10 11/5 F

1. Do bathtub vortices have opposite spin in the southern hemisphere as they have in the northern one? Derive some ballpark number for the exit speed of a bathtub vortex at the north pole and one at the south pole, assuming the bath water is initially at rest compared to the rotating earth. Use Kelvin’s theorem. Note that the theorem applies to an inertial frame, not that of the rotating earth. What do you conclude about the starting question?

2. A Boeng 747 has a maximum take-off weight of about 400,000 kg and take-off speed of about 75 m/s. The wing span is 65 m. Estimate the circulation around the wing from the Kutta-Joukowski relation. This same circulation is around the trailing wingtip vortices. From that, ballpark the typical circulatory velocities around the trailing vortices, assuming that they have maybe a thickness of a quarter of the span. Compare to the typical take-off speed of a Cessna 52, 50 mph.

3. Find boundary conditions for the streamfunction for transverse ideal flow around a circular cylinder. The velocity far away from the cylinder is $U{\widehat \imath}$ and the radius of the cylinder is $r_0$. Note that appendix D.2 has an error. The correct equation is
   
   $$v_\theta = - \frac{\partial \psi}{\partial r}$$
   
   

4. Following similar lines as in class, but watching the new boundary conditions, solve the equation for the streamfunction around the circular cylinder. Before continuing, check your results for the radial and tangential velocity components at the surface of the cylinder against the one from the velocity potential solution obtained in class. Is the velocity at the top and bottom points $2U$? Are the stagnation points correct?

5. Find the pressure on the surface of the cylinder.