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\begin{document}


\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Problem Set 2}}

%{\large{\bf Due in class Wednesday Feb 7}}

\end{center}

\vspace{0.1in}
\ni
Readings: Pringle \& King 1.3--1.4, 1.6.1, 1.7.1, 1.7.3; Shu pages 20--24, 30--33, 49--50; Tritton 10.4--10.7; Landau and Lifshitz Chapter 1, section 5, 6; Movies: e.g., Eulerian/Lagrangian, Fluid Quantity and Flow, Flow Visualization

\vspace{0.1in}
\ni
{\large{\bf Problem 1.}}
The Many Guises of the Energy Equation

\ni Start from the {\it total energy equation} (Shu equation 2.32; we
derived this in class):

\begin{equation}
\frac{\partial}{\partial t} \left( \frac{\rho}{2} |u|^2  + \rho \varepsilon \right) + \frac{\partial}{\partial x_k} \left[ \frac{\rho}{2} |u|^2 u_k + u_i \left( P\delta_{ik} - \tau_{ik} \right) + \rho \varepsilon u_k + F_k \right] = -\rho u_k \frac{\partial\phi}{\partial x_k} 
\end{equation}

\ni and derive the {\it internal energy equation} (Shu equation 2.36):

\begin{equation}
\rho \frac{D\varepsilon}{Dt} = -P \nabla \cdot \vec{u} - \nabla \cdot \vec{F}_{\rm con} + \tau_{ik}\frac{\partial u_i}{\partial x_k}
\end{equation}

\ni The notation above follows that in lecture.

\vspace{0.1in}
\ni
{\large{\bf Problem 2.}} Torricelli's Tank or, ``A Hole in My Bucket''

\ni A tall cylindrical tank having radius $R_1$
is filled with water. A nozzle of radius $R_2 \ll R_1$
is opened at the bottom of the tank, and water comes pouring out.
Take the height of water in the tank to be $h$ at some time.

\ni (a) Derive an expression for the velocity with which
water emerges from the nozzle, assuming the flow is steady
and inviscid.

\ni (b) The assumption of steady flow is not justified over arbitrarily
long times, since
the water level is continuously dropping. Estimate the timescale
over which the flow can be approximated as steady.

%\vspace{0.1in}
%\ni
%{\large{\bf Problem 4.}} Steady As She Blows
%Steady radiation driven winds

\end{document}




