Navier-Stokes-ligningene

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
        
          (
          
            
              
                
                  ∂
                  
                    v
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              v
            
            ⋅
            ∇
            
              v
            
          
          )
        
        =
        −
        ∇
        p
        +
        ∇
        ⋅
        
          T
        
        +
        
          f
        
        .
      
    
    {\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 
  
    
      
        ∇
        ⋅
        
          T
        
      
    
    {\displaystyle \nabla \cdot \mathbb {T} }
   erstattes med 
  
    
      
        μ
        
          ∇
          
            2
          
        
        
          v
        
      
    
    {\displaystyle \mu \nabla ^{2}\mathbf {v} }
  , der 
  
    
      
        μ
      
    
    {\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,

  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  u
                
                
                  ∂
                  t
                
              
            
            +
            u
            
              
                
                  ∂
                  u
                
                
                  ∂
                  x
                
              
            
            +
            v
            
              
                
                  ∂
                  u
                
                
                  ∂
                  y
                
              
            
            +
            w
            
              
                
                  ∂
                  u
                
                
                  ∂
                  z
                
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              x
            
          
        
        +
        μ
        
          (
          
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  u
                
                
                  ∂
                  
                    x
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  u
                
                
                  ∂
                  
                    y
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  u
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
          
          )
        
        +
        ρ
        
          g
          
            x
          
        
      
    
    {\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}}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  v
                
                
                  ∂
                  t
                
              
            
            +
            u
            
              
                
                  ∂
                  v
                
                
                  ∂
                  x
                
              
            
            +
            v
            
              
                
                  ∂
                  v
                
                
                  ∂
                  y
                
              
            
            +
            w
            
              
                
                  ∂
                  v
                
                
                  ∂
                  z
                
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              y
            
          
        
        +
        μ
        
          (
          
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  v
                
                
                  ∂
                  
                    x
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  v
                
                
                  ∂
                  
                    y
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  v
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
          
          )
        
        +
        ρ
        
          g
          
            y
          
        
      
    
    {\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}}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  w
                
                
                  ∂
                  t
                
              
            
            +
            u
            
              
                
                  ∂
                  w
                
                
                  ∂
                  x
                
              
            
            +
            v
            
              
                
                  ∂
                  w
                
                
                  ∂
                  y
                
              
            
            +
            w
            
              
                
                  ∂
                  w
                
                
                  ∂
                  z
                
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              z
            
          
        
        +
        μ
        
          (
          
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  w
                
                
                  ∂
                  
                    x
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  w
                
                
                  ∂
                  
                    y
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  w
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
          
          )
        
        +
        ρ
        
          g
          
            z
          
        
      
    
    {\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:

  
    
      
        
          
            
              ∂
              u
            
            
              ∂
              x
            
          
        
        +
        
          
            
              ∂
              v
            
            
              ∂
              y
            
          
        
        +
        
          
            
              ∂
              w
            
            
              ∂
              z
            
          
        
        =
        0
      
    
    {\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:

  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  θ
                
              
            
            +
            
              u
              
                z
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  z
                
              
            
            −
            
              
                
                  u
                  
                    θ
                  
                  
                    2
                  
                
                r
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              r
            
          
        
        +
        μ
        
          [
          
            
              
                1
                r
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                r
                
                  
                    
                      ∂
                      
                        u
                        
                          r
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  
                    θ
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
            −
            
              
                
                  u
                  
                    r
                  
                
                
                  r
                  
                    2
                  
                
              
            
            −
            
              
                2
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  θ
                
              
            
          
          ]
        
        +
        ρ
        
          g
          
            r
          
        
      
    
    {\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}}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  θ
                
              
            
            +
            
              u
              
                z
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  z
                
              
            
            +
            
              
                
                  
                    u
                    
                      r
                    
                  
                  
                    u
                    
                      θ
                    
                  
                
                r
              
            
          
          )
        
        =
        −
        
          
            1
            r
          
        
        
          
            
              ∂
              p
            
            
              ∂
              θ
            
          
        
        +
        μ
        
          [
          
            
              
                1
                r
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                r
                
                  
                    
                      ∂
                      
                        u
                        
                          θ
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  
                    θ
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
            +
            
