Continuity:
∂ρ/∂t + (1/r)∂_r(rρu_r) + (1/r)∂_θ(ρu_θ) = 0
Momentum (vector form, polar components):
ρ(∂_t u + u·∇u) = −∇p + ∇·τ + ρf
=> Radial: ρ(∂_t u_r + u_r∂_r u_r + (u_θ/r)∂_θ u_r − u_θ^2/r) = −∂_r p + (∇·τ)_r + ρf_r
=> Azimuthal: ρ(∂_t u_θ + u_r∂_r u_θ + (u_θ/r)∂_θ u_θ + u_r u_θ/r) = −(1/r)∂_θ p + (∇·τ)_θ + ρf_θ
Energy:
ρ(∂_t e + u·∇e) = −p∇·u + τ:∇u + ∇·(k∇T)
(τ is the Newtonian viscous stress tensor; expand τ and ∇·τ into r,θ Laplacian/derivative terms for the full componentwise viscous expressions.)