Introduction

When droplets are immersed in immiscible liquid and contact with each other, under the interfacial tension, they will coalesce into a single droplet with smaller surface area. Droplet coalescence is an important physical process in microgravity experiments and applications (Banhart et al. 2008; Antar et al. 2003; Chen and Deng 2017; Han and Chen 2019). Likewise, such liquid-liquid interface evolution during coalescence of the droplets is common in practical practices like microfluidic chip (Christopher et al. 2009; Wang et al. 2018; Shlegel et al. 2019), microreactor (Wang et al. 2013; Chen et al. 2016; Li et al. 2020), chemical extraction (Okubo et al. 2004; Jildeh et al. 2013; Khoobi et al. 2013), fusion energy(Lan et al. 2016; Liu et al. 2016) and petrochemical processes (Amini et al. 2012), and has a great influence on heat and mass transfer(Zhang et al. 2011; Zhang et al., 2019a, b; Zhang et al., 2019a, b). Therefore, knowing unstable evolution of the interface under interfacial tension will help manipulating and optimizing practical uses and has significant scientific and practical values.

Droplet coalescence is very complicated under the impact of external fluid and subject to many factors(Liu and Zhang, 2011). According to experimental study (Gilet et al. 2007), the coalescence process was mainly subject to Bond number (Bo, relating gravitational force to surface tension force), Ohnesorge number (Oh, relating viscous force to surface tension force), as well as viscosity ratio and density ratio. When Bo and Oh are both less than critical values, the coalescence process was mainly controlled by surface tension forces and the coalesced droplets were smaller than the original ones; when Bo and Oh are both greater than the critical values, the droplets would be fully coalesced. As Bo and Oh decreased, the radius ratio between coalesced droplet and the original droplets increased until reaching a constant value, which was about 0.5. Charles et al. (Charles and Mason 1960) studied the droplet coalescence in liquid-liquid interface very early. They found that liquid columns appeared after the coalescence, and then shrunk and pinched off at the bottom; in the meanwhile, small droplets were generated. The radius ratio between generated small droplet and original droplets was related to the viscosity ratio of internal and external fluids. When the viscosity ratio is 1, the generated small droplets had the maximum radius; when the viscosity ratio is less than 0.02 or greater than 11, there’ll be no small droplets generated. Small droplets were more likely to be generated by adding surfactant or in electric field. (Mohamed-Kassim and Longmire 2004) adopted particle image velocimetry (PIV) method to study the droplet coalescence through an even liquid-liquid interface. When the droplets contact the interface and before they coalesce, there’s a liquid film with ambient liquid as the working medium. The droplets usually coalesce when the liquid film ruptures unsymmetrically.

Additives and additional force field also have critical impact on the coalescence process. For example, even immiscible oil droplets will coalesce in water under the stimulation of electro-hydrodynamic flows (Vigo and Ristenpart 2010). In shear flow field within confined space, there’s critical capillary number, which is subject to droplet size, confinement degree and shear flow field strength, making the droplets coalesce (Al-Mulla and Gupta 2000; Chen et al. 2009). When droplets collide with each other in external fluid, the coalescence type will be influenced by the collision way. (Borrell et al. 2004) taken polystyrene polyethylene (PS-PE) added to PS/HDPE mixture for instance, it can effectively restrain coalescence of droplets floating in the mixture. (Lyu et al. 2002; Charles and Mason 1960) found by experiment that small droplets tended to be generated with surfactant or in electric field. They studied the mechanism of such condition to clarify the complicated coalescence process of droplets.

