Numerical Simulation of the Dynamic Behavior Exhibited by Charged Droplets Colliding with Liquid Film

Abstract

Since droplet collision with walls has become a research hotspot, scholars have conducted a large number of studies on the dynamic behavior of electrically neutral droplets colliding with dry walls. However, with the rapid development of electrostatic spray technology, there is an increasingly urgent need to study the dynamic process of collision between charged droplets and walls. In this paper, considering the actual working conditions of electrostatic spray, an electric field model is introduced based on the two-phase flow field. Through the coupling of a multiphase flow field and electric field and a multiphysics field, the dynamic numerical calculation method is used to explore the collision electrodynamic behavior of charged droplets and liquid film. The dynamic evolution process of the formation and development of the liquid crown in the collision zone was clarified, and the critical velocity and critical Weber number of the rebound, spreading, and splashing of charged droplets were tracked. The distribution characteristics of electrostatic field, pressure field, and velocity field under different working conditions are analyzed, and the dynamic mechanism of the charged droplet collision liquid film under multi-physics coupling is revealed based on the electro-viscous effect. It is confirmed that the external electric field can increase the critical velocity of droplet splashing and fragmentation and promote the spreading and fusion behavior of droplets and liquid films. The influence of the impact angle of charged droplets on the collision behavior was further explored. It was found that the charged droplets not only have a smaller critical angle for fragmentation and splashing, but also have a faster settling and fusion speed.

1. Introduction

At present, the behavior of droplets colliding with the wall has become a research hotspot in many fields such as rocket propulsion burners, nuclear energy cooling devices, microelectronics heat dissipation, and atomization spraying [1,2,3]. The process of droplet colliding with the wall is usually accompanied by dynamic behaviors such as droplet deformation, rebound, spreading, and splashing fragmentation [4]. Existing studies have shown that droplet velocity, initial particle size, liquid physical properties, and wall characteristics all have an impact on the behavior of droplets colliding with walls [5,6]. As a two-phase flow dynamics process, the collision of liquid droplets with the wall not only involves the interaction between impact inertia force, interfacial tension, viscous force, and contact line force, but also has obvious nonlinear and strong transient characteristics, which undoubtedly poses great difficulties for the study of such problems [7,8,9].

Currently, researchers mainly study the behavior of droplets colliding with the wall through high-speed camera experiments and numerical calculation methods. These researchers mainly focus on the influence of droplet motion, geometry, and physical and chemical properties on droplet collision behavior, in order to effectively control the droplet collision process and reveal its dynamic mechanism. Zhang et al. [10,11] investigated the influence of droplet velocity and morphology on collision force and found that the impact force of droplets increases with the increase in droplet velocity and diameter. Under the same speed conditions, the droplet curvature radius becomes the key factor affecting the initial impact force and wall fusion process. The research results of Yao et al. [12] indicate that apart from the significant influence of droplet velocity on impact dynamics, the droplet impact angle cannot be ignored. In addition, liquid viscosity and surface tension play a decisive role in the maximum spreading diameter and center thickness. Higher viscosity may lead to insufficient droplet spreading, and lower surface tension may lead to overspreading or splashing of droplets [13]. As an extremely complex physical process, the droplet collision with the wall is not only affected by the parameters of the droplet, but also has an inseparable relationship with the properties of the wall [14]. Different wall affinity properties will have an impact on the collision process. When the droplet collides with the affinity wall, the collision usually ends in the form of spreading. When a droplet collides with a sparse wall, due to the low surface energy between the two, it is difficult for the wall to be wetted by the droplet, which can easily lead to droplet rebounding, bouncing, and sliding behavior [15]. In addition, the roughness of the wall also has a certain influence on the collision behavior. Qin et al. [16] investigated the spreading and splashing behavior of droplets on surfaces with different roughness. The results show that compared with smooth surfaces, the viscous resistance of droplets spreading on rough surfaces is greater, so the maximum spreading factor of droplets after collision is smaller. When the droplet collides with the inclined wall, in addition to the above behavior, there is also a droplet slip phenomenon. The greater the inclination of the wall, the farther the droplet slip distance [17]. Jin et al. [18] studied the process of droplets colliding with cold walls at different inclination angles. The experimental results showed that when the surface inclination angle is large enough, the droplets are difficult to maintain a single droplet shape and usually break up to form two or more smaller droplets, accompanied by the generation of satellite droplets. It can be seen that the inclined wall will accelerate the droplet breaking behavior. The walls described in the above studies are all dry walls. However, in most engineering applications, a layer of thick or thin liquid film will be formed after the droplets continue to collide with the wall, which is actually the collision process between the droplets and the wet surface [19]. The existence of the liquid film will undoubtedly have an important impact on the collision behavior of the droplets and ultimately change the contact state of the droplets [20]. Hong et al. [21] showed that a thicker liquid film may lead to faster spreading of droplets and reduced splash behavior, while a thinner film may increase the rebound and splash behavior of droplets. At present, the academic community has established a relatively complete theoretical framework for the collision dynamics of electrically neutral droplets. In contrast, the study of charged droplet collision fluid films is relatively rare. Guo et al. [22] studied the effects of charges and electric fields on droplet collision coalescence and droplet size distribution evolution and calculated the collision efficiency of droplets with different radii and charges under the action of downward electric fields of different intensities. This only proves that charges and electric fields can increase collision efficiency.

With the rapid development of electrostatic spray technology, there is an increasingly urgent need to study the collision behavior between charged droplets and walls. The collision behavior between charged droplets and liquid films is a multi-physics coupling problem of multi-phase flow fields and electric fields. Under the action of an external electric field, the directional transfer and distribution of charges inside the liquid will not only change its surface tension, viscosity, and other physical properties, but also have a relatively complex impact on the shape of the droplet itself. This article focuses on the numerical calculation of the dynamic process of charged droplets colliding with a liquid film, tracking the rebound, spreading, and splashing fragmentation behaviors that occur during the collision of charged droplets with a liquid film, and exploring the critical conditions for the corresponding behaviors. The dynamic mechanism of collision between charged droplets and liquid films was discussed based on the electro-viscous effect. The influence of the droplet impact angle on the collision behavior was further explored, and the dynamic evolution process of the single-sided liquid line was analyzed. The work in this paper has theoretical significance for the in-depth study of the electrodynamic behavior of charged droplet collision liquid films.

2. Materials and Methods

2.1. Theoretical Modeling of Charged Droplet Collision Behavior

(1)

Phase field method and model assumptions

Considering that the size of the investigated droplet is below 350 μm, two-phase flow phase field method is used to simulate the dynamic behavior of charged droplet colliding with the liquid film surface to ensure the accuracy of two-phase interface calculation. Compared with the horizontal set method, the phase field method does not directly track the two-phase flow interface. Instead, the tension is equivalent to the chemical potential product and the variable gradient, which is then incorporated into the fluid equation as the volume force term, and the phase field variable evolution process is controlled by the Cahn-Hilliard 4th order partial differential equation. However, because the calculation amount of phase field method is large, in order to simplify the calculation model and reduce computational complexity, and to ensure that the subsequent model has a solution, the following assumptions need to be made:

(A)

The velocity of the calculated fluid is less than the sonic velocity, so the fluid is assumed to be incompressible.

(B)

In the calculation process, the fluid is Newtonian fluid, and its viscosity and surface tension are assumed to be constant during the collision process.

(C)

The charged droplet is assumed to be regular sphere before collision.

(D)

Assuming that there is no phase change process at the gas-liquid collision interface, the heat transfer behavior between phases is not considered.

(E)

Since the dynamic current is small and the magnetic field effect is negligible, the applied electric field can be considered as an irrotational field.

(2)

Flow field control equation

For incompressible Newtonian fluid, the mass conservation equation is as follows:

$$\nabla ·\mathit{u}=0$$

(1)

The mass conservation equation, also known as the fluid continuity equation, where u is the velocity vector (m/s), is based on the assumption that the fluid is incompressible; the velocity gradient can be set to 0.

Based on Newton’s second law, the total momentum is kept unchanged during the interaction of liquid. Considering pressure, viscous force, gravity, interfacial tension, and applied electric field force, the conservation equation of momentum can be obtained as follows:

$$\rho (\partial \mathit{u}/\partial t)+\rho \left(\mathit{u}·\nabla \right)\mathit{u}=\nabla ·\left[-p\mathit{I}+\mathit{K}\right]+{\mathit{F}}_{\mathrm{s}\mathrm{t}}+\rho \mathbf{g}+{\mathit{F}}_{\mathrm{e}\mathrm{c}}$$

(2)

$$\mathit{K}=\mu \left(\nabla \mathit{u}+{\left(\nabla \mathit{u}\right)}^{\mathit{T}}\right)$$

(3)

where $\rho$ is the density (kg/m³), $p$ is the pressure (Pa), $\mathit{I}$ is the unit matrix, $\mathbf{g}$ is the gravity vector (m/s²), $\mathit{K}$ is the viscous force term, $\mu$ is the dynamic viscosity (Ns/m²), ${\mathit{F}}_{\mathrm{s}\mathrm{t}}$ is the interfacial tension term, and ${\mathit{F}}_{\mathrm{e}\mathrm{c}}$ is the electric field force term.

The physical significance of phase field variables is mainly to describe and quantify the microstructure of materials, as well as reflect various physical properties of the system. In the phase field method, the dimensionless phase field variable $\mathsf{\Phi }$ is used to reflect the two-phase interface diffusion state, and its range and corresponding characterization phase are as follows:

$$\mathsf{\Phi }(X,\mathrm{t})=\left\{\begin{array}{c}1,X\in {\mathsf{\Omega }}_{\mathrm{o}\mathrm{i}\mathrm{l}}\\ 0,X\in {\mathsf{\Gamma }}_{\mathrm{b}\mathrm{o}\mathrm{u}}\\ -1,X\in {\mathsf{\Omega }}_{\mathrm{a}\mathrm{i}\mathrm{r}}\end{array}\right.$$

(4)

When the phase field variable $\mathsf{\Phi }$ is 1, it corresponds to the pure liquid phase (vegetable oil) ${\mathsf{\Omega }}_{\mathrm{o}\mathrm{i}\mathrm{l}}$; when the phase field variable $\mathsf{\Phi }$ is −1, it corresponds to the pure gas phase ${\mathsf{\Omega }}_{\mathrm{a}\mathrm{i}\mathrm{r}}$; when the phase field variable $\mathsf{\Phi }$ is between −1 and 1, it corresponds to the two-phase diffusion interface; when the phase field variable $\mathsf{\Phi }$ is 0, it corresponds to the two-phase interface ${\mathsf{\Gamma }}_{\mathrm{b}\mathrm{o}\mathrm{u}}$, which is calculated by the Cahn–Hilliard equation:

$$(\partial \mathsf{\Phi }/\partial t)+u·\nabla \mathsf{\Phi }=\nabla ·(\gamma \lambda /{\epsilon }^2)\nabla \mathsf{\Psi }$$

