Navier-Stokes-ligningene

Mal:Kildeløs 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

$ρ (\frac{\partial 𝐯}{\partial t} + 𝐯 \cdot \nabla 𝐯) = - \nabla p + \nabla \cdot 𝕋 + 𝐟 .$

For et newtonsk fluid kan leddet $\nabla \cdot 𝕋$ erstattes med $μ \nabla^{2} 𝐯$ , der $μ$ 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,

ρ (\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) = - \frac{\partial p}{\partial x} + μ (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) + ρ g_{x}

ρ (\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}) = - \frac{\partial p}{\partial y} + μ (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}) + ρ g_{y}

ρ (\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}) = - \frac{\partial p}{\partial z} + μ (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}) + ρ g_{z}

For en ikke-kompressibel væske gir kontinuitetsligningen:

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

Et variabelskifte på ligningssettet i kartesiske koordinater gir impulsligningene for r, θ, og z:

ρ (\frac{\partial u_{r}}{\partial t} + u_{r} \frac{\partial u_{r}}{\partial r} + \frac{u_{θ}}{r} \frac{\partial u_{r}}{\partial θ} + u_{z} \frac{\partial u_{r}}{\partial z} - \frac{u_{θ}^{2}}{r}) = - \frac{\partial p}{\partial r} + μ [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u_{r}}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} u_{r}}{\partial θ^{2}} + \frac{\partial^{2} u_{r}}{\partial z^{2}} - \frac{u_{r}}{r^{2}} - \frac{2}{r^{2}} \frac{\partial u_{θ}}{\partial θ}] + ρ g_{r}

ρ (\frac{\partial u_{θ}}{\partial t} + u_{r} \frac{\partial u_{θ}}{\partial r} + \frac{u_{θ}}{r} \frac{\partial u_{θ}}{\partial θ} + u_{z} \frac{\partial u_{θ}}{\partial z} + \frac{u_{r} u_{θ}}{r}) = - \frac{1}{r} \frac{\partial p}{\partial θ} + μ [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u_{θ}}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} u_{θ}}{\partial θ^{2}} + \frac{\partial^{2} u_{θ}}{\partial z^{2}} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial θ} - \frac{u_{θ}}{r^{2}}] + ρ g_{θ}

ρ (\frac{\partial u_{z}}{\partial t} + u_{r} \frac{\partial u_{z}}{\partial r} + \frac{u_{θ}}{r} \frac{\partial u_{z}}{\partial θ} + u_{z} \frac{\partial u_{z}}{\partial z}) = - \frac{\partial p}{\partial z} + μ [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u_{z}}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} u_{z}}{\partial θ^{2}} + \frac{\partial^{2} u_{z}}{\partial z^{2}}] + ρ g_{z}

Kontinuitetsligningen gir:

\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{1}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{\partial u_{z}}{\partial z} = 0 .

ρ (\frac{\partial u_{r}}{\partial t} + u_{r} \frac{\partial u_{r}}{\partial r} + \frac{u_{ϕ}}{r \sin (θ)} \frac{\partial u_{r}}{\partial ϕ} + \frac{u_{θ}}{r} \frac{\partial u_{r}}{\partial θ} - \frac{u_{ϕ}^{2} + u_{θ}^{2}}{r}) = - \frac{\partial p}{\partial r} + ρ g_{r}

μ [\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial u_{r}}{\partial r}) + \frac{1}{r^{2} \sin (θ)^{2}} \frac{\partial^{2} u_{r}}{\partial ϕ^{2}} + \frac{1}{r^{2} \sin (θ)} \frac{\partial}{\partial θ} (\sin (θ) \frac{\partial u_{r}}{\partial θ}) - 2 \frac{u_{r} + \frac{\partial u_{θ}}{\partial θ} + u_{θ} \cot (θ)}{r^{2}} + \frac{2}{r^{2} \sin (θ)} \frac{\partial u_{ϕ}}{\partial ϕ}]

ρ (\frac{\partial u_{θ}}{\partial t} + u_{r} \frac{\partial u_{θ}}{\partial r} + \frac{u_{ϕ}}{r \sin (θ)} \frac{\partial u_{θ}}{\partial ϕ} + \frac{u_{θ}}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{u_{r} u_{θ} - u_{ϕ}^{2} \cot (θ)}{r}) = - \frac{1}{r} \frac{\partial p}{\partial θ} + ρ g_{θ}

μ [\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial u_{θ}}{\partial r}) + \frac{1}{r^{2} \sin (θ)^{2}} \frac{\partial^{2} u_{θ}}{\partial ϕ^{2}} + \frac{1}{r^{2} \sin (θ)} \frac{\partial}{\partial θ} (\sin (θ) \frac{\partial u_{θ}}{\partial θ}) + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial θ} - \frac{u_{θ} + 2 \cos (θ) \frac{\partial u_{ϕ}}{\partial ϕ}}{r^{2} \sin (θ)^{2}}]

ρ (\frac{\partial u_{ϕ}}{\partial t} + u_{r} \frac{\partial u_{ϕ}}{\partial r} + \frac{u_{ϕ}}{r \sin (θ)} \frac{\partial u_{ϕ}}{\partial ϕ} + \frac{u_{θ}}{r} \frac{\partial u_{ϕ}}{\partial θ} + \frac{u_{r} u_{ϕ} + u_{ϕ} u_{θ} \cot (θ)}{r}) = - \frac{1}{r \sin (θ)} \frac{\partial p}{\partial ϕ} + ρ g_{ϕ}

μ [\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial u_{ϕ}}{\partial r}) + \frac{1}{r^{2} \sin (θ)^{2}} \frac{\partial^{2} u_{ϕ}}{\partial ϕ^{2}} + \frac{1}{r^{2} \sin (θ)} \frac{\partial}{\partial θ} (\sin (θ) \frac{\partial u_{ϕ}}{\partial θ}) + \frac{2 \frac{\partial u_{r}}{\partial ϕ} + 2 \cos (θ) \frac{\partial u_{θ}}{\partial ϕ} - u_{ϕ}}{r^{2} \sin (θ)^{2}}]

Kontinuitetsligningen gir:

\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} u_{r}) + \frac{1}{r \sin (θ)} \frac{\partial u_{ϕ}}{\partial ϕ} + \frac{1}{r \sin (θ)} \frac{\partial}{\partial θ} (\sin (θ) u_{θ}) = 0