Vorticity, circulation and Kelvin's circulation theorem

The vorticity is the curl of the velocity, \(\boldsymbol{\omega} = \nabla\times\mathbf{u}\). To see what it measures, split the velocity gradient into symmetric and antisymmetric parts, \(\nabla\mathbf{u} = S + W\) with \(S = \tfrac12(\nabla\mathbf{u} + \nabla\mathbf{u}^T)\) and \(W = \tfrac12(\nabla\mathbf{u} - \nabla\mathbf{u}^T)\). Near a point, \(\mathbf{u}(\mathbf{x}+\mathbf{h}) \approx \mathbf{u}(\mathbf{x}) + S\mathbf{h} + W\mathbf{h}\), and a direct calculation gives \(W\mathbf{h} = \tfrac12\boldsymbol{\omega}\times\mathbf{h}\). So a small fluid element translates, is strained by \(S\), and rotates rigidly with angular velocity \(\boldsymbol{\omega}/2\). For solid-body rotation \(\mathbf{u} = \Omega(-y,x,0)\) the vorticity is \((0,0,2\Omega)\), twice the rotation rate.

Vorticity is not the same as streamlines curving. The shear flow \(\mathbf{u} = (y,0,0)\) has straight streamlines and vorticity \((0,0,-1)\): a small paddle wheel placed in it spins, because the top is pushed harder than the bottom. The point vortex \(\mathbf{u} = \frac{\Gamma}{2\pi r^2}(-y,x)\) has circular streamlines and zero vorticity away from the origin: the paddle wheel goes round the centre without turning about its own axis.

The circulation around a closed curve \(C\) is \(\Gamma = \oint_C \mathbf{u}\cdot d\mathbf{l}\). By Stokes' theorem it is the flux of vorticity through any surface spanning the loop, \(\Gamma = \iint \boldsymbol{\omega}\cdot\mathbf{n}\,dS\). For the point vortex every loop around the origin has circulation \(\Gamma\), even though the curl vanishes on the loop, because the region inside contains the singular centre and is not simply connected.

Kelvin's circulation theorem (W. Thomson, 1869). Suppose the fluid is inviscid, the density is constant (or more generally a function of pressure alone), and the body force is the gradient of a potential. Then the circulation around any closed curve that moves with the fluid is constant in time. The proof is short. For a material loop, \[ \frac{d\Gamma}{dt} = \oint_C \frac{D\mathbf{u}}{Dt}\cdot d\mathbf{l} + \oint_C \mathbf{u}\cdot d\mathbf{u}. \] The second integral is \(\oint d(|\mathbf{u}|^2/2) = 0\). Under the hypotheses, the Euler equations make \(D\mathbf{u}/Dt\) a gradient, \(-\nabla(p/\rho + \Phi)\), and the integral of a gradient round a closed loop is zero.

The consequences are strong. Helmholtz (1858) had shown that vortex lines move with the fluid; Kelvin's theorem implies that a flow which starts irrotational stays irrotational, so vorticity in an ideal fluid can be neither created nor destroyed, only moved and deformed. Real vorticity is born at solid walls through viscosity, which is exactly the hypothesis Kelvin's theorem drops.

Picture it: a smoke ring keeps its circulation as it travels; shrinking its radius would speed it up. Think it: vorticity is the part of the velocity gradient invisible to strain, and circulation is its integrated form; Kelvin's theorem says the flow map of an ideal fluid transports the one-form \(\mathbf{u}\cdot d\mathbf{x}\) up to an exact form.