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.