Resonant and chaotic advection in a curved pipe

Abstract

The Lagrangian description is used to study the motion of fluid particles in a curved pipe, when laminar fully developed flow is generated by an oscillatory longitudinal pressure gradient. The (viscous) fluid is assumed to be incompressible, so that the transverse motion is governed by a Hamiltonian dynamical system, which is near-integrable when the flow is quasi-steady. Islands and chaotic regions are seen in a Poincaré section, in accordance wtih Kolmogorov-Arnol'd-Moser (KAM) theory. Resonance between the longitudinal and transverse motion causes ballistic longitudinal transport of particles whose trajectories lie on KAM tori within certain islands. Intermittent longitudinal transport is seen in the chaotic regions bordering such islands, as trajectories are trapped by cantori in the vicinity of both resonant and non-resonant KAM tori. The interaction of advection and molecular diffusion is discussed. When the Péclet number is very large, particles are transported longitudinally in an intermittent manner which resembles transport in chaotic regions. Here the trapping of trajectories is caused by islands, not cantori.