Compute the solution (Psi) of the pde (20.26) for the Chaplygin-Lamb dipole streamfunction [ frac{partial^{2} Psi}{partial r^{2}}+frac{1}{r}

Question:

Compute the solution $\Psi$ of the pde (20.26) for the Chaplygin-Lamb dipole streamfunction

\[ \frac{\partial^{2} \Psi}{\partial r^{2}}+\frac{1}{r} \frac{\partial \Psi}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} \Psi}{\partial \theta^{2}}=-\omega \]

for $\omega(r, \theta)=n^{2}(\Psi(r, \theta)-\lambda$, where $\lambda$ is an arbitrary constant, requiring continuity of velocity across the cylinder surface $r=a$. Plot streamlines and vorticity within the cylinder and the vorticity profiles for several values of $\lambda$ along the vertical axis $\theta=\frac{\pi}{2}$.

pde (20.26)