Projects per year
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 twodimensional profiles; the onedimensional 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 twodimensional. Consideration of the highfrequency modes allows us to compute an estimate for the analytic extensibility of solutions in C2. We also examine the addition of a physically derived thirdorder 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 spatiotemporal 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 language  English 

Article number  20170687 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  474 
Issue number  2211 
Early online date  28 Mar 2018 
DOIs  
Publication status  Published  Mar 2018 
Keywords
 Kuramoto–Sivashinsky equation
 spatiotemporal chaos
 active dissipative–dispersive nonlinear PDE
Projects
 2 Finished

Controlling the complex behaviour of microfluidic flows using surfactants. (Leverhulme Early Career Fellowship)
Kalogirou, A. & Blyth, M.
4/01/17 → 30/04/19
Project: Fellowship

The Mathematics of Multilayer Microfluidics: Analysis, Hybrid Modelling and Novel Simulations Underpinning New Technologies at the Microscale.
Blyth, M., Papageorgiou, D., Crowdy, D. & Tseluiko, D.
Engineering and Physical Sciences Research Council
1/02/14 → 31/01/17
Project: Research