Vorticity and Circulation Preliminaries

Vorticity, \( \mathbf{\omega} \), is defined to be the curl of velocity and so is given by \begin{equation} \mathbf{\omega} \equiv \nabla \times \mathbf{v} \end{equation}

Circulation, \( C \) (or \( \Gamma \)), is defined to be the integral of velocity around a closed fluid loop and so is given by \begin{equation} C \equiv \oint \mathbf{v} \cdot d\mathbf{r} = \int_S \mathbf{\omega} \cdot d\mathbf{S} \end{equation}

where the second expression uses Stokes’ theorem and \( S \) is any surface bounded by the loop. The circulation around the path is equal to the integral of the normal component of vorticity over any surface bounded by that path. The circulation is not a field like vorticity and velocity; rather, we think of the circulation around a particular material line of finite length, and so its value generally depends on the path chosen

If \( \delta S \) is an infinitesimal surface element whose normal points in the direction of the unit vector \( \hat{\mathbf{n}} \) \begin{equation} \hat{\mathbf{n}} \cdot (\nabla \times \mathbf{v}) = \frac{1}{\delta S} \oint_{\delta r} \mathbf{v} \cdot d\mathbf{r} \end{equation}

where the line integral is around the infinitesimal area. Thus at a point the component of vorticity in the direction of \( \mathbf{n} \) is proportional to the circulation around the surrounding infinitesimal fluid element, divided by the elemental area bounded by the path of the integral

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