Kelvin's circulation theorem

In fluid mechanics, Kelvin's circulation theorem states:^[1][2]

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.^[3] 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.