(5)

$$\mathsf{\Psi }=-\nabla {\epsilon }^2\nabla \mathsf{\Phi }+\left({\mathsf{\Phi }}^2-1\right)\mathsf{\Phi }$$

(6)

A further solution is as follows:

$$\gamma =\chi {\epsilon }^2$$

(7)

$$\lambda =3\epsilon \sigma /\sqrt{8}$$

(8)

where $\mathsf{\Psi }$ is the phase field auxiliary variable, $\gamma$ is the mobility (m³·s/kg), $\lambda$ is the mixed energy density (J/m³), $\epsilon$ is the interface thickness control parameter (m), $\chi$ is the mobility adjustment parameter (m·s/kg), and $\sigma$ is the surface tension coefficient at the interface (N/m). Generally, the interface thickness $\epsilon$ can be set to be half of the size of the feature grid passed by the two-phase flow interface, so the feature grid shall be refined to ensure the smoothness of the diffusion interface. Mobility $\gamma$ is a measurement parameter of diffusion time, and its selection range needs to be reasonably applicable. If it is too large, excessive inhibition of convection will occur, and if it is too small, it is difficult to maintain the stability of interface thickness.

As an important evaluation parameter, the volume fraction of each phase fluid in the phase field method can be calculated as follows:

$$V_{f1}=(1-\mathsf{\Phi })/2$$

(9)

$$V_{f2}=(1+\mathsf{\Phi })/2$$

(10)

$$V_{f1}+V_{f2}=1$$

(11)

In this model, gas phase is defined as fluid phase 1 and liquid phase is defined as fluid phase 2.

To ensure a smooth transition of the model, the density and viscosity in the diffusion layer are calculated as follows:

$$\rho ={\rho }_1{V}_{f1}+{\rho }_2{V}_{f2}$$

(12)

$$\mu ={\mu }_1{V}_{f1}+{\mu }_2{V}_{f2}$$

(13)

In the phase field method, based on the two-phase flow interface diffusion expression method, the interface tension term can be calculated as follows:

$$F_{st}=G\nabla \mathsf{\Phi }$$

(14)

$$G=\lambda \left[-{\nabla }^2\mathsf{\Phi }+\mathsf{\Phi }\left({\mathsf{\Phi }}^2-1\right)/{\epsilon }^2\right]=(\lambda /{\epsilon }^2)\mathsf{\Psi }$$

(15)

where G is the chemical potential (J/m³).

According to Formulas (14) and (15), the phase field method calculates interface tension in the form of diffusion interface distribution force, avoiding the complex calculation of surface curvature and surface normal vector.

(3)

Electric field control equation

For the study of charged micro-droplet colliding with the liquid film, the corresponding interface of the electric field module should be introduced in the model. In order to track the charge density carried by the liquid phase before and after the collision, the model selects the current interface. In the process of charged droplet collision, the current conservation equation should be satisfied, which is expressed as follows:

$$\nabla ·J=Q_{j,v}$$

(16)

$$J={\sigma }^{\prime }E+J_e$$

(17)

$$E=-\nabla V$$

(18)

where $\mathit{J}$ is the fluid current density (A/m²), $Q_{j,v}$ is the charge density (C/m³), E is the electric field intensity (V/m), $J_e$ is the external current density (A/m²), $V$ is the potential (V), and ${\sigma }^{\prime }$ is conductivity (S/m).

In addition, the conduction and movement of charges at the two-phase interface under the action of an electric field also need to satisfy the following constitutive relationship:

$$J_c={\sigma }^{\prime }E$$

(19)

$$D={\epsilon }_0{\epsilon }_{r}E$$

(20)

where ${\mathit{J}}_{\mathit{c}}$ is the conducted current density (A/m²), and $\mathit{D}$ is the potential shift (C/m²), ${\epsilon }_0$ is the vacuum dielectric constant (F/m), and ${\epsilon }_r$ is the relative dielectric constant of the computational domain.

(4)

Physical field coupling equation

From the above analysis, it can be concluded that the process of charged liquid droplet colliding with a liquid film is a multi-physics field coupling problem, in which the electric field and flow field interact, so it is necessary to couple the electric field equation with the flow field equation. The electric field force can be written into the fluid motion equation in the form of a Maxwell stress tensor, as a volume force participating in the fluid calculation process. The specific expression is as follows:

$$F_{ec}=\nabla ·\mathrm{T}$$

(21)

where $\mathbf{T}$ is the Maxwell stress tensor, and the relationship between other electric field strengths and potentials is as follows:

$$\mathbf{T}=\mathit{E}{\mathit{D}}^{\mathrm{T}}-\frac{1}{2}\left(\mathit{E}·\mathit{D}\right)\mathbf{I}$$

(22)

The matrix component form is:

$$\begin{gathered} \mathbf{T}=\left[\begin{array}{cc}{\mathbf{T}}_{\mathrm{x}\mathrm{x}}& {\mathbf{T}}_{\mathrm{x}\mathrm{y}}\\ {\mathbf{T}}_{\mathrm{y}\mathrm{x}}& {\mathbf{T}}_{\mathrm{y}\mathrm{y}}\end{array}\right]= \\ \left[\begin{array}{cc}{\epsilon }_0{\epsilon }_r{E}_x^2-\frac{1}{2}{\epsilon }_0{\epsilon }_r\left(E_x^2+E_y^2\right)& {\epsilon }_0{\epsilon }_r{E}_x{E}_y\\ {\epsilon }_0{\epsilon }_r{E}_y{E}_x& {\epsilon }_0{\epsilon }_r{E}_y^2-\frac{1}{2}{\epsilon }_0{\epsilon }_r\left(E_x^2+E_y^2\right)\end{array}\right] \end{gathered}$$

