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1. Unlike the ideal point vortex you analyzed in the 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. Integrate 5.1a, e, and f, and explain their physical meaning.

3. 5.14

4. 5.11

5. 5.12. As always, both mass and momentum conservation are needed.