10 HW 10

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Differentiate the equation obtained in the previous question twice; if your equation is correct, $f''$ should satisfy the same equation as $f_{\rm Stokes}$ for Stokes' first problem. So $f''$ too should be a combination of a multiple of the complementary error function and a constant. Then two integrations should produce your desired function $f$, if you take account of the boundary conditions. Your task is now to find a simple expression for $f$ in terms of the (complementary) error function and elementary functions. Note: you might want to google “repeated integrals of the error function.” Note: Another way of doing this is to change the order of integration in the double integrals you encounter. For example, to find $f'$ from $f''$, the integral of erfc can be written:
   
   $$g(\eta) = \int_{\eta_1=\eta}^\infty {\rm erfc}(\eta_1) { \... ...rt{\pi}}e^{-\eta_2^2} { \rm d}\eta_2\right) { \rm d}\eta_1$$
   
   
   the second equality from the definition of the complementary error function. If you change the order of integration to integrate $\eta_1$ first, you can do the first integral since the integrand does not depend on $\eta_1$. To figure out the new limits of integration, simply draw the original region of integration in the $\eta_1,\eta_2$ plane and look at it. This trick will give you $f'$ in terms of a single integral. Repeat the trick to find $f$ from $f'$ and identify in terms of erfc. Write the velocity field.

2. For an ideal point vortex at the origin, the velocity field is given in cylindrical coordinates $r,\theta,z$ by
   
   $$\vec v = \frac{\Gamma}{2\pi r} \hat\imath _\theta$$
   
   
   Show that the vorticity $\vec\omega=\nabla\times\vec v$ of this flow is everywhere zero. Now sketch a contour (closed curve) $C$ that loops once around the vortex at the origin, in the counter-clockwise direction. In fluid mechanics, (for any flow, not just this one), the circulation $\Gamma$ of a contour is defined as
   
   $$\bar\Gamma = \oint_C \vec v \cdot { \rm d}\vec r$$
   
   
   Here the integration starts from an arbitrary point on the contour and loops back to that point in the counter-clocwise direction. Evaluate the circulation of your contour around the vortex. Do not take a circle as contour $C$; take a square or a triangle or an arbitrary curve. Of course you know that in polar coordinates an infinitesimal change ${\rm d}\vec r$ in position is given by
   
   $${\rm d}\vec r = \hat\imath _r {\rm d}r + \hat\imath _\theta r {\rm d}\theta$$
   
   
   (If not, you better also figure out what it is in spherical.) You should find that $\Gamma$ has a nonzero value for your contour.

3. So far so good. But the Stokes theorem of Calculus III says
   
   $$\oint \vec v \cdot { \rm d}\vec r = \int_A \nabla \times \vec v \cdot \vec n { \rm d}A$$
   
   
   where $A$ is an area bounded by contour $C$. You just showed that the left hand side in this equation is not zero, but that the right hand side is because $\nabla\times\vec v$ is. Something is horribly wrong???! To figure out what is going on, instead of using an ideal vortex, use the Oseen vortex from your notes. To simplify this, now take your contour C to be (the perimeter of) a circle around the origin in the $x,y$-plane, and take area $A$ to be the inside of that circle in the $x,y$-plane. Do both the contour integral and the area integral. In this case, they should indeed be equal. Now in the limit $t\downarrow 0^+$, the Oseen vortex becomes an ideal vortex. So if you look at a very small time, you should be able to figure out what goes wrong for the ideal vortex with the Stokes theorem. You might want to plot the vorticity versus $r$ for a few times that become smaller and smaller. Based on that, explain what goes wrong for $t\downarrow 0^+$. Is the area integral of the ideal vortex really zero?

4. Do bathtub vortices have opposite spin in the southern hemisphere as they have in the northern one? Derive some ballpark number for the exit speed and angular velocity of a bathtub vortex at the north pole and one at the south pole, assuming that the bath water is initially at rest compared to the rotating earth. Use Kelvin’s theorem. Note that the theorem applies to an inertial frame, not that of the rotating earth. So assume you look at the entire thing from a passing star ship. (But define the direction of rotation as the one someone on earth looking at the bathtub sees.) What do you conclude about the starting question? In particular, how do you explain the bathtub vortices that we observe?

5. Consider a two-dimensional cylindrical balloon of radius $R$ surrounded by an incompressible fluid with an ideal vortex flow field. If we lower pressure inside the balloon, its radius decreases. Then there is also an ideal sink flow field proportional to $-\dot R$. The complete ideal flow field is then:
   
   $$\vec v = \frac{\bar\Gamma}{2\pi r} \hat\imath _\theta + \frac{R \dot R}{r} \hat\imath _r$$
   
   
   Now if this is a Newtonian fuid, over time viscous boundary layers would develop around the surface of the balloon that would propagate outwards and the rotational motion would slow down. But suppose we apply just enough of an axial moment on the balloon to keep it rotating with the ideal flow fluid velocity? Then we have a viscous no-slip flow around a body that is also an ideal one, i.e. an irrotational flow, i.e. one with zero vorticity. Sounds interesting?
   1. Is the above flow indeed irrotational?
   2. Integrate the circulation along a circular fluid contour around the cylinder. What is it?
   3. Is the ring of fluid particles right at the expanding balloon surface a material contour? Why?
   4. Suppose $R$ decreases in time like $R=R_0 t_0/t$, what happens when time increases from $t_0$ to 10 $t_0$? In particular, what does the Kelvin theorem say about what velocity component of the fluid particles at the surface? And what about the other velocity component? What happens to the angular velocity $\Omega$ at which the cylinder must rotate?
   5. What is the moment per unit axial length that must be exerted on the cylinder to keep it rotating at the right speed? (Ignore the inertia of the balloon.) Does the moment become infinite when $R$ tends to zero? Should it not take more and more effort to keep the flow rotating when total dissipation $\int\varepsilon{ \rm d} V$ increases.