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Based on the volume of fluid (VOF) method, a theoretical model of compound droplet deformation in curved minichannel is developed. The effects of curved angle, continuous phase, radius ratio between the inner and integral droplets, and viscosity of the middle phase are examined to reveal the underlying mechanism of compound droplet deformation. The results indicate that the deformation process of the compound droplets in the curved minichannel can be divided into three stages, namely, the initial stage, the turning stage, and the adjustment stage. Both large curved angle and high capillary number of the continuous phase result in the large shear force and high eccentricity of the compound droplet. However, as the radius ratio increases, the influence of the inner droplet on the deformation of the compound droplet transits from enhancing to suppressing.

Core fluid encapsulated in the shell fluid, referred as compound droplet, is useful in the storage, transportation, and controlled release of the functional materials, which has a great potential in microreactor [

Several attempts have been applied to research the multiphase flow phenomena in the curved channel [

To provide a guideline for the design of the curved minichannel and the manipulation of the flow field in the curved minichannel, it is extremely important to develop a quantitative analysis that considers the effect of the confinement and the parameters of the liquid. Therefore, based on the VOF method, a theoretical model of the deformation behaviors of the compound droplets in the curved minichannel is developed to investigate the hydrodynamics of the compound droplets in the curved minichannel. The effects of curved angle, continuous phase, radius ratio, and viscosity of the middle phase are analyzed to reveal the deformation mechanism of the compound droplets in the curved minichannel. The current simulation provides a deep understanding of the hydrodynamic behaviors of the compound droplet in the curved minichannel.

To investigate the hydrodynamics of the compound droplet in the curved minichannel, a two-dimensional theoretical model of compound droplets flowing in a curved minichannel is developed. As shown in Figure _{m} and viscosity _{m}) and dispersed phase (density _{d} and viscosity _{d}) are initially set in a rectangular shape and repeat periodically in minichannels, shown in Figure _{c} and viscosity _{c}).

Schematics and computational domain of the mathematical model: (a) compound droplet deformation in the curved minichannel; (b) compound droplet formation.

In this study, the VOF method [_{c}, _{m}, and _{d} are the volume fractions of the continuous phase, middle phase, and dispersed phase, respectively. The density

The densities of the three phases are all 1000 kg/m^{3}, so the gravity force is neglected. Moreover, the viscosities of the three phases are _{c} = _{d} = 0.005 Pa·s, _{m} = 0.05 Pa·s, for the typical condition. The densities and viscosities of each phase are assumed to be constant through the process. The continuity equation is as follows:

And the momentum equation is given by_{i} and the outer interface _{o} are 0.001 N/m.

The finite volume-based commercial software Fluent 6.3 is used to solve the governing equations. The computational domains of the curved minichannel presented in Figure

To verify the grid independence, several different sizes of mesh are used. Figure

Grid independence test of different cell numbers _{c}: (a) _{c} = 23604; (b) _{c} = 73360; (c) _{c} = 106970.

To validate the theoretical model, the numerical simulation is compared with the experimental results. Figure

Schematic of the validation experiment setup.

Comparison of the numerical simulation and the experimental results of compound droplet deformation in the curved minichannel: (a)

Based on the above model, the deformation behaviors of the compound droplets in the curved minichannel are predicted. The effects of curved angle, continuous phase, radius ratio, and viscosity of the middle phase are analyzed to reveal the deformation mechanism of the compound droplets in the curved minichannel.

Figure

Time revolution of compound droplet deformation in the curved minichannel: (a) interface shape; (b) pressure distribution; (c) velocity distribution.

There are diversities of the microfluidic device structures, especially the curved channels with different angles. Therefore, it is urgent to study the effect of the curved angle on the droplet deformation. Figure

Effect of curved angle on the compound droplet deformation: (a)

The external fluid has a significant impact on the hydrodynamic behavior of the compound droplet. Hence, we introduce capillary number of the continuous phase Ca to generally analyze the effect of the continuous phase on the hydrodynamic behavior of the compound droplet:_{c} is the dynamic viscosity of the continuous phase and _{c} is the average inlet velocity in the continuous phase.

As shown in Figure

Effect of capillary number of continuous phase on the interface shape during compound droplet deformation: (a) Ca = 0.05; (b) Ca = 0.025.

Effect of capillary number on the velocity distribution during compound droplet deformation: (a) Ca = 0.05; (b) Ca = 0.025.

Effect of capillary number on the pressure distribution during compound droplet deformation: (a) Ca = 0.05; (b) Ca = 0.025.

The radius ratio between inner and integral droplets

Effect of radius ratio on the compound droplet deformation: (a) interface shape; (b) pressure distribution.

Figure

Effect of the viscosity of the middle phase on the compound droplet deformation: (a) interface shape; (b) pressure distribution.

Based on the VOF method, a theoretical model of the deformation behaviors of the compound droplets in the curved minichannel is developed to investigate the hydrodynamics of the compound droplets in the curved minichannel. The effects of curved angle, continuous phase, radius ratio, and viscosity of the middle phase are analyzed to reveal the deformation mechanism of the compound droplets in the curved minichannel. The results indicate that

The motion of the compound droplets in the curved minichannel can be divided into three stages, i.e., the initial stage, the turning stage, and the adjustment stage.

As the curved angle increases, the eccentricity of the compound droplet is higher and the breakup is easier to occur, due to the higher shear force.

With a relatively large capillary number, the compound droplet exhibits greater deformation due to the increasing shear force from the continuous phase, eventually leading to the breakup of the compound droplet.

At a low radius ratio, the deformation of the integral droplet is promoted by the motion and deformation of the inner droplet. However, at a high radius ratio, the deformation of the integral droplet is suppressed by the inner droplet.

This simulation not only provides a deep understanding of hydrodynamic behaviors of the compound droplet flowing in the curved minichannels but also contributes to precisely controlling the concentricity and sphericity of the compound droplet, which is of significance for the application of the microspheres. Three-dimensional simulation with a more complex channel structure may be considered in the future research work.

Capillary number

Radius ratio between the inner and integral droplets

_{c}:

Cell numbers

Pressure (Pa)

Time (s)

Velocity of the fluid (m/s)

Curved angle (°)

Volume fraction

Dynamic viscosity (Pa·s)

Density (kg m^{−3})

Surface tension (N/m)

Continuous phase

Middle phase

Dispersed phase

Inner interface

Outer interface.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 51776037).