A mathematical model and mesh-free numerical method for contact-line motion in lubrication theory

Abstract

We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading–including Tanner’s Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here. Article highlights We formulate a mathematical model to regularize the contact-line singularity for droplet spreading. The model can be solved using a fast, accurate mesh-free numerical method. Numerical simulations confirm that the model describes the quantitative aspects of droplet spreading well.

Flow stability in shallow droplets subject to localized heating of the bottom plate

“…A grid-refinement study has been carried out in the context of off-centerd heating, which we describe below in Section IV. The discretization produces a system of nonlinear equations for each time step which is solved using a Newton's method [25,26]. The initial condition is radially symmetrical with the form…”

Role of volatility and thermal properties in droplet spreading: a generalisation to Tanner's law

“…The wetting dynamics of non-volatile liquids are known to be governed by capillary force and viscous dissipation. Specifically, the spreading rate of a non-volatile Newtonian droplet in perfect wetting cases can be quantitatively described by Tanner's law, R(t) ∼ t 1/10 (Tanner 1979;Lelah & Marmur 1981;Blake 2006), which can be derived mathematically by employing the lubrication approximation and balancing the effect of capillary forces with the resisting viscous forces generated by the droplet motion (Voinov 1976;Carlson 2018;Pang & Náraigh 2022). The spreading law has then been validated by successive experiments (Marmur 1983) and further extended to non-Newtonian liquids (Rafaï, Bonn & Boudaoud 2004;Jalaal, Stoeber & Balmforth 2021), gravity-influenced cases (Cazabat & Stuart 1986), near-critical cases (Saiseau et al 2022), liquids on non-rigid solids and on thin liquid films (Carré, Gastel & Shanahan 1996;Cormier et al 2012), as well as under-liquid systems (Goossens et al 2011;Mitra & Mitra 2016).…”