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

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