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

{\displaystyle \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 {\displaystyle \nabla \cdot \mathbb {T} } erstattes med {\displaystyle \mu \nabla ^{2}\mathbf {v} }, der {\displaystyle \mu } er den dynamiske viskositetskonstanten for fluidet.

Kartesiske koordinater
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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}}
{\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}}
{\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}}
For en ikke-kompressibel væske gir kontinuitetsligningen:

{\displaystyle {\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*:

{\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}}
{\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 }}
{\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}}
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.}
### Kulekoordinater

{\displaystyle \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}}
{\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]}
{\displaystyle \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 }}
{\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]}
{\displaystyle \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 }}
{\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]}
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}

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