Numerical Solution for Both Steady and Unsteady State of Fluid Flow Between Two Heated Parallel Walls

The study of the steady and unsteady state of any system is very important for which it continues to exist or not; for this reason, the unsteady state cannot stay longer for some time under circumstances conditions and if the system continues working without affecting whatever the time, that means the steady state. For this and more, the study of the steady and unsteady solution has gained the impotency. The attention in this paper has been given to solving a problem starting from an unsteady state by using an ADI technique for some iterations to reach a steady state simultaneously. The results have been plotted to explain the situation. Ganish and Krishnambal [1] studied unsteady magneto hydrodynamic flow between parallel porous plates. They discussed the problem and gave some analysis of their work with the help of graphs. Gnana Prasuna et al. [2] investigated the unsteady flow. They solved the problem using two stages, steady and unsteady, by using the Laplace transformation method to explain the results and deduced that the velocity profiles are parabolic and symmetric about the channel. They also noticed that the porosity varies linearly during time. Attia et al. [3] discussed unsteady non-Darcian viscous incompressible fluid surrounded by two parallel porous plates, they applied uniform and constant pressure gradient the viscous dissipation is considered in the energy equation, it is found that porosity, internal effects and suction have a remarked effect on decreasing the velocity distribution with inverse proportionally manner. Uwanta and Hamza [4], presented the natural convection for an unsteady state of heat generating and absorbing fluid flow in a vertical channel. The problem has been solved using the semi-implicit finite difference method. The steady state was also obtained by expressing the velocity, concentration, and temperature and interpreting the results graphically for some parameters such as suction, injection and Soret number. Moses et al. [5] reported unsteady magneto hydrodynamic coutte flow with the lower plate considered porous. They solved the government equations by using separable method, the effect of various parameters such as Hartman, Prandtl numbers have been taken into account on the flow, also deduced that the velocity profile and temperature distribution and the skin friction decrease with high Hartman number, the convection increased with large Nusselt number. The magnetic field significantly affects the flow of unsteady coutte flow between two infinite parallel porous plates in an inclined magnetic field with heat transfer. Hamza et al. [6] suggested a study of two steady and unsteady states of natural convection flow in a vertical channel with the presence of a uniform magnetic field. The partial differential equations were solved approximately using a semi-implicit finite difference scheme, and the computed results for velocity, temperature, and skin friction were discussed and presented graphically. A comparison has been made with previously published work. It was found that the fluid velocity and temperature increase with increasing variable thermal conductivity, while the magnetic parameter retards the motion of the fluid. Uddin et al. [7] analyzed unsteady laminar free convection fluid flow numerically, and a mixed method has been adopted to the solution of the problem, fourth-order Runge-Kutta, shooting methods were used. Sattar and Subbhni [8] considered a non-Newtonian incompressible fluid under the effect of couple stress and magnetic field using a finite element technique. They assumed the pulsatile pressure gradient in the direction of motion with the effect of different parameters, and they concluded that the flow is damping with increasing stress, which is used in some cases, such as blood diseases. The studies [9, 10] investigated the exact solution of unsteady flow using the integral transform method based on Laplace and sine Fourier transformation. The effect of various parameters on fluid velocity is presented graphically, and the time of steady state has been evaluated. The studies [11-14] presented unsteady flow and heat transfer of a viscous, incompressible electrically conducting fluid through a porous horizontal channel. The associated equations of the problem were transformed to dimensionless form and treated analytically using the perturbation method. The present work provides a mathematical technique for solving scientific problems that is used to solve them separately by introducing some assumptions that lead to solving the problem twice, whoever it could be solved once using this method (ADI), starting in an unsteady state until reaching a steady state for limited iterations under this method.

Consider a laminar fluid confined between two heated conducting parallel walls at a vertical position with different temperature. The horizontal walls are taken to be isolated. Imagine that the x-axis is parallel to non-conducting walls while the y- axis is parallel to the conducting walls. The distance between the wall is L with height h, as it is shown in Figure 1 below.

where, $\alpha=\frac{k}{\rho c_p}$ are dissipation function and thermal conductivity, respectively. Introducing the following non-dimensional terms:

$p r=\frac{\vartheta}{\alpha}$ Prandtl number

$G r=\frac{g \beta L^3 \Delta T}{\vartheta^2}$ Grashof number

$\mathrm{Ra}=\mathrm{Gr} \cdot \operatorname{Pr}=\frac{g \beta L^3 \Delta T}{\vartheta \alpha}$ Rayleigh number

$X=\frac{x}{L}, \Upsilon=\frac{y(\sqrt{G r})^{1 / 2}}{L}$, and $\theta=\frac{T-T_0}{\Delta T}, \Delta T=T_1-T_0$

$U=\frac{u L}{\vartheta \sqrt{G r}}, V=\frac{v L}{\vartheta(\sqrt{G r})^{1 / 2}}, \tau=\frac{t \vartheta \sqrt{G r}}{L^2}$

With the boundary conditions given by

$u=0,0 \leq x \leq l$ and $v=0,0 \leq y \leq h$

$T=T_1, T_0$ atx $=0,1, \mathrm{t}=0$

$0 \leq x \leq L, 0 \leq y \leq h: u=v=$ constant,$T=T_1=10.0$

$\frac{\partial T}{\partial y}=0$ at $y=0, h$

$y=0: u=v=\mathrm{constan} t, T=T_0=0.0$

$y=h: u=v=$ constant,$T=T_1=10.0$

By using these conditions, the governing equations becomes:

$\xi=-\nabla^2 \psi$