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

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.