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

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 scriptL2 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 double-struckC2. 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.

Point-actuated feedback control of multidimensional interfaces

“…Analogously, with the addition of point actuators, we find that once stability of the fourth-order scheme has been achieved (the scheme is not stable for ∆t = 2 × 10 −4 ), then C 1 (2) is accurate to 3 decimal places (2 significant figures). For the unforced equation, numerical solutions showed that the spectrum of solutions decays exponentially, indicating analyticity (Tomlin et al, 2018). It was also observed that the spectrum decays faster in the transverse k2 modes than the streamwise k1 modes; this is expected given the asymmetry of the linear part of (1.2).…”

“…we have a fixed number of actuators per unit area, and set α = 150 (this is chosen to be large so that the control strength is not a limiting factor -see the previous subsection). A finite energy density is observed for solutions of the uncontrolled system with κ = 0 on large domains (Tomlin et al, 2018), thus it is expected that the number of actuators required per unit area for successful control should be constant in the limit of large periodicities, L 1 , L 2 → ∞.…”

“…(2.3) Equation (1.2) may thus be replaced by an equivalent system of ODEs for the Fourier coefficients The unbounded growth for κ < 0 is due to a linear (Rayleigh-Taylor) instability in transverse modes (assuming L 2 is sufficiently large) which is not saturated by the nonlinearity -this can be seen from the ODE system (2.4) where the nonlinear (summation) term has no contribution if k1 = 0. Further details of the dynamical regimes of (1.2,2.4) can be found in Tomlin et al (2019); in particular, the vertical falling film case has been considered extensively both in analytical and numerical studies (Nepomnyashchy, 1974a,b;Pinto, 1999Pinto, , 2001Akrivis et al, 2016;Tomlin et al, 2018). We focus our attention on stabilising the dynamics in the unstable regimes, κ < 1.…”

Optimal Control of Thin Liquid Films and Transverse Mode Effects

“…In this case, the transverse modes of uncontrolled solutions are damped for any choice of Q with chaotic dynamics emerging for sufficiently large L 1 and L 2 . In [85], the authors found that the time-averaged energy density behaves as η := lim…”

Global solutions of the two-dimensional Kuramoto-Sivashinsky equation with a linearly growing mode in each direction

“…The fourth-order nature of the parabolic evolution (1) of course implies the absence of a maximum principle. Other authors have shown that related systems with maximum principles do have global solutions [20], [22], or have also modified the nonlinear term, showing that related equations have finite-time singularities [8] or global solutions [11], [26]; see also [30] for a numerical study of a modified equation. The second author and Feng have shown that a modification of (1) with additional advection also has global solutions [13].…”