Kelvin's circulation theorem

In fluid mechanics, Kelvin's circulation theorem states:

In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time.

The theorem is named after William Thomson, 1st Baron Kelvin who published it in 1869.

Stated mathematically:

${\displaystyle {\frac {\mathrm {D} \Gamma }{\mathrm {D} t}}=0}$

where ${\displaystyle \Gamma }$ is the circulation around a material moving contour ${\displaystyle C(t)}$ as a function of time ${\displaystyle t}$. The differential operator ${\displaystyle \mathrm {D} }$ is a substantial (material) derivative moving with the fluid particles. Stated more simply, this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour remains constant.

This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example the Coriolis force) or non-barotropic pressure-density relations.

In the simplifying special case of steady flow the theorem can be applied to a closed curve that has a fixed position so that fluid elements flow through the closed curve. In the study of airfoils producing lift, it is often instructive to examine the circulation around any closed curve that fully encloses the airfoil; in the steady flow of an inviscid fluid past a stationary airfoil, the theorem can be applied to this closed curve.