1. Introduction
In general, compound droplets exist in two types [
1]. The first type of compound droplets is available with the outer interface partially enclosing its inner droplets [
2]. Such a compound droplet is not considered in the present study. In the second type, a compound droplet consists of an outer interface (i.e., outer droplet) that completely encapsulates one or many inner droplets [
3,
4,
5]. When the droplet is small, it is all called a “double or multiple emulsion droplet” [
6]. The second type has shown its applications in many academic problems and industrial processes, for example, biotechnological processes [
7], delivery of drugs [
8] and some other applications [
9]. In these processes and applications, a compound droplet can collide with one or many other ones and the dynamical behaviors of collisions have been classified into three main modes—reversing, passing-over and merging [
10,
11]. Before coming into contact with others, a compound droplet may carry one or many daughter droplets [
6]. In this study, to simplify we pay attention to the compound droplets with one inner core (i.e., single-core compound droplets) that are suspended in a simple shear flow [
11,
12,
13].
The dynamics of one single-core compound droplet in the simple shear flow has been investigated carefully for many years [
14,
15,
16,
17]. For example, Luo and his co-workers [
17] considered an initially concentric droplet with its finite deformation caused by the shear of the continuous outer flow and the authors claimed that the compound droplets exhibit less deformation than those without any inner core (i.e., single-phase droplets). Chen et al. [
15] numerically showed that the shear flow may cause the compound droplet to decompose into smaller droplets in three mechanisms—instability breakup, necking breakup and end-breakup. The transitions between the breakup and the finite deformation of the compound droplet were presented by Vu et al. [
12]. However, as mentioned, these studies focused on the dynamical deformation and breakup of one droplet, which does not interact with any other. In addition, these works have not considered the effects of the compound droplet eccentricity. In a few other works, for example, References [
18,
19], the authors considered the role of the location of the inner droplet, that is, the eccentric compound droplet, in the dynamical behaviors of the droplet. Once again, these works [
18,
19] have not extended to the cases of binary droplets collisions as done in our present study.
Recently, few attempts have been made to investigate the dynamical interaction of two single-core compound droplets [
10,
11,
20]. For instance, Liu and co-workers [
11,
20] considered two colliding modes (passing-over and reversing) of PVA (polyvinyl alcohol) solution/PS (polystyrene) solution/PVA solution droplets and compared with single-phase droplets. The authors found that the compound droplet moves in a similar trajectory but deforms differently as compared with the single-phase droplet. Revisiting this problem, our previous work [
10] showed that, in addition to the passing-over and reversing modes, two compound droplets may become one as they are in contact with each other. This mode of collisions is merging. However, these works have not considered the effects of the position of the inner droplets on the collision behaviors of the droplets. This missing gap is the focus of our present work.
As aforementioned, in various applications of compound droplets including single-core compound droplets, droplet collisions may appear and the colliding behaviors are affected by the presence of the inner droplets [
10,
11]. Many applications indicated that when compound droplets are carried out by the continuous flow, their inner droplets are not concentric (e.g., Figure 1b in Reference [
21], Figures 4 and 5 in References [
22,
23]) before or during the contact stage. Accordingly, the presence of the inner droplet and its position within its enclosing outer interface (i.e., droplet eccentricity) before the contacting stage should have an important role in the colliding behaviors. However, so far, such a detailed investigation has not been found. Hence, in this study, we consider eccentric compound droplets and investigate how the eccentricity leaves its impact on the dynamics of the collision. The rest of the paper is as follows. In the following section, we describe the numerical problem and its solving method. We then present the results of colliding behaviors under the influence of eccentricity. Finally, some concluding remarks are provided. To ease understanding, some animations of droplet collisions are also provided in the
Supplementary Materials.
