Circulation, vorticity and rotational flow

Definitions

Circulation $\Gamma$ is the line integral of the tangential velocity component around a closed curve:

$$\Gamma = \oint \vec V \cdot d\vec s$$

Vorticity $\vec\zeta$ is twice the angular velocity of a fluid particle, $\vec\zeta = 2\vec\omega = \nabla \times \vec V$. For 2D flow in the $xy$-plane:

$$\zeta_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$$

Circulation round a rectangle

Take an element $dx \times dy$ with velocity components $u$, $v$ at the lower left corner, and go anticlockwise.

• Bottom side (to the right): $u\,dx$
• Right side (up): $\left(v + \dfrac{\partial v}{\partial x}dx\right)dy$
• Top side (to the left): $-\left(u + \dfrac{\partial u}{\partial y}dy\right)dx$
• Left side (down): $-v\,dy$

Adding:

$$d\Gamma = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) dx\,dy = \zeta_z\, dA$$

So circulation per unit area equals vorticity. This is the basis of Stokes' theorem, $\Gamma = \iint \zeta\, dA$.

Rotational and irrotational flow

For irrotational flow a velocity potential $\phi$ exists, with $u = \partial\phi/\partial x$ and $v = \partial\phi/\partial y$.