Navier-Stokes-ligningene

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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

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

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

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:

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