next up previous
Next: 8.2.1 Vorticity Method Up: 8. Hydrodynamics Previous: 8.1.3 Smoothed Particle Hydrodynamics

8.2 Incompressible Flow with Viscosity

In 8.3 we put $d \rho / dt \equiv \partial \rho + \vec{v} \cdot \nabla \rho=0$ to find
\nabla \cdot \mbox{$\bf v$}=0
\end{displaymath} (8.52)

$\Longrightarrow$Flow is source-free!

A consequence of this is that
\nabla \cdot (\nabla \mbox{$\bf v$})+\nabla \cdot (\nabla \mbox{$\bf v$})^{T}=
\nabla^{2} \mbox{$\bf v$}
\end{displaymath} (8.53)

by which the Navier-Stokes equation reduces to the form
\frac{\partial \mbox{$\bf v$}}{\partial t} + (\mbox{$\bf v$}...{$\bf v$}=
- \nabla \bar{p} + \nu \nabla^{2}\mbox{$\bf v$}
\end{displaymath} (8.54)

with $\nu \equiv \mu/\rho$ and $\bar{p}\equiv p/\rho$.

Two techniques to solve 8.52,8.54:
- vorticity method
- pressure method


Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001