A new solving procedure for the Kelvin&Kirchhoff equations in case of falling a rotating torus

TL;DR

This paper introduces a new analytical method for solving Kelvin&Kirchhoff equations describing a rotating torus falling in an ideal fluid, considering symmetry and boundary layer effects, and classifies resulting motion trajectories.

Contribution

The paper presents a novel solving procedure for Kelvin&Kirchhoff equations that incorporates boundary layer effects and symmetry assumptions, providing analytical approximations for torus dynamics.

Findings

Derived analytical expressions for fluid torque components.

Identified three classes of motion trajectories.

Validated the approach with calculations of angular velocity components.

Abstract

We present in this communication a new solving procedure for Kelvin&Kirchhoff equations, considering the dynamics of falling the rigid rotating torus in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the rotating body, I_1 = I_2. Fundamental law of angular momentum conservation is used for the aforementioned solving procedure. The system of Euler equations for dynamics of torus rotation is explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of torus rotation). The second finding is associated with the fact that the Stokes boundary layer phenomenon on the boundaries of the torus is also been assumed at formulation of basic Kelvin&Kirchhoff equations (for which analytical expressions for the components of fluid torque vector {T_2, T_3} were obtained earlier). The results of calculations for the components of angular velocity should then be used for full solving the momentum equation of Kelvin&Kirchhoff system. Trajectories of motion can be divided into, preferably, 3 classes: zigzagging, helical spiral motion, and the chaotic regime of oscillations.