User:Eguitar62/sandbox

For a flow to be considered incompressible, the divergence of the fluid must be zeros at all points in the flow.

$\nabla \cdot \mathbf {v} =0$

By divergence theorem we can write the following

$\iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {v} \right)\,dV=$ $\scriptstyle S$ $\mathbf {v} \cdot \mathbf {n} \,dS$

Substituting into this equation

$\iiint _{V}\left(0\right)\,dV=$ $\scriptstyle S$ $\mathbf {v} \cdot \mathbf {n} \,dS$

Which implies that

$0=$ $\scriptstyle S$ $\mathbf {v} \cdot \mathbf {n} \,dS$

Also going back to earlier:

$\nabla \cdot \mathbf {v} ={\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}=0$