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Travelling waves in axisymmetric pipe flows

The weakly nonlinear dynamics of axisymmetric Poiseuille pipe flows is investigated. It is shown that small perturbations of the laminar flow with amplitude ϵ ∼ O(Re−2.5) obey a coupled system of nonlinear Korteweg–de Vries-type equations. To leading order, these support inviscid soliton-type solutions and periodic waves in the form of toroidal vortex tubes that, due to viscous effects, slowly decay in time on a longer time scale t ∼ O(ϵ−2.5).