A SELF-SUSTAINED SPHEROIDAL-SPIRALOIDAL VORTEX WITH AN AXIAL FUNNEL

Abstract

This paper proposes a mathematical model of a new type of self-consistent vortex structure - a hybrid spheroidal-spiraloidal vortex with an internal spiral funnel. In contrast to classical models such as Hill’s vortex and toroidal vortices [1-3], the proposed configuration represents a single continuous elliptical vortex region in which no distinct thin toroidal vortex tube is present. Instead, the internal flow is organized as a continuous axial channel - a funnel smoothly connected to the upper and lower parts of the vortex. It is shown that, within this framework, the source of motion, pressure, and density is transferred from the external flow to the internal structure: the axial flow and the associated angular momentum are responsible for the translational motion of the entire system. The velocity field is represented via a stream function decomposed into a Hill-type component and a funnel-induced component, and a Clebsch representation of the rotational potential is introduced, localized within the internal flow and the corresponding spiral vortex lines. Expressions are derived for the hydrodynamic impulse, circulation, streamline equations, angular momentum, and vortex moment. It is demonstrated that the existence of a self-consistent vortex is ensured by a closed cycle of fluid particles (micro-vortices), which acquire momentum in the funnel and subsequently move along single continuous three-dimensional spiral trajectories, encircling the internal region and returning to the base of the funnel without discontinuities or trajectory switching. This leads to the formation of a stable self-consistent structure in which flow closure, momentum transport, and pressure maintenance are governed by the internal geometry of motion, without the need to prescribe an external potential flow with continuous pressure [5-9].