Universal evolution of a viscous–capillary spreading drop

Abstract

The rate of spreading or retraction of a drop on a flat substrate is determined through a balance of surface tension and hydrodynamic flow. While asymptotic regimes are known, no general rate equation has hitherto been available. Here, we revisit this classic problem, in a regime governed by capillary and viscous forces, by performing an exhaustive numerical study of drop evolution as a function of the contact angle with the substrate. Our study reveals a universal evolution of the drop radius parameterised only by the substrate wettability. Two limits of this evolution recover the familiar exponential and algebraic regimes. Our results show quantitative comparison with the evolution derived from lubrication theory, indicating that dissipation at the contact line is the key determinant in drop evolution. Our work, both numerical and theoretical, provides a foundation for studying the full temporal dynamics of droplet evolution under the influence of external fields and thermal fluctuations, which are of importance in nanofluidics.