In the previous studies, the initial stage of the coalescence process is of important focus. (Eggers et al. 1999) studied the morphology and radius changes of liquid bridge connecting two droplets in the early coalescing stage. The result indicated that, like droplet coalescence in the air, the liquid bridge connecting two droplets in external fluid also had radius changing meeting Rb/R ∝ t0.5 during the early coalescing stage. When the droplets were coalesced, the external fluid formed vacuoles with meniscus radius meeting R∝Rb1.5 within the droplets. (Ishiguro et al. 2004) studied the droplet coalescence controlled by curvature and mobility of interface instead of viscous dissipation and flow inertia of working medium, and found that the bridge radius changing during early coalescing stage met Rb/R ∝ t1/3 rather than existing study result (Rb/R ∝ t0.5). The study of (Aarts et al. 2005; Aarts and Lekkerkerker 2008) on the early stage of droplet coalescence showed that, with low and high Reynolds, the droplet coalescence driven by surface tension gradually weakened due to viscosity force and inertia respectively. As for droplet coalescence with big enough viscosity or small enough surface tension in viscosity area, the bridge radius expanded in constant capillary speed. According to study on droplet coalescence with extremely low surface tension, the coalescence process can be divided into three stages: (1) draining of the liquid film between droplets; (2) rupture of the liquid film between droplets; (3) expanding of liquid bridge between droplets. In the third stage, the bridge radius expanded in constant capillary speed.

The scholars also paid attention to coalescence of nonspherical droplets in external fluid. For example, (Yokota and Okumura 2011) observed the coalescence of droplet immersed in different liquid and the liquid-liquid interface between two plates with distance less than the droplet radius. They found that the early coalescence stage can be described by existing 3D viscous hydrodynamic theory of spherical droplet coalescence. However, in confined space, since the droplet radius was larger than the distance between the plates, the later coalescence stage can be described by Quasi-2d viscous hydrodynamic theory. The result is significant for droplet coalescence in confined space like microfluidics. (Burton and Taborek 2007) observed the coalescence of circular oil firm (could be considered as a 2D structure) floating on water surface. They found that the coalescence process was similar to coalescence of 3D liquid droplets. But small droplets generated due to pinch-off after liquid droplet coalescence were hardly seen in oil firm coalescence.

The researches above mostly adopted experimental methods to observe the liquid-liquid interface evolution during droplet coalescence and analyzed the impacts of difference factors on the coalescence. However, these researches are still insufficient for revealing the mechanism of droplet coalescence under the impact of external fluid. For this purpose, based on the VOF method (Krause et al. 2011; Chen et al. 2015; Chu et al. 2018; Mohammadi et al. 2012) and taking no account of the influence of the gravity, the numerical simulation is carried out to study the coalescence process of droplets in external fluid, and the effects of liquid-liquid interface characteristics, external fluid viscosity and relative size of droplets on the coalescence process are concerned.

Numerical Model

Computational Domain

As shown in Fig. 1, an axisymmetric rectangular region with L = 10 mm and W = 4 mm is used as the computational domain (i.e. the domain enclosed by black solid line and green solid line). There are two droplets suspending in the external fluid. They are both located at the centre of the computational domain. Since the droplets and the needles are axisymmetric, and the needles at two sides are coaxial, only half of the needles and droplets are included in the computational domain.

Fig. 1
figure 1

Schematic of computational domain

Full size image

Theoretical Model

The volume of fluid (VOF) method is utilized to track the movement of liquid-liquid interface(Chen et al. 2010). The volume fraction functions defining liquid phase of the external and the internal fluid phase (droplet) are αo and αi respectively, therefore, αo + αi = 1. The computational domain of the whole flow field satisfies the following transport equation:

$$ \frac{{\partial \alpha}_{\mathrm{o}}}{\partial t}+\nabla \cdotp \left(\mathbf{v}\cdotp {\upalpha}_{\mathrm{o}}\right)=0 $$
(1)
$$ \frac{{\partial \alpha}_{\mathrm{i}}}{\partial t}+\nabla \cdotp \left(\mathbf{v}\cdotp {\upalpha}_{\mathrm{i}}\right)=0 $$
(2)

