Viscous Fluid Conduits as a Prototypical Nonlinear Dispersive Wave Platform

LOWMAN, NICHOLAS K. Viscous Fluid Conduits as a Prototypical Nonlinear Dispersive Wave Platform. (Under the direction of Mark Hoefer.) This thesis is devoted to the comprehensive characterization of slowly modulated, nonlinear waves in dispersive media for physically-relevant systems using a threefold approach: analytical, long-time asymptotics, careful numerical simulations, and quantitative laboratory experiments. In particular, we use this interdisciplinary approach to establish a two-fluid, interfacial fluid flow setting known as viscous fluid conduits as an ideal platform for the experimental study of truly one dimensional, unidirectional solitary waves and dispersively regularized shock waves (DSWs). Starting from the full set of fluid equations for mass and linear momentum conservation, we use a multiple-scales, perturbation approach to derive a scalar, nonlinear, dispersive wave equation for the leading order interfacial dynamics of the system. Using a generalized form of the approximate model equation, we use numerical simulations and an analytical, nonlinear wave averaging technique, Whitham-El modulation theory, to derive the key physical features of interacting large amplitude solitary waves and DSWs. We then present the results of quantitative, experimental investigations into large amplitude solitary wave interactions and DSWs. Overtaking interactions of large amplitude solitary waves are shown to exhibit nearly elastic collisions and universal interaction geometries according to the Lax categories for KdV solitons, and to be in excellent agreement with the dynamics described by the approximate asymptotic model. The dispersive shock wave experiments presented here represent the most extensive comparison to date between theory and data of the key wavetrain parameters predicted by modulation theory. We observe strong agreement. Based on the work in this thesis, viscous fluid conduits provide a well-understood, controlled, table-top environment in which to study universal properties of dispersive hydrodynamics. Motivated by the study of wave propagation in the conduit system, we identify four new admissibility criteria required for proper application of the Whitham-El DSW closure method for a general class of scalar dispersive hydrodynamic equations. Further, we explore regularization distinguishing characteristics of dissipative versus dispersive smoothing in a bidirectional system, Fermi gas at unitarity, in which the appropriate physical mechanism is unclear. It is shown that key differences in the resolution of nonlinear wave breaking allow one to design regularization determining experiments.