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}

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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)

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