Interfacial pattern selection in defiance of linear growth

Abstract

In this study, we revisit Rayleigh’s visionary hypothesis (Rayleigh, Proc. R. Soc. Lond., vol. 29, 1879a, pp. 71–97), that patterns resulting from interfacial instabilities are dominated by the fastest-growing linear mode, as we study nonlinear pattern selection in the context of a linear growth (dispersion) curve that has two peaks of equal height. Such a system is obtained in a physical situation consisting of two liquid layers suspended from a heated ceiling, and exposed to a passive gas. Both interfaces are then susceptible to thermocapillary and Rayleigh–Taylor instabilities, which lead to rupture/pinch off via a subcritical bifurcation. The corresponding mathematical model consists of long-wavelength evolution equations which are amenable to extensive numerical exploration. We find that, despite having equal linear growth rates, either one of the peak-modes can completely dominate the other as a result of nonlinear interactions. Importantly, the dominant peak continues to dictate the pattern even when its growth rate is made slightly smaller, thereby providing a definite counter-example to Rayleigh’s conjecture. Although quite complex, the qualitative features of the peak-mode interaction are successfully captured by a low-order three-mode ordinary differential equation model based on truncated Galerkin projection. Far from being governed by simple linear theory, the final pattern is sensitive even to the phase difference between peak-mode perturbations. For sufficiently long domains, this phase effect is shown to result in the emergence of coexisting patterns, wherein each peak-mode dominates in a different region of the domain.