Navier-Stokes equations
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Equations (1)-(9), supplemented with gas data for <math>\gamma</math>, <math>Pr</math>, <math>\mu</math> and perhaps <math>R</math>, form a closed set of partial differential equations, and need only be complemented with boundary conditions. | Equations (1)-(9), supplemented with gas data for <math>\gamma</math>, <math>Pr</math>, <math>\mu</math> and perhaps <math>R</math>, form a closed set of partial differential equations, and need only be complemented with boundary conditions. | ||
- | == | + | == Boundary conditions == |
- | ==External | + | ==Existence and uniqueness== |
+ | |||
+ | ==External links== | ||
*[http://www.navier-stokes.net Navier-Stokes.net] | *[http://www.navier-stokes.net Navier-Stokes.net] | ||
*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com] | *[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com] | ||
*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem] | *[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millenium Problem] |
Revision as of 04:14, 14 September 2005
The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newtons Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations.
The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:
| (1) |
| (2) |
| (3) |
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
| (4) |
Where the trace-less viscous strain-rate is defined by:
| (5) |
The heat-flux, , is given by Fourier's law:
| (6) |
Where the laminar Prandtl number is defined by:
| (7) |
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
| (8) |
Where , , and are constant.
The total energy is defined by:
| (9) |
Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.
Equations (1)-(9), supplemented with gas data for , , and perhaps , form a closed set of partial differential equations, and need only be complemented with boundary conditions.