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

$$\nabla \cdot \mbox{$\bf v$}=0$$ (8.52)

$\Longrightarrow$Flow is source-free!

A consequence of this is that

$$\nabla \cdot (\nabla \mbox{$\bf v$})+\nabla \cdot (\nabla \mbox{$\bf v$})^{T}= \nabla^{2} \mbox{$\bf v$}$$ (8.53)

by which the Navier-Stokes equation reduces to the form

$$\frac{\partial \mbox{$\bf v$}}{\partial t} + (\mbox{$\bf v$}... ...box{$\bf v$}= - \nabla \bar{p} + \nu \nabla^{2}\mbox{$\bf v$}$$ (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