2 9/11 W

1. The velocity field of shallow water waves is just below the surface $y\approx0$ given by
   
   $$u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t)$$
   
   
   Find the pathlines for these water waves. Since this would be a messy process if done exactly, simplify it by assuming that $\epsilon$ is small. In that case the particle displacements are small, and that allows you to approximate $x$ in the sine and cosine by the $x$-value $\xi$ of the initial particle position, which is constant for a given particle:
   
   $$u = \epsilon \sin(k\xi+\omega t) \qquad v = - \epsilon \cos(k\xi+\omega t)$$
   
   
   Find and draw a representative collection of particle paths under that assumption. Show the features.

2. As noted in the previous question, the velocity field of shallow water waves is near the surface given by
   
   $$u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t)$$
   
   
   where amplitude $\epsilon$, wave number $k$, and frequency $\omega$ are all positive constants. Find and draw the streamlines of the flow. Do not approximate in this case. Compare with the pathlines. Why are they not the same?

3. Continuing the previous question, draw the streakline coming from a generator at the origin, which is turned on at time $t=0$. Draw the streakline for times $\omega t = 0$, $\frac14\pi$, $\frac12\pi$, $\pi$, $2\pi$, $4\pi$, $6\pi$, .... In a separate graph, draw the particle path of the particle that was released at time zero. Compare its position at the given times with that in the streaklines. Hint: Read up on how to get streaklines in your notes.

4. For stagnation point flow, write down the velocity derivative tensor, then its antisymmetric part, and then find the angular velocity of the fluid rotation produced by that part. Repeat for the solid body rotation flow
   
   $$u = - d y \qquad v = d x$$
   
   
   with $d$ a constant.