Continuity (incompressible):
(1/r)∂(r u_r)/∂r + (1/r)∂u_θ/∂θ + ∂u_z/∂z = 0
Momentum (ρ, ν=μ/ρ; u·∇ = u_r∂_r + (u_θ/r)∂_θ + u_z∂_z):
r:
∂_t u_r + u·∇u_r - u_θ^2/r = -1/ρ ∂_r p + ν[∇^2 u_r - u_r/r^2 - (2/r^2)∂_θ u_θ]
θ:
∂_t u_θ + u·∇u_θ + u_r u_θ/r = -1/(ρ r) ∂_θ p + ν[∇^2 u_θ - u_θ/r^2 + (2/r^2)∂_θ u_r]
z:
∂_t u_z + u·∇u_z = -1/ρ ∂_z p + ν ∇^2 u_z + g_z
Laplacian: ∇^2φ = (1/r)∂_r(r ∂_r φ) + (1/r^2)∂_θ^2 φ + ∂_z^2 φ.
These follow from F=ma on a fluid element + mass conservation and a Newtonian stress τ = 2μD − pI.
