# Difference between revisions of "Navier-Stokes equations"

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where <math>\mathbf{f} </math> is a volumetric force (e.g. <math>\rho g</math> for gravity), and <math>\mathbb{T} </math> is the stress tensor. | where <math>\mathbf{f} </math> is a volumetric force (e.g. <math>\rho g</math> for gravity), and <math>\mathbb{T} </math> is the stress tensor. | ||

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+ | ==Stress== | ||

The vector quantity <math> \nabla \cdot\mathbb{T} </math> is the ''shear stress''. For a Newtonian incompressible fluid, | The vector quantity <math> \nabla \cdot\mathbb{T} </math> is the ''shear stress''. For a Newtonian incompressible fluid, |

## Revision as of 17:25, 17 May 2010

## 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: