Lie Subalgebras and Invariant Solutions to the Equation of Fluid Flows in Toroidal Field

TL;DR

This paper uses Lie symmetry methods to analyze invariant solutions of a scalar form of the Navier-Stokes equation for toroidal flows, revealing the structure of Lie subalgebras and their solutions.

Contribution

It uncouples the 3D Navier-Stokes equations into a scalar equation for toroidal flows and analyzes invariant solutions via Lie symmetry methods, providing new insights into flow structure.

Findings

Identification of Lie subalgebras for the scalar equation

Derivation of invariant solutions for toroidal flows

Reduction of Navier-Stokes equations to scalar form

Abstract

In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The reduction and invariant solutions to the equation are analyzed by the Lie symmetry method subsequently. The Lie subalgebras for the equation are discussed and the corresponding invariant solutions are also presented.