Part I. On the structure in separatrix-swept regions of slowly-modulated Hamiltonian systems. Part II. On the quantification of mixing in chaotic Stokes' flows : the eccentric journal bearing

Citation

Kaper, Tasso J. (1992) Part I. On the structure in separatrix-swept regions of slowly-modulated Hamiltonian systems. Part II. On the quantification of mixing in chaotic Stokes' flows : the eccentric journal bearing. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MDD9-3Z11. https://resolver.caltech.edu/CaltechETD:etd-08062007-100540

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. PART I: In this work, we establish the structure of the large [...]-sized separatrix-swept regions in Hamiltonian systems which depend on a slow-varying parameter z: H = H(p,q,z = [...]), where 0 < [...] 1. These regions are complementary to those in which the theory of adiabatic invariance and Arnold's extension of the KAM Theorem to adiabatic systems apply. We prove the following theorem about the structure of lobes, which, being regions of phase space bounded by segments of intersecting stable and unstable manifolds terminating on principal homoclinic points, are the fundamental building blocks of homoclinic tangles. Theorem: The area of a lobe in (1.1) is: A = [...], where Z0 and Z1 are two adjacent simple zeroes of MA(z), the adiabatic Melnikov function of the system. We also derive a 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, [...] and [...]. The remaining terms are [...]. Theorem 1 and the corollary establish several important results: First, the area occupied by the homoclinic tangles formed by the intersection of the stable and unstable manifolds is [...] to leading order. Second, Theorem 1 implies that the flux between regions separated by instantaneous separatrices is [...] asymptotically, see Part II for an application. Third, for systems in which H depends periodically or quasiperiodically on z, Theorem 1 states that the region in which orbits evolve chaotically is [...] in the limit of [...]. This result stands in marked contrast to the known examples of chaotic systems in which the "stochastic" regions are either of [...] or [...] and vanish as [...]. Finally, since islands must lie outside of the lobes, Theorem 1 shows that the phase space area in which islands must lie vanishes with [...] as [...]. We remark that we lower the upper bounds on island size presented in Elskens and Escande [1991] using asymptotic expansions of the exact resonance-zone area formula of MacKay and Meiss [1986]. We also derive an exact lobe area formula for general time-dependent Hamiltonian systems, which eliminates the need for the existence of a recurrent p - q section in the extended phase space assumed in the previous work. A direct measurement of the type used on weakly-perturbed Hamiltonian systems is not possible in adiabatic systems since the pieces of stable and unstable manifold defining the boundary of the lobe cannot be expressed as graphs over the unperturbed separatrix. Therefore, the shape-independence of this exact formula is needed. We illustrate our results on the adiabatic pendulum: H = [...] cos q and on a model due to Hastings and McLeod. Finally, for z-periodic H, we show for the first time in an example that a Smale horseshoe map can be created in one iteration of the Poincare map. 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. In stark contrast to the flows usually studied with dynamical systems, however, these slowly-varying systems are singular-perturbation problems in which the nonintegrability is due to the slow [...] modulation of the position of the saddle stagnation point and the two streamlines stagnating on it. We establish an analytical technique to determine the location and size of the region in which mixing occurs. This technique gives us explicit control over the mixing process. We also develop a transport theory based on the lobes formed by the segments of stable and unstable manifolds of the fixed points of the Poincare map, which are responsible for the transport of tracer in the mixing zone. In particular, we show that the radically different shape of these lobes, as compared to the shape of the lobes studied in the usual flows, readily makes them identifiable as the mechanism by which the modulation causes the patches of tracer to develop into elaborately striated and folded lamellar structures. 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 transport theory for the first time. From the measurement of these quantities, we determine the combination of the flow parameters with which one achieves the most efficient mixing possible. Furthermore, we use an extension of the KAM theory to explain the highly-regular appearance of islands in quasi-steady Stokes' flows for the first time. Finally, we 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.

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