10 11/15

1. 7.17a corrected. Repeat the Stokes problem analysis, but now assume that instead of the plate velocity for $t>0$ constant being a constant $V_0$, the plate velocity is $V_0(t)=A t^n$. (So that Stokes' 2nd problem corresponds to the special case $n=0$.) Assume the velocity profiles become similar in terms of $f_n=u/V_0(t)$ and $\eta=y/\delta(t)$ where $\delta(t)$ is a typical boundary layer thickness to be determined and $f_n(\eta)$ the scaled velocity profile. Plug the similarity assumption $u=V_0(t) f_n(\eta)$ into the PDE, and note that it does not separate if $n$ is not zero. But show that $\delta$ is still the same as for Stokes 2nd by writing the PDE at $\eta=0$ and noting that $f''(0)/n f(0)$ is just a constant that you can take to be 4 (any other value only changes the definition of $\delta$.) With $\delta$ found, clean up the PDE to a simple ODE for $f_n(\eta)$. Verify that for $n=0$ it is Stokes' one ( $f_0''+2\eta f_0'=0$.)

2. 7.17b Differentiate the Stokes ODE and verify that it produces the ODE for $f_{-1/2}$ if $f_0'=f_{-1/2}$. Since the general solution for the Stokes problem was
   
   $$f_0 = C_1 + C_2 {\rm erfc}(\eta),$$
   
   
   where ${\rm erfc}(\infty)$ was zero, its derivative provides a solution for $f_{-1/2}$. Show that it can satisfy both boundary conditions for $f_{-1/2}$, at $\eta=0$ and $\eta=\infty$. What does $C_2$ have to be?

3. 7.17c. Now go the other way, to get the requested solution at $n=1/2$. Differentiate the equation for $f_{1/2}$, and you will get an equation for $f_{1/2}'$ for which you know the solution. Integrate to find $f_{1/2}$ itself, and make sure that the boundary condition at $\eta=\infty$ is satisfied. Don't worry too much about the boundary condition at $\eta=0$. (This process can be repeated to find solutions for any half-integer value of $n$.)

4. 7.17d. Is there a value of $n$ for which the shear stress that the plate applies to the fluid is constant? If so, sketch the plate velocity for that case as a function of time.

5. It is sometimes claimed that bathtub vortices rotate counterclockwise in the northern hemispere and clockwise in the southern one. Assume you are on the north pole and fill a cylindrically symmetric bathtub of radius 1 m with water. When a circular contour of water particles of initial radius 1 m goes out the drain of radius 1 cm, the tangential rotating velocity of the cicular contour increases according to Kelvin's theorem. Find out how much the tangential velocity was when the water was at rest compared to the tub with the drain closed, and from that, the tangential velocity when it is going in the drain. Express in terms of the revolutions per second the contour makes.

6. Solve the incompressible irrotational flow around an expanding cylinder of radius $r_0(t)$. Write the partial differential equation and boundary conditions. Solve after assuming that the tangential velocity component $v_\theta$ is zero by symmetry. (Actually, you might notice that the tangential velocity component does not have to be zero, but anyway.)