Kelvin’s Theorem Unphysical

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Kelvin’s Circulation Theorem states that the vorticity of incompressible inviscid flow is preserved in a fluid without exterior forcing (subject only to inertial and pressure forces). Kelvin’s Theorem is presented as a corner stone of the Kutta-Zhukovsky circulation theory of lift. The proof consists in taking the curl of the momentum equation of the incompressible Navier-Stokes equations, to obtain the following equation for the vorticity \omega =\nabla\times u in the domain of the fluid with u the flow velocity :

  • \dot\omega + (u\cdot\nabla )\omega - (\omega\cdot\nabla )u = 0.

Formally it then follows that the vorticity at a later time is determined by the vorticity at initial time, so that if the velocity at initial time vanishes, so does the vorticity for later times.

But this argument is unphysical because the vorticity equation is unstable with a possibility of exponential growth which means that an infinitesimal perturbation at initial time can give rise to substantial vorticity at later time. This is exhibited in the 3d rotational separation of the flow around a circular cylinder.

The corner stone of circulation theory is thus unphysical, because stability aspects must be taken into account in a physically meaningful statement, also referred to as wellposedness/illposedness in mathematical literature.   This is one of several arguments showing that circulation theory is unphysical; the corner stone of the criticism of this theory exposed on The Secret of Flight.

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