## Incompressible Fluids

If a fluid is incompressible andis the velocity vector field that describes the velocity of the fluid at each point then

This equation follows from the continuity equation

First write the continuity equation as

Since the fluid is incompressinble

There is still the possibility that the fluid can has different densities at different points, but if the fluid is incompressible a fluid element cannot change in density as it moves so

Hence

Ifthenand the fluid is incompressible.