Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows

A thin liquid film coating a planar horizontal substrate may be unstable to perturbations in the film thickness due to unfavourable intermolecular interactions between the liquid and the substrate, which may lead to finitetime rupture. The self-similar nature of the rupture has been studied before by utilising the standard lubrication approximation along with the Derjaguin (or disjoining) pressure formalism used to account for the intermolecular interactions, and a particular form of the disjoining pressure with exponent n = 3 has been used, namely, Π(h) ∝ −1/h3, where h is the film thickness. In the present study, we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent n (not necessarily equal to 3), which has not been previously performed. We focus on axisymmetric point-rupture solutions and show for the first time that pairs of solution branches merge as n decreases, starting at nc ≈ 1.485. We verify that this observation also holds true for plane-symmetric line-rupture solutions for which the critical value turns out to be slightly larger than for the axisymmetric case, nplane c ≈ 1.499. Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading, increasingly small structures

Axisymmetric self-similar rupture of thin films with general disjoining pressure

A thin film coating a dewetting substrate may be unstable to perturbations in the thickness, which leads to finite time rupture. The self-similar nature of the rupture has been studied by numerous authors for a particular form of the disjoining pressure, with exponent n = 3. In the present study we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent n. Pairs of solution branches merge when n is close to unity, indicating that a more detailed examination of the dynamics of a thin film in this regime is warranted. We also numerically evaluate the power law behaviour of characteristic quantities of solutions in the limit of large branch number

Healing capillary films

We investigate, by means of theoretical arguments, numerical simulation and numerics, the closing of a circular cavity (healing) in a thin liquid film. We assume that the process is dominated by capillary forces. The final stages of the evolution can be described by means of self-similar solutions to the problem. A comparison with experimental data is also presented

Healing capillary films

Consider the dynamics of a healing film driven by surface tension, that is, the inward spreading process of a liquid film to fill a hole. The film is modelled using the lubrication (or thin-film) approximation, which results in a fourth-order nonlinear partial differential equation. We obtain a self-similar solution describing the early-time relaxation of an initial step-function condition and a family of self-similar solutions governing the finite-time healing. The similarity exponent of this family of solutions is not determined purely from scaling arguments; instead, the scaling exponent is a function of the finite thickness of the prewetting film, which we determine numerically. Thus, the solutions that govern the finite-time healing are self-similar solutions of the second kind. Laboratory experiments and time-dependent computations of the partial differential equation are also performed. We compare the self-similar profiles and exponents with both measurements in experiments and time-dependent computations near the healing time, and we observe good agreement in each case