Navier-Stokes-ligningene

Navier–Stokes-ligningene, oppkalt etter Claude-Louise Navier og George Gabriel Stokes, er ligninger som beskriver bevegelse av viskøse væsker og gasser. Som oftest gir ligningene opphav til ikke-lineære differentialligninger som demonstrerer fenomenet kaos.

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

Kartesiske koordinater

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

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

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

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

Kontinuitetsligningen gir:

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

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:

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