What is Vorticity?

Example 2.15 (Three plane flows)

(i) Stagnation flow $\vect v = k(x, -y)$: $\operatorname{div}\vect v = 0$, $\vect\omega = \vect 0$, $\varphi = \tfrac12k(x^2 - y^2)$; streamlines $xy =$ const, hyperbolas — a jet hitting a wall, near the axis. (ii) Point vortex $\vect v = \dfrac{\Gamma}{2\pi r}\vect e_\theta$ (plane polar coordinates): incompressible and irrotational everywhere except at $r = 0$, $\varphi = \Gamma\theta/2\pi$ (multivalued), and the circulation on any circle around the centre is $\Gamma$ — the vorticity is concentrated on the axis. (iii) Rankine vortex: a core $r < a$ in solid rotation $v_\theta = \Omega r$ (vorticity $2\Omega$) matched to the point vortex $v_\theta = \Omega a^2/r$ outside: the model of a tornado, a bathtub swirl or the eddy behind the bridge pier — the velocity peaks at the edge of the core.