In spherical coordinates, the r , ϕ ϕ , and θ θ momentum equations are (note the convention used: θ θ is polar angle, or colatitude, 0 ≤ ≤ θ θ ≤ ≤ π π ): r : ρ ρ ( ∂ ∂ t u r + u r ∂ ∂ r u r + u φ φ r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u r + u θ θ r ∂ ∂ θ θ u r − − u φ φ 2 + u θ θ 2 r ) = − − ∂ ∂ r p + μ μ ( 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r u r ) + 1 r 2 sin 2 ⁡ ⁡ θ θ ∂ ∂ φ φ 2 u r + 1 r 2 sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( sin ⁡ ⁡ θ θ ∂ ∂ θ θ u r ) − − 2 u r + ∂ ∂ θ θ u θ θ + u θ θ cot ⁡ ⁡ θ θ r 2 − − 2 r 2 sin ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ ) + 1 3 μ μ ∂ ∂ r ( 1 r 2 ∂ ∂ r ( r 2 u r ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( u θ θ sin ⁡ ⁡ θ θ ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ ) + ρ ρ g r φ φ : ρ ρ ( ∂ ∂ t u φ φ + u r ∂ ∂ r u φ φ + u φ φ r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ + u θ θ r ∂ ∂ θ θ u φ φ + u r u φ φ + u φ φ u θ θ cot ⁡ ⁡ θ θ r ) = − − 1 r sin ⁡ ⁡ θ θ ∂ ∂ φ φ p + μ μ ( 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r u φ φ ) + 1 r 2 sin 2 ⁡ ⁡ θ θ ∂ ∂ φ φ 2 u φ φ + 1 r 2 sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( sin ⁡ ⁡ θ θ ∂ ∂ θ θ u φ φ ) + 2 sin ⁡ ⁡ θ θ ∂ ∂ φ φ u r + 2 cos ⁡ ⁡ θ θ ∂ ∂ φ φ u θ θ − − u φ φ r 2 sin 2 ⁡ ⁡ θ θ ) + 1 3 μ μ 1 r sin ⁡ ⁡ θ θ ∂ ∂ φ φ ( 1 r 2 ∂ ∂ r ( r 2 u r ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( u θ θ sin ⁡ ⁡ θ θ ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ ) + ρ ρ g φ φ θ θ : ρ ρ ( ∂ ∂ t u θ θ + u r ∂ ∂ r u θ θ + u φ φ r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u θ θ + u θ θ r ∂ ∂ θ θ u θ θ + u r u θ θ − − u φ φ 2 cot ⁡ ⁡ θ θ r ) = − − 1 r ∂ ∂ θ θ p + μ μ ( 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r u θ θ ) + 1 r 2 sin 2 ⁡ ⁡ θ θ ∂ ∂ φ φ 2 u θ θ + 1 r 2 sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( sin ⁡ ⁡ θ θ ∂ ∂ θ θ u θ θ ) + 2 r 2 ∂ ∂ θ θ u r − − u θ θ + 2 cos ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ r 2 sin 2 ⁡ ⁡ θ θ ) + 1 3 μ μ 1 r ∂ ∂ θ θ ( 1 r 2 ∂ ∂ r ( r 2 u r ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ θ θ ( u θ θ sin ⁡ ⁡ θ θ ) + 1 r sin ⁡ ⁡ θ θ ∂ ∂ φ φ u φ φ ) + ρ ρ g θ θ .