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1. If the density of air at sea level is 1.225 kg/m$^3$, what is the average spacing of the molecules? The molecular mass of air is 28 g/mol. For what size of bodies would you expect major problems in trying to define continuum values of density, velocity, and pressure?

   If the free path length of air at sea level is 6.6 $10^{-8}$ m, at what size would you expect that normal equations of motion (like the Euler and Navier-Stokes equations) become unusuable? When the body size becomes comparable to the mean molecular spacing or to the mean free path length?

   Repeat for 200 km height, where the number of molecules is 8 $10^{15}$/m$^3$ and the free path length 200 m. Can you define a continuum air velocity and density for the flow around a rocket? Can you use continuum equations?

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

3. For ideal stagnation point flow, what is the relation between the position vector $\vec{r}$ and the acceleration vector $\vec{a}$? Graphically show, by drawing a pathline in the first quadrant and acceleration vectors at points on that line, that at the point where $x=y$, the acceleration is all centripetal acceleration. Also show graphically that for $x<y$ the acceleration has a tangential component that slows the fluid down, while for $x>y$, the fluid speeds up again.

4. For ideal stagnation point flow, find the pressure in terms of $x$ and $y$ from the Bernoulli law (ideal stagnation point flow is inviscid, steady, and all streamlines have the same stagnation pressure). Now verify that the force $\rho\vec{a}$ per unit volume equals $-\nabla{p}$ where $\nabla{p}$ is the pressure gradient $(\partial{p}/\partial{x},\partial{p}/\partial{y})$. It is true for any inviscid flow that minus the pressure gradient gives the net force per unit volume on the particles.

5. A velocity field is given by $\vec v={\widehat \imath}\cos t+{\widehat \jmath}\sin t$. Is this a steady flow? Find the particle paths and draw a few of them. Find the streamlines and draw planes of streamlines at a couple of times. Find the streakline coming from a smoke generator at the origin that is turned on at time $t=0$; sketch the streakline at time $t=\pi$.

6. 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.)

   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 $f_2$?