1 9/06 W

1. If the density of air at sea level is 1.225 kg/m$^3$, and the molecular mass 28 g/mol, then what is the number of molecules per unit volume? What is the average spacing $\ell$ of the molecules?

   Consider a molecule of diameter $d$ that moves over one free path length $\lambda$. During that motion it will hit another molecule if the center of the other molecule is within a radius $d$ from the path of the molecule. In other words, the center of the other molecule must be inside a cylinder of radius $d$ around the path $\lambda$ of the first molecule. There should be about one collision in a free path, so there should be about one other molecule within the cylinder. So the free path can be ballparked from setting the volume of the cylinder equal to the average volume per particle:
   

   $$\pi d^2 \lambda = \mbox{average volume occupied per particle}$$
   
   
   Take the average diameter of the molecules to be 0.3 nm and compute $\lambda$. A more careful analysis says you still need to divide this by $\sqrt{2}$, so do so.

2. Suppose you have a body of typical size $L$. Which length, $\ell$ or $\lambda$, relative to $L$, determines whether you can define a meaningful pointwise density and velocity? (Here pointwise means using small volumes much smaller than $L$.) Which length determines whether you can define a pointwise density and velocity that you can also use to compute the flow development? In particular, take sea-level air. Then for what body size $L$ can you no longer use the normal (Navier-Stokes) equations to compute the flow at the points around the body? For what size body can you no longer find a meaningful velocity for the points around the body even if you used molecular dynamics?

3. For ideal stagnation point as discussed in class, compute the pressure field (Eulerian) from the Bernoulli law. Then verify Newton's second law $\rho\vec{a}=-{\nabla}p$ using the Lagrangian expressions for the particle paths to get $\vec{a}$. Hints: Note that the pressure is a scalar, not a vector. And that you need to write it in terms of $x$ and $y$, to take the gradient. The gradient of the pressure is a vector. The acceleration is the second derivative of the, Lagrangian, particle positions; read you notes.

4. The velocity field of small water waves near the surface is 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. You can assume that the water surface is (approximately) at $y=0$ and the water is below that surface.

5. The pathlines for water waves are more difficult to find. Therefor, assume 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:
   
   $$u = \epsilon \sin(k\xi+\omega t) \qquad v = - \epsilon \cos(k\xi+\omega t)$$
   
   
   Find and draw the particle paths under that assumption. Compare with the streamlines. Why are they not the same?

6. 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.