By computing volume fraction functions in the grid, we get to know the phase distribution in the grid, i.e.:

$$ \left\{\begin{array}{c}{\alpha}_o=0\kern2em Droplet\\ {}0<{\alpha}_v<1\kern1em Interface\\ {}{\alpha}_o=1\kern2em External\kern0.33em fluid\end{array}\right. $$
(3)

The entire flow field satisfies the continuity equation:

$$ \frac{\partial }{\partial t}\left(\rho \right)+\nabla \cdotp \left(\rho v\right)=0 $$
(4)

and momentum equation:

$$ \frac{\partial \left(\rho \mathbf{v}\right)}{\partial t}+\nabla \cdotp \left(\rho \mathbf{v}\mathbf{v}\right)=-\nabla p+\nabla \cdotp \left[\mu \left(\nabla \mathbf{v}+\nabla {\mathbf{v}}^{\mathrm{T}}\right)\right]+{\mathbf{F}}_{\mathbf{v}\mathbf{ol}} $$
(5)

The physical property parameters in Eqs. (1)-(5) can be described as follows:

$$ \rho ={\alpha}_o{\rho}_o+{\alpha}_i{\rho}_i $$
(6)
$$ \mu ={\alpha}_o{\mu}_o+{\alpha}_i{\mu}_i $$
(7)

To take the impact of surface tension into full consideration, continuous surface force model is introduced into the theoretical model. The volume force Fvol is as follows:

$$ {F}_{vol}=\sigma \frac{\rho k\nabla {\alpha}_o}{0.5\left({\rho}_o+{\rho}_i\right)} $$
(8)

where the curvature of liquid-vapor interface is as follows:

$$ k=-\left[\nabla \cdotp \left(\frac{1}{\left|\nabla {\alpha}_o\right|}\nabla {\alpha}_o\right)\right] $$
(9)

Numerical Solution Methods and Boundary Conditions

Numerical Solution Methods

PLIC method is adopted to reconstruct the liquid-liquid interface. Laminar model is used for simulation, while PISO method for proximity correction and skewness correction is adopted for coupling of pressure field and velocity field. The momentum equation is discretized by second-order up-wind differencing scheme, while the rest equations are discretized by first-order up-wind differencing scheme. For better convergence, under-relaxation factors (pressure: 0.2; density: 0.5; body force: 0.4; momentum: 0.2) are used for certain parameters in the governing equations. The computational time step during unsteady solution is set as 10−6 s., and 10−5 is the residual convergence criteria for parameters in the fields during computation.

Boundary Conditions

As shown in Fig. 1, the bottom edge of the computational domain is set as the axisymmetric boundary condition. Non-slip boundary condition is imposed for contact surface between two fluid phases and the needle walls (inner and outer walls):

$$ {u}_w={v}_w=0 $$
(10)

Constant-velocity inlet condition is imposed for two needle inlets:

$$ x=0,\kern1em y\in \left(\mathrm{0,0.1}\right)\kern1.43em u=0 $$
(11)
$$ x=10,\kern1em y\in \left(\mathrm{0,0.1}\right)\kern1.43em u=0 $$
(12)

Outflow boundary condition is imposed for other boundaries:

$$ x=0\kern2.00em y\in \left(\mathrm{0.1,4}\right)\kern2.10em \frac{\partial u}{\partial x}=\frac{\partial v}{\partial x}=0 $$
(13)
$$ x=10,\kern1em y\in \left(\mathrm{0.1,4}\right)\kern1.43em \frac{\partial u}{\partial x}=\frac{\partial v}{\partial x}=0 $$
(14)
$$ x\in \left(0,10\right),\kern1em y=4\kern1.33em \frac{\partial u}{\partial y}=\frac{\partial v}{\partial y}=0 $$
(15)

Grid Partition

Figure 2 shows grid partitioning of computational domain in Fig. 1. As shown in the figure, the computational domain is partitioned into structured quadrilateral grid. The area with droplet coalescence in is partitioned into uniform quadrilateral grid. Since the external fluid far away from the droplets has very little influence on the coalescence and to reduce the computational work, grids in other area becomes scanty towards the computational boundary. After testing the grid independence, the grid number adopted in computation is determined as 43,790.