(23)

To ensure a smooth transition of the electric field force within the calculation domain, the relative dielectric constant in the calculation domain of the two-phase interface can be expressed as follows:

$${\epsilon }_r={\epsilon }_{r1}{V}_{f1}+{\epsilon }_{r2}{V}_{f2}$$

(24)

where ${\epsilon }_{r1}$ is the relative dielectric constant of gas and ${\epsilon }_{r2}$ is the relative dielectric constant of the liquid.

2.2. Physical Modeling of Charged Droplet Collision Behavior

(1)

Geometric model and boundary setting

Analysis is based on the actual operating conditions of electro-spraying, where a charged liquid droplet collides with the surface of the workpiece at a certain initial velocity. In the initial lubrication stage, the dry wall is formed, and after the wall is wetted, a liquid film is formed. The liquid film always exists in the vast majority of processing steps. Therefore, the behavior of charged liquid droplet colliding with the liquid film is mainly studied. The two-dimensional geometric modeling method is used, as shown in Figure 1. Based on the assumed conditions, the droplet can be set as a circle with a radius of r₀, and its fluid domain is limited by the physical properties of soybean oil, represented by Ω_oil−1. The initial velocity u_y and initial height H₂ of the droplet can be given according to the actual physical meaning of the calculation. Setting the total computational domain height to H₁ = 0.8 mm and width to W = 3 mm, including the soybean oil liquid film region with a height of H₃ = 50 um, represented by Ω_oil−2. Except for the domain limited by droplet and liquid film, set the other computational domains as the air domain, represented by Ω_air. Based on the actual physical meaning of the model and the conditions for solving multiple physical fields, set the boundary conditions of the model. Firstly, based on the flow field properties, the left and right sides and upper boundaries of the air domain are set as the fluid outlet. To avoid the influence of liquid surface tension on the liquid surface morphology, the left and right boundaries of the liquid film are set as the liquid phase inlet, and the geometric lower boundary is set as the Navier sliding wall. Furthermore, based on the electric field properties, simplify the electric field model (it is reasonable to simplify it into a uniform electric field because the geometric size of the model is small enough), and set the initial voltage at the upper boundary of the model to V₀, which can be set according to the calculation conditions of the model. Set the left and right boundaries of the air domain and liquid film as electrical insulation, and the lower boundary of the liquid film as the grounding terminal.

The model is a two-dimensional geometric model, divided into triangular meshes. The minimum unit mass is 0.7082, the average unit mass is 0.9879, and the number of triangles is 447344. Considering that the model mainly tracks the collision behavior between droplet and liquid surface, the boundary of droplet and liquid surface needs to be refined, the maximum unit size at the collision boundary is 2, and the minimum unit size is 0.8, as shown in Figure 1a,b. In addition, there is a phenomenon of fragmentation when droplet collides with the liquid surface, resulting in smaller sizes of generated small droplet. In order to accurately track the newly formed small droplet, the air domain grid division also needs to be refined, otherwise it will lead to the loss of the newly formed droplet.

(2)

Setting of model input parameters

The model calculation requires specific settings for the materials of the two-phase flow. The properties of the gas-phase materials can be imported from the platform data, while the liquid phase materials are soybean oil, and their properties need to be defined by themselves, as shown in

Table 1. Physical attribute parameters of soybean oil.

Density	Dynamic Viscosity	Interfacial Tension	Relative Dielectric Constant	Electrical Conductivity
ρ = 916.8 kg/m3	μ = 0.061 Pa · s	σ = 0.032 N/m	εr = 3.5	σ′ = 0.54 mS/cm

:

In the model, the initial temperature of the air domain, droplet, and liquid film is 293.15 K, the outlet pressure of the fluid is normal pressure, the radius of the droplet is r₀, the initial velocity is u_y, and the height of the droplet is H₂, which can be set according to the model comparison requirements. In order to couple the electric field force to the flow field calculation, the component form of Maxwell stress tension to be defined in the model is as follows:

$${\mathbf{T}}_{\mathrm{x}\mathrm{x}}={\epsilon }_0{\epsilon }_r{E}_x^2-\frac{1}{2}{\epsilon }_0{\epsilon }_r\left(E_x^2+E_y^2\right)$$

(25)

$${\mathbf{T}}_{\mathrm{x}\mathrm{y}}={\epsilon }_0{\epsilon }_r{E}_x{E}_y$$

(26)

$${\mathbf{T}}_{\mathrm{y}\mathrm{y}}={\epsilon }_0{\epsilon }_r{E}_y^2-\frac{1}{2}{\epsilon }_0{\epsilon }_r\left(E_x^2+E_y^2\right)$$

(27)

$${\mathbf{T}}_{\mathrm{y}\mathrm{x}}={\epsilon }_0{\epsilon }_r{E}_y{E}_x$$

(28)

The electrostatic volume force formed by the electrostatic force in the X and Y directions can be further defined as follows:

$$F_{ec-x}=\partial {\mathbf{T}}_{\mathrm{x}\mathrm{x}}/\partial \mathrm{x}+\partial {\mathbf{T}}_{\mathrm{x}\mathrm{y}}/\partial \mathrm{y}$$

