AN ANALYTICAL STUDY OF TRANSPORT IN STOKES FLOWS EXHIBITING LARGE-SCALE CHAOS IN THE ECCENTRIC JOURNAL BEARING

In the present work, we apply new tools from the field of adiabatic dy namical systems theory to make quantitative predictions of various imp ortant mixing quantities in quasi-steady Stokes flows which possess sl owly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed us ing only knowledge of the streamlines in steady-state flows and the ex ternally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the s lowly sweeping instantaneous stagnation streamlines. Tracer patches ge t stretched by an amount O(1/epsilon) during each period, and the aver age striation thickness of the patch decreases by a factor of epsilon in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially grow ing 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 transp ort of passive scalars in a low Reynolds number flow in the two-dimens ional eccentric journal bearing when the angular velocities of the cyl inders are slowly modulated, continuously and periodically in time, wi th frequency epsilon. In contrast to the flows usually studied with dy namical systems, these slowly varying systems are singular perturbatio n (apparently far from integrable) problems exhibiting large-scale cha os, 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.