5. Let's consider some particle paths in the flow past the cylinder of problem 1.
(a) Write a material particle position in polar coordinates as $oldsymbol{x}(oldsymbol{X}, t)=r(t) hat{oldsymbol{r}}+ heta(t) hat{oldsymbol{ heta}}$. Determine the initial value problem for $r(t)$ and $ heta(t)$ for this material.
(b) The points $r=a, heta in{0, pi}$ are called stagnation points, where the velocity is zero. The material that begins there stays there forever. What is the speed of the fluid along the rest of the surface? How can we still have mass conservation if the speed is non-uniform along this curve?
(c) For $ heta=pi / 2$ what is the speed of the fluid as a function of $r$? The rate of decay to the background flow speed in $r$ gives us a sense of how far away from the cylinder its presence is 'noticeable'.
(d) Let's focus on the particles on the cylinder again. Since there is no normal component of velocity on a boundary, any particles which start on the boundary stay there. Consider a point which begins at $t=0$ at a point $(r, heta)=left(a, pi- heta_0
ight)$. Show that the time required until the particle is at its symmetrically located position, where $ heta= heta_0$, is
$$
T=frac{1}{2 U} log left[frac{1+cos left( heta_0
ight)}{1-cos left( heta_0
ight)}
ight].
$$
(e) If $ heta_0=pi / 2+alpha$, where $|alpha| ll 1$, estimate the transit time. Give the answer in terms of $ heta_0$.
(f) If $ heta_0=pi-alpha$, where $|alpha| ll 1$, estimate the transit time. Give the answer in terms of $ heta_0$. Describe this result, physically.