(29)

$$F_{ec-y}=\partial {\mathbf{T}}_{\mathrm{y}\mathrm{x}}/\partial x+\partial {\mathbf{T}}_{\mathrm{y}\mathrm{y}}/\partial y$$

(30)

(3)

Selection of solvers

In the phase field method, after the phase field variable $\mathsf{\Phi }$ is solved by the transport equation (Formula (5)), the characteristics of the calculated distance function will disappear. Therefore, the phase field function $\mathsf{\Phi }$ of the next layer shall be initialized during the solution process to make it have the characteristics of the distance function again. The specific equations for initialization are as follows:

$$\partial \mathsf{\Phi }/\partial t+\left(\omega ·\nabla \right)\mathsf{\Phi }={\mathsf{\Phi }}_0/\sqrt{{\mathsf{\Phi }}_0^2+\left(\Delta \mathrm{x}+\Delta y\right)/2}$$

(31)

$$\omega ={(\mathsf{\Phi }}_0/\sqrt{{\mathsf{\Phi }}_0^2+\left(\Delta \mathrm{x}+\Delta y\right)/2})·(\nabla \mathsf{\Phi }/|\nabla \mathsf{\Phi }|)$$

(32)

where ${\mathsf{\Phi }}_0$ is the $\mathsf{\Phi }$ value of the previous layer, $\Delta \mathrm{x}$ and $\Delta y$ are the lateral and longitudinal dimensions of the current calculation unit, respectively, and $\omega$ is the virtual expansion speed of the phase interface normal direction.

The study is required to track the dynamic and kinematic behavior of the droplet collision process, so a transient solver with phase initialization is selected for the fully coupled solution.

3. Results and Discussions

3.1. Results of Model Calculation

(1)

Tracking of phase field variables

Related studies show that there are generally three basic forms of droplet rebounding, droplet spreading, and droplet splashing and fragmentation when the droplet collides with the liquid film. However, the current study only focuses on the non-charged droplet, and there is no report on the phenomenon and mechanism of charged droplet colliding with the liquid film. Therefore, in combination with the actual working condition of electro-spraying, the numerical calculation of charged/non-charged droplet impingement on liquid film is carried out. Through the phase field variable, velocity field, and pressure field, the interaction behavior between droplet and liquid film is tracked and observed, so as to reveal the influence of electrical parameters and jet parameters on the impingement behavior of droplet. The research shows that with the increase of initial velocity, the collision behavior between liquid and liquid film appears three states in turn, namely, droplet rebounding, droplet spreading, and droplet splashing and fragmentation, as shown in Figure 2, Figure 3 and Figure 4:

Figure 2a,b show the processes of non-charged and charged droplet colliding with the liquid film surface at low speed, which are different from the processes of droplet colliding with the dry wall. When the droplet collides with the dry wall, the droplet continues to move downward under the effect of inertia force after contacting with the wall. The droplet is compressed to form an oval shape. When the droplet reaches the lowest point, the droplet retracts upward to form a spherical shape and separate from the wall, forming a single bounce or multiple bounce process. When the droplet collides with the liquid film, the surface of the liquid film will appear as pits under the action of the pressure layer, which can also be called the crater. At the same time, the droplet will also deform under the action of the pressure and become oval. At this time, the droplet is not in contact with the liquid surface. When a droplet reaches the maximum deformation, it will rebound, and the droplet will gradually return to the spherical shape. At the same time, the pressure between the droplet and the liquid film will gradually decrease. When the upward lifting force of the pressure layer is less than the sum of the gravity of the droplet and the downward volume force, the droplet begins to settle to the liquid film and gradually fuse. It can be seen that the droplet is in contact with the wall in the recovery phase when the droplet collides with the dry wall; in the recovery phase when the droplet collides with the liquid film, the droplet is not in contact with the liquid film, but there is an air pressure layer in the middle. By comparing the low-velocity collision of non-charged droplet and charged droplet on the liquid film, it is found that there is no significant difference between the droplets in the falling process and the interaction with the liquid film under the two conditions. However, under the charged condition, the droplet appears as an ellipsoid under the action of the electric field before the collision. In addition, the liquid phase distribution is more uniform and the fusion speed with the liquid film is better than that of the non-charged state. The droplet rebound phenomenon is defined as the following deformation process of the droplet before reaching the liquid film: spherical–ellipsoid–spherical. The droplet does not contact the liquid surface before returning to the spherical state, and there is always a pressure layer between the droplet and the liquid film. If the droplet contacts the liquid surface before fully restoring the spherical shape, it is considered that the droplet does not rebound during the collision process.

Figure 3a,b show the spreading process of a non-charged droplet and a charged droplet colliding with the liquid film, respectively; the droplet spreading is defined as the intermediate process of droplet rebound and splash breaking, the intermediate state of the droplet being neither interface rebound nor splash breaking. Comparing with the droplet colliding with the dry wall, it is easier to form a liquid crown when colliding with the liquid film, and the greater the collision speed, the higher the liquid crown. During the spreading phase, the droplet impinges on the liquid film and initially forms a crater on the surface of the liquid film, but the crater is significantly deeper and wider than the rebound behavior. Under the action of inertia force and pressure, the droplet is deformed obviously and appear as shoe-shaped gold ingot. At the same time, the crater on the surface of the liquid film will be presented as a bowl structure under the action of the pressure layer. Further, under the action of the pressure and the viscous force of the liquid itself, the liquid at the bowl mouth will be squeezed and move upwards to form a liquid crown, and then the liquid crown will expand outward in the form of a wave, integrate into the liquid film, and gradually recover to be flat. Comparing with the non-charged droplet, the charged droplet has wider and shallower crater on the surface of the liquid film, a more uniform distribution of the liquid phase of the droplet and the pressure layer below the droplet. The fusion process between the droplet and the liquid film is significantly faster, and the liquid surface has been basically leveled at 1000 $\mathsf{\mu }\mathrm{s}$.

