2 9/10

1. A steady stream of air enter a pipe with a diameter of 1'' at a velocity of 20 m/s at a pressure of 1 bar. The pipe has a contraction in diameter to $\frac12$''. What are the velocity and pressure after the contraction? Ignore viscous effects. This is a steady flow, $\partial\vec{v}/\partial{t}=0$, so explain how it is possible for the fluid to change velocity in a steady flow.

2. Write the velocity derivative tensor for ideal stagnation point flow. From this tensor, decide whether or not ideal stagnation point flow is an incompressible flow.

3. Find the strain rate tensor for ideal stagnation point flow. Diagonalize it. What are the principal strain rates? What are the principal strain axes (i.e. the directions of ${\widehat \imath}'$, ${\widehat \jmath}'$, and ${\widehat k}'$)?

4. Continuing that flow, based on the strain rate tensor, 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.

5. Laminar flow through a long pipe is called Poisseuille flow. The velocity profile in cylindrical coordinates $r$, $\theta$ and $z$, with $z$ along the pipe axis, is
   
   $$\vec v = {\widehat \imath}_z v_{\rm max} \left(1-\frac{r^2}{r_0^2}\right)$$
   
   
   where $r_0$ is the radius of the pipe and $v_{\rm {max}}$ the center line velocity. Determine whether this is an incompressible flow field by looking up the divergence in cylindrical coordinates in Appendix B. Also look up the velocity derivative tensor and use it to evaluate the strain rate tensor at $r=\frac12r_0$. Compare your answer to Appendix C. What is the strain rate tensor on the axis? So, how do small fluid regions at the axis deform?