9 11/6

All problems are for incompressible flow (constant density and coefficient of viscosity.)

1. 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. Steady 2D flow means in this case that velocity and pressure are independent of $z$ and time $t$ and that the velocity in the $z$-direction $w=0$. Do not ignore gravity. Assume ``developed flow'' in which the velocity components have become independent of $x$ (taken to be downwards, with $y$ the distance from the cylinder surface). 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. Do not make any other assumptions than listed above; they should be all you need.

2. What is the wall shear for the previous flow? Explain this value physically.

3. 7.5 Ignore gravity. To be original, assume that, in cylindrical coordinates, the streamlines are circles around the axis (which tells you that some velocity components are zero) and that the velocity and pressure are steady and independent of the axial coordinate $z$. Do not assume that the velocity components are independent of $\theta$. Skip the $r$-momentum equation for now, you can do without by showing that the pressure gradient in the $\theta$ direction must be zero. To show that the pressure gradient in the $\theta$ direction is zero, first show that it must be independent of $\theta$, (like we showed in class for duct flow). Next integrate to find the form of the pressure and use the fact that the pressure at a given point must have a unique value, so that if $\theta$ increases by $2\pi$, the pressure must return to the same value. (p 785 has the NS equations you need and 781 the continuity equation.)

4. Show that the skipped $r$-momentum equation in the previous problem can also be satisfied, and give the most general pressure if all equations are satisfied.

5. (noncredit question) Consider the below graph for the minor head losses due to sudden changes in pipe diameter:
   
   Discuss the following issues as well as possible from the sort of flow you would expect.
   1. How come the head loss become zero for an area ratio equal to 1?
   2. Why would the head loss be exactly one for a large expansion? Coincidence?
   3. Why would the head loss be less than one if the expansion is less? If the expansion is less, is not the pipe wall in the expanded pipe closer to the flow, so should the friction with the wall not be more??
   4. Why is there a head loss for a sudden contraction? The mechanism cannot be the same as for the sudden expansion, surely? Or can it?