Scaling Laws of Droplet Coalescence: Theory and Numerical Simulation

Abstract

When two Newtonian liquid droplets are brought into contact on a solid substrate, a highly curved meniscus neck is established between the two which transforms the bihemispherically shaped fluid domain to a hemispherically shaped domain. The rate at which such topological transformation, called coalescence phenomenon, evolves results from a competition between the inertial force which resists the transformation, the interfacial force which promotes the rate, and the viscous force which arrests it. Depending on the behaviour of these forces, different scaling laws describing the neck growth can be observed, predicted theoretically, and proved numerically. The twofold objective of the present contribution is to propose a simple theoretical framework which leads to an Ordinary Differential Equation, the solution of which predicts the different scaling laws in various limits, and to validate these theoretical predictions numerically by modelling the phenomenon in the commercial Finite Element software COMSOL Multiphysics.

On droplets coalescence in a quasi-2D fluid

“…The surface energy contributes to the pressure inside the droplets and an additional line tension at the droplet-film contact line. The full 3D model is computationally challenging, especially because of the ill-posed nature of the boundary condition at the contact line and the singularity at the initial stage of the bridge formation 18,23 . The problem can be significantly simplified if we adopt the lubrication approximation and reduce the problem to two dimensions by averaging the flow profile across the thickness of the film and neglecting the inertia.…”

“…The problem can be significantly simplified if we adopt the lubrication approximation and reduce the problem to two dimensions by averaging the flow profile across the thickness of the film and neglecting the inertia. Such approach has been used to describe the coalescence of sessile droplets on a solid substrate 18,20,25 . In this model, the contact line singularity is relieved by the introduction of a precursor film stabilised by the disjoining pressure.…”

“…The coalescence of sessile droplets was extensively studied in 8,18,20,25 . Due to the non-slip boundary conditions at the bottom interface of the droplets and the free slip at the top interface, the governing equation for h in lubrication approximation is 25…”

“…Pioneering studies of the droplet coalescence date back to the works of Lord Rayleigh in the nineteenth century 16,17 . Many experimental and theoretical investigations of the 3D and 2D coalescence of liquid objects have been performed during the last two decades to fully understand the merging process of liquid objects 2,8,[18][19][20][21][22][23][24][25][26] .…”

“…The coalescence of 3D droplets exhibits of three different dynamic regimes such as the inertial limited viscous regime at the very early stage of the neck formation, viscous regime dominating at the sufficiently early time of the neck formation, the inertial regime occurring at later times 19,23,27 . Interactions with solid and viscoelastic substrate strongly affect the dynamics of the coalescing droplets 5,18,[28][29][30] .…”