Final Report Summary - TFE (Investigation of mathematical models for thin-film flows) The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that we experience in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nano-fluidics and micro-fluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the scientific literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, rapidly rotating substrates and evaporating films, as well as those manipulated via use of electric fields to produce nanoscale patterns.Our project had the following objectives • studying exact asymptotic behaviour of travelling wave solutions for the thin film equation with non-zero contact angles; • constructing generalised weak solutions for multi-dimensional coating flow models without surfactant and investigating of the behaviour of support of these solutions; • analysing the asymptotic behaviour of generalised weak solutions for multi-dimensional coating flows contaminated by surfactant. We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike all previously known results, the surface diffusion coefficient is not assumed to be a constant and its value depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations. We also prove the existence of global non-negative weak solutions for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown. We proceed by examining surface-induced ordering in confined binary alloys. The hyperbolic initial boundary value problem is used to describe a scenario of spatio-temporal ordering in a disordered phase. As a result, a highly anisotropic surface-induced bulk separation is discovered. In long-time dynamics, asymptotically periodic spatia-temporal distributions of non-conserved order parameter and of one binary alloy component are found. It is shown that the order parameter admits decomposition into the sum of two waves travelling in opposite directions with non-decaying amplitudes and the number of "discontinuity points", where the phase transition from one ordered phase to another takes place, is finite or infinite. In the first case, we obtain solutions of relaxation type for the mathematical problem. In the second case, we obtain solutions of pre-turbulent or turbulent type. The corresponding bifurcation diagram for long time solutions is constructed. For the non-ideal binary alloys asymptotically quasi-periodic spatiotemporal distributions are found on torus. Existence of invariant solutions that are elements of strange chaotic and strange non-chaotic attractors is also shown.In summary, our main results achieved so far are(i) sufficient conditions for finite speed of support propagation and for waiting time phenomena for a generalised solution of a system of two coupled parabolic equations that model the spreading of a drop of an insoluble surfactant on a thin liquid film;(ii) the existence theorem for weak solutions to the initial-boundary value problem for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films;(iii) bifurcation diagram for long time behaviour of travelling-wave solutions to hyperbolic initial boundary value problem describing a scenario of spatio-temporal ordering in confined binary alloys.We developed much needed alternative ways to study accurately and in detail nonlinear degenerate parabolic equations arising, among others, in the lubrication theory. These equations have variety of applications: in biological and biomedical contexts, for instance, medical implants, lung airways and linings, tear-film flows; in geophysical settings, for example, mud, granular and debris flows; industrial coating (spin coating, liquid thin-film flows on vibrating substrate)Importantly the modern and emerging areas of micro-fluidics and nano-fluidics naturally call upon techniques associated with thin fluid films. Thus the field is currently thriving with new discoveries and applications occurring almost daily. In parallel with technologically important applications, the techniques available to tackle the nonlinear equations that arise have dramatically improved. As a result, our theoretical results are interesting from an applied point of view as they give qualitative information about the behaviour of a thin film. Because UK applied mathematics is closely connected with end-users in the industrial, engineering and biomedical sectors we believe that these results could help toward solving open applied problems.
