# Difference between revisions of "Navier-Stokes equations"

m (New section) |
(Other form for the mom. eq.) |
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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. | ||

+ | 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 <math>\alpha=1,2,3</math>: | ||

+ | |||

+ | :<math> \frac{\partial \rho v_\alpha}{\partial t} + | ||

+ | \nabla \cdot {\rho v_\alpha \mathbf{v}} = | ||

+ | -\frac{\partial p}{\partial x_\alpha} + | ||

+ | \sum_\beta \frac{\partial }{\partial x_\beta} \mathbb{T}_{\beta\alpha} + f_\alpha. </math> | ||

+ | |||

+ | In vector form: | ||

+ | |||

+ | :<math> \frac{\partial \rho v_\alpha}{\partial t} + | ||

+ | \nabla \cdot {\rho \mathbf{v} \mathbf{v}} = | ||

+ | -\nabla p + \nabla\cdot\mathbb{T} + \mathbf{f}. </math> | ||

+ | |||

+ | The term <math> \mathbf{v} \mathbf{v} </math> is a dyad (direct tensor product). | ||

==Stress== | ==Stress== |

## Latest revision as of 17:32, 17 May 2010

## 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: