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Navier-Stokes equations

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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 | continuity equation]]'' and the ''energy equation''. Usually, the term Navier-Stokes equations is used to refer to all of these equations.
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 | continuity equation]]'' and the ''energy equation''. Usually, the term Navier-Stokes equations is used to refer to all of these equations.
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The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:
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<table width="100%">
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<tr><td>
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:<math>
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\frac{\partial \rho}{\partial t} +
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\frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0
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</math>     
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</td><td width="5%">(1)</td></tr>
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<tr><td>
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:<math>
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\frac{\partial}{\partial t}\left( \rho u_i \right) +
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\frac{\partial}{\partial x_j}
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\left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0
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</math>
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</td><td>(2)</td></tr>
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<tr><td>
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:<math>
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\frac{\partial}{\partial t}\left( \rho e_0 \right) +
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\frac{\partial}{\partial x_j}
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\left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0
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</math>
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</td><td>(3)</td></tr>
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</table>
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For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
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<table width="100%">
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<tr><td>
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:<math>
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\tau_{ij} = 2 \mu S_{ij}^*
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</math>
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</td><td width="5%">(4)</td></tr>
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</table>
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Where the trace-less viscous strain-rate is defined by:
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<table width="100%">
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<tr><td>
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:<math>
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S_{ij}^* \equiv
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\frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} +
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                \frac{\partial u_j}{\partial x_i} \right) -
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                \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}
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</math>
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</td><td width="5%">(5)</td></tr>
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</table>
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The heat-flux, <math>q_j</math>, is given by Fourier's law:
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<table width="100%">
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<tr><td>
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:<math>
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q_j = -\lambda \frac{\partial T}{\partial x_j}
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    \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}
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</math>
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</td><td width="5%">(6)</td></tr>
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</table>
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Where the laminar Prandtl number <math>Pr</math> is defined by:
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<table width="100%">
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<tr><td>
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:<math>
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Pr \equiv \frac{C_p \mu}{\lambda}
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</math>
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</td><td width="5%">(7)</td></tr>
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</table>
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To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
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<table width="100%">
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<tr><td>
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:<math>
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\gamma \equiv \frac{C_p}{C_v} ~~,~~
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p = \rho R T ~~,~~
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e = C_v T ~~,~~
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C_p - C_v = R
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</math>
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</td><td width="5%">(8)</td></tr>
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</table>
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Where <math>\gamma</math>, <math>C_p</math>, <math>C_v</math> and <math>R</math> are constant.
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The total energy <math>e_0</math> is defined by:
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<table width="100%">
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<tr><td>
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:<math>
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e_0 \equiv e + \frac{u_k u_k}{2}
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</math>
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</td><td width="5%">(9)</td></tr>
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</table>
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Note that the corresponding expression (15) for [[Favre_averaged_Navier-Stokes_equations | Favre averaged turbulent flows]] contains an extra term related to the turbulent energy.
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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.
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==Existence and Uniqueness==
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==External Links==
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*[http://www.navier-stokes.net Navier-Stokes.net]
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*[http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html Navier-Stokes equations at mathworld.com]
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*[http://www.claymath.org/millennium/Navier-Stokes_Equations/ Millemium Problem]

Revision as of 10:07, 8 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:

\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0
(1)
\frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0
(2)
\frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0
(3)

For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:

\tau_{ij} = 2 \mu S_{ij}^*
(4)

Where the trace-less viscous strain-rate is defined by:

S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}
(5)

The heat-flux, q_j, is given by Fourier's law:

q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}
(6)

Where the laminar Prandtl number Pr is defined by:

Pr \equiv \frac{C_p \mu}{\lambda}
(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:

\gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ e = C_v T ~~,~~ C_p - C_v = R
(8)

Where \gamma, C_p, C_v and R are constant.

The total energy e_0 is defined by:

e_0 \equiv e + \frac{u_k u_k}{2}
(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 \gamma, Pr, \mu and perhaps R, form a closed set of partial differential equations, and need only be complemented with boundary conditions.

Existence and Uniqueness

External Links

Retrieved from "https://openfoam.cfd-online.com/Wiki/Navier-Stokes_equations"
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