vorticity

Drop a tiny paddle-wheel into a flowing fluid. At some places it just drifts along; at others it spins. Vorticity is the precise measure of that local spinning — how fast and about which axis a fluid element is rotating at each point. It is what makes whirlpools, smoke rings, tornadoes, and the swirl behind a moving spoon. Even a flow that looks like it is going in straight lines can be full of vorticity if neighbouring layers slide past each other.

Mathematically, vorticity is the curl of the velocity field: omega = curl u. In two dimensions it reduces to a single scalar, omega = u2_x - u1_y (the difference of cross-derivatives of the velocity components), which is exactly twice the local angular velocity of a fluid blob. A flow with omega = 0 everywhere is called irrotational; potential flow (where u = grad of a potential) is irrotational by construction. A powerful move is to take the curl of the Navier-Stokes equations to get the vorticity equation: in 2D it reads omega_t + (u dot grad) omega = nu Laplacian omega, which says vorticity is simply advected by the flow and diffused by viscosity — the pressure has vanished entirely, which is why this form is so beloved.

Vorticity is the natural language of turbulence and of why wings generate lift. In two dimensions vorticity is conserved along particle paths (ignoring viscosity), but in three dimensions a new term, vortex stretching ((omega dot grad) u), lets vortex tubes stretch and intensify — and this 3D-only stretching is widely believed to be the mechanism that could, in principle, drive a Navier-Stokes blow-up. So vorticity is not a side character: it sits at the heart of the deepest open question about fluids.