The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate

We consider a viscous drop, loaded with an insoluble surfactant, spreading over a flat plane that is covered initially with a thin liquid film. Lubrication theory allows the flow to be modelled using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the multi-region asymptotic structure of the spatially one-dimensional spreading flow and derive a simplified ODE model that captures its dominant features at large times. The model includes a version of Tanner's law accounting for a Marangoni flux through the drop's effective contact line, the magnitude of which is influenced by a rarefaction wave in the film ahead of the contact line. Focusing on the neighbourhood of the contact line, we then examine the stability of small-amplitude disturbances with spanwise variation, using long-wavelength asymptotics and numerical simulations to describe the growth-rate/wavenumber relationship. In addition to revealing physical mechanisms and new scaling properties, our analysis shows how initial conditions and transient dynamics have a long-lived influence on late-time flow structures, spreading rates and contact-line stability

An adaptive moving mesh method for two-dimensional thin film flow equations with surface tension

In this paper, we extend our previous work [A. Alharbi and S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Computational and Applied Mathematics, 319 (2017), pp. 365-384.] on a one-dimensional r-adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations (MMPDEs) to two dimensions. As a test problem, we consider the gravitydriven thin film flow down an inclined and pre-wetted plane including surface tension and a moving contact line. This technique accurately captures and resolves the moving contact line and associated fingering instability. Moreover, the computational effort is hugely reduced in comparison to a fixed uniform mesh

Liquid film dynamics in horizontal and tilted tubes: dry spots and sliding drops

Using a model derived from lubrication theory, we consider the evolution of a thin viscous film coating the interior or exterior of a cylindrical tube. The flow is driven by surface tension and gravity and the liquid is assumed to wet the cylinder perfectly. When the tube is horizontal, we use large-time simulations to describe the bifurcation structure of the capillary equilibria appearing at low Bond number. We identify a new film configuration in which an isolated dry patch appears at the top of the tube and demonstrate hysteresis in the transition between rivulets and annular collars as the tube length is varied. For a tube tilted to the vertical, we show how a long initially uniform rivulet can break up first into isolated drops and then annular collars, which subsequently merge. We also show that the speed at which a localized drop moves down the base of a tilted tube is non-monotonic in tilt angle