2. Numerical Problem and Method
In the present study, we deal with the collision of two compound droplets with only one core in each droplet (known as “single-core compound droplets”), which are suspended in a simple shear flow (
Figure 1). As mentioned below, we focus on small-sized droplets with a little density difference and ignored buoyancy effects and thus the motions of the droplets during the collision are in the plane through their centers of mass [
10,
11]. Accordingly, a two-dimensional configuration to facilitate high-resolution computations with a reasonable computational cost, as considered in this study, can be accepted [
10] even though three-dimensional simulations should be conducted. Initially, the compound droplets are placed symmetrically to the center (
xc,
yc) of the domain and the inner and outer droplets are circular and eccentric with the horizontal (
εx0) and vertical (
εy0) eccentricities defined as:
Here, ∆x and ∆y are the horizontal and vertical distances between two droplets with the subscripts i and o denoting respectively the inner and outer droplets. (xci1, yci1) and (xco1, yco1) are respectively the coordinates of the centroid of the inner droplet 1 and that of the outer droplet 1. Accordingly, for example, εx0 < 0 and εy0 = 0 correspond, in terms of droplet centroids, to the outer droplets closer than the inner droplets in the horizontal direction. The initial radii of the inner and outer droplets are R1 and R2, respectively. The inner droplet contains the inner fluid denoted by “fluid 1,” the fluid in between the inner and outer interfaces, that is, the middle fluid, is called “fluid 2” and the rest is the outer, carrying fluid denoted by “fluid 3” Three fluids are assumed immiscible and each fluid has a constant density (denoted by ρ) and a constant viscosity (denoted by µ). The interfacial tension coefficients of the inner and outer droplet interfaces are respectively denoted by σ1 and σ2.
To handle the droplet interface and its movement, a front-tracking method [
24,
25] is used. The interface is modeled by a finite number of straight-line segments whose point coordinates
xf are updated by integrating the following equation:
In Equation (3), the velocity
Vn is interpolated from the nearest velocities solved from the following governing equations [
10], on a background, rectangular and fixed grid:
Equations (4)–(6) are for Newtonian and incompressible fluids with
u = (
u, v)—velocity vector,
p—pressure,
t—time,
Fb = (
Fbx,
Fby)—body force (e.g., gravity) and
Fs = (
Fsx,
Fsy)—interfacial tension force defined as:
In Equation (7), κ is the mean curvature,
nf is the unit vector normal to the droplet interface and the subscript
f represents the interface.
nf points outward from the droplet. δ is the Dirac delta function. Like our previous paper [
10], we consider small droplets with the size from a few micrometers to a few millimeters [
11,
20] and thus the effect of gravity is neglected (i.e.,
Fb =
0).
x = (
x,
y) is the position vector and
xf indicates the position of the interface.
Equations (4)–(6) are discretized by a finite difference method and the discretized equations are solved on the uniformly distributed, staggered background grid. A second-order predictor-corrector scheme is used to integrating the equations in time. Such discretization and integration are to give the highest possible accuracy [
24]. We use the location of the inner droplet to construct the indicator functions to specify three fluids and their fluid properties at every location in the domain [
10].
The boundary conditions are as follows. At the left and right boundaries, we set a periodic condition. At the top boundary,
u = (
U, 0) is specified and
u = (–
U, 0) is set at the bottom boundary. Accordingly, a shear rate γ is created:
where
H is the height of the domain and is chosen as 6
R2. The domain width is denoted by
W and equal to 24
R2 (
Figure 1) [
10]. The dimensionless time is
.
When the droplets move, the length of interface segments is changed in time due to the movement of the interface points (Equation (3)) but it is always in the range of 0.2
h–0.8
h by deleting or adding front elements, where
h is the grid spacing of the background grid [
24]. To handle the merging of two droplets, we calculate the distance from one segment of one droplet interface to another segment of the other droplet interface. As this distance is smaller than 0.5
h, the droplets perform merging. This coalescence technique has been widely used in front-tracking-based simulations [
24]. Like other front-capturing techniques, for example, the level set method [
26] and the volume of fluid method [
27], the coalescence is dependent on the grid spacing. However, we can easily add a physics-based coalescence treatment to our method but it is not considered in this study. The method was implemented in Fortran (i.e., an in-house serial code) originally developed by Unverdi and Tryggvason [
28]. The code was run on a 3.6 GHz Intel Core i9-9900 K workstation. The CPU time, for example, for a typical case, using a 1536 × 384 grid resolution (discussed below) was about fifty-three hours.
