3 9/27 W

1. In Poiseuille flow (laminar flow through a pipe), the velocity field is in cylindrical coordinates given by
   
   $$\vec v = {\widehat \imath}_z v_{\rm max} \left(1 -\frac{r^2}{R^2}\right)$$
   
   
   where $v_{\rm {max}}$ is the velocity on the centerline of the pipe and $R$ the pipe radius. Use Appendices B and C to find the velocity gradient and strain rate tensors of this flow. Do not guess. Evaluate the strain rate tensor at $r=0$, $\frac12R$ and $R$. What can you say about the straining of small fluid particles on the axis? Is Poisseuille flow an incompressible flow? Also find the vorticity. Do particles on the axis rotate? If not, what do they do?

2. (20pt) For the Poisseuille flow of the previous question, derive the principal strain rates and the principal strain directions for an arbitrary radial position $r$ using class procedure.

3. Make a neat picture of a $r,z$ plane through the axis showing, for a point at an arbitrary $r,z$, a small cubic fluid particle, in cross section, that is aligned with the principal strains at that point. Using arrows as appropriate, show the translational, straining, and average rotational motions that this particle is performing. If the particle was a small solid sphere, would you expect it to rotate, and if so, in which direction?