Calculation of the pressure field

Both of these, of course, match Middleman.  We now need to get the pressure field.  It would have been easy to have obtained the angular form from the equation just by substituting in the angular expressions for and .  However, we did not do this, so we go back to the last time we saw the pressure in the momentum equations.  This was the θ momentum equation.      

We can then substitute for G[s] and F[s],

It should be apparent that the angular dependence is cos(θ) (because we take a θ derivative of p in this equation).  We need the solution to match a boundary condition on pressure (recall that pressure appears as first derivatives).   The obvious one for the r direction is that the pressure much match the far away pressure for the undisturbed flow, .  

We can transform this back to dimensional pressure for later calculation.