# Hydrodynamic Viscous Fluid Materials (LAW6)

This law is specifically designed to model liquids and gases.

The equations used to describe the material are:(1)
${S}_{ij}=2\rho v{\stackrel{˙}{e}}_{ij}$
(2)
$p={C}_{0}+{C}_{1}\mu +{C}_{2}{\mu }^{2}+{C}_{3}{\mu }^{3}+\left({C}_{4}+{C}_{5}\mu \right){E}_{n}$
Where,
${S}_{ij}$
Deviatoric stress tensor
$\stackrel{\to }{V}$
Kinematic viscosity
${\stackrel{˙}{e}}_{ij}$
Deviatoric strain rate tensor
The kinematic viscosity $\stackrel{\to }{V}$ is related to the dynamic viscosity, $\eta$ by:(3)
$v=\frac{\eta }{\rho }$

## Perfect Gas Model

To model a perfect gas, all coefficients C0, C1, C2, C3 must be set to equal zero. Also:(4)
${C}_{4}={C}_{5}=\gamma -1$
(5)
${E}_{n0}=\frac{{P}_{0}}{\gamma -1}$

A perfect gas allows compressibility and expansion and contraction with a rise in temperature. However, for many situations, especially very slow subsonic flows, an incompressible gas gives accurate and reliable results with less computation.

## Incompressible Gas Model

To model an incompressible gas, the coefficients should be set to:(6)
${C}_{0}={C}_{1}={C}_{2}={C}_{3}={C}_{4}={C}_{5}={E}_{0}=0$
(7)
${C}_{1}={\rho }_{0}\cdot {c}^{2}$
Where,
$c$
Speed of sound

Incompressibility is achieved via a penalty method. The sound speed is set to at least 10 times the maximum velocity.

This classical assumption is not valid when fluid and structures are coupled. In this case, set the sound speed in the fluid so that the first eigen frequency is at least 10 times higher in the fluid than in the structure.