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Lie scale invariance is used to reduce the incompressible Navier-Stokesequations to non-linear ordinary equations. This yields a formulation in termsof logarithmic spirals as independent variables. We give the equations when thespirals lie on cones as well as in planes. The theory gives a locus incylindrical coordinates of singularities as they arise in the reducedNavier-Stokes equations. We give two formal examples aimed at discoveringsingularities in the flow; another example is related to a Hele-Shaw cell, andfinally we explore the flow through propellers comprised of blades made fromcongruent logarithmic spirals.

A Lie Scale Invariance in Fluids with Applications