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Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions

Tomlin, Ruben J.; Kalogirou, Anna; Papageorgiou, Demetrios T.

Authors

Ruben J. Tomlin

Demetrios T. Papageorgiou



Abstract

A Kuramoto–Sivashinsky equation in two space dimensions arising in thin film flows is considered on doubly periodic domains. In the absence of dispersive effects, this anisotropic equation admits chaotic solutions for sufficiently large length scales with fully two-dimensional profiles; the one-dimensional dynamics observed for thin domains are structurally unstable as the transverse length increases. We find that, independent of the domain size, the characteristic length scale of the profiles in the streamwise direction is about 10 space units, with that in the transverse direction being approximately three times larger. Numerical computations in the chaotic regime provide an estimate for the radius of the absorbing ball in ℒ2 in terms of the length scales, from which we conclude that the system possesses a finite energy density. We show the property of equipartition of energy among the low Fourier modes, and report the disappearance of the inertial range when solution profiles are two-dimensional. Consideration of the high-frequency modes allows us to compute an estimate for the analytic extensibility of solutions in ℂ2. We also examine the addition of a physically derived third-order dispersion to the problem; this has a destabilizing effect, in the sense of reducing analyticity and increasing amplitude of solutions. However, sufficiently large dispersion may regularize the spatio-temporal chaos to travelling waves. We focus on dispersion where chaotic dynamics persist, and study its effect on the interfacial structures, absorbing ball and properties of the power spectrum.

Journal Article Type Article
Publication Date Mar 31, 2018
Journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
Print ISSN 1364-5021
Electronic ISSN 1471-2946
Publisher Royal Society, The
Peer Reviewed Peer Reviewed
Volume 474
Issue 2211
Pages 20170687
APA6 Citation Tomlin, R. J., Kalogirou, A., & Papageorgiou, D. T. (2018). Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2211), 20170687. doi:10.1098/rspa.2017.0687
DOI https://doi.org/10.1098/rspa.2017.0687
Publisher URL https://royalsocietypublishing.org/doi/10.1098/rspa.2017.0687

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