Figure 2 presents the results of our grid convergence study for a merging mode using four grid resolutions 0512 × 128, 1024 × 256, 1536 × 384 and 2048 × 512. The parameters defined below are
Re = 1.58,
Ca = 0.01,
R12 = 0.65,
σ12 = 1.0, ∆
xo0/
R2 = 3.0, ∆
yo0/
R2 = 0.6,
εx0 =
εy0 = 0.0. This merging case is selected for the grid study because the merging mode is sensitive to the grid resolution. The shapes of the droplets plotted at τ = 2.0, 4.0 and 12.0 are compared among four grids.
Figure 2 indicates that no difference in the droplet shape is available before merging (τ = 2.0 and 4.0). However, because of merging at different resolutions, the shapes of the droplets are different after merging. The most difference is for the inner droplet and the coarsest grid, while we see almost the same results for the outer droplet shape with a little difference in the inner droplet shape when 1536 × 384 and 2048 × 512 were used. Accordingly, we believe that the grid resolution of 1536 × 384 used for the computational results presented below is acceptable. The grid independence study for a passing mode with a higher
Ca (i.e.,
Ca = 0.1) was described in Reference [
10]. It is noting that the grid resolution used in the present study is finer than that in our previous [
10]. The method accuracy was also confirmed in our previous work [
10] as illustrated in
Figure 3. For this comparison, the values of the Capillary and Reynolds numbers are 0.25 and 1.0. Three fluids of the compound droplets are identical. Initially, the droplets are concentric. The inner-to-outer radius ratios
R12 (defined below) of 0.5 is computed and the steady-state shape of the droplet and its outer deformation parameter are compared with those of Hua et al. [
16]. The comparison indicates clearly that the method predicts very well the deformation of the compound droplets in the simple shear flow. A more detailed description of this validation can be found in Reference [
10]. More comparisons supporting the method accuracy can be found in References [
12,
13,
24,
29].
4. Conclusions
We have presented the computational results of two single-core compound droplets colliding in the simple shear flow undergoing the influence of eccentricity. The Capillary number Ca is varied in the range of 0.005 to 0.32 with Re = 1.6, R12 = 0.65 and the ratios of material properties set to unity. The eccentricities in terms of εx0 (x direction) and εy0 (y direction) are varied from −0.3 to 0.3. The results are three modes of collisions (merging, reversing and passing-over) recognized. The collision modes for Ca ≤ 0.01 resulting in the merging mode and Ca ≥ 0.16 leading to the formation of the passing-over mode are almost independent of the location of the inner droplets. In contrast, when the value of Ca is varied from 0.02 to 0.16, the eccentricity becomes important and affects the modes of collisions. Particularly, εx0 varying from −0.3 to 0.3 can cause the collision behavior to change from a passing-over mode to a reversing mode and ends at a merging mode, for example, for Ca = 0.02 with εy0 = 0.0. While varying in this range (−0.3–0.3), εy0 induces the transitions just between the merging and reversing modes (e.g., for Ca = 0.01 and εx0 = 0.15), or between the passing-over and reversing modes (e.g., for Ca = 0.04 and εx0 = 0.15), or between the passing-over and merging modes (e.g., for Ca = 0.02 and εx0 = 0.0).
The present study (or our previous work [
10]) is limited to two-dimensional results and single-core compound droplets. Accordingly, three-dimensional computations are necessary for more accurate predictions and to consider the effect of the eccentricity caused by the inner droplet located out of the plane through the centers of two outer droplets. In addition, to find out what will happen when the compound droplets encapsulate many inner droplets.