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{\bf CME306 / CS205B Theory Homework 8}
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\section*{Euler equations}
For incompressible flow the inviscid 1D Euler equations decouple to:
\begin{align*}
  \rho_t + u\rho_x = 0 \\
  u_t + \frac{p_x}{\rho} = 0 \\
  e_t + ue_x = 0
\end{align*}

The 3D Euler equations are given by
\begin{equation}
  \begin{pmatrix}
    \rho \\ \rho u \\ \rho v \\ \rho w \\ E
  \end{pmatrix}_t + 
  \begin{pmatrix}
    \rho u \\ \rho u^2 + p \\ \rho uv \\ \rho uw \\ (E+p)u
  \end{pmatrix}_x + 
  \begin{pmatrix}
    \rho v \\ \rho uv \\ \rho v^2 + p \\ \rho vw \\ (E+p)v
  \end{pmatrix}_y + 
  \begin{pmatrix}
    \rho w \\ \rho uw \\ \rho vw \\ \rho w^2 + p \\ (E+p)w
  \end{pmatrix}_z = 0
\end{equation}
where $\rho$ is the density, $\u = (u,v,w)$ are the velocities, $E$ is the total energy per unit volume and $p$ is the pressure.  The total energy is the sum of the internal energy and
the kinetic energy.
\begin{align*}
  E &= \rho\left(e + \frac{1}{2}\|\u\|^2\right) \\
  &= \rho e + \rho(u^2 + v^2 + w^2)/2
\end{align*}
where $e$ is the internal energy per unit mass.  The assumption of incompressiblity gives
\begin{equation}
  \nabla\cdot\u = u_x + v_y + w_z = 0,
\end{equation}
Show that in 3D the inviscid Euler equations with the assumption of incompressible flow decouple to:
\begin{align*}
  \rho_t+\u\cdot{\nabla}\rho &=0 \\
  u_t+\u†\cdot{\nabla}u+\frac{p_x}{\rho} &=0 \\
  v_t+\u\cdot{\nabla}v+\frac{p_y}{\rho} &=0 \\
  w_t+\u\cdot{\nabla}w+\frac{p_z}{\rho} &=0\\
  e_t+\u\cdot{\nabla}e &=0
\end{align*}

\pagebreak
\section*{Compressible Flow}
Find the Jacobian and the right eigenvectors for Euler's equations in 1-D, {\em (hint: it is useful, in the calculation of the eigenvectors, to consider the enthalpy $H = \frac{E + p}{\rho}$, and the sound speed $c = \sqrt{\frac{\gamma p}{\rho}}$).
\begin{equation}
  \begin{pmatrix}
    \rho \\ \rho u \\ E
  \end{pmatrix}_t + 
  \begin{pmatrix}
    \rho u \\ \rho u^2 + p \\ Eu + pu
  \end{pmatrix}_x = 0.
\end{equation}
You should assume the ideal gas law as your equation of state,
\begin{equation}
  p(\rho,e) = (\gamma - 1)\rho e.
\end{equation}

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