Navier-Stokes-ligningene

**Navier–Stokes-ligningene**, oppkalt etter Claude-Louis Navier og George Gabriel Stokes, er en ligning som beskriver bevegelse av viskøse væsker og gasser. Ligningen er en ikke-lineær, partiell differensialligning.

Vektorligningen er

\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}.

For et newtonsk fluid kan leddet  \nabla \cdot\mathbb{T}  erstattes med   \mu \nabla^2 \mathbf{v} , der  \mu  er den dynamiske viskositetskonstanten for fluidet.

## Kartesiske koordinater

Ved å skrive ut komponentene i vektorligningen over får vi følgende ligninger for impulsen i 3-D,

: \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}+ w \frac{\partial u}{\partial z}\right) =  -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) + \rho g_x

: \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}+ w \frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) + \rho g_y

: \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y}+ w \frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) + \rho g_z

For en ikke-kompressibel væske gir kontinuitetsligningen:

:{\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} = 0

### Sylinderkoordinater

Et variabelskifte på ligningssettet i kartesiske koordinater gir impulsligningene for r, θ, og *z*:

:
\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\theta}^2}{r}\right) = 
-\frac{\partial p}{\partial r} +
\mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \theta^2} + \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2} - \frac{2}{r^2}\frac{\partial u_{\theta}}{\partial \theta}\right] + \rho g_r

:
\rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + u_z \frac{\partial u_{\theta}}{\partial z} + \frac{u_r u_{\theta}}{r}\right) = 
-\frac{1}{r}\frac{\partial p}{\partial \theta} +
\mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\theta}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\theta}}{\partial \theta^2} + \frac{\partial^2 u_{\theta}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}}{r^2}\right] + \rho g_{\theta}

:
\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_z}{\partial \theta} + u_z \frac{\partial u_z}{\partial z}\right) = 
-\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_z}{\partial \theta^2} + \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z

Kontinuitetsligningen gir:

:
\frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) + 
\frac{1}{r}\frac{\partial u_\theta}{\partial \theta} + 
\frac{\partial u_z}{\partial z} = 0.

### Kulekoordinater

:
\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_r}{\partial \phi} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\phi}^2 + u_{\theta}^2}{r}\right) = -\frac{\partial p}{\partial r} + \rho g_r
::
\mu \left[
\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_r}{\partial r}\right) + 
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_r}{\partial \phi^2} + 
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_r}{\partial \theta}\right) - 
2 \frac{u_r + \frac{\partial u_{\theta}}{\partial \theta} + u_{\theta} \cot(\theta)}{r^2} + 
\frac{2}{r^2 \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi}
\right]


:
\rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\theta}}{\partial \phi} + \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_r u_{\theta} - u_{\phi}^2 \cot(\theta)}{r}\right) = -\frac{1}{r} \frac{\partial p}{\partial \theta} + \rho g_{\theta}
::
\mu \left[
\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\theta}}{\partial r}\right) + 
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\theta}}{\partial \phi^2} + 
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\theta}}{\partial \theta}\right) + 
\frac{2}{r^2} \frac{\partial u_r}{\partial \theta} - 
\frac{u_{\theta} + 2 \cos(\theta) \frac{\partial u_{\phi}}{\partial \phi}}{r^2 \sin(\theta)^2}
\right]


:
\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi} + \frac{u_{\theta}}{r} \frac{\partial u_{\phi}}{\partial \theta} + \frac{u_r u_{\phi} + u_{\phi} u_{\theta} \cot(\theta)}{r}\right) = -\frac{1}{r \sin(\theta)} \frac{\partial p}{\partial \phi} + \rho g_{\phi}
::
\mu \left[
\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\phi}}{\partial r}\right) + 
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\phi}}{\partial \phi^2} + 
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\phi}}{\partial \theta}\right) + 
\frac{2 \frac{\partial u_r}{\partial \phi} + 2 \cos(\theta) \frac{\partial u_{\theta}}{\partial \phi} - u_{\phi}}{r^2 \sin(\theta)^2}
\right]


Kontinuitetsligningen gir:

:
\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 u_r\right) + 
\frac{1}{r \sin(\theta)}\frac{\partial u_\phi}{\partial \phi} + 
\frac{1}{r \sin(\theta)}\frac{\partial}{\partial \theta}\left(\sin(\theta) u_\theta\right) = 0