Part I. O the Structure in Separatrix-Swept Regions of Slowly-Modulated Hamiltonian Systems. Part II. O the Quantification of Mixing in Chaotic Stokes' Flows: the Eccentric Journal Bearing

Part I. We establish the structure of the large {cal O} (1)-sized separatrix -swept regions in Hamiltonian systems H = H(p,q,z = epsilon t), where 0 < epsilon << 1. We prove Theorem: The area of a lobe in these systems is A = int _sp{Z _0}{Z _1} M_{A }(z)dz + {cal O}(epsilon), where Z_0 and Z _1 are two adjacent simple zeroes of M _{A}(z), the adiabatic Melnikov function of the system. Corollary: The area of a lobe in these systems is given to leading order by the difference between the areas enclosed by two sequential extremal instantaneous separatrices, Gamma ^{Z _0 } and Gamma ^{Z_1 }. The remaining terms are { cal O} (epsilon). Thus, the area occupied by the homoclinic tangles is {cal O} (1) to leading order. Second, the flux between regions separated by instantaneous separatrices is {cal O} (1) asymptotically. Third, for systems in which H depends periodically or quasiperiodically on z, the region in which orbits evolve chaotically is {cal O}(1) asymptotically. This result stands in marked contrast to the known examples of chaotic systems in which the "stochastic" regions vanish as epsilon to 0. We also lower the upper bounds on island size presented in Elskens and Escande (1991) and give an exact lobe area formula for general time-dependent systems. We illustrate our results on the adiabatic pendulum: H = {p ^2 over 2} + (1 - gamma {rm cos}(z = epsilon t)) {rm cos} q.. Part II. We study the transport of tracer dye in a low Reynolds number flow in the two-dimensional eccentric journal bearing. Modulation of the angular velocities of the cylinders continuously, slowly, and periodically in time causes the integrable steady-state flow to become nonintegrable. We establish an analytical technique to determine the location and size of the region in which mixing occurs. Using the transport theory we develop, we show that the radically different shape of these lobes, as compared to the shape of the lobes studied in the usual (weakly-perturbed) flows, makes them identifiable as the mechanism by which the patches of tracer develop into elaborately striated and folded lamellar structures covering large areas. When the modulation frequency is small we apply the tools developed in Part I to analytically predict several important quantities associated with the lobes and mixing for the first time. Finally, we show that diffusion enhances stretching, discuss the robustness of our model by analyzing the influence of the inertial terms, and compare our results to those obtained experimentally using so-called blinking protocols.