Poincaré Mapping Method for Hydrodynamic Systms. Dynamic Chaos in a Fluid Layer between Eccentrically Rotating Cylinders

Abstract

Investigation of the plane–parallel motion of particles of an incompressible medium reduces to investigation of a Hamiltonian system. The Hamiltonian function is a stream function. The time‐periodic mixing of an incompressible medium is described by a time‐periodic Hamiltonian function. The mixing of the medium is associated with dynamic chaos. Transition to dynamic chaos is studied by analysis of the positions of Lagrangian particles at times divisible by the period — Poincaré recurrence points. The set of Poincaré recurrence points is studied with the use of Poincaré mapping on the phase flow. A method for constructing Poincaré maps in parametric form is proposed. A map is constructed as a series in a small parameter. It is shown that the parametric method has a number of advantages over the generating function method is shown. The proposed method is used to examine the motion of particles of an incompressible viscous fluid layer between two circular cylinders. The outer cylinder is immovable, and the inner cylinder rotates about a point that does not coincide with the centers of both cylinders. An optimal mode for the motion is established, in which the area of the chaotic region is maximal.