Pattern formation on time-dependent domains

Abstract

In the quest to understand the dynamics of distributed systems on time-dependent spatial domains, we study experimentally the response to domain deformations by Faraday wave patterns – standing waves formed on the free surface of a liquid layer due to its vertical vibration – chosen as a paradigm owing to their historical use in testing new theories and ideas. In our experimental set-up of a vibrating water container with controlled positions of lateral walls and liquid layer depth, the characteristics of the patterns are measured using the Fourier transform profilometry technique, which allows us to reconstruct an accurate time history of the pattern three-dimensional landscape and reveal how it reacts to the domain dynamics on various length and time scales. Analysis of Faraday waves on growing, shrinking and oscillating domains leads to a number of intriguing results. First, the observation of a transverse instability – namely, when a two-dimensional pattern experiences an instability in the direction orthogonal to the direction of the domain deformation – provides a new facet to the stability picture compared to the one-dimensional systems in which the longitudinal (Eckhaus) instability accounts for pattern transformation on time-varying domains. Second, the domain evolution rate is found to be a key factor dictating the patterns observed on the path between the initial and final domain aspect ratios. Its effects range from allowing the formation of complex sequences of patterns to impeding the appearance of any new pattern on the path. Third, the shrinkage–growth process turns out to be generally irreversible on a horizontally evolving domain, but becomes reversible in the case of a time-dependent liquid layer depth, i.e. when the dilution and convective effects of the domain flow are absent. These experimentally observed enigmatic effects of the domain size variations in time are complemented here with appropriate theoretical insights elucidating the dynamics of two-dimensional pattern evolution, which proves to be more intricate compared to one-dimensional systems.