3 9/20

1. Consider the following flow of water through a two dimensional duct:
   
   
   
   If the length of the duct is 10 m and the total vertical height of the duct is
   
   $$h(x) = h(0) - 0.05 x \qquad (0\le x\le 10)$$
   
   
   with $h(0)=1$ m and the fluid enters at $x=0$ with a velocity $u_0=3$ m/s, show that the centerline velocity at arbitrary $x$ equals
   
   $$u = \frac{\displaystyle 3}{\displaystyle 1-0.05 x} \mbox{ m/s} \qquad v=0 \mbox{ m/s}$$
   
   
   and that the pressure is
   
   $$p=p(0) + 4500 - \frac{\displaystyle 4500}{\displaystyle (1-0.05 x)^2} \mbox{ Pa}$$
   
   
   where $p(0)$ is the pressure at $x=0$, which you can take to be zero. Use mass conservation and Bernoulli. What are the exit velocity and pressure at $x=10$ compared to the ones at the entrance $x=0$?

2. See whether or not Euler's differential momentum equations are satisfied on the centerline:
   
   $$\rho\left( \frac{\partial u}{\partial t} + u \frac{\parti... ...ial u}{\partial y} \right) = -\frac{\partial p}{\partial x}$$
   
   
   
   
   $$\rho\left( \frac{\partial v}{\partial t} + u \frac{\parti... ...ial v}{\partial y} \right) = -\frac{\partial p}{\partial y}$$
   
   
   (By symmetry, $v$ and $\partial p/\partial y$ are zero on the symmetry line.)

3. Use differential mass conservation,
   
   $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
   
   
   to determine the sign of $\partial v/\partial y$ on the centerline. So, will $v$ be positive or negative above the centerline? And is that what you would expect?

4. 4.5 ($v_3=0$. Note that this is our old stagnation point flow.)

5. 4.8 ($v_3=0$. Note that this is our old stagnation point flow.)

6. 4.2 (Note that you can find expressions for the vorticity in cylindrical coordinates in the appendices at the back of the book.)

7. 4.9 (Note that you can find expressions for the strain rate tensor in cylindrical coordinates in the appendices at the back of the book.)