1 9/1 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.

   Suppose you have a body of typical size $L$. Which length, $\ell$ or $\lambda$, relative to $L$, determines whether you can define a continuum density and velocity? Which length determines whether you can define a continuum density and velocity that you can use to compute the flow development?

2. A carbon nanotude has a diameter of 1 nm. There is a steady flow of standard air around it. Can you define a continuum density and velocity? Do not answer to quickly. Note that in a steady flow you can average over both space and time.

   Now suppose the flow about the nanotube is truly unsteady. Can you define a continuum density and velocity in that case? Comment in particular about the ``ensemble average.''

   Will you be able to use the Euler or Navier-Stokes to find these continuum fields? If not, will you be able to write modified equations for the continuum quantities that you can use instead?

3. Is a rain droplet in saturated air a Lagrangian region / material region / control mass? How about a droplet in dry air? Explain.

4. For ideal stagnation point as discussed in class, compute the pressure field (Eulerian) from the Bernoulli law. Then verify Newton's second law $\rho\vec{a}=-{\nabla}p$.

5. Convert the velocities of ideal stagnation point flow to polar coordinates using
   
   $$v_r = u \cos\theta + v \sin\theta \qquad v_\theta = - u \s... ... v \cos\theta \qquad x = r\cos\theta \qquad y = r\sin\theta$$
   
   
   Then find the streamlines in polar coordinates. Show that you get the same answer.

6. The velocity field of water waves near the surface is given by
   
   $$u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t)$$
   
   
   where $\epsilon$, $k$, and $\omega$ 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.

7. 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 $x_0$ of the initial particle position which is constant:
   
   $$u = \epsilon \sin(kx_0+\omega t) \qquad v = - \epsilon \cos(kx_0+\omega t)$$
   
   
   Find and draw the particle paths under that assumption. Compare with the streamlines. Why are they not the same?