beavers and Joseph [

Furthermore, the study of fully developed forced convection in porous matrix is applied and in progressed by Vafai and Tien [

The proposed study of MHD multi fluid flow and heat transfer are through two vertical plates partially filled with porous matrix when both the plates are moving in opposite directions.

Consider a channel of an incompressible, viscous, steady and electrically conducting MHD multi fluid flow past between two vertical plates partially filled with porous media and partially with a clear fluid having an interface is discussed, when both the plates are moving in opposite directions and one plate is heated and the other is cooled. The

Under Bousinesque approximation, the flow of clear fluid and porous medium is governed by the following equations:

Clear Fluid Region:

(a) Momentum equation

(b) Energy equation

Pore Fluid Region:

(a) Momentum equation

(b) Energy equation

The corresponding boundary conditions are [boundary at interface

Introducing dimensions by using the following transformation:

And

So Equation (1) to Equation (4) become:

where

The Brinkman-extended Darcy law (Darcy [

The boundary conditions on velocity are no-slip conditions requiring that the velocity must be same as that at the plate. In addition, to maintain the continuity of velocity, shear stress, temperature and heat flux at the interface is assumed so boundary conditions (5) in non-dimension form are:

where _{u} the Buoyancy parameter, β the coefficient of thermal expansion, μ is the viscosity, β_{0} is the magnetic field intensity, σ is the electric conductivity, d is the distance of interface from the plate y = 0, ʋ is kinematic viscosity, g the acceleration due to gravity, θ is the temperature at any point of the fluid flow. L is the distance of vertical plates, u the dynamic velocity, U_{0} the velocity of the plate. A is the plate temperature at y = 0 and B is the plate temperature at y = 1. The subscripts f represents a clear fluid layer, p porous layer, h hot plate and c the cool plate.

Solutions:

It is observed that above governing equations are coupled non-linear. Accordingly, we assume for small but (=1), a very small Buoyancy parameter in most of the practical problems:

Substituting the Equation (12) into the Equations (7)-(10), we have:

The corresponding boundary conditions are:

Solving Equations (15) and (19) using boundary conditions (21) gives the following temperature component:

Now solving Equations (13) and (17) by the homotopy perturbation technique with boundary conditions (21), construct homotopy (He [

where

Let

Boundaries and matching conditions are:

And

Then the solution of Equations (13) and (17):

With the help of a solution of Equations (22), (27) and (28), (29), we get the solution of Equations (14), (16), (18), and (20):

And

The constants are dropped for the sake of brevity.

The numerical values of Skin-friction are exposed in _{0} = 0.4. t_{1}, t_{2} is the skin friction when A = 1, B = 0 and A = 0, B = 1 respectively. It is evident from

In order to mull over the configuration of magneto-hydrodynamic two-phase flow in vertical plates, channels partially filled with porous substrate when plates are moving in opposite directions has been discussed. The closed form of solutions of nonlinear-coupled momentum and energy equations have been analytically obtained by the homotopy perturbation technique and the solutions of ordinary differential equation method. The findings

βu | M | Da | d | A = 1_{ } | B = 0_{ } | A = 0 | B = 1 | ||
---|---|---|---|---|---|---|---|---|---|

