7.3.1 Differential equations

The behavior of a compressible fluid in a bounded domain of the plain can be described similarly to (7.12) and is called compressible Navier-Stokes equations :

$$\boxed{ \begin{split}& \partial_t \begin{pmatrix}\varrho \ \v... ...d \qquad \quad \text{+ initial values\qquad + B.C.} \end{split} }$$ (7.19)

In addition to the values known from section 7.2.1 we have :

$\tau_{ij}$ - components of the viscous part of the stress tensor
$\lambda$, $\mu$ - viscosity coefficients
$\Theta$ - absolute temperature
$c_v$ - specific heat at a constant volume
$k$ - heat conductivity
$\delta_{ij} \;=\; \begin{cases}1 &\; \text{if}\; i=j \\ 0 &\; \text{if}\; i\neq j \end{cases}$ - Dirac's delta function.

We assume $c_v, k, \mu > 0$ and $\lambda = -\tfrac{3}{2}\mu$.

Equations (7.22) include the Navier-Stokes (7.1) and the Euler equations (7.12) as special cases.