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\begin{center}
{\bf EGN 5456 \hfill Computational Mechanics \hfill 10/27/97}\\
{\it Open book \hfill Van Dommelen \hfill 1:05-2:20pm}
\end{center}

Show all reasoning and intermediate results leading to your answer.

\begin{enumerate}

\item Consider steady heat conduction in a plate with the
elliptical boundary $x^2 + y^2/4 = 1$.  The temperature
on the elliptical boundary is $T=x$.  Draw the isotherms
$T=0, \pm 0.25, \pm 0.5, \pm 0.75$ inside the plate.
Now assume that we increase the boundary temperature
on the right side of the boundary, to $T=1+x$, and reduce
it on the left side of the boundary, to $T=-1+x$.
Sketch the isotherms $T= 0, \pm 0.5, \pm 1, \pm 1.5$
inside the plate.

\item Consider entropy convection inside a pipe bend into a closed
ring of unit radius. The nonconducting inviscid entropy equation is
$s_t + u s_x = 0$. Assume $u$ is a constant velocity.  Show that a
solution to this problem is
\begin{displaymath}
    s = \sum_{k=-\infty}^\infty C_k e^{ikx} e^{-ikut},
\end{displaymath}
where $i=\sqrt{-1}$. What determines the constants $C_k$?
Why is this form of solution with complex exponentials
$e^{ikx}$ more convenient than the real form
\begin{displaymath}
    s = \sum_{k=-\infty}^\infty A_k(t) \cos(kx) + B_k(t) \sin(kx),
\end{displaymath}
aside from the fact that there are now two different types of terms?

\item The one-dimensional inviscid momentum equation for flow
in a pipe is
\begin{displaymath}
    (\rho u)_t + (\rho u^2)_x + p_x = 0.
\end{displaymath}
From this equation, derive a conservation law for an
arbitrary piece of pipe with moving end points.  From it,
show that you can get the momentum equation in the
way expressed in your physics classes.

\item Consider one-dimensional inviscid nonconducting
isentropic flow in a pipe. The governing equations
can be expressed in terms of the flow velocity $\bar u$,
the speed of sound $\bar a$, and the ratio of specific heats
$\gamma=1.4$ as:
\begin{displaymath}
  \begin{array}{rcl}
    \bar a_t + \bar u \bar a_x + \frac{\gamma-1}2 \bar a \bar u_x & = & 0 \\
    \bar u_t + \bar u \bar u_x + \frac2{\gamma-1} \bar a \bar a_x & = & 0
  \end{array}
\end{displaymath}
To simplify, we will linearize this to
\begin{displaymath}
  \begin{array}{rcl}
    {a}_t + u_0 {a}_x + \frac{\gamma-1}2 a_0 {u}_x & = & 0 \\
    {u}_t + u_0 {u}_x + \frac2{\gamma-1} a_0 {a}_x & = & 0
  \end{array}
\end{displaymath}
where $u_0$ and $a_0$ are now constant reference values of the
velocity and speed of sound, and $u$ and $a$ are the perturbation
velocity and speed of sound to be found. Classify this system of
equations for $u$ and $a$.  Find the Riemannian invariants and solve
if the initial velocity $u(x,0)=\sin(x)$ and the speed of
sound $a(x,0)=0$.

\end{enumerate}

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