t_{1} | t_{2} | ||||||||

u_{f } | u_{p} | u_{f} | u_{p} | ||||||

0 | 1 | 0.1 | 0.5 | −0.5013 | 1.5671 | −0.6714 | 1.7053 | ||

0.7 | −0.4909 | 1.5702 | −0.6591 | 1.7116 | |||||

0 | 1.5 | 0.1 | 0.5 | −0.6078 | 1.6118 | −0.7738 | 1.7485 | ||

0.7 | −0.6013 | 1.6613 | −0.7655 | 1.7509 | |||||

0 | 1 | 0.01 | 0.5 | −0.5170 | 4.0395 | −0.6908 | 4.1185 | ||

0.7 | −0.3656 | 4.0741 | −0.5442 | 4.1529 | |||||

0 | 1.5 | 0.01 | 0.5 | −0.6167 | 4.0633 | −0.7862 | 4.1420 | ||

0.7 | −0.4973 | 4.0928 | −0.6644 | 4.1715 | |||||

0.1 | 1 | 0.1 | 0.5 | −0.4906 | 1.5684 | −0.6565 | 1.7035 | ||

0.7 | −0.4789 | 1.5717 | −0.6411 | 1.7097 | |||||

0.1 | 1.5 | 0.1 | 0.5 | −0.6060 | 1.6139 | −0.7654 | 1.7495 | ||

0.7 | −0.5993 | 1.6236 | −0.7566 | 1.7521 | |||||

0.1 | 1 | 0.01 | 0.5 | −0.5063 | 4.0400 | −0.6724 | 4.1190 | ||

0.7 | −0.3514 | 4.0746 | −0.5256 | 4.1534 | |||||

0.1 | 1.5 | 0.01 | 0.5 | −0.6157 | 4.0638 | −0.7779 | 4.1425 | ||

0.7 | −0.4898 | 4.0934 | −0.6565 | 4.1721 | |||||

are presented graphically and discussed in detail. The velocity distribution (zeroth and first order) and temperature distribution (only first order) are depicted in the figures. The zeroth order temperature profile is not shown since it is linear. The effect of Darcy dissipation and joule dissipation is taken into account. There domains are viewed such as clear fluid domain (near y = 0), interface domain (at y = d) and pore domain (near the plate y = 1).

The zeroth order velocity profile flow is plotted in

Figures 13-15 represent the temperature profile in the boundary layer and momentum of various value of Hartmann number (M). It is encountered from these figures that temperature distribution increases in thermal

boundary layer adjacent to the plates with growing the Hartmann number (M) for both of the Darcy parameters (0.01, 0.001). It is attributed to the fact that kinetic energy lost from the fluid flow due to the magnetic field effect is manifested as joule (ohmic) heating so Hartmann contributed significantly in generation of temperature. Temperature profile remains almost constant with increasing the value of Hartmann number (M) in porous media and it is greater in the clear fluid in close proximity to the plate. Therefore, it is imperative to conclude that magnetic parameter associated with Darcy parameter fails to contribute much temperature distribution in momentum. It is interesting to note of these figures that giving raises to temperature in the flow field either A or B exists.

1) Presence of magnetic field (M) decreases the velocity (magnitude) of flow field in the clear and porous fluid region. It is so as a retarding effect of Lorentz force.

2) Velocity (magnitude) of fluid falls in whole region with decreasing Darcy parameter (Da).

3) Temperature coupled with Darcy parameter decelerates the motion flow and decreases the temperature profile of fluid.

4) Inclusion of magnetic field with Darcy parameter is the beneficial gaining temperature in the thermal boundary layer adjacent to the plates.

5) Low Darcy number enhances the temperature distribution near the plates.

6) Shear stress on both of the plates (y = 0 and y = 1) increases due to increasing the width of the clear fluid layer.

7) The skin friction decreases with the increasing Hartmann number (M) in clear fluid, but the reversal effect shows in a porous medium and skin friction on both the plates ( comparing at y = 0 and at y = 1 for A or B) increases when those are heated.

8) Giving raises to temperature in the flow field either A or B exists.

Author Ajay Jain is thankful to Prof (Dr.) R.S. Tiwari Director (R & D), RCERT for valuable suggestions and discussion for improvement of manuscript of the paper.

Author Ajay Jain is thankful to Referee for valuable suggestions.

V. G.Gupta,AjayJain,Abhay KumarJha, (2016) Convective Effects on MHD Flow and Heat Transfer between Vertical Plates Moving in Opposite Direction and Partially Filled with a Porous Medium. Journal of Applied Mathematics and Physics,04,341-358. doi: 10.4236/jamp.2016.42041