Camassa-Holm type equations for axisymmetric Poiseuille pipe flows

Abstract

We present a study on 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 toroidal vortices or vortexons. The inviscid vortexon is similar to the nonlinear neutral structures found by Walton (2011) and it may be a precursor to puffs and slugs observed at transition, since most likely it is unstable to non-axisymmetric disturbances.

11 pages, 4 figures, 31 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/

References in corpus (4)

• A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
• Hamiltonian form and solitary waves of the spatial Dysthe equations
• Solitary wave interaction in a compact equation for deep-water gravity waves
• Vortexons in axisymmetric Poiseuille pipe flows