Figure 4a,b show the splashing and fragmentation processes of the non-charged droplet and the charged droplet colliding with the liquid film, respectively. The splashing and fragmentation of droplet is defined as the phenomenon whereby the droplet is excited by the collision between the droplet and the liquid film, and the liquid crown extends upward and small droplet is separated at the top. When the droplet collides with the liquid film at a high speed, the crater formed on the surface of the liquid film is deep, even reaching the bottom of the liquid film. Then, the liquid crown is excited, and under the action of inertial force, it continuously expands outwards to form a thin wall. Comparing with the non-charged condition, the edge of the bowl-shaped structure formed under the charged condition is closer to the liquid surface, and it then cracks at the top of the liquid film. The separated liquid part forms the droplet under the action of inertial force and flies to both sides. For the non-charged droplet, the flight direction is inclined upward. When it reaches 530 $\mathsf{\mu }\mathrm{s}$, the droplet escapes from the computational domain, and then the unseparated liquid phase begins to gradually return to the liquid film. Under the charged condition, the liquid mass separated from the liquid crown migrates downward under the action of electric field force. When it reaches 490 $\mathsf{\mu }\mathrm{s}$, the liquid droplet starts to fuse with the liquid film. Meanwhile, it can be seen that the recovery speed of the liquid film under the charged condition is significantly faster.

(2)

Charge density and space potential distribution

For the study of the charged droplet colliding with liquid film, the analysis of electric field characteristics is essential. Based on the dielectric property analysis of liquid droplet and liquid film, the surface charge is distributed on the surface of liquid droplet and liquid film during the charged liquid droplet colliding with the liquid film. The spatial charge density distribution is also different at each stage of droplet drop, impingement, and fusion, as shown in Figure 5, which shows the charge distribution at the time nodes 0, 5, 10, and 500 $\mathsf{\mu }\mathrm{s}$ along the axis of the model for the 20 kV condition. It is evident that the induced charge above and below the droplet is reversed with a non-uniform longitudinal distribution as the spatial charge is subsequently migrated down the droplet. The upper and lower charge distribution of the droplet is symmetric at 0 $\mathsf{\mu }\mathrm{s}$, and the lower charge distribution is higher than the upper charge distribution at 10 $\mathsf{\mu }\mathrm{s}$ as the symmetric morphology of the droplet drop is broken. When the time reaches 500 $\mathsf{\mu }\mathrm{s}$, the droplet begins to fuse with the liquid film. It can be seen that there is only a charge distribution near the surface of the liquid film, and that the rest of the space charge density is zero. When the initial voltage is 25, 30, 35, and 40 kV, the spatial charge density distribution is consistent with that at 20 kV, but the value is different.

The spatial potential distribution in the model is shown in Figure 6 when the initial voltage is 20 kV. The potential at the top of the model is 20 kV, and the potential at the bottom of the liquid film is 0 kV. The potential in the whole model space is linearly reduced from top to bottom. It is important to note that the potential distribution around the droplet is distorted by the charge distribution on the surface of the droplet, as seen from the Figure 6a potential distribution positions and the Figure 6b potential lines along the central axis, and that the deformation of the potential distribution above the droplet is larger than that below the droplet. As the droplet drops, the potential distortion area also moves downward. When the time reaches 1000 $\mathsf{\mu }\mathrm{s}$, the potential disappears in the entire space potential distortion area after the droplet integrates into the liquid film. It can be seen from the potential distribution line at the 500 $\mathsf{\mu }\mathrm{m}$ vertical intercept in Figure 6c that the change rate of the potential in the liquid film in the air domain is significantly smaller than that in the air domain, which is caused by the different dielectric properties of the gas–liquid two-phase materials. In addition, it can be seen from the local potential amplification diagram in the air domain that the surface charge of droplet also has a certain influence on the potential distribution at 500 $\mathsf{\mu }\mathrm{m}$ away from the center of droplet, but the influence is relatively small and can be ignored.

3.2. Behavior of Charged Droplet Colliding with Liquid Film

3.2.1. Dynamic Behavior Analysis of Droplet Rebound

As described in Section 3.1, a droplet rebound occurs when it impinges on the liquid film at a lower velocity. From the perspective of dynamics, when the droplet approaches the liquid film, it will compress the air between the droplet and the liquid film, so as to form a pressure layer, as shown in Figure 7. Under the action of the pressure layer, the droplet presents an ellipsoid shape, and a crater is formed on the surface of the liquid film facing the droplet. The droplet with a certain initial velocity is influenced by the downward inertia force and gravity in the falling process, and the downward electric field force also exists under the charged condition. In addition, the droplet is influenced by the upward air resistance under both conditions. If the droplet is subjected to a downward resultant force less than the upward pressure produced by the pressure layer, the droplet cannot be brought into contact with the liquid film and, thus, the droplet deformation and rebound occur. As the air in the pressure layer flows to the surrounding environment, the pressure in the pressure layer gradually decreases. When the pressure is less than the downward force of the droplet, the droplet continues to move through the pressure layer and contacts and fuse with the liquid film. When the initial velocity of droplet is 2 m/s, the pressure distribution is shown in Figure 7b. The pressure in the center of the pressure layer is the maximum. The pressure above the pressure layer decreases with the increase of distance and recovers to the atmospheric pressure when reaching the upper interface of the droplet. The pressure below the pressure layer decreases first and then increases with the increase of distance, which is caused by the solid interface below the liquid film, and the reverse pressure is formed when the pressure is transmitted to the interface.