Fig. 2
figure 2

Grid partition

Full size image

Model Validation

To verify the theoretical model adopted in the current study, coalescence of droplets (R* = 1.22, Oh = 0.31) immersed in water is simulated (the working medium of droplet is the mixture of silicone oil and 1,2-dichloroethane with volume ratio of 0.88:0.12). The numerical simulation results are compared with the experimental data to verify if the theoretical model is correct. Figure 3(a) and (b) show visualized experimental results of the droplets coalescence process and its comparison with the numerical simulation results respectively. According to the figures, under the simulated experiment condition, the numerical simulation results are quite consistent with the visualized results of flow patterns observed during experiment, indicating that the theoretical model is rational and reliable and can be used to predicate the behavior characteristics of liquid-liquid interface evolution during droplet coalescence.

Fig. 3
figure 3

Comparison of experimental results and numerical simulation results of droplet coalescence (R* = 1.22, Oh = 0.31) (a) images observed during experiment, (b) numerical simulation results

Full size image

Data Analysis

The equivalent radius of the droplets is treated as the characteristic size and can be calculated with the following equation:

$$ R=\frac{2{R}_l{R}_s}{R_l+{R}_s} $$
(16)

where Rl and Rs are radiuses of the bigger and smaller droplets respectively. When the droplets coalesce in external fluid, evolution of the liquid-liquid interface is mainly influenced by surface tension and viscosity force. The relative severity of these two forces can be presented by Ohnesorge number as follows:

$$ Oh=\frac{\mu_d}{\sqrt{\rho \sigma R}} $$
(17)

The viscosity ratio of droplet and external fluid and relative size of droplets can also influence the coalescence process and can be presented as follows:

$$ K=\frac{\mu_d}{\mu_o} $$
(18)
$$ R\ast =\frac{R_l}{R_s} $$
(19)

Dimensionless time is adopted to represent coalescence progress of the droplets and can be calculated as follows:

$$ t\ast =\frac{t}{t_c} $$
(20)

where tc is the capillary time(Chen et al. 2015). In Eqs. (17) to (20), σ, μd, μo, ρ and t are the surface tension coefficient of the interface between droplet and water, droplet viscosity, external fluid viscosity, droplet density and coalescence time after the droplets contacting each other, respectively.

Numerical Simulation Results and Analysis

Three types of coalescence include coalescence with no pinch-off (Type I), coalescence with single pinch-off (type II) and coalescence with double pinch-off (type III) are obtained. However, comparing with the droplet coalescence in air, the impact of gravity on the coalescence process is neglected and the influence of the viscous surrounding fluid is non-ignorable in the current study. These differences result in different types of droplet coalescence in surrounding liquid and air. In these coalescence types with pinch-off, the neck pinches off in its first necking stage in this study. It is different from that of droplet coalescence in air that there is the coalescence type of “coalescence with pinch-off during the second necking stage”. Furthermore, without the effect of gravity pushing liquid to converge from one droplet to the other one, coalescence type with double pinch-off is obtained in the current investigation. The process of droplet coalescence in surrounding liquid with non-ignorable viscosity is influenced by viscosity ratio of internal and external fluids, relative size of the droplet and Oh number, etc.

Impact of K on the Coalescence

Unlike droplet coalescence in the air, for which the impact of external fluid can be ignored, coalescence of droplets immersed in other external fluid with much higher viscosity could be influenced by the external fluid due to viscous dissipation.

