1 9/05 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 (i.e. not nonsensical) pointwise density and velocity at a given time? (Here pointwise means using small volumes much smaller than $L$.) Which length determines whether you can define a meaningful pointwise density and velocity that would be enough info, say, to compute the further 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 continuum 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 $\rho\vec{a}=-{\nabla}p$ (mass per unit volume times acceleration equals force per unit volume) 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 your notes.

4. If the surface temperature of a river is given by $T=2x+3y+ct$ and the surface water flows with a speed $\vec v={\widehat \imath}-{\widehat \jmath}$, then what is $c$ assuming that the water particles stay at the same temperature? (Hint: ${\rm {D}}T/{\rm {D}}t=0$ if the water particles stay at the same temperature. Write this out mathematically.)

5. A boat is cornering through this river such that its position is given by $x_b=f_1(t)$, $y_b=f_2(t)$. What is the rate of change ${\rm d}{T}/{\rm d}{t}$ of the water temperature experienced by the boat in terms of the functions $f_1$ and $f_2$?

6. The velocity field of shallow water waves is near the surface given by
   
   $$u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t)$$
   
   
   Find the pathlines for these water waves. Since this is a messy process, 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.