Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem (named after the Irish scientist who published this theorem in 1869) states In an inviscid, barotropic flow 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 was developed by William Thomson, 1st Baron Kelvin. Stated mathematically:
where is the circulation around a material contour . 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 are equal.
This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations.
The circulation around a closed material contour is defined by:
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
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 owing to barotropicity.
For the second term, we note that evolution of the material line element is given by
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 , if the circulation is modified thus:
Here is the position of the area of fluid. From Stoke's theorem, this is:
- Katz, Plotkin: Low-Speed Aerodynamics
- Kundu, P and Cohen, I: Fluid Mechanics, page 130. Academic Press 2002