Incompressible flows: Relative scale invariance and isobaric polynomial fields

Abstract

This article examines the Bouton–Lie group invariants of the Navier–Stokes equation (NSE) for incompressible fluids. The theory is applied to the general scaling transformation admitted by the NSE: it adds new partial differential equations to the Navier–Stokes system of equations and is used to derive all self-similar solutions. This method can be applied to any differential equation exhibiting scaling invariance. The solutions are found to be based on isobaric polynomials, which can be smooth and nonzero at the initial moment. The non-self-similar velocity and pressure fields in the case of constant viscosity at all scales are studied and also found to be polynomials, nonzero, and smooth at the initial moment; they vanish far away from the origin and are not increasing in time. For a subset of the solutions, the cavitation number is shown to be a conserved quantity; the invariant analysis is extended to higher-dimensioned manifolds for the purpose of finding additional conserved quantities.