8 11/03 F

1. 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. (And that the fluid is Newtonian with constant density and viscosity, of course.) Do not assume that the radial velocity is zero, derive it. Do not assume that 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 $2\pi$ because physically it is the same point. Answer for $v_\theta$:
   
   $$\frac{\Omega r_0^2r_1}{r_1^2-r_0^2}\left(\frac{r_1}{r}-\frac{r}{r_1}\right)$$
   
   

2. 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?

3. 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 this minor head loss becomes zero for an area ratio equal to 1?
   2. Why do they use different scales and reference velocities for a sudden contraction than for a sudden expansion?
   3. Why would the head loss be exactly one for a large expansion? Coincidence?
   4. 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??
   5. 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?
   6. Any other observations you can offer?
   In answering this, think of where the head loss comes from, what its source is. What is lost?