Advances in Mathematical Physics

Volume 2018, Article ID 4906016, 16 pages

https://doi.org/10.1155/2018/4906016

## Scaling Laws of Droplet Coalescence: Theory and Numerical Simulation

^{1}Department of Mathematics, Faculty of Science, University of Mauritius, Réduit 80837, Mauritius^{2}Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand^{3}Department of Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Correspondence should be addressed to M. Irshad Khodabocus; moc.liamg@obadohki

Received 15 July 2018; Accepted 24 September 2018; Published 18 October 2018

Academic Editor: Luigi C. Berselli

Copyright © 2018 M. Irshad Khodabocus et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

When two Newtonian liquid droplets are brought into contact on a solid substrate, a highly curved meniscus bridge is established and thereafter driven by surface tension so as to transform the bulk flow region from bihemispherically shaped at early-time regimes to more hemispherically shaped at late-time regimes. Such topological transformation, involving the merging of two or more sessile droplets into a single one on a solid surface, is called* droplet coalescence phenomenon*.

Physically speaking, the coalescence process consists of three regimes, called the* viscous*,* visco-inertia,* and* inertial coalescence regimes* [1]. In each regime, a range of surprisingly complex behaviours can be observed, so to speak topological changes of the interface [2]. Thus, the subject of droplets coalescence is a fascinating, multifaceted subject of inquiry with many interesting phenomena to learn and discover. Indeed, the subject has intrigued many physicists [3–6] and mathematicians [7, 8].

In actual fact, authors of [9] appear to be the first scientists who have addressed the subject. The authors study the formation of vortex rings by droplets falling into liquids and some allied phenomena. For a miscible droplet falling from not too great a height into its bulk fluid, it is observed that the droplet descends through the bulk fluid as a ring, whereas for the case of an immiscible droplet, [9] observed the droplet to descend through the bulk fluid as a spherical droplet. Thereafter, there have been many attempts on experimental, theoretical, and numerical sides to elucidate the underlying physics governing the dynamics of coalescence.

Authors of [10] have studied the partial coalescence of a water droplet, with radius mm, with the free-surface of a bulk fluid composed of water-glycerol mixtures experimentally. For cases , mm, the authors drew attention that the coalescence process is arrested by gravitational and viscous effects and, depending on these two effects, three coalescence regimes were observed, which [10] termed* gravity*,* inertio-capillary,* and* viscous* regimes. Furthermore, the authors argued that the coalescence time is strongly dependent upon the Bond number and the Ohnesorge number , where , , and denote the density, dynamic viscosity, and surface tension of the droplet, respectively; and denote the properties of its vapour phase surrounding, and denotes the gravitational acceleration.

Reference [7] addressed the more general case of the coalescence of droplets surrounded by a viscous outer fluid, both analytically and numerically, using asymptotic methods and the integral equation given by [11]. Describing the initial coalescence regime by the Stokes equations, they showed that the three-dimensional solution is asymptotically equivalent to the two-dimensional one reported in the works of [12–16]. When the viscosity of the outer fluid is disregarded, the radius and width of the highly curved meniscus bridge were found to obey the following laws: and , respectively, where is the time, and properties , , and are the radius, surface tension, and viscosity constant of the droplet; when the viscosity of the outer fluid is taken into account, a toroidal bubble of radius is formed by the outer fluid, and . Employing the Euler equations in the inertial regime, they showed that , where denotes the density of the coalescing droplets. (In the present study, the property will stand for ; designates the set of property constants upon which it depends.)

The paper of [7] is one classic paper which treats the coalescence of droplets both theoretically and numerically. The method used by [7] is termed* conventional* by [8]. The work of [8] is another classic paper on the subject. They addressed the subject using the conventional method and a different method called* interface formation/disappearance method* and comparing their corresponding results with experimental findings. The authors argued that their proposed model shows better agreement with experimental data than the conventional one.

In the literature, [4] has given a combined theoretical and experimental account of the coalescence of water droplets on a solid substrate surrounded by an atmosphere of nitrogen saturated with water. For small contact angle and large contact angle , the bidroplet undergoes very fast coalescence. Moreover, the relaxation time characterising the coalescence flow is found to be (or, according to [4], ), and the bulk capillary relaxation time, , where , , and are the radius, surface tension, and viscosity constant of the bidroplet system.

From a hydrodynamical viewpoint, [3] found that the time evolution of the characteristic scale of that small liquid bridge—where the meniscus of the bidroplet system is highly curved—obeys a time-dependent scaling/power law: , where . For instance, in the* inertial coalescence regime* it was found that , whereas in the* viscous coalescence regime *. The confirmation of these laws as regards the growth of the property in those coalescence regimes was also reported by earlier scientists. Reference [2] addressed the early-time evolution of the highly curved meniscus bridge experimentally. They found that , where is the inverse of the curvature of the meniscus bridge; by comparison, [7].