According to Table 1, which shows type distribution of coalescence in internal and external fluids with different viscosities when Oh is constant, as K decreases, the viscous dissipation of external fluid imposes increasingly obvious impact on the droplet coalescence, while the coalescence process is subject to increasing flow resistance. As the coalescence process evolves and with smaller K, kinetic energy of droplet for overcoming the interfacial energy barrier gets fewer. Therefore, the coalescence type transforms from coalescence with pinch-off to coalescence with no pinch-off.

Table 1 Coalescence type distribution under different viscosity ratios of internal and droplet when Oh = 0.1.
Full size table

Figure 4 compares the pressure nephograms and velocity vectors at the liquid bridge under different viscosity ratios of droplet and external fluids at the t* = 0.023 when Oh = 0.1 and R* = 1. Like droplet coalescence in the air, (Chen et al. 2015) there’s low pressure area in the liquid bridge. Due to the capillary pressure difference between the droplet and the liquid bridge, (Chen et al. 2015) the working fluid flows from the droplet to the liquid bridge, making volume of liquid bridge increases. By comparing pressure nephogram at the liquid bridge during droplet coalescence in the air and that in external fluid (Fig. 4), it’s found that, for droplet coalescence in external fluid, there’s high pressure area (the area indicated by an ellipse in Fig. 4) inside external fluid near the liquid bridge, which indicates that the external fluid could impede the droplet coalescence. By comparing Fig. 4(a-1) and (b-1), it can be concluded that the smaller the K is, the larger the high pressure area could be. Besides, larger pressure difference between external fluid and the droplet could lead to more obvious impediment to the droplet coalescence.

Fig. 4
figure 4

Comparing pressure and velocity distribution at the liquid bridge under different viscosity ratios of droplet and external fluids (Oh = 0.1, R* = 1, t* = 0.023). (a): K = 1000, (a-1) pressure nephogram, (a-2) velocity vector; (b): K = 0.1, (b-1) pressure nephogram, (b-2) velocity vector

Full size image

By comparing pressure nehograms at the liquid bridge under the two wording conditions, it’s found that, with smaller the viscosity ratio between droplet and external fluid, both high pressure area in the liquid bridge and capillary pressure difference with the droplet are smaller. According to the velocity vector of flowing fluid at the liquid bridge, with great viscosity ratio between droplet and external fluid, the fluid flows fast to the liquid bridge and the liquid-liquid interface between droplet and external fluid also changes fast. It indicates that the smaller the external fluid viscosity is, the faster the droplet coalescence process will be, which means greater external fluid viscosity will lead to more obvious impediment to the droplet coalescence.

Figure 5 shows the evolution of bridge radius under different viscosity ratios of droplet and external fluid within the same time period, when K = 0.001 and t* = 0.023, the neck pinches off. As indicated in Fig. 5, as the liquid bridge evolves, the smaller the K (i.e. viscosity of the external fluid) is, the faster the bridge radius increases. In other words, with smaller external fluid viscosity, the liquid bridge changes faster, the droplets coalesce faster and the external fluid is less impeditive to the coalescence process. Furthermore, the faster the coalescence process is, the less time will be spent and the coalescence will be subject to less impact of viscous dissipation. Consequently, coalescence type with pinch-off is more likely to appear, as shown in Table 1.

Fig. 5
figure 5

Comparing liquid bridge radius changes after droplet coalescence under different viscosity ratios of internal and external fluids

Full size image

Like droplet coalescence in the air, as volume of the liquid bridge increases, the droplet decreases in volume and forms the neck which either pinches off or does not pinch off. (Chen et al. 2015) As for the pinch-off, the neck change mechanism is just like that of droplet coalescence in the air. When shrinking to thin enough, high pressure area forms in the neck owing to the net capillary force at the neck (Chen et al. 2015). Due to the pressure difference between high pressure neck and the droplet, the liquid gradually flows out as shown by the velocity vector in Fig. 6 (b). Finally, the neck will pinch off. However, for the cases of the neck with no pinch-off, due to the impact of viscous dissipation of droplet and surrounding liquid, the shrinking of the neck slows down gradually, and cannot be thin enough. Instead of high pressure, low pressure region shown in the dashed box in Fig. 7 arises in the neck. Then the liquid in droplet will flow into the neck under the pressure difference, as shown by the velocity vector in Fig. 7 (b). Ultimately, the neck disappears during the coalescence process and no pinch-off of neck is brought out.

