3 9/16

1. Is the Poisseuille flow of the previous question an incompressible flow? Derive the principal strain rates and the principal strain directions as a function of the radial position.

2. For the Poisseuille flow of the previous question, draw a thin ring of fluid of radius $\frac12R$ at a time $t$ and a slightly later time $t+{\rm d}{t}$. Assume the ring has a square cross section at time $t$. Next in the $rz$ cross section, using very neat drawings show graphically that the motion of the ring cross section consists of two uni-axial strainings plus a solid-body rotation.

3. 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 $f_2$?

4. Write the velocity derivative tensor for 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 square particle (aligned with the principal strain axes), during a small time interval. Also sketch the deformation of an initially circular particle.