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

Ruben Tomlin, Anna Kalogirou, Demetrios Papageorgiou

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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.
Original languageEnglish
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume474
Issue number2211
Early online date28 Mar 2018
DOIs
Publication statusPublished - Mar 2018

Keywords

  • Kuramoto–Sivashinsky equation
  • spatio-temporal chaos
  • active dissipative–dispersive nonlinear PDE

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