Study of heat transfers problem of dissipative fluid flow in a porous walls channel

A model of heat transfer by natural convection of dissipative fluid in a channel of porous walls has been discussed, the solution of governing partial differential equations was obtained using Alternating Direction Implicit method. The unsteady state as well as steady state solutions are founds simultaneously during the successive iteration of (ADI ) for the first time .


1-Introduction:
The study of fluids flow in a channel of porous walls is very important because it has a wide range of implementation, and this is due to the fact that these flows have many engineering and geophysical applications which include geothermal resources, Blood flow inside human being bodies, building insulation, Oil extraction, heat salt leaching in soils, flow system for transporting lymph , urinary circulatory system, transpiration cooling and many more.
In previous works, Beithou [2] has studied the effect of variable porosity on the free convection flow along a vertical plate embedded in porous medium numerically. The results showed when porosity increases , temperature variation becomes steepr and the Nusselt number increase utmost linearly with increasing porosity. Al-Odat [3] investigated transient MHD double diffusive of an electrically conducting fluid by free convection over a flat plate embedded in Darcy and non-Darcy porous medium in the presence of surface suction or blowing and magnetic field effects, he found that the presence magnetic field lowers both the Nusselt and Sherwood numbers in Darcy as well as Fochheimer flow regimes. Makinde [5], presented the unsteady twodimensional laminar flow of a viscous incompressible and electrically conducting fluid through a channel with the one wall impermeable and the other porous under the influence of a transverse magnetic field, he used the integral method in his investigation. Bukhari [6] has analyzed a linear stability by using the spectral Chebyshev polynomial method to obtain the numerical solution of multi-layer system consisting of the finger convection onset in a fluid layer overlying a porous layer. Mhone [7], presented the investigation of combined effects of a transverse magnetic field and irradiative heat transfer on unsteady flow of a conducting optically thin fluid through a channel filled with saturated porous medium and a non-uniform walls temperature, his results showed that increasing magnetic field intensity reduces wall shear stress while increasing radiation parameter through heat obseption causes an increase in the magnetic of wall shear stress. Das and Sahoo [8], considered the unsteady free convection and mass transfer boundary layer flow past an accelerated infinite vertical porous flat plate with suction when the plate accelerates in its own plane. The governing equations are solved both analytically and numerically using finite difference scheme and finally El-Kabeir [9] discussed an investigation to the thermal dispersion effect on non-Darcy MHD natural convection flow over a permeable sphere ] 57 [ maintained at uniform heat flux in a variable porosity porous medium. In this paper, we study the natural convection in a channel with porous walls, the governing differential equations were solved using (ADI ) method.

2-Mathematical Model:
Consider the unsteady flow of a dissipative fluid passing through a long channel with porous walls, the Cartesian coordinate system ( ) has been taken as the x -axis lay in the center of the channel, yaxis represents the width of the channel while the z -axis is the normal of xy plane. Let The governing equations in dimensional form are given by: where v u, are the velocity components, t is the time and are the temperature, gravitational acceleration, thermal expansion coefficient, concentration expansion coefficient, Kinematics viscosity, permeability of the medium, thermal diffusivity, dynamical viscosity, density, specific heat at constant pressure, mass diffusion coefficient, respectively. With the following boundary conditions,
1d) Simplifying the above equations, the governing equations under these non-dimensional quantities becomes,

4-Method of solution:
In order to solve the system of equations (3.2)-(3.5) with the boundary conditions (3.6), we resort to ADI finite difference method [1], and to achieve this we have to start with last equation              The coefficients V U , are treated as constants during any one time-step of the computation [4], each of the equations (diffusion, heat, motion) creating a tridiagonal system which are solved by using Gauss ] 67 [ elimination method, all details are given in reference no. [1].

5-Results and figures:
We present in this section some of the results obtained from the computation done, and these results have been expressed by figures to illustrate how the solution for different cases becomes as well as the effects of different parameters as follows: The Diffusion Function

6-Conclusions:
In this work we have used ADI method in the solution of the governing equations completely without reducing or changing and from the results obtained we conclude that the steady state can be reached after some iterations for all equations and this is clear from the figures given previously in last section, due to this fact figure (5.1) represents the results obtained by the solution of diffusion equation which showed that for different point the values of the diffusion function goes to steady state for some iterations and remains fixed to the end, the same things happened to the temperature and velocity functions which is obvious from figures (5.4) and (5.5) respectively. Other remarks can be noticed for some parameters like grashof and prandtl numbers. It is also noticed that the parameters Sc (Schmidt number) and pr (Prandtl number) has effects into motion equation only through diffusion factor φ and heat factor θ equation (3.3).