Free Convection Flow of Viscous Dissipative Fluid in a Rectangular Cavity

79 Free Convection Flow of Viscous Dissipative Fluid in a Rectangular Cavity  Ahmed M. Jassim Tagread H. Shuker College of Computer Sciences and Mathematics College of Education University of Mosul, Iraq Received on: 18/12/2005 Accepted on: 05/03/2007 ABSTRACT Free convection flow of incompressible viscous fluid with dissipation in a rectangular cavity has been studied, a finite difference technique based on ADI scheme is adopted in the solution of the problem. The effect of dissipation parameter ( ), which usually appears as a term in the energy equation, has been taken into account.The results indicated that the effect of dissipation number ( 004 . 0  =  ) was very small which is accepted with the fact of neglecting the dissipation function in the energy equation of most convection problems.

, are the density, specific heat at constant volume, pressure, thermal conductivity, temperature, dissipation function and internal heat generation rate, respectively.
The effect of viscous dissipation represented by  has not been studied till 1999, when Soundalgekar [7] has presented a transient free convection flow of viscous incompressible fluid past a semi-infinite vertical plate by taking into consideration viscous dissipative heat and he solved the governing non-linear equations by using the implicit finite-difference method of Crank-Nicolson type. In this paper we study the effect of viscous dissipative heat convection and we solve the problem by using ADI scheme.

2-Mathematical Model:
A two-dimensional unsteady flow of a viscous, incompressible fluid in a rectangular cavity is considered, the flow under the usual Boussinesq's approximation can be shown to be governed by the following boundary layer equations, eliminating the pressure terms from equations (2-2a)-(2-2b), the set of governing equations becomes:

3-Non-dimensional form:
In order to solve the governing equations (2-1)-(2-3) with the boundary conditions (2-4), we have to convert these equations to the nondimensional form and this will occur by introducing the non-dimensional quantities [10], The governing equations under these non-dimensional quantities become,

4-Method of solution:
We shall use the ADI finite difference method in the solution of the heat equation (3.4) as follows:, see [ 3,4,5,7] .. (4.5) followed by, The coefficients are treated as constants during any one time-step of the computation, each of the equations (4.4) and (4.6) are creating a tridiagonal system which are solved by using Gauss elimination method, [1].

5-Results and Conclusions:
We investigate in this work the effect of dissipation number  into the non-dimensional temperature  , so that we have concentrated our effort only on heat equation .It is necessary to say that Rayleigh number Ra play a very important rule in the solution of motion equation, refer to [6,8,9], but in this paper we have presented a numerical solution of a non-linear equation (3.4) by using ADI finite difference method which leads to conclude the following remarks: 1-In order to implement the ADI in the solution of heat equation as it has appeared in the model under discussion, we assumed that the velocity components V U , are constants with small quantities [10], for this reason we don't need to discuss the Rayleigh number Ra .
2-We have presented some results for the values of the temperature  in table-1, it shows the non-dimensional temperature  at the left side of the cavity for different values of dissipation number  , figures 1 to 3 represent the curve of temperature  at different positions of the cavity, the first one is showing the effect of the parameter  into the curve of the temperature near the left side of the enclosure. It seems that the effect of  is very small, same remark has been noticed for figure -2 which represents the temperature at the core of the enclosure. Finally, figure -3 shows the effect of dissipation parameter  into the temperature near the right side of the region. 3-There is no contradiction between the results which we have obtained and the fact of neglecting the dissipation function of the heat equation, which means that the effect of number  is so small that it can be neglected in most convection problems.