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.
is a substantial (material) derivative moving with the fluid particles.
This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example the Coriolis force) or non-barotropic pressure-density relations.
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 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.
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 gradient theorem.
Since both terms are zero, we obtain the result A similar principle which conserves a quantity can be obtained for the rotating frame also, known as the Poincaré–Bjerknes theorem, named after Henri Poincaré and Vilhelm Bjerknes, who derived the invariant in 1893[4][5] and 1898.
From Stokes' theorem, this is: The vorticity of a velocity field in fluid dynamics is defined by: Then: