Numerics of thin-film free boundary problems for partial wetting

We present a novel framework to solve thin-film equations with an explicit non-zero contact angle, where the support of the solution is treated as an unknown. The algorithm uses a finite element method based on a gradient formulation of the thin-film equations coupled to an arbitrary Lagrangian-Eulerian method for the motion of the support. Features of this algorithm are its simplicity and robustness. We apply this algorithm in 1D and 2D to problems with surface tension, contact angles and with gravity

Dynamics of liquid nanofilms

The van der Waals forces across a very thin liquid layer (nanofilm) in contact with a plane solid wall make the liquid nonhomogeneous. The dynamics of such flat liquid nanofilms is studied in isothermal case. The Navier-Stokes equations are unable to describe fluid motions in very thin films. The notion of surface free energy of a sharp interface separating gas and liquid layer is disqualified. The concept of disjoining pressure replaces the model of surface energy. In the nanofilm a supplementary free energy must be considered as a functional of the density. The equation of fluid motions along the nanofilm is obtained through the Hamilton variational principle by adding, to the conservative forces, the forces of viscosity in lubrication approximation. The evolution equation of the film thickness is deduced and takes into account the variation of the disjoining pressure along the layer.Comment: 13 pages. International Journal of Engineering Science / International Journal of Engineering Sciences 46 (2008) to appea

Dewetting on porous media with aspiration

We consider a porous solid covered with a water film (or with a drop) in situations where the liquid is pumped in, either spontaneously (if the porous medium is hydrophilic) or mechanically (by an external pump). The dynamics of dewetting is then strongly modified. We analyse a few major examples: a) horizontal films, which break at a certain critical thickness, b) the "modified Landau-Levich problem" where a porous plate moves up from a bath and carries a film: aspiration towards the plate limits the height H reached by the film, c) certain situation where the hysteresis of contact angles is important.Comment: Revised version: The analysis of the 'modified Landau-Levich problem' (section 3) has been significantly revised. It is now treated as a singular perturbation problem (using boundary-layer techniques), leading to a more accurate physical pictur

Thermal fluctuations of an interface near a contact line

The effect of thermal fluctuations near a contact line of a liquid interface partially wetting an impenetrable substrate is studied analytically and numerically. Promoting both the interface profile and the contact line position to random variables, we explore the equilibrium properties of the corresponding fluctuating contact line problem based on an interfacial Hamiltonian involving a "contact" binding potential. To facilitate an analytical treatment we consider the case of a one-dimensional interface. The effective boundary condition at the contact line is determined by a dimensionless parameter that encodes the relative importance of thermal energy and substrate energy at the microscopic scale. We find that this parameter controls the transition from a partially wetting to a pseudo-partial wetting state, the latter being characterized by a thin prewetting film of fixed thickness. In the partial wetting regime, instead, the profile typically approaches the substrate via an exponentially thinning prewetting film. We show that, independently of the physics at the microscopic scale, Young's angle is recovered sufficiently far from the substrate. The fluctuations of the interface and of the contact line give rise to an effective disjoining pressure, exponentially decreasing with height. Fluctuations therefore provide a regularization of the singular contact forces occurring in the corresponding deterministic problem.Comment: 40 Pages, 12 Figure

Effects of confinement and surface enhancement on superconductivity

Within the Ginzburg-Landau approach a theoretical study is performed of the effects of confinement on the transition to superconductivity for type-I and type-II materials with surface enhancement. The superconducting order parameter is characterized by a negative surface extrapolation length $b$. This leads to an increase of the critical field $H_{c3}$ and to a surface critical temperature in zero field, $T_{cs}$, which exceeds the bulk $T_c$. When the sample is {\em mesoscopic} of linear size $L$ the surface induces superconductivity in the interior for $T T_{cs}$. In analogy with adsorbed fluids, superconductivity in thin films of type-I materials is akin to {\em capillary condensation} and competes with the interface delocalization or "wetting" transition. The finite-size scaling properties of capillary condensation in superconductors are scrutinized in the limit that the ratio of magnetic penetration depth to superconducting coherence length, $\kappa \equiv \lambda/\xi$, goes to zero, using analytic calculations. While standard finite-size scaling holds for the transition in non-zero magnetic field $H$, an anomalous critical-point shift is found for H=0. The increase of $T_c$ for H=0 is calculated for mesoscopic films, cylindrical wires, and spherical grains of type-I and type-II materials. Surface curvature is shown to induce a significant increase of $T_c$, characterized by a shift $T_c(R)-T_c(\infty)$ inversely proportional to the radius $R$.Comment: 37 pages, 5 figures, accepted for PR

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues