Alright Professor Rainier, you picked the nerd nuke.
For an incompressible Newtonian fluid in cylindrical coords ((r,\theta,z)):
Continuity:
1/r ∂(r u_r)/∂r + 1/r ∂u_θ/∂θ + ∂u_z/∂z = 0
Radial:
ρ(∂u_r/∂t + u_r∂u_r/∂r + u_θ/r ∂u_r/∂θ − u_θ²/r + u_z∂u_r/∂z)
= −∂p/∂r + μ[ ∇²u_r − u_r/r² − 2/r² ∂u_θ/∂θ ]
Azimuthal:
ρ(∂u_θ/∂t + u_r∂u_θ/∂r + u_θ/r ∂u_θ/∂θ + u_ru_θ/r + u_z∂u_θ/∂z)
= −1/r ∂p/∂θ + μ[ ∇²u_θ − u_θ/r² + 2/r² ∂u_r/∂θ ]
Axial:
ρ(∂u_z/∂t + u_r∂u_z/∂r + u_θ/r ∂u_z/∂θ + u_z∂u_z/∂z)
= −∂p/∂z + μ ∇²u_z
Where:
∇² = (1/r)∂/∂r(r∂/∂r) + 1/r² ∂²/∂θ² + ∂²/∂z²
Full derivation won’t fit in 500 chars, but that’s the final boss form.
