6 10/8 postponed to 10/10

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 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.

3. 7.5. Use the appendices. You may assume that $\vec{v}=\vec{v}(r)$. with $v_z=0$, and $p=p(r,\theta)$ in cylindrical coordinates. Do not assume that the radial velocity is zero, derive it. Do not assume the 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 $2\pi$. 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?

5. 7.9. You can assume that the film thickness is so small that the curvature of the pipe wall can be ignored. In that case, it becomes 2D steady flow along a flat wall of spanwise length $2\pi r_0$ in the $z$-direction. Take the $x$-axis downwards. Assume $v=0$ (vertical streamlines), $u=u(x,y)$ and $w=0$ (two-dimensional flow), and that $p=p(x,y,z)$. Everything else must be derived; derive both pressure and velocity field. Do not ignore gravity. For the boundary conditions at the free surface, assume that the liquid meets air of zero density and constant pressure $p_a$ there. Also write appropriate boundary conditions where the fluid meets the cylinder surface.