AN ANATOMY OF LAGRANGIAN CHAOS IN LOW REYNOLDS NUMBER FLOW BETWEEN TWO ECCENTRIC ROTATING CYLINDERS

Abstract

Chaotic motion of fluid particles in low Reynolds number flow between two eccentric cylinders rotating alternately is studied by investigating the tangling between invariant manifolds of the underlying dynamical system. Such tangles are revealed by a numerical method carrying out the global search of periodic points of the associated Poincaré map and the global tracking of stable and unstable manifolds to the hyperbolic points thus found. Along with such an anatomy the breakup of some elliptic orbits as predicted by Poincaré–Birkhoff theorem is clearly shown. It is found that several drastic changes either in the characteristic of ellipticity and hyperbolicity or in the pattern of homoclinic tangles take place when the cylinders' eccentricity is increased towards its maximum. The results are confirmed either by other numerical approaches or by a rigorous shadowing to a certain extent, and are expected to be of use to some geometric understanding about mixing.