2 HW 2

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Two-dimensional ideal stagnation point flow is given by
   
   $$\vec v = \hat\imath c x - \hat\jmath c y$$
   
   
   where $c$ is some constant. Find the vorticity (in 3D). Find the strain rate tensor. Diagonalize it by rotating the axes suitably. What are the principal strain axes (i.e. the directions of $\hat\imath '$, $\hat\jmath '$, and $\hat k'$)? What are the principal strain rates? Neatly sketch the deformation of a small initially square particle (aligned with the principal strain axes), during a small time interval. Also sketch the deformation of a small initially circular particle. Also show the complete particle changes when you include the solid body rotation.

2. If you put a cup of coffee at the center of a rotating turn table and wait, eventually, the coffee will be executing a “solid body rotation” in which the velocity field is, in cylindrical coordinates:
   
   $$\vec v = \hat\imath _\theta \Omega r$$
   
   
   where $\Omega$ is the angular velocity of the turn table, $r$ the distance from the axis of rotation, and $\theta$ the angular position around the axis. Find the vorticity and the strain rate tensor for this flow, using the expressions in appendix B. Do not guess. (The book might different, bad, symbols for $r$ and $\theta$.) Show mathematically that indeed the coffee moves as a solid body, i.e. the fluid particles do not deform, and that for a solid body motion like this, indeed the vorticity is twice the angular velocity.

3. In Poiseuille flow (laminar flow through a pipe), the velocity field is in cylindrical coordinates given by
   
   $$\vec v = \hat\imath _z v_{\rm max} \left(1 -\frac{r^2}{R^2}\right)$$
   
   
   where $v_{\rm {max}}$ is the velocity on the centerline of the pipe and $R$ the pipe radius. Use Appendix B to find the strain rate tensor 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?

4. Is the Poisseuille flow of the previous questions an incompressible flow?

5. Find the vorticity of the Poisseuille flow of the previous questions. Do the fluid particles on the axis rotate?

6. For the Poisseuille flow of the previous questions, given that the principal directions of the strain rate tensor are everywhere
   
   $$\hat\imath ' = \frac{1}{\sqrt{2}}\left(\hat\imath _r+\hat\i... ...h _r-\hat\imath _z\right)\qquad \hat k' = \hat\imath _\theta$$
   
   
   find the principal strain rates. Sketch the deformation of a fluid particle at an arbitrary radius $r$. In particular, in the $r,z$-plane, neatly sketch a particle that was spherical at time $t$ at time $t+{\rm d}t$. What happens in the $\theta$ direction with the particle?