11 11/22

1. The potential of irrotational transverse flow around a circular cylinder of radius $r_0$ is
   
   $$\phi = U \left(r +\frac{r_0^2}{r}\right)\cos(\theta) - \frac{\bar\Gamma}{2\pi} \theta$$
   
   
   where $U$ is the magnitude of the velocity at large distances, and $\bar\Gamma$ the circulation around the cylinder. Verify that the correct boundary conditions at the surface of the cylinder and at large distances are satisfied.

2. Verify that the circulation around the cylinder is indeed $\bar\Gamma$ by integrating around a suitable contour.

3. For the flow of the previous question, find the pressure on the cylinder surface as a function of angular location.

4. For the flow of the previous question, find the force on the cylinder assuming the viscosity is zero, so there are no viscous stresses on the cylinder surface. Is d'Alembert satisfied? Is Kutta-Joukowski satisfied?

5. Derive the streamfunction of ideal stagnation point flow. The steps are similar to the ones used to derive the corresponding potential flow in class. From the streamfunction, determine the mathematical form of the streamlines.