3 9/15 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. Is the Poisseuille flow of the previous homework an incompressible flow? Derive the principal strain rates and the principal strain directions as a function of the radial position. Also find the vorticity.

4. For the Poisseuille flow of the previous question, describe what motions small particles perform. Neatly sketch a particle, that was spherical at time $t$, at time $t+{\rm d}t$. Show both the individual motions and the combined motion. Repeat for an initially cubical particle.

5. Write the velocity derivative tensor for two-dimensional ideal stagnation point flow. (With the velocity field as discussed in class.) From this tensor, decide whether or not ideal stagnation point flow is an incompressible flow. Find the strain rate tensor. Diagonalize it. What are the principal strain rates? What are the principal strain axes (i.e. the directions of ${\widehat \imath}'$, ${\widehat \jmath}'$, and ${\widehat k}'$)? Neatly sketch the deformation of an initially cubical particle (aligned with the principal strain axes), during a small time interval. Also sketch the deformation of an initially spherical particle. Also show the complete particle changes when you include the solid body rotation.