# Project description¶

## Given task¶

In this exercise you calculate the viscous flow between two parallel paltes.

Such flow is described by the diffusion equation:

Each plage is a distance L apart and the boundary conditions are \(u(y=0) = 0\) and \(u(y=L) = 1\). The exact solution for this equation for any location in space and time can be written as:

where the constants, \(a_{n}\), in the infinite series depend on the initial condition specified. For this project you must solve the flow for the following initial condition:

The following combined implicit-explicit difference formulation (Combined Method A in section 4.2.5 in Tannehill, Anderson and Pletcher) should be used:

First, non-dimensionalize \(t\) and \(y\) by \(\tau\) and \(L\), respectively. Select an expression for \(\tau\) that essentially removes \(\nu\) from the governing flow equation. Write out the non-dimensional for this problem. You will numerically solve the non-dimensional form of the problem on a uniformly spaced mesh (i.e. constant \(\Delta y\)) with jmax grid points (including the bottom and top wall).

Compute the time-dependent and steady state solution using a direct solution technique (i.e. non-iterative) at each time step. To do this, first rearrange the discretized equation (non-dimensional form) so that it is in tri-diagonal form. You will be able to solve for all jmax points simultaneously at each time step by writing a computer program which uses the Thomas Algorithm.