As shown in Figure 7c,d, the center pressure of the pressure layer reaches the maximum value of 2404 Pa when the time is 216 $\mathsf{\mu }\mathrm{s}$ under the charged condition and 2363 Pa when the time is 212 $\mathsf{\mu }\mathrm{s}$ under the non-charged condition. It can be seen that the penetration force of the charged droplet is stronger due to the electric field force, so it takes longer to reach the maximum pressure point and the central pressure formed is larger. However, from the numerical point of view, the difference between the central pressure values under the two conditions is not significant, so the movement of the low-speed droplet is basically the same. Numerical calculation results show that the initial speed of the droplet is 2 m/s, which is the critical point of the droplet rebound. When the initial speed of the droplet increases to 2.5 m/s, the droplet can easily contact the liquid film before it completely recovers into a sphere. It should be noted that due to the large amount of simulation calculation, the speed increase is 0.5 m/s. If the critical Weber number is used to define the droplet rebound, the expression is as follows:

$${\mathrm{W}\mathrm{e}}_{cr}={\rho }_1{v}_0^2{d}_0/\sigma$$

(33)

where ${\mathrm{W}\mathrm{e}}_{cr}$ is the critical Weber number of the droplet, ${\rho }_1$ is the liquid phase density (kg/m³), v₀ is the initial velocity of the droplet (m/s), and d₀ is the initial particle size of the droplet (m). The initial speed of the droplet is 2 m/s, and the critical Weber number of rebound when the initial particle size is 150 μm is 19.1. Therefore, it can be seen that the critical pressure of the pressure layer for the rebound of a droplet with a particle size of 150 μm is approximately 2400 Pa. Furthermore, it can be seen from the pressure contours shown in Figure 7c,d that the pressure layers under charge are respectively more flat, corresponding to the ellipsoidal shape of the droplets themselves before colliding with the liquid film as previously described. The smaller the initial speed of the droplet, the greater the rebound amount of the droplet; the longer it stays above the liquid film, the easier it is for the droplet to fly and drift, causing pollution, and it is not conducive to the effective use of lubricant. If the distance between the nozzle and the workpiece is calculated as 20 mm, it can be seen from the initial velocity and acceleration of droplet at the outlet of the nozzle that when the air pressure is 0.1 MPa, the minimum velocity of droplet reaches the workpiece surface is about 7.58 m/s. It can be seen that under electro-spraying condition, the droplet velocity is higher than the rebound critical velocity even if the minimum air pressure of 0.1 MPa is used. When the initial velocity is 7.5 m/s, the two-phase interaction in the collision process under the charged and non-charged conditions is shown in Figure 8.

It can be seen that when the droplet velocity reaches 7.5 m/s, the droplet does not rebound. The inertia force and gravity of the droplet are enough to break through the pressure layer below and realize the adsorption and spreading of the droplet. Comparing with the non-charged condition, the charged droplet is more flat under the action of surface charge, which leads to more shallow craters on the surface of the liquid film, and the faster the mutual fusion speed.

3.2.2. Dynamic Analysis of Droplet Splashing and Fragmentation

Based on the droplet dynamics calculation at the outlet of the nozzle [1], when the air pressure is 0.4 MPa and the voltage is 40 kV, the droplet has the maximum initial velocity and acceleration. If the distance between the nozzle and the target workpiece is 20 mm, the collision velocity of the droplet arriving at the liquid film on the workpiece surface is about 25.5 m/s. It should be noted that the collision velocity calculated at this time is under the full action of aerodynamic force and electric field force and does not take into account the air resistance during the movement of the droplet. Therefore, comprehensive analysis believes that the actual collision speed of the droplet will not exceed 25 m/s. Combined with the previous analysis, it can be seen that there are two main collision behaviors of droplet spreading and splashing in the process of droplet and liquid film collision. Droplet splashing will undoubtedly reduce the effective utilization rate of the micro-lubricant and cause environmental pollution. Therefore, it is particularly important to reveal the dynamic nature and critical conditions of droplet splashing and fragmentation for determining the aerodynamic parameters and electric field parameters in the electro-spraying process.