On increasing the viscosity of the fluid, [17] has shown the linear dependence of on time. Reference [18] has studied the influence of the geometrical properties of low viscosity liquid droplets experimentally. For contact angles , they deduced , while for their analysis yielded to . Most interestingly, they demonstrated the exponent scaling law in the limit that , thereby unifying the coalescence of liquid droplets and freely suspended droplets in the inertial regime. The exponent scaling law was also argued by [19] who attributed it to the self-similar description of the shape of inviscid pinchoff of a liquid droplet in a still air.

On the other hand, [20] has considered that the influence of surface tension gradient on droplet coalescence was addressed on a combined experimental and numerical basis. The situation considered by these authors was the coalescence of a droplet deposited on the surface of a miscible liquid reservoir. Their results reveal three distinct coalescence regimes based on the reservoir-to-droplet surface tension ratio . For instance, the situation results in the ejection of a small daughter droplet from the top of the top of the coalescing droplet. Contrarily, it was only when that total coalescence could be observed.

Reference [21] has addressed the early-time coalescence of viscous droplets on a flat, wettable substrate. The authors have shown in terms of their respective radii , initial heights , surface tensions , and dynamics viscosities that , which led them to conclude that the evolution of is highly sensible to the geometrical properties of the liquid droplets. Reference [22] considered the coalescence of air bubbles and salt water droplets systems in silicone oils experimentally. The proportionality constant and the -exponent of the scaling law are analysed; the authors argued that the evolution of is independent of the viscosity of its outer fluid, changing only the proportionality constant by a factor roughly equal to and leaving the -exponent unaffected.

The above references reveal extensive studies of the characters of the property with respect to some properly/carefully chosen pair of scalar fields , paying little attention to the characters of the coalescence flow fields in three-dimensional space and on cutting planes and leaving the simultaneous establishment of these power law regimes from one and only one generic equation untouched. Moreover, should an attempt be made to study a class of physical situations ranging from droplets spreading to self-propulsion and coalescence, the several modelling frameworks proposed by those scientists are inconvenient to employ without reconstructing either the mathematical framework or the numerical framework, though the outcomes of their works are no doubt striking. Thus, on the one hand, we must admit that a -evolution equation which permits the deduction of those power laws has never been derived and, on the other hand, the models proposed by those authors make it difficult, if not impossible, to display the global characters of the coalescence flow fields as well as to adapt from one physical situation to another.

In this part, we develop a sound framework which upon variations of an interdependent leading parameter permits to study either the coalescence of two liquid droplets or the spreading of a single droplet on a three-dimensional solid substrate. Taking advantage of the proposed geometrical model, we then establish through theoretical analysis that -evolution equation and derive from it the power law relationships (, , ).

To demonstrate that the proposed model might be adopted with advantage, the , , -exponent scaling laws are proved numerically, and the characters of the coalescence flow fields are illustrated in three-dimensional space and on cutting planes.

#### 2. Theory

##### 2.1. Mathematical Formulations

To formulate the problem mathematically, we reason in the following manner. Suppose two liquid droplets , coalesce on a solid substrate , at some time they will occupy a* master* domain satisfying the geometrical conditionFollowing [8], equality follows from (1) only when the size of the* interfacial circular sector* centered at the point (see Figure 1) is zero, and strict inequality whenever it is nonzero, as is easily understood. In the present work, the domain , which can vary in time, will be termed* coalescence domain*, though the term* bidroplet system* is also attributed to; the set (with ) denotes the period of time over which the study is carried out. Mathematically, the coalescence domain is bounded by two disjoint sets called the* interface * and the* footprint * of the coalescence domain. In Figure 1, we illustrate the coalescence of two viscous liquid droplets , conditioned by* zero-flux* of momentum, mass and energy across the (free) surface and the (contact) surface ; the null flux vectors (with , , ) stand for those boundary conditions. Figure 2 exposes to view the projection of the coalescence domain in the -plane (top) and in the -plane (bottom). Points , designate the rear and centroid of the bidroplet system; and, points , the centroids of its first and second droplets, both taken at the instant of time . Quantities , (with ) are the initial radii of the coalescing droplets, and the characteristic size at their corresponding points of contact. In particular, it is assumed in Figure 2 that the coalescence domain is initially* bihemispherical*, units in length in the direction of , with respective radii , (with , by hypothesis), and satisfying the condition . (We agree that necessarily implies that and, implies that the coalescing droplets have not yet been brought into contact; see Figure 2.)