For (u_r,u_φ,u_z) in cylindrical (r,φ,z), incompressible Navier–Stokes is:
Continuity:
(1/r)∂(ru_r)/∂r + (1/r)∂u_φ/∂φ + ∂u_z/∂z = 0
Radial:
∂u_r/∂t + u_r∂u_r/∂r + (u_φ/r)∂u_r/∂φ + u_z∂u_r/∂z - u_φ²/r
= -(1/ρ)∂p/∂r + ν(∇²u_r - u_r/r² - 2/r² ∂u_φ/∂φ) + f_r
Azimuthal:
∂u_φ/∂t + u_r∂u_φ/∂r + (u_φ/r)∂u_φ/∂φ + u_z∂u_φ/∂z + u_ru_φ/r
= -(1/ρr)∂p/∂φ + ν(∇²u_φ - u_φ/r² + 2/r² ∂u_r/∂φ) + f_φ
Axial:
∂u_z/∂t + u_r∂u_z/∂r + (u_φ/r)∂u_z/∂φ + u_z∂u_z/∂z
= -(1/ρ)∂p/∂z + ν∇²u_z + f_z
with
∇² = (1/r)∂/∂r(r∂/∂r) + (1/r²)∂²/∂φ² + ∂²/∂z²
If you want, I can do the full derivation from momentum balance step-by-step instead of just dropping the final spicy equations 