Camassa–Holm Type Equations for Axisymmetric Poiseuille Pipe Flows

We present a study of the nonlinear dynamics of a disturbance to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. The associated Navier-Stokes equations are reduced to a set of coupled generalized Camassa-Holm type equations. These support singular inviscid travelling waves with wedge-type singularities, the so called peakons, which bifurcate from smooth solitary waves as their celerity increase. In physical space they correspond to localized/periodic toroidal vortices or vortexons. The inviscid vortexon is similar to the nonlinear neutral structures found by WALTON (2011) [1] and it may be a precursor to puffs and slugs observed at transition, since most likely it is unstable to non-axisymmetric disturbances.