8 10/24 W

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 moment needed to keep the rod rotating, per unit axial length? What is the power needed? What is the pressure difference between the surfaces of the pipe and the rod?

3. 7.6. Do not ignore gravity, but assume the pipe is horizontal. And that the $y$-axis of the Cartesian coordinate system with $z$ along the axis of the pipe is pointing upwards. Careful, the gravity vector is not constant in polar coordinates. Find the components using geometry or from $\vec{g}=-g\nabla{h}$. Assume $x$ is horizontal like $z$ and $y$ vertically upwards. Do not ignore the pressure gradients: assume the pressure can be any function $p=p(r,\theta,z,t)$ and derive anything else. Merely assume that the pressure distribution at the end of the pipe and rod combination is the same as the one at the start. For the velocity assume $v_r=v_\theta=0$ and $v_z=v_z(r,z)$. Anything else must be derived. Give both velocity and pressure field. Check that your answer is the same as you would get from using a kinetic pressure. What is the force required to pull the rod through the axis, per unit length?

4. 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 coefficient 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 coefficient be exactly one for a large expansion? Coincidence?
   4. Why would the head loss coefficient 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 coefficient 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?