10 / CHAPTER 10
The Navier–Stokes equations
Assemble mass, momentum, pressure, viscosity, and body-force terms in local form.
ANCHOR EQUATION
Navier–Stokes
In symbols: ρDu/Dt = −∇p + μ∇²u + ρg
Local momentum balance for a Newtonian fluid with pressure, viscosity, and body force.
Vorticity distinguishes spinning from turning flow
Vorticity ω = ∇×u measures local rotation of a tiny fluid element. A streamline can curve without local spin, while shear can spin a small paddle wheel even if streamlines are straight.
Take the curl of velocity
The circulation per area of velocity around a shrinking loop tends to (∇×u)·n. Thus ω is twice the local angular velocity for rigid-body rotation. In inviscid irrotational regions, ∇×u = 0 and a velocity potential may exist locally.
For solid-body rotation u = (−Ωy, Ωx,0), ω_z = ∂u_y/∂x − ∂u_x/∂y = Ω − (−Ω) = 2Ω. At Ω = 5 rad/s, vorticity is 10 s⁻¹.
Irrotational does not mean no motion
A uniform flow has zero vorticity and carries matter; potential flow can have nonzero circulation if a singular vortex or multiply connected domain is present.
Work it: Calculate vorticity for u = (2y, 3x, 0). Is the flow locally rotating? Find circulation around a small square of area 0.01 m².