Vorticity

Definition

Let $\Omega \subseteq \R^3$ be open and let $\mathbf{u} = (u_1, u_2, u_3) \colon \Omega \to \R^3$ be a continuously differentiable velocity field. The vorticity of $\mathbf{u}$ is the vector field $\boldsymbol{\omega} = \nabla \times \mathbf{u}$, the curl of $\mathbf{u}$: $$\boldsymbol{\omega} = \left( \frac{\partial u_3}{\partial x_2} - \frac{\partial u_2}{\partial x_3},\ \frac{\partial u_1}{\partial x_3} - \frac{\partial u_3}{\partial x_1},\ \frac{\partial u_2}{\partial x_1} - \frac{\partial u_1}{\partial x_2} \right).$$ For a plane flow $\mathbf{u} = (u, v)$ the vorticity is the scalar $\omega = \partial v / \partial x - \partial u / \partial y$. The flow is irrotational if $\boldsymbol{\omega} = 0$ on $\Omega$.