              
                2
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  θ
                
              
            
            −
            
              
                
                  u
                  
                    θ
                  
                
                
                  r
                  
                    2
                  
                
              
            
          
          ]
        
        +
        ρ
        
          g
          
            θ
          
        
      
    
    {\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 }}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  θ
                
              
            
            +
            
              u
              
                z
              
            
            
              
                
                  ∂
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  z
                
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              z
            
          
        
        +
        μ
        
          [
          
            
              
                1
                r
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                r
                
                  
                    
                      ∂
                      
                        u
                        
                          z
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  
                    θ
                    
                      2
                    
                  
                
              
            
            +
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      z
                    
                  
                
                
                  ∂
                  
                    z
                    
                      2
                    
                  
                
              
            
          
          ]
        
        +
        ρ
        
          g
          
            z
          
        
      
    
    {\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:

  
    
      
        
          
            1
            r
          
        
        
          
            ∂
            
              ∂
              r
            
          
        
        
          (
          
            r
            
              u
              
                r
              
            
          
          )
        
        +
        
          
            1
            r
          
        
        
          
            
              ∂
              
                u
                
                  θ
                
              
            
            
              ∂
              θ
            
          
        
        +
        
          
            
              ∂
              
                u
                
                  z
                
              
            
            
              ∂
              z
            
          
        
        =
        0.
      
    
    {\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 ===

  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    ϕ
                  
                
                
                  r
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  ϕ
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  θ
                
              
            
            −
            
              
                
                  
                    u
                    
                      ϕ
                    
                    
                      2
                    
                  
                  +
                  
                    u
                    
                      θ
                    
                    
                      2
                    
                  
                
                r
              
            
          
          )
        
        =
        −
        
          
            
              ∂
              p
            
            
              ∂
              r
            
          
        
        +
        ρ
        
          g
          
            r
          
        
      
    
    {\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}}
  

  
    
      
        μ
        
          [
          
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                
                  r
                  
                    2
                  
                
                
                  
                    
                      ∂
                      
                        u
                        
                          r
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  
                    )
                    
                      2
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  
                    ϕ
                    
                      2
                    
                  
                
              
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                ∂
                
                  ∂
                  θ
                
              
            
            
              (
              
                sin
                ⁡
                (
                θ
                )
                
                  
                    
                      ∂
                      
                        u
                        
                          r
                        
                      
                    
                    
                      ∂
                      θ
                    
                  
                
              
              )
            
            −
            2
            
              
                
                  
                    u
                    
                      r
                    
                  
                  +
                  
                    
                      
                        ∂
                        
                          u
                          
                            θ
                          
                        
                      
                      
                        ∂
                        θ
                      
                    
                  
                  +
                  
                    u
                    
                      θ
                    
                  
                  cot
                  ⁡
                  (
                  θ
                  )
                
                
                  r
                  
                    2
                  
                
              
            
            +
            
              
                2
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  ϕ
                
              
            
          
          ]
        
      
    
    {\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]}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    ϕ
                  
                
                
                  r
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  ϕ
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  θ
                
              
            
            +
            
              
                
                  
                    u
                    
                      r
                    
                  
                  
                    u
                    
                      θ
                    
                  
                  −
                  
                    u
                    
                      ϕ
                    
                    
                      2
                    
                  
                  cot
                  ⁡
                  (
                  θ
                  )
                
                r
              
            
          
          )
        
        =
        −
        
          
            1
            r
          
        
        
          
            
              ∂
              p
            
            
              ∂
              θ
            
          
        
        +
        ρ
        
          g
          
            θ
          
        
      
    
    {\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 }}
  

  
    
      
        μ
        
          [
          
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                
                  r
                  
                    2
                  
                
                
                  
                    
                      ∂
                      
                        u
                        
                          θ
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  
                    )
                    
                      2
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      θ
                    
                  
                
                
                  ∂
                  
                    ϕ
                    
                      2
                    
                  
                
              
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                ∂
                
                  ∂
                  θ
                
              
            
            
              (
              
                sin
                ⁡
                (
                θ
                )
                
                  
                    
                      ∂
                      
                        u
                        
                          θ
                        
                      
                    
