Abstract
Drop coalescence occurs through the rapid growth of a liquid bridge that connects the two drops. At early times after contact, the bridge dynamics is typically self-similar, with details depending on the geometry and viscosity of the liquid. In this paper we analyse the coalescence of two-dimensional viscous drops that float on a quiescent deep pool; such drops are called liquid lenses. The analysis is based on the thin-sheet equations, which were recently shown to accurately capture experiments of liquid lens coalescence. It is found that the bridge dynamics follows a self-similar solution at leading order, but, depending on the large-scale boundary conditions on the drop, significant corrections may arise to this solution. This dynamics is studied in detail using numerical simulations and through matched asymptotics. We show that the liquid lens coalescence can involve a global translation of the drops, a feature that is confirmed experimentally.
Highlights
Coalescence of drops is one of the most common capillarity-driven phenomena which can be observed in multiphase fluid dynamics
We provide a detailed analysis of the coalescence of highly viscous lenses and elucidate the coupling between the inner ‘bridge’ solution and the global dynamics of the drops
(Ristenpart et al 2006; Hernández-Sánchez et al 2012) the flow in the bridge region is quasi-two-dimensional in the early stage of coalescence; (ii) the equilibrium contact angle θ is small, such that a slender body approximation can be employed; (iii) the influence of the bath on the dynamics is negligible, i.e. free slip boundary conditions can be employed at both interfaces of the two-dimensional lenses; (iv) due to negligible differences in surface tension between the bath and liquid lens and the liquid lens and air, the liquid lenses are assumed to be symmetrical with respect to the bath–air interface – asymmetric surface tensions can be mapped to an ‘effective’ symmetric surface tension
Summary
Coalescence of drops is one of the most common capillarity-driven phenomena which can be observed in multiphase fluid dynamics. The viscous similarity analysis, contains a salient feature that remains to be explained: the obtained self-similar velocity profile does not decay at large distance from the thin bridge region, but reaches a finite value This is rather unusual for problems involving coalescence (or drop breakup, cf Eggers & Fontelos 2015). These floating drops do not remain stationary, but their centres of mass exhibit an inward motion as soon as the drops establish contact. The thin-sheet equations admit an outer solution where the drop’s centre of mass can migrate freely, closely resembling the motion observed in figure 2(b) In this latter case, the corrections to the leading-order result are much smaller.
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