6 10/6 W

1. Unlike the ideal point vortex you analyzed in a previous homework, a true vortex diffuses out with time, and its velocity field is given by
   
   $$\vec v = {\widehat \imath}_\theta \frac{C}{2\pi r} \left[1 - \exp\left(-\frac{r^2}{4\nu t}\right)\right]$$
   
   
   Find the vorticity of this flow field. Also find the circulation along a circle of an arbitrary radius $r$. Then show that Stokes theorem does work for this flow. (The velocity is zero at $r=0$; just apply l’Hopital.) Finally show that if the coefficient of viscosity is very small, the vorticity is only nonzero in some narrow spike near the origin, so that it looks almost like a an ideal vortex. (But the vorticity still integrates to $\Gamma$, despite the small radius of the region with appreciable vorticity.)

2. 5.2 Make sure to first write the full equations for the special case that the density is constant before assuming a radial flow. Make that a neat graph, and include the streamlines.

3. 5.3. This is two-dimensional Poiseuille flow (in a duct instead of a pipe). $T_{ij}$ is the book’s notation for the complete surface stress including the pressure,
   
   $$T_{ij} = -p \delta_{ij} + \tau_{ij}$$
   
   
   where $\delta_{ij}$ is called the Kronecker delta or unit matrix. So the book is really saying the pressure is $-5$ and there is an additional viscous stress $\tau_{xy}=-2\mu v_0y/h^2$. Watch it, the expression $n_j\tau_{ji}$ gives the stress components in the $x,y,z$-axis system.

4. 5.6. $Z$ is the height $h$. The final sentence is to be shown by you based on the obtained result. Hints: take the curl of the equation and simplify. Formulae for nabla are in the vector analysis section of math handbooks. If there is a density gradient, then the density is not constant. And neither is the pressure. $T_{ij}$ is the book’s notation for the complete surface stress, so the book is saying there is no viscous stress. (That is self-evident anyway, since a still fluid cannot have a strain rate to create viscous forces.)