                    
                      ∂
                      θ
                    
                  
                
              
              )
            
            +
            
              
                2
                
                  r
                  
                    2
                  
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      r
                    
                  
                
                
                  ∂
                  θ
                
              
            
            −
            
              
                
                  
                    u
                    
                      θ
                    
                  
                  +
                  2
                  cos
                  ⁡
                  (
                  θ
                  )
                  
                    
                      
                        ∂
                        
                          u
                          
                            ϕ
                          
                        
                      
                      
                        ∂
                        ϕ
                      
                    
                  
                
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  
                    )
                    
                      2
                    
                  
                
              
            
          
          ]
        
      
    
    {\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]}
  
  
    
      
        ρ
        
          (
          
            
              
                
                  ∂
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  t
                
              
            
            +
            
              u
              
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  r
                
              
            
            +
            
              
                
                  u
                  
                    ϕ
                  
                
                
                  r
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                
                  ∂
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  ϕ
                
              
            
            +
            
              
                
                  u
                  
                    θ
                  
                
                r
              
            
            
              
                
                  ∂
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  θ
                
              
            
            +
            
              
                
                  
                    u
                    
                      r
                    
                  
                  
                    u
                    
                      ϕ
                    
                  
                  +
                  
                    u
                    
                      ϕ
                    
                  
                  
                    u
                    
                      θ
                    
                  
                  cot
                  ⁡
                  (
                  θ
                  )
                
                r
              
            
          
          )
        
        =
        −
        
          
            1
            
              r
              sin
              ⁡
              (
              θ
              )
            
          
        
        
          
            
              ∂
              p
            
            
              ∂
              ϕ
            
          
        
        +
        ρ
        
          g
          
            ϕ
          
        
      
    
    {\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 }}
  

  
    
      
        μ
        
          [
          
            
              
                1
                
                  r
                  
                    2
                  
                
              
            
            
              
                ∂
                
                  ∂
                  r
                
              
            
            
              (
              
                
                  r
                  
                    2
                  
                
                
                  
                    
                      ∂
                      
                        u
                        
                          ϕ
                        
                      
                    
                    
                      ∂
                      r
                    
                  
                
              
              )
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  
                    )
                    
                      2
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      2
                    
                  
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  ∂
                  
                    ϕ
                    
                      2
                    
                  
                
              
            
            +
            
              
                1
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  )
                
              
            
            
              
                ∂
                
                  ∂
                  θ
                
              
            
            
              (
              
                sin
                ⁡
                (
                θ
                )
                
                  
                    
                      ∂
                      
                        u
                        
                          ϕ
                        
                      
                    
                    
                      ∂
                      θ
                    
                  
                
              
              )
            
            +
            
              
                
                  2
                  
                    
                      
                        ∂
                        
                          u
                          
                            r
                          
                        
                      
                      
                        ∂
                        ϕ
                      
                    
                  
                  +
                  2
                  cos
                  ⁡
                  (
                  θ
                  )
                  
                    
                      
                        ∂
                        
                          u
                          
                            θ
                          
                        
                      
                      
                        ∂
                        ϕ
                      
                    
                  
                  −
                  
                    u
                    
                      ϕ
                    
                  
                
                
                  
                    r
                    
                      2
                    
                  
                  sin
                  ⁡
                  (
                  θ
                  
                    )
                    
                      2
                    
                  
                
              
            
          
          ]
        
      
    
    {\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:

  
    
      
        
          
            1
            
              r
              
                2
              
            
          
        
        
          
            ∂
            
              ∂
              r
            
          
        
        
          (
          
            
              r
              
                2
              
            
            
              u
              
                r
              
            
          
          )
        
        +
        
          
            1
            
              r
              sin
              ⁡
              (
              θ
              )
            
          
        
        
          
            
              ∂
              
                u
                
                  ϕ
                
              
            
            
              ∂
              ϕ
            
          
        
        +
        
          
            1
            
              r
              sin
              ⁡
              (
              θ
              )
            
          
        
        
          
            ∂
            
              ∂
              θ
            
          
        
        
          (
          
            sin
            ⁡
            (
            θ
            )
            
              u
              
                θ
              
            
          
          )
        
        =
        0
      
    
    {\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}