Extension of Kelvin's minimum energy theorem to incompressible fluid domains with open regions

Abstract

Kelvin's minimum energy theorem predicts that the irrotational motion of a homogeneously incompressible fluid in a simply connected region will carry less kinetic energy than any other profile that shares the same normal velocity conditions on the domain's boundary. In this work, Kelvin's analysis is extended to regions with boundaries on which the normal velocity requirements are relaxed. Given the ubiquity of practical configurations in which such boundaries exist, the question of whether Kelvin's theorem continues to hold is one of significant interest. In reconstructing Kelvin's proof, we find it useful to define a net rotational velocity as the difference between the generally rotational flow and the corresponding potential motion. In Kelvin's classic theorem, the normal component of the net rotational velocity at all domain boundaries is zero. In contrast, the present analysis derives a sufficient condition for ensuring the validity of Kelvin's theorem in a domain where the normal component of net rotational velocity at some or all of the boundaries is not zero. The corresponding criterion requires the evaluation of a simple surface integral over the boundary.