This coupled with equation 4, the Poission equation that relates the stream function to vorticity, gives us the self-contained system we will need to solve for the flow given the appropriate boundary and initial conditions. The boundary values for the stream function can be found along the wall by use of equation 5. This as well shows that for a stationary object inside our rectangular domain we will need a region where both the vorticity and stream function is zero. We will also create repeated boundaries at the top and bottom by solving for the stream function at the bottom as if the next increment down is the first zone from the top and then by resolving the stream function at the top layer with the solutions from the bottom.

Numerical Approximations

The process for our numerical approximation will be as stated in Fluid Mechanics section 6.3
1. Given \(\omega_{i,j}\) at all interior points, solve for \(\psi_{i,j}\).
2. Find the boundary vorticity, \(\omega_{wall}\).
3. Calculate the vorticity at the new time,  \(\omega_{i,j}^{n+1}\)  for all interior points.
4. Set  \(t\ =t\ +\ \Delta t\)  and go back to first step.
For step one we will use the equation,\(\)