Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh-Plesset Cavitation Dynamics

The following article extends a decomposition to the Navier-Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman's presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh-Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh-Plesset equations. The article then proceeds with the demonstration that the Raleigh-Plesset equations may be obtained from this energy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh-Plesset functional in the greater scheme of fluid mechanics is commented upon.