Denne artikkelen mangler
kildehenvisninger, og opplysningene i den kan dermed være vanskelige å
verifisere. Kildeløst materiale kan bli
fjernet. Helt uten kilder. (10. okt. 2015)
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
For et newtonsk fluid kan leddet
erstattes med
, der
er den dynamiske viskositetskonstanten for fluidet.
Ved å skrive ut komponentene i vektorligningen over får vi følgende ligninger for impulsen i 3-D,
![{\displaystyle \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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c41a27a7b1f5a788a0099be1d46b1f4333a1dd)
![{\displaystyle \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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f6352d522e9bcb684d4673778dc2fbf49a5acc)
![{\displaystyle \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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d950081b9635ee068c7e380bf349ad9923a0c441)
For en ikke-kompressibel væske gir kontinuitetsligningen:
![{\displaystyle {\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb75881d8b51313934cb45d6d59dd8e2a4cdb09)
Et variabelskifte på ligningssettet i kartesiske koordinater gir impulsligningene for r, θ, og z:
![{\displaystyle \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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8cb65ed9b00e87b23784dab8103a2168f1302b)
![{\displaystyle \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 ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/c654c4bdd92031e5b1f7776f301ee55491a8283b)
![{\displaystyle \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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/957fabebb8da99d88d99927044d37d8c1a441b9e)
Kontinuitetsligningen gir:
![{\displaystyle {\frac {1}{r)){\frac {\partial }{\partial r))\left(ru_{r}\right)+{\frac {1}{r)){\frac {\partial u_{\theta )){\partial \theta ))+{\frac {\partial u_{z)){\partial z))=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06a929deebefcaea8a95ee5b54a6fd6f67053ca)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf5ba424b55eb68f83b3d8d6465c79a88684bfa)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad9ba530919fe3e533c8cf1d92c9d0faa15b1e9)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094cd5c34f2ea5826c69373777e7a631b3a1a3b7)
Kontinuitetsligningen gir:
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa4903686afab8aa3af657b9b70be2d89db0c52)