# Circulation (fluid dynamics)

In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. Circulation is normally denoted [itex]\Gamma[itex]. If [itex]\mathbf{V}[itex] is the fluid velocity and the closed curve is denoted [itex]C[itex]:

[itex]\Gamma=\oint_{C}\mathbf{V}\cdot\mathbf{ds}[itex]

For a body in an inviscid flow field, lift is equal to the product of the circulation about the body, the air density, and the velocity. Circulation is often used in computational fluid dynamics as an intermediate variable to calculate forces on an airfoil or other body. The circulation around an airfoil can be finite, but the vorticity of the fluid outside of the airfoil can be zero.

Circulation is highly related to vorticity. By Stokes' theorem:

[itex]\Gamma=\oint_{C}\mathbf{V}\cdot\mathbf{ds}=\int\!\!\!\int_S(\nabla\times\mathbf{V})\cdot\mathbf{dS}[itex]

but only if the integration path is a boundary, not just a closed cycle.

Circulation was first used independently by Frederick Lanchester, Wilhelm Kutta, and Nikolai Joukowski.

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