2 9/20 W

1. 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.)

2. 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 $f_2$?

3. Substitute the Eulerian velocity field of stagnation point flow into the Euler equations. In the force per unit volume, include the gravity force per unit volume. Assume that gravity is in the minus $y$-direction. You get three equations for the pressure, one giving its $x$-derivative, one its $y$ derivative, and the third its $z$-derivative. More than one equation for a single scalar unknown $p$ is usually too much, but show that in this case, there is indeed a solution $p$ that satisfies all three equations. Find out what it is. Does it satisfy the Bernoulli law?