Derivation and analysis of lubrication models for two-layer thin-films

Abstract

The subject of this thesis is mathematical modeling and analysis of dewetting dynamics and equilibrium patterns of two-layer thin liquid films. We derive systems of coupled thin-film equations for two immiscible liquid layers on a solid substrate that take interfacial slip and intermolecular forces into account. By using these thin-film models, we study the stability of two-layer systems and investigate its dependence on the order of magnitude of slip. The resulting dispersion relations exhibit two local maxima. One appears for small wavenumbers and mainly affects the liquid-gas interface, the other arises for moderate wavenumbers and its associated perturbations have higher amplitudes at the liquid-liquid interface. Varying the slip lengths at the interfaces influences the two maxima, in particular, it may lead to a transition of the dominant wavelength and thereby change the spinodal patterns significantly. Then, we investigate stationary states of flows of thin liquid two-layers via the derived models. Assuming a negative spreading coefficient, which emerges from the intermolecular potential, the energy to the system favors the lower liquid layer to be only partially covered by the upper liquid. On the other hand, the intermolecur forces lead to an ultra-thin layer of thickness h∗. For the stationary problem to the thin-film models we prove existence of solutions. Moreover, in the limit h∗ to 0 we apply matched asymptotic analysis to derive sharp-interface models and corresponding contact angles, i.e. the Neumann triangle. We use Γ-convergence to derive a sharp-interface energy rigorously in this limit and determine the minimizers to this energy. These minimizers agree with the solutions obtained by matched asymptotics. Furthermore, we present comparisons of numerical simulations with experimental results. We also show existence of non-negative global weak solutions to the derived thin-film models for small and moderate slip lengths in the case that intermolecular forces are neglected. In addition, we prove existence of positive smooth solutions when intermolecular forces between the liquids and between the lower liquid and the solid substrate are taken into account. For the thin-film model which allows for large slip we show that weak solutions are stable under perturbations of the initial data.