A fluid velocity field can contain local rotational motion even when the flow is not a rigid body. The quantity that measures this local rotational tendency is the vorticity

$$\boxed{\boldsymbol\omega=\nabla\times\mathbf v}.$$

Because velocity has units of length per time and curl introduces one spatial derivative,

$$[\boldsymbol\omega]=\mathrm{s^{-1}}.$$

Vorticity is a vector. Its direction gives the local rotation axis according to the right-hand rule, and its sign distinguishes the sense of rotation.

Two-dimensional flow

For a planar velocity field

$$\mathbf v=u(x,y,t)\hat{\mathbf x}+v(x,y,t)\hat{\mathbf y},$$

vorticity points perpendicular to the plane:

$$\boxed{\omega_z= \frac{\partial v}{\partial x}

\frac{\partial u}{\partial y}}.$$

A positive $\omega_z$ corresponds to counterclockwise local rotational tendency when viewed from the positive $z$ direction.

Vorticity and the angular velocity of a fluid element

For a sufficiently small fluid element, the angular velocity associated with its local rigid-body rotation is

$$\boxed{\boldsymbol\Omega_{\rm local}=\frac12\boldsymbol\omega}.$$

The factor of two is important. Vorticity is twice the local angular velocity of an infinitesimal material element.

This relation does not mean that all motion near the point is rigid rotation. A fluid element can rotate while also deforming.

Example: solid-body rotation

Consider planar rigid rotation at angular speed $\Omega$:

$$u=-\Omega y,$$

$$v=\Omega x.$$

Then

$$\frac{\partial v}{\partial x}=\Omega,$$

and

$$\frac{\partial u}{\partial y}=-\Omega.$$

Therefore

$$\omega_z =\Omega-(-\Omega) =\boxed{2\Omega}.$$

Hence

$$\Omega_{\rm local}=\frac{\omega_z}{2}=\Omega,$$

as required for a fluid rotating like a rigid body.

Example: simple shear

Now consider

$$u=Gy,$$

$$v=0,$$

where $G$ is a constant shear rate. Then

$$\omega_z =0-G =\boxed{-G}.$$

The local rotational angular velocity is

$$\Omega_{\rm local}=-\frac G2.$$

A tiny fluid element in simple shear therefore rotates while it also changes shape. Vorticity captures the rotational part of the local velocity gradient; it is not a measure of deformation by itself.

Vorticity is local

Vorticity is evaluated at a point from spatial derivatives of the velocity field. It should not be confused with statements such as

• whether a streamline is curved;
• whether the entire flow circles around some distant object;
• whether a particle follows a closed path.

A particle can travel along a curved path through a locally irrotational velocity field, and a globally circulating flow can have zero vorticity throughout part of its domain.

The distinction is analogous to curl for a general vector field: vorticity measures infinitesimal local circulation density, not the visual shape of the complete flow pattern.

Why vorticity matters

Taking the curl of a velocity field converts the kinematics of fluid motion into a compact measure of local rotation. Vorticity is central to

• shear layers and boundary layers;
• wakes and vortices;
• rotating and geophysical flows;
• the generation and diffusion of rotation by viscosity;
• circulation and aerodynamic lift.

It provides the local quantity whose accumulated flux through a surface is related to circulation around that surface's boundary.