# Navier-Stokes equations

## Contents

## Continuity[edit]

or, using the substantive derivative:

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

## Momentum[edit]

(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.

Another form of the equation, more similar in form to the continuity equation, stresses the fact that the **momentum density** is conserved. For each of the three Cartesian coordinates :

In vector form:

The term is a dyad (direct tensor product).

## Stress[edit]

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: