7.2.1 Differential equations

The Euler equations for gas dynamics are

$$\boxed{ \begin{split}& \partial_t \begin{pmatrix}\varrho \ \v... ...d \qquad \quad \text{+ initial values\qquad + B.C.} \end{split} }$$ (7.9)

The meaning of the symbols is

$\varrho$ - density
$u$, $v$ - velocity in x- and y-direction
$p$ - pressure
$e$ - energy
$\gamma = c_p / c_v \enspace$.

If the density is time independent then we achieve by neglecting the 4.th equation in (7.12) the incompressible Navier-Stokes equations (7.1).

A more compact form of the equations above is ($p$ has already been substituted by the additional equation) :

$$\boxed{ \begin{split}& \partial_t \vec{u} + \partial_x f_1(\vec... ...quad \text{+ initial values\qquad + B.C.} \enspace, \end{split} }$$ (7.10)

where $\vec{u} = \begin{pmatrix}\varrho , & \varrho u , & \varrho v , & e \end{pmatrix}^T$ denotes the conservative variables.
The vector of primary variables would be $\vec{w}= \begin{pmatrix}\varrho, & u, & v, & e \end{pmatrix}^T$ .