7 10/15 F

1. 6.1. Use the appendices. Based on the results, discuss whether this is incompressible flow, and in what direction the viscous stresses on the surface of the sphere are. Also state in which direction the inviscid stress on the surface is.

2. 6.2 Discuss your result in view of the fact, as stated in (6.1), that the Reynolds number must be small for Stokes flow to be valid. What does it physically mean?

3. 7.5. Use the appendices. You may only assume that $v_r=v_r(r)$, $v_\theta=v_\theta(r)$, $v_z=0$, and $p=p(r,\theta)$ in cylindrical coordinates. Do not assume that the radial velocity is zero, derive it. Do not assume that he pressure is independent of $\theta$, derive it. Ignore gravity as the question says. Note that $p$ must have the same value at $\theta=0$ and $2\pi$ because physically it is the same point. Answer for $v_\theta$:
   
   $$\frac{\Omega r_0^2r_1}{r_1^2-r_0^2}\left(\frac{r_1}{r}-\frac{r}{r_1}\right)$$
   
   

4. In 7.5, what is the power needed to keep the rod rotating, per unit axial length? What is the pressure difference between the surfaces of the pipe and the rod?