# Kelvin's circulation theorem

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## Kelvin's circulation theorem

A theorem in fluid dynamics that pertains to the dynamics of vortices and the use of ideal-fluid potential-flow equations. The theorem states that the circulation (defined as the line integral of the component of velocity tangential to the closed contour) in an inviscid and incompressible fluid subject to only conservative forces is constant. By using Stokes' theorem of integral calculus, it may be shown that the circulation is also related to the flux of vorticity (defined as the curl of the velocity field) normal to the area transcribed by the contour.

The principal use of Kelvin's theorem is in the study of incompressible, inviscid fluid flows. If a body is moving through such a fluid, the vorticity far from the body is, by definition, zero. Then according to Kelvin's theorem, the vorticity in the fluid will everywhere be zero and the flow will be irrotational. This permits the reduction of the governing equations from the Euler equations to the Laplace equation and presents the many mathematical techniques of potential theory for solving fluid-flow problems. *See* Laplace's irrotational motion, Vortex