Fig. 6
figure 6

Comparing pressure and velocity distribution at the neck under the droplet coalescence type with pinch-off. (a) Pressure nephogram, (b) Velocity vector

Full size image
Fig. 7
figure 7

Comparing pressure and velocity distribution at the neck under the droplet coalescence type with no pinch-off. (a) Pressure nephogram, (b) Velocity vector

Full size image

With greater viscosity ratio of the droplet and external fluid, the external fluid will be less impeditive to the droplet coalescence. As for coalescence type with pinch-off, the neck may pinch off in shorter time. As shown in Fig. 8, dimensionless pinch-off time of the neck after coalescence decreases with the increase of the viscosity ratio (K) of the droplet and external fluid. Besides, when K is larger than 10, the viscosity ratio of the droplet and external fluid has little impact on dimensionless pinch-off time of the neck.

Fig. 8
figure 8

The impact of K on neck pinch-off time after droplet coalescence in the water

Full size image

Impact of Oh on the Droplet Coalescence

When two droplets contact each other, they start to coalesce due to interfacial tension. Surface energy of the droplets transforms into kinetic energy of the flowing fluid. However, due to the impact of viscous dissipation, kinetic energy of the flowing fluid is gradually consumed during coalescence process. As Oh represents the strength of viscosity force comparing to interfacial tension, when Oh gets larger, viscous dissipation will have more obvious impact on coalescence process of the droplets and consumes more kinetic energy of the flowing fluid; besides, after the neck is formed, the kinetic energy of remaining fluid in it for overcoming interfacial energy barrier is less. Therefore, as Oh increases, the droplet coalescence type transforms from coalescence with pinch-off to coalescence with no pinch-off, as shown in Table 2.

Table 2 Droplet coalescence type distribution under different Oh
Full size table

When the droplets contact each other, they coalesce rapidly under the impact of interfacial tension. With large Oh and relatively small interfacial tension, there will be relatively small force driving the coalescence. Besides, under the larger viscosity force, flow resistance during coalescence is more obvious. Therefore, the larger Oh is, the slower the changing interface is during coalescence process. In addition, the bridge radius increases more slowly. As indicated in Fig. 9, it compares the changing of bridge radius under different Oh. When the droplets have the same equivalent radiuses and R* is 1, at the same time, the larger Oh is, the smaller the bridge radius will be. Furthermore, with large Oh, there will be relatively small interfacial tension driving the bridge expansion but large viscosity force impeding the expansion. Therefore, the bridge radius increases more slowly with larger Oh, shown as the curve of Oh = 1 in Fig. 9.

Fig. 9
figure 9

Comparing bridge radius changes during droplet coalescence under different Oh

Full size image

Figure 10 shows the impact of Oh on the dimensionless pinch-off time of the neck. It indicates that, as the viscous dissipation getting increasingly obvious with the enlarging Oh, dimensionless pinch-off time of the neck increases. Meanwhile, the results suggest that, if Oh is less than 0.01, it has little impact on pinch-off time. The hollow triangle symbol in the figure represents the dimensionless pinch-off time of the neck at the smaller droplet side with “coalescence type with double pinch-offs” when R* = 1.75、Oh = 0.001.

