# Navier-Stokes equations

## Contents

## Continuity

or, using the substantive derivative:

For an incompressible fluid, is constant, hence the velocity field must be divergence-free:

## Momentum

(Also known as *the* Navier-Stokes equation.)

or, using the substantive derivative:

where is a volumetric force (e.g. for gravity), and is the stress tensor.

## Stress

The vector quantity is the *shear stress*. For a Newtonian incompressible fluid,

with being the (dynamic) viscosity.

For an inviscid fluid, the momentum equation becomes Euler's equation for ideal fluids: