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\title{Homework 6}

\markboth{CS 237C, Spring 2006} {Shell \MakeLowercase{\textit{et
al.}}: Bare Demo of IEEEtran.cls for Journals} \maketitle

\section{}
\noindent Compute the Jacobian for the $1D$ Euler equations presented in class. You may assume the ideal gas equation of state given by
$p(\rho,e) = (\gamma-1) \rho e$.

\section{}
\noindent We show in class that for 1D incompressible flow the inviscid 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*}

\noindent The $3D$ Euler equations are given by
\begin{eqnarray*}
\left( \begin{array}{c}
\rho \\ \rho u \\ \rho v \\ \rho w \\ E \\ 
\end{array} \right)_t +
\left( \begin{array}{c}
\rho u \\ \rho u^2 + p \\ \rho u v \\ \rho u w \\ (E+p) u \\
\end{array} \right)_x +
\left( \begin{array}{c}
\rho v \\ \rho u v \\ \rho v^2 + p \\ \rho v w \\ (E+p) v \\
\end{array} \right)_y+
\left( \begin{array}{c}
\rho w \\ \rho u w \\ \rho v w \\ \rho w^2 + p \\ (E+p) w \\
\end{array} \right)_z = 0
\label{3D_Euler_equations}
\end{eqnarray*}
where $ \rho $ is the density, $ \mathbf{v} = (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} \left \Vert \textbf{v} \right \Vert ^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 incompressibility gives
\begin{eqnarray*}
\nabla \cdot \textbf{v} = u_x + v_y + w_z = 0.
\end{eqnarray*}
Show that in 3D the inviscid Euler equations with the assumption of incompressible flow decouple to:
\begin{align*}
\rho_t+\mathbf{v}\cdot{\nabla}\rho &=0 \\
u_t+\mathbf{v}\cdot{\nabla}u+\frac{p_x}{\rho} &=0 \\
v_t+\mathbf{v}\cdot{\nabla}v+\frac{p_y}{\rho} &=0 \\
w_t+\mathbf{v}\cdot{\nabla}w+\frac{p_z}{\rho} &=0\\
e_t+\mathbf{v}\cdot{\nabla}e &=0
\end{align*}
where $\mathbf{v}=\left(\begin{array}{c}u \\ v \\ w\end{array}\right)$.

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