Nonlinear pattern formation in parametric surface waves in weakly viscous fluids is studied by using both analytical and numerical means. The investigation is based on a quasi-potential approximation valid for weakly viscous incomprehensible fluids. The three-dimensional quasi-potential equations are then written in a two-dimensional nonlocal form (2D-QPEs). Standing wave and traveling wave amplitude equations are derived by using a multiple scale perturbation method. The 2D-QPEs are also solved numerically by using a pseudospectral method. In addition, analytical and numerical studies are also performed on a two-dimensional order parameter model to further describe the nonlinear dynamics of parametric surface waves away from onset. On the conceptual level, the findings of our investigation include (i) an amplitude-limiting effect by the driving force in parametrically forced systems; (ii) the importance of three-wave resonant interactions among capillary-gravity waves to pattern selection. Our results provide explanations to a number of recent experimental observations, as well as a number of predictions that await experimental verification. The main results include: (1) We explain why standing wave patterns of square symmetry are observed experimentally near onset of capillary Faraday waves with a sinusoidal forcing. (2) We predict that hexagonal or triangular patterns, and patterns of quasicrystalline symmetry can be stabilized in certain mixed capillary-gravity waves with a sinusoidal forcing. (3) Analytical results for a bicritical line for two-frequency forced Faraday waves are obtained. The results are in qualitative agreement with the available experimental results. (4) The triad resonant condition for capillary-gravity waves is modified for two-frequency forced Faraday waves of frequency ratio 1:2 compared to the case of single frequency forcing. / As a result, square patterns can be unstable for subharmonic responses of the fluid surface even in the capillary wave limit. (5) An order parameter equation (OPE) for a two-dimensional complex field is proposed for weakly damped Faraday waves. Stationary solutions of this OPE become unstable to transverse amplitude modulation (TAM) at a finite value of the reduced driving amplitude. For larger values of the driving force, TAM defects appear, and the system appears to be spatiotemporally chaotic due to the erratic motions of TAM defects. / Source: Dissertation Abstracts International, Volume: 55-11, Section: B, page: 4910. / Co-Major Professors: Jorge Vinals; Dennis W. Duke. / Thesis (Ph.D.)--The Florida State University, 1994.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_77300 |
| Contributors | Zhang, Wenbin., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 225 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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