Applications of Group Analysis to the Three-Dimensional Equations of Fluids with Internal Inertia

TL;DR

This paper applies group analysis to classify three-dimensional fluid flow equations with internal inertia, identifying special cases and invariant solutions, including the Ovsyannikov vortex, for models like bubbly fluids and shallow water.

Contribution

It provides a comprehensive group classification of fluid equations with internal inertia and constructs invariant solutions, including the Ovsyannikov vortex, for specific potential functions.

Findings

15 cases of the potential function W are classified.

Invariant solutions are constructed based on rotational symmetry.

Complete analysis of solutions for a special potential type is provided.

Abstract

Group classification of the three-dimensional equations describing flows of fluids with internal inertia, where the potential function $W= W(\rho,\dot{\rho})$, is presented. The given equations include such models as the non-linear one-velocity model of a bubbly fluid with incompressible liquid phase at small volume concentration of gas bubbles, and the dispersive shallow water model. These models are obtained for special types of the function $W(\rho,\dot{\rho})$. Group classification separates out the function $W(\rho,\dot{\rho})$ at 15 different cases. Another part of the manuscript is devoted to one class of partially invariant solutions. This solution is constructed on the base of all rotations. In the gas dynamics such class of solutions is called the Ovsyannikov vortex. Group classification of the system of equations for invariant functions is obtained. Complete analysis of invariant solutions for the special type of a potential function is given.