Localised states in the Zhang-Vinals equations

Can the Zhang–Vinals (ZV) equations be used to understand the underlying mechanism that has led, in certain experimental settings, to highly localised, oscillating states within the Faraday wave system?

The Faraday wave system has been found to be quite versatile in terms of the patterns that can be formed on the surface of a fluid undergoing vertical vibrations. A simple Faraday wave experiment places a layer of fluid on a rigid, horizontal plate that is vibrated vertically at a certain frequency and acceleration (in a sinusoidal manner). When a critical acceleration (or critical forcing) is surpassed, the patternless surface loses stability to patterns whose symmetry depends on the parameters of the system. The contribution of the Faraday system to the field of fluid dynamics can be measured by the longevity of interest in its rich dynamics, dating from the early recordings of Faraday (1831) to more recent experiments that display a range of fascinating surface patterns.

The stability of various patterns that have been observed has been investigated theoretically, and it is evident (see Cross and Hohenberg 1993 and Miles and Henderson 1990 for reviews) that the types of models that aim to describe the Faraday wave system exhibit interesting nonlinear behaviour regarding pattern formation. Most analytical investigations have focused on global patterns (patterns that fill the experimental domain, for example). However, highly localised patterns have been found in the Faraday system that have so far received less attention. Localised patterns that oscillate in time have been termed oscillons (Gleiser, 1994). These oscillons can exist in both a homogeneous and a patterned background, and form as peaks and craters on the fluid surface. Experimentally they have been shown to exist in a variety of situations in both Newtonian (Arbell and Fineberg, 2000) and non-Newtonian (Lioubashevski et al., 1999) fluids. The experiments of Umbanhowar et al. (1996) show that oscillons also exist in granular media with similar characteristics to those excited in fluids.

The Zhang–Vinals (ZV) model is a fluid dynamics model that is derived from first principles in the limit of small viscosity (via a reduction of the Navier–Stokes equations) and has been shown to include properties critical to global pattern formation. The ZV model’s potential contribution to the understanding of localised states within the Faraday system has not previously been explored in detail. A derivation is presented in this thesis that closely accounts for the relative sizes of the fluid properties near onset of instability which is supported by results from a linear stability analysis of the Navier–Stokes equations. A previously unidentified scaling assumption was highlighted from the derivation. In order to neglect nonlinear viscous terms in the derivation of the ZV equations, the size of the surface displacement must be small relative to the thickness of the viscous boundary layer near the surface. This may be indirectly related to the “uncontrolled approximation” present in the original derivation (Zhang and Vinals, 1997a,b; Chen and Vinals, 1999).

Results from a combination of analytical and numerical techniques are presented to outline a methodology for searching for localised states in the ZV equations. Guided by the experiments of Arbell and Fineberg (2000), the new methodology is presented for localised hexagonal patterns which oscillate harmonically with respect to a two-frequency forcing in the ratio 2:3. A parameter range was found where solutions to numerical simulations of the ZV equations converged to temporally harmonic, localised hexagonal patterns existing among a flat (patternless) background. The localised patterns were distinguished by the number of fully formed peaks present on a local hexagonal lattice. Distinct patterns with 31, 43, and 55 localised peaks were found. The existence of localised solutions in a system describing the Faraday wave phenomenon that is derived from first principles is a new and important result which aids further investigation regarding localised states in the ZV equations. Localised oscillating states have been found in model PDEs which incorporate periodic forcing (Alnahdi et al., 2018), the theory of which may be extended to the ZV system within the parameter range highlighted in the presented work.