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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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/5fa4903686afab8aa3af657b9b70be2d89db0c52)