Fig. 10
figure 10

Impact of Oh on neck pinch-off time

Full size image

Impact of R* on the Droplet Coalescence

Relative size of the droplet also has great influence on the droplet coalescence process. Table 3 shows the coalescence type distribution under different relative sizes of the droplet. With large R*, which means large size difference between droplets, the neck typically pinches off once after the droplet coalescence, and the pinch-off takes places at the smaller droplet side. As R* decreases, under the small Oh number (for example, Oh = 0.001), the coalescence type transforms from “coalescence with single pinch-off” (Type II) to “coalescence with double pinch-offs” (Type III). If the Oh number is large (for example, Oh = 1), the coalescence type transforms from “coalescence with single pinch-off” (Type II) to “coalescence with no pinch-off” (Type I). As R* decreases, size difference between droplets reduces and the neck could be formed at the both sides of the liquid bridge during coalescence.

Table 3 Coalescence type distribution under different R*
Full size table

When Oh is small, viscous dissipation has little impact on the coalescence process. The fluid in necks at both sides has sufficient kinetic energy to overcome the interfacial energy barrier, resulting in double pinch-offs. Moreover, under the experimental conditions of R* = 1.75, Oh = 0.05 and R* = 1.5, Oh = 1, it takes a long time for the fluid to flow from larger droplet to the liquid bridge, hence the remaining kinetic energy is insufficient to overcome the interfacial energy barrier due to viscous dissipation and the neck doesn’t pinch off. On the other hand, there’s less fluid in the smaller droplet and it takes longer time for it to flow to the liquid bridge. Being subject to less impact from viscous dissipation, the kinetic energy of remaining fluid is able to overcome the interfacial energy barrier and the neck pinches off. It suggests that, when there’s size difference between the droplets, the fluid at two sides flows out and forms neck asynchronously. Besides, viscous dissipation has different impacts on the coalescence processes, which also indicates that relative size of the droplet has great influence on the coalescence.

Figure 11 shows how dimensionless neck pinch-off time changes with R*. With large R*, the volume of smaller droplet is relatively small. Therefore, there’s less fluid and it could be drained from the smaller droplet in shorter time, which could facilitate the formation of the neck and its pinch-off and lead to shorter dimensionless neck pinch-off time after coalescence. Therefore, dimensionless pinch-off time of the neck after coalescence decreases with increase of R*. The hollow box in the Fig. 9 represents dimensionless pinch-off time of the neck at the smaller droplet side under the “coalescence type with double pinch-offs.

Fig. 11
figure 11

The dimensionless pinch-off time versus R*

Full size image

Conclusions

To study the internal mechanism of the liquid-liquid interface evolution during droplet coalescence under the impact of external fluid, in this paper, numerical simulation is carried out by taking no account of the influence of the gravity. The impacts of viscosity ratio of droplet and external fluids, as well as the relative size and Oh number of the droplets on the coalescence process are analyzed. According to the numerical simulation results:

  1. 1.

    As the viscosity ratio of the droplet and external fluid decreases, the droplet coalescence process is subject to increasing impacts from the viscous dissipation, and the droplet coalescence type transforms from coalescence with no pinch-off to coalescence with pinch-off.

  2. 2.

    The greater the viscosity ratio of droplet and external fluid is, the faster the fluid flows to the liquid bridge and the faster the bridge radius increases over time. Dimensionless pinch-off time of the neck decreases with the increase of the viscosity ratio of the droplet and external fluid. When the viscosity ratio of the droplet and external fluid is greater than 10, it has little impact on dimensionless pinch-off time of the neck.

  3. 3.

    As the radius radio of the larger and smaller droplets decreases, the coalescence type transforms from “coalescence with single pinch-off” to “coalescence with double pinch-offs” or “coalescence with no pinch-off”. Dimensionless pinch-off time of the neck decreases with increase of the radius radio.

  4. 4.

    With the increase of Oh, the droplet coalescence type transforms from coalescence with pinch-off to coalescence with no pinch-off. Under greater Oh, due to greater viscous resistance during coalescence process, the bridge radius increases more slowly. Dimensionless pinch-off time of the neck increases as the increase of Oh. Moreover, when Oh is less than 0.01, it does not have much impact on pinch-off time.