linguistic:voice_change, linguistic:translation
Spherical coordinates

In spherical coordinates, the r {\textstyle r}, ϕ {\textstyle \phi }, and θ {\textstyle \theta } momentum equations are (note the convention used: θ {\textstyle \theta } is polar angle, or colatitude, 0 ≤ θ ≤ π {\textstyle 0\leq \theta \leq \pi }): r: ρ (∂ t u r + u r ∂ r u r + u φ r sin θ ∂ φ u r + u θ r ∂ θ u r − u φ 2 + u θ 2 r) = − ∂ r p + μ (1 r 2 ∂ r (r 2 ∂ r u r) + 1 r 2 sin 2 θ ∂ φ 2 u r + 1 r 2 sin θ ∂ θ (sin θ ∂ θ u r) − 2 u r + ∂ θ u θ + u θ cot θ r 2 − 2 r 2 sin θ ∂ φ u φ) + 1 3 μ ∂ r (1 r 2 ∂ r (r 2 u r) + 1 r sin θ ∂ θ (u θ sin θ) + 1 r sin θ ∂ φ u φ) + ρ g r {\displaystyle {\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{r}}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{r}}-{\frac {u_{\varphi }^{2}+u_{\theta }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{r}}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{r}}\right)-2{\frac {u_{r}+{\partial _{\theta }u_{\theta }}+u_{\theta }\cot \theta }{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{r}\\\end{aligned}}} φ: ρ (∂ t u φ + u r ∂ r u φ + u φ r sin θ ∂ φ u φ + u θ r ∂ θ u φ + u r u φ + u φ u θ cot θ r) = − 1 r sin θ ∂ φ p + μ (1 r 2 ∂ r (r 2 ∂ r u φ) + 1 r 2 sin 2 θ ∂ φ 2 u φ + 1 r 2 sin θ ∂ θ (sin θ ∂ θ u φ) + 2 sin θ ∂ φ u r + 2 cos θ ∂ φ u θ − u φ r 2 sin 2 θ) + 1 3 μ 1 r sin θ ∂ φ (1 r 2 ∂ r (r 2 u r) + 1 r sin θ ∂ θ (u θ sin θ) + 1 r sin θ ∂ φ u φ) + ρ g φ {\displaystyle {\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\\end{aligned}}} θ: ρ (∂ t u θ + u r ∂ r u θ + u φ r sin θ ∂ φ u θ + u θ r ∂ θ u θ + u r u θ − u φ 2 cot θ r) = − 1 r ∂ θ p + μ (1 r 2 ∂ r (r 2 ∂ r u θ) + 1 r 2 sin 2 θ ∂ φ 2 u θ + 1 r 2 sin θ ∂ θ (sin θ ∂ θ u θ) + 2 r 2 ∂ θ u r − u θ + 2 cos θ ∂ φ u φ r 2 sin 2 θ) + 1 3 μ 1 r ∂ θ (1 r 2 ∂ r (r 2 u r) + 1 r sin θ ∂ θ (u θ sin θ) + 1 r sin θ ∂ φ u φ) + ρ g θ. {\displaystyle {\begin{aligned}\theta :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}}

Mass continuity will read: ∂ t ρ + 1 r 2 ∂ r (ρ r 2 u r) + 1 r sin θ ∂ φ (ρ u φ) + 1 r sin θ ∂ θ (sin θ ρ u θ) = 0. {\displaystyle {\partial _{t}\rho }+{\frac {1}{r^{2}}}\partial _{r}\left(\rho r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }(\rho u_{\varphi })}+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(\sin \theta \rho u_{\theta }\right)=0.}

These equations could be (slightly) compacted by, for example, factoring 1 r 2 {\textstyle {\frac {1}{r^{2}}}} from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.