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\begin{document}


\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 3}}

%{\large{\bf Due in class Wednesday Feb 14}}

\end{center}

\vspace{0.1in}
\ni
Readings: Tritton 5.8, chapter 2, chapter 8; Movies: Pressure Fields and Fluid Acceleration; whatever text you would like to read about the Taylor-Proudman Theorem (e.g., Acheson pages 279--280 works).

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}}
Running on All Cylinders

\ni Consider a {\it barotropic} star---one whose
pressure $P$ depends only its density $\rho$.
The star is self-gravitating and
rotates about a central axis. The flow is steady,
axisymmetric, and purely rotational (in the azimuthal direction only).
Furthermore, the Rossby number of the flow is small;
i.e., $u/(\Omega_0 L) \ll 1$, where the fluid velocity $u$ is evaluated
in a frame that rotates with the characteristic
angular velocity of the star $\Omega_0$,
and $L$ is a characteristic lengthscale of the flow.
The Rossby number being small means that the star is nearly (but not
exactly) rigidly rotating.

\ni Prove that for such rotating barotropes,
the angular velocity $\vec{\Omega} \equiv \Omega \hat{z}$
is constant on cylinders.
That is, $\Omega = \Omega(r)$ is a function of cylindrical
radius $r$ only, measured from the rotation axis.
(Picture the fluid to be a rotating oblate spheroid,
and intersect the spheroid with an imaginary cylinder
whose axis coincides with the spheroid's rotation axis.
Then every fluid element on the surface
of that cylinder rotates with the same angular velocity.
Cylinders of different radii will correspond
to different angular velocities [i.e., the fluid
is not necessarily in rigid body rotation. We say the body
is in {\it differential rotation} (on cylinders, in this case).]

\ni Perhaps the most straightforward way to proceed is to take
the curl of the momentum equation in a rotating frame. Be clear
about your approximations.

\ni This result is essentially a re-statement of the Taylor-Proudman theorem.
However, most textbooks that give the T-P theorem assume that the {\it fluid}
is incompressible.
That assumption is not valid here. (Also, it is
not necessary to understand the T-P theorem to do this problem.)

\ni (Helioseismology tells us that in the outer convective
shell of the Sun, the angular velocity is {\it not} constant
on cylinders, but constant rather on surfaces of constant latitude.
Explaining this observation
remains an unsolved problem. By contrast,
giant planet interiors are thought to be rotating more-or-less on
cylinders.)

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} The Venturi Tube

\ni The Venturi tube is a horizontal tube that is pinched (constricted)
in the middle. Take the cross-sectional area of the unpinched part of the
tube to be $A_1$ and the cross-sectional area of the pinched part
of the tube to be $A_2$.

\ni Atomic hydrogen gas flows steadily, inviscidly, and highly subsonically
through the tube. Furthermore, the tube is insulated so that heat is neither
lost nor gained from the tube to the outside world.

\ni In the unpinched part of the flow, the gas
density is measured to be $\rho$.

\ni The pressure difference between the unpinched and
pinched parts of the tube is measured to be $\Delta P$.

\ni Derive an expression for $\dot{m}$, the rate of mass flow
through the tube [mass per time].

\vspace{0.1in}
\ni
{\large{\bf Problem 3.}} Reynolds Numbers

\ni Estimate the Reynolds numbers characterizing

\ni (a) water flow around a bluefin tuna

\ni (b) air flow around a falling raindrop

\ni (c) gas flows in a solar photospheric granule

\end{document}




