1 9/04 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? (First compute the average volume per molecule, then think of the molecules as being equally spaced in all three directions of a $1\times1\times1$ m$^3$ box)

   Consider a molecule of average 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. Continuing the previous question, suppose you have a body of typical size $L$ in the flow of some gas. Which length, $\ell$ or $\lambda$, relative to $L$, determines whether you can define a meaningful pointwise density and velocity at a given time and position (by taking a small volume around the point and then taking the ratio of mass inside to volume, respectively momentum inside divided by mass inside at a given time)? (Of course, the small volume must be small compared to the body for it to be considered a point instead of a region in the flow.)

   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 other words, the pointwise density and velocity (and temperature) must fully determine the state of the gas at the points.

   In particular, take sea-level air like in the previous question. 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 by averaging mass and momentum in small volumes even if you used molecular dynamics?

3. The Lagrangian equation for inviscid flow is Newton's second law in the form $\rho\vec{a}=-{\nabla}p+\rho\vec{g}$, i.e. mass per unit volume times acceleration equals pressure force per unit volume plus gravity force per unit volume.
   1. In this question you need to write that in Eulerian form first to give the Euler equations. Compare with the equations in the appendices in the back of the book.

   2. Now substitute the ideal stagnation point as discussed in class into your Euler equations, and verify that the Euler equations contain the same acceleration vector as the Lagrangian solution already discussed in class.

   3. Then compute the pressure field (Eulerian) from these Euler equations. Make sure you do that properly, as discussed in class. Include gravity, producing a force per unit volume $\rho\vec{g}$ where the acceleration of gravity $\vec{g}$ points down in the minus $y$-direction. Compare the result with the Bernoulli law.

   4. Conversely, starting with the Bernoulli law, take minus the gradient and verify that that gives the 2D Euler equations, assuming that $\partial{u}/\partial{y}$ equals $\partial{v}/\partial{x}$. (For ideal stagnation point flow, they are both zero). In writing the Bernoulli law, you can assume that the constant in it is the same for all streamlines.

   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 Lagrangian time derivative of the particle velocity; read your notes. In finding the pressure, put the solution of the first equation into the second.

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 in Eulerian form.)

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 $f_2$?