An analytical study of transport in Stokes flows exhibiting large-scale chaos in the eccentric journal bearing

Abstract

In the present work, we apply new tools from the field of adiabatic dynamical systems theory to make quantitative predictions of various important mixing quantities in quasi-steady Stokes flows which possess slowly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed using only knowledge of the streamlines in steady-state flows and the externally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the slowly sweeping instantaneous stagnation streamlines. Tracer patches get stretched by an amount O(1/ε) during each period, and the average striation thickness of the patch decreases by a factor of ε in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially growing lower bound. Finally, the highly regular appearance of islands in quasi-steady Stokes’ flows is explained using an extension of the KAM theory. As an example to illustrate these results, we study the transport of passive scalars in a low Reynolds number flow in the two-dimensional eccentric journal bearing when the angular velocities of the cylinders are slowly modulated, continuously and periodically in time, with frequency ε. In contrast to the flows usually studied with dynamical systems, these slowly varying systems are singular perturbation (apparently far from integrable) problems exhibiting large-scale chaos, in which the non-integrability is due to the slow, continuous O(1) modulation of the position of the saddle stagnation point and the two streamlines stagnating on it.