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Investigation of mathematical models for thin-film flows

Final Report Summary - TFE (Investigation of mathematical models for thin-film flows)

1.Final publishable summary report

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 are familiar 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 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, and 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;
- proving the existence of 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 with surfactant.

We proved the existence, uniqueness and asymptotic behaviour of travelling wave solutions to the thin film equation with non-zero contact angle of the interface. By studying qualitative properties of the relevant nonlinear third-order ordinary differential equations, we obtained explicit lower and upper bounds for these solutions. Moreover, we analysed the stability of stationary travelling wave solutions. We also proved the uniqueness of nonnegative entropy solutions of the thin film equation. Uniqueness was obtained under assumptions that the initial data satisfy some finite entropy condition and that the solution is locally monotone at the touchdown point. The new dissipated functional was used to prove an auxiliary energy equality, and application of Gronwall's lemma then led to the proof of uniqueness. We investigated energy solutions of a Cauchy problem for the p-Laplace evolution equation with nonlinear gradient absorption and nonnegative compactly supported initial data. We obtained sufficient local asymptotic conditions on the initial data that imply backward motion and waiting time phenomena. We also considered an elastic deformation effect on the character of the magnetic-field penetration into high-temperature type-II superconductors. The effective creep activation barrier was assumed to depend nonlinearly upon the transport current density.

The elastic properties of the vortex lattice and the influence of the elastic modulus on the depth of the magnetic field penetration into a sample were taken into account. We considered the formation of self-organized spatial-temporal oscillating structures in symmetric binary polymer blends confined by two flat walls. This phenomenon was modelled by an initial boundary-value problem for the conserved order parameter (or the concentration of one of the components in a binary mixture). Under a special choice of the dynamical boundary conditions, these structures look like lamellar structures. We studied the existence of weak and strong solutions to the initial-boundary value problem for the thin film type equation with unstable diffusion in multi-dimensional domains. Depending on the initial data and the parameter values, we proved the local and global in time existence of the nonnegative weak and strong solutions.

We studied the initial-boundary value problem for the fourth order Mullins equation. It was proved that weak solutions existed globally in time and we could also show that the energy minimizing steady state is the global attractor. We proved local and global in time existence of nonnegative weak solutions to a coupled system of two degenerate parabolic equations which models the spreading of an insoluble surfactant on a thin liquid film. We considered a nonlinear fourth order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. Sufficient conditions for finite speed of support propagation and for waiting time phenomena were obtained. We also considered pinning effect on the character of the magnetic field penetration in high-temperature type-II superconductors in the case of a rigid vortex lattice. Depth and velocity of a magnetic wave were defined for superconductors which were in the viscous flow vortex phase with a nonlinear dependence of the critical current density on the magnetic field induction. To the best of our knowledge this is the first result that takes into account the dependence of the critical current on the value of hydrostatic pressure which changes the character of the magnetic field penetration into a superconductor.

In summary, our main results achieved so far are
(i)the theorem about the existence, uniqueness and asymptotic behaviour of travelling wave solutions to the thin film equation with non-zero contact angle of the interface;
(ii)the theorem about the existence of weak and strong solutions to the initial-boundary value problem for the thin film type equation with unstable diffusion in multi-dimensional domains;
(iii)sufficient conditions for finite speed of support propagation and for waiting time phenomena for a generalised solution of a nonlinear fourth-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity.

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 many applied problems.