11 11/13

1. Sketch streamlines for the potential flow
   
   $$F = z^{2/3}$$
   
   
   Explain why such a flow might be relevant to flow about a corner. What is the (effective) pressure on the positive $x$-axis? Comment on what happens to the pressure when $x=0$.

2. Find the polar velocity components and pressure of the source flow
   
   $$F =\frac{Q}{2\pi}\ln z$$
   
   
   where $Q$ is a constant. Show that this is a valid solution of the Navier Stokes equations for the flow outside a cylindrical balloon whose radius $R$ is expanding according to the relationship
   
   $$\frac{{\rm d}R}{{\rm d}t}= \frac{Q}{2\pi R}$$
   
   
   Then show that this means that the cross sectional area of the balloon is linearly increasing with time.

3. Show that the potential flow
   
   $$F =\frac{Q}{2\pi}\ln z$$
   
   
   where $Q$ is not a constant but equal to $2\pi t$ is an exact solution of the viscous Navier-Stokes equation for flow around a balloon whose radius expands as $R=t$. Then find the pressure, using the correct Bernoulli equation for an unsteady potential flow. Comment on the pressure far from the balloon.

4. In the familiar potential flow around a cylinder,
   
   $$F = U\left(z+ \frac{r_0^2}{z}\right)$$
   
   
   the $Uz$ term produces the incoming uniform flow and the $Ur_0^2/z$ term produces the flow induced by the cylinder. That means that if the fluid is at rest at infinity and it is the cylinder that moves, the potential is given by
   
   $$F = - \dot x_0 \frac{r_0^2}{z-x_0}$$
   
   
   where $x_0(t)$ is the position of the center of the cylinder on the $x$-axis. Find the time derivative $\partial F/\partial t$ and the spatial derivative $W=\partial F/\partial z$. Watch it: both $x_0$ and $\dot x_0$ in $F$ depend on time. Now evaluate these derivatives on the surface of the cylinder where $z-x_0=r_0e^{i\theta}$. (Here the angle $\theta$ is measured from the center of the cylinder, not from the origin.) Then find $\partial\phi/\partial t$ as the real part of $\partial F/\partial t$. Also find the square magnitude of the velocity as $W\bar W$, where $\bar W$ is the complex conjugate of $W$. Use this to find the pressure on the surface of the cylinder. Answer:
   
   $$p_{\rm eff} = p_\infty - \frac12\rho\dot x_0^2 + \rho r_0 \ddot x_0\cos\theta + \rho \dot x_0^2\cos2\theta$$
   
   

5. From the pressure of the previous question, find the force on the cylinder. Show that it implies that to accelerate the cylinder, in addition to the force required to accelerate the cylinder itself, there will be an additional force as if an additional mass equal to an amount of fluid with the volume of the cylinder also must be accelerated. Explain why an apparent mass effect must be there on behalf of the second law of thermodynamics.