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Interfacial pattern selection in defiance of linear growth

Published online by Cambridge University Press:  19 September 2017

Jason R. Picardo Open the ORCID record for Jason R. Picardo [Opens in a new window]  and
R. Narayanan
Jason R. Picardo
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, TN 600036, India Department of Chemical Engineering, University of Florida, Gainesville, Fl 32608, USA
R. Narayanan
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, Fl 32608, USA
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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.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 829 , 25 October 2017 , pp. 345 - 363
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Copyright
© 2017 Cambridge University Press 

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