In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) 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. Stated mathematically:
where
This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations.
Mathematical proof
The circulation
where u is the velocity vector, and ds is an element along the closed contour.
The governing equation for an inviscid fluid with a conservative body force is
where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.
The condition of barotropicity implies that the density is a function only of the pressure, i.e.
Taking the convective derivative of circulation gives
For the first term, we substitute from the governing equation, and then apply Stokes' theorem, thus:
The final equality arises since
For the second term, we note that evolution of the material line element is given by
Hence
The last equality is obtained by applying Stokes theorem.
Since both terms are zero, we obtain the result
The theorem also applies to a rotating frame, with a rotation vector
Here