r:ρ(
∂t
∂u
r
​
​ +u
r
​
∂r
∂u
r
​
​ +
rsinθ
u
ϕ
​
​
∂ϕ
∂u
r
​
​ +
r
u
θ
​
​
∂θ
∂u
r
​
​ −
r
u
ϕ
2
​ +u
θ
2
​
​ )=
displaystyle ; ; ; ; -frac{partial p}{partial r}+rho g_{r}+muleft[frac{1}{r^{2}}frac{partial}{partial r}left(r^{2}frac{partial u_{r}}{partial r}right)+frac{1}{r^{2}sin^{2}theta}frac{partial^{2}u_{r}}{partialphi^{2}}+frac{1}{r^{2}sintheta}frac{partial}{partialtheta}left(sinthetafrac{partial u_{r}}{partialtheta}right)-2frac{u_{r}+frac{partial u_{theta}}{partialtheta}+u_{theta}cottheta}{r^{2}}-frac{2}{r^{2}sintheta}frac{partial u_{phi}}{partialphi}right]−
∂r
∂p
​ +ρg
r
​ +μ[
r
2

1
​
∂r
∂
​ (r
2

∂r
∂u
r
​
​ )+
r
2
sin
2
θ
1
​
∂ϕ
2

∂
2
u
r
​
​ +
r
2
sinθ
1
​
∂θ
∂
​ (sinθ
∂θ
∂u
r
​
​ )−2
r
2

u
r
​ +
∂θ
∂u
θ
​
​ +u
θ
​ cotθ
​ −
r
2
sinθ
2
​
∂ϕ
∂u
ϕ
​
​ ]

displaystyle phi: rholeft(frac{partial u_{phi}}{partial t}+u_{r}frac{partial u_{phi}}{partial r}+frac{u_{phi}}{rsintheta}frac{partial u_{phi}}{partialphi}+frac{u_{theta}}{r}frac{partial u_{phi}}{partialtheta}+frac{u_{r}u_{phi}+u_{phi}u_{theta}cottheta}{r}right)=ϕ:ρ(
∂t
∂u
ϕ
​
​ +u
r
​
∂r
∂u
ϕ
​
​ +
rsinθ
u
ϕ
​
​
∂ϕ
∂u
ϕ
​
​ +
r
u
θ
​
​
∂θ
∂u
ϕ
​
​ +
r
u
r
​ u
ϕ
​ +u
ϕ
​ u
θ
​ cotθ
​ )=
displaystyle ; ; ; ; -frac{1}{rsintheta}frac{partial p}{partialphi}+rho g_{phi}+muleft[frac{1}{r^{2}}frac{partial}{partial r}left(r^{2}frac{partial u_{phi}}{partial r}right)+frac{1}{r^{2}sin^{2}theta}frac{partial^{2}u_{phi}}{partialphi^{2}}+frac{1}{r^{2}sintheta}frac{partial}{partialtheta}left(sinthetafrac{partial u_{phi}}{partialtheta}right)+frac{2sinthetafrac{partial u_{r}}{partialphi}+2costhetafrac{partial u_{theta}}{partialphi}-u_{phi}}{r^{2}sin^{2}theta}right]−
rsinθ
1
​
∂ϕ
∂p
​ +ρg
ϕ
​ +μ[
r
2

1
​
∂r
∂
​ (r
2

∂r
∂u
ϕ
​
​ )+
r
2
sin
2
θ
1
​
∂ϕ
2

∂
2
u
ϕ
​
​ +
r
2
sinθ
1
​
∂θ
∂
​ (sinθ
∂θ
∂u
ϕ
​
​ )+
r
2
sin
2
θ
2sinθ
∂ϕ
∂u
r
​
​ +2cosθ
∂ϕ
∂u
θ
​
​ −u
ϕ
​
​ ]

displaystyle theta: rholeft(frac{partial u_{theta}}{partial t}+u_{r}frac{partial u_{theta}}{partial r}+frac{u_{phi}}{rsintheta}frac{partial u_{theta}}{partialphi}+frac{u_{theta}}{r}frac{partial u_{theta}}{partialtheta}+frac{u_{r}u_{theta}-u_{phi}^{2}cottheta}{r}right)=θ:ρ(
∂t
∂u
θ
​
​ +u
r
​
∂r
∂u
θ
​
​ +
rsinθ
u
ϕ
​
​
∂ϕ
∂u
θ
​
​ +
r
u
θ
​
​
∂θ
∂u
θ
​
​ +
r
u
r
​ u
θ
​ −u
ϕ
2
​ cotθ
​ )=
displaystyle ; ; ; ; -frac{1}{r}frac{partial p}{partialtheta}+rho g_{theta}+muleft[frac{1}{r^{2}}frac{partial}{partial r}left(r^{2}frac{partial u_{theta}}{partial r}right)+frac{1}{r^{2}sin^{2}theta}frac{partial^{2}u_{theta}}{partialphi^{2}}+frac{1}{r^{2}sintheta}frac{partial}{partialtheta}left(sinthetafrac{partial u_{theta}}{partialtheta}right)-frac{2}{r^{2}}frac{partial u_{r}}{partialtheta}-frac{u_{theta}+2costhetafrac{partial u_{phi}}{partialphi}}{r^{2}sin^{2}theta}right]−
r
1
​
∂θ
∂p
​ +ρg
θ
​ +μ[
r
2

1
​
∂r
∂
​ (r
2

∂r
∂u
θ
​
​ )+
r
2
sin
2
θ
1
​
∂ϕ
2

∂
2
u
θ
​
​ +
r
2
sinθ
1
​
∂θ
∂
​ (sinθ
∂θ
∂u
θ
​
​ )−
r
2

2
​
∂θ
∂u
r
​
​ −
r
2
sin
2
θ
u
θ
​ +2cosθ
∂ϕ
∂u
ϕ
​
​
​ ]

Last edited on 14 September 2018, 17:54.

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