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Analysis of droplet splashing and fragmentation behavior under non-charged conditions

As described in Section 3.1, when the droplet collides with the liquid film at a speed of 10 m/s and 20 m/s, two kinds of interaction behaviors of droplet spreading and splash breaking appear, respectively. With a speed increase of 0.5 m/s, the critical speed of splash breaking of non-charged droplet with a particle size of 150 μm is tracked. The tracking results show that the splash fragmentation phenomenon occurs for the first time when the initial velocity of the droplet reaches 17.5 m/s. In order to further reveal the influence of velocity on splash fragmentation, Figure 9 shows the morphological changes of the liquid crown at the initial velocity of 17 m/s, 17.5 m/s, 18 m/s, 20 m/s, and 25 m/s. When the initial velocity of the droplet is 17 m/s, the crater is formed after the droplet collides with the liquid film. A local triangular-like bulge is formed on both sides of the crater, and then the bulge continues to move upward to form a liquid crown. At this time, the droplet has been integrated into the liquid film inside the liquid crown. Under the action of inertial force, the liquid crown continues to expand upward and the thickness becomes thinner, and it gradually forms a liquid mass with a thickness greater than the average wall thickness of the liquid crown at the edge of the liquid crown. A small liquid bridge is formed between the liquid mass at the top of the liquid crown and the liquid at the bottom of the liquid crown. As the inertial force of the liquid is gradually offset, under the action of liquid viscosity and surface tension, the liquid mass and liquid bridge begin to retract toward the bottom of the liquid crown, and eventually merge into the liquid film. When the initial velocity of the droplet reaches 17.5 m/s, the early interaction behavior between the droplet and the liquid film is similar to that at 17 m/s, but the thickness of the whole liquid crown and the liquid bridge is thinner, so that the viscous retraction force of the liquid in the liquid bridge fails to balance the inertial force of the liquid mass at the top of the liquid crown, which makes the liquid crown break at the liquid bridge. After the liquid bridge breaks, the top liquid mass continues to move to both sides under the action of inertial force and retracts into a spherical shape under the action of surface tension. At the same time, the liquid below the liquid bridge begins to move to the liquid film and gradually returns to the inside of the liquid film. According to the collision behavior of the droplet and the liquid film when the initial velocity of the droplet is 18, 20, and 25 m/s, it can be found that with the increasing initial velocity of droplet, the crater formed on the surface of the liquid film becomes wider and deeper, and the thickness and height of the excited liquid cap becomes thinner and higher. Compared with the velocity of 17.5 m/s, the droplet has a similar collision behavior to that of the liquid film at a velocity of 18 m/s, but the top liquid mass moves farther to both sides after separating from the liquid crown. When the droplet velocity reaches 20 m/s, the droplets formed by the fragmentation of the liquid cap have reached the edge of the calculation area at 500 μs. When the droplet velocity reaches 25 m/s, and the droplets formed by the fragmentation of the liquid cap have reached the edge of the calculation area at 170 μs. In summary, when the initial velocity of the droplet reaches 17.5 m/s, the liquid crown will spatter and break up, and the critical Weber number is 1309.15. With the further increase of the velocity, the faster the droplet moves to both sides, and the higher the height from the liquid film, so it is more likely to fly and disperse, and the greater the threat regarding the surrounding environmental pollution and human health.

Figure 10 shows the state of the droplet colliding with the liquid film at an initial velocity of 17.5 m/s for 26 μs. From the phase field variable distribution in Figure 10a, it can be seen that at this time, the droplet has not been contacted with the liquid film, and there is a pressure layer between the droplet and the liquid film; an obvious crater appears on the surface of the liquid film, and the liquid at the edge of the crater has protruded from the surface of the liquid film to form the initial form of a liquid crown. Combined with the pressure distribution contour at 26 μs in Figure 10b, it can be seen that the air below the droplet is compressed to form a pressure layer, and the maximum pressure in the center can reach 1.338 × 10⁵ Pa. The liquid film under pressure is compressed downward and flows to both sides, thus forming a crater structure on the surface of the liquid film. Figure 10c shows the velocity distribution contour at 26 μs. It can be observed that air rapidly escapes to both sides within the pressure layer, forming a high-speed airflow layer with a maximum speed of 27.24 m/s. The high-speed airflow layer generates gas drag on the liquid surface inside the crater, driving the liquid to move upward along the surface of the crater, thereby causing the liquid at the edge of the crater to bulge out of the liquid film surface. In addition, when the liquid directly below the crater moves downward and encounters the solid surface, the flow direction changes to form a horizontal flow. When the horizontal flow encounters the surrounding stationary fluid, it pushes the liquid upward, so the liquid at the edge of the crater continues to rise. The analysis of the force on the liquid area to the left of the crater Figure 10d shows that under the pressure layer, the liquid on the inside of the crater is subjected to a downward and leftward pressure, causing the liquid in that area to have a tendency to move downward and to the left. Due to the viscosity of the liquid, when it moves to the left, it experiences resistance from its own viscosity, which causes the liquid near the crater to be compressed. The gas drag drives the liquid on the inside of the crater to move upward along the contour of the crater, causing the liquid at the edge of the crater to protrude from the liquid film surface. The protruding liquid surface is affected by the resistance of the air on the left, which, to some extent, hinders the expansion of the liquid crown to the left, thereby forming the initial form of the liquid crown.

From the above analysis, it can be seen that the formation of liquid crown is caused by the pressure difference and velocity difference on both sides of the liquid near the crater. Figure 11 shows the velocity and pressure curve of the liquid film surface when the time is 22–30 μs. The abscissa arc length of 1500 μm in the figure corresponds to the abscissa zero point in Figure 11a. The area between arc length 1300–1400 μm can be preliminarily defined as the research fluid area. From the liquid film surface velocity curve in Figure 11a, it can be seen that the velocity on the left and right sides of the crater increases rapidly first and then decreases gradually with time, and the velocity reaches the maximum value at 26 μs. At this time, the velocity on the liquid film surface at an arc length of 1400 μm is 9.4 m/s greater than that at an arc length of 1300 μm. According to the pressure curve of the liquid film surface in Figure 11b, the pressure also shows a trend of increasing first and then decreasing with time, and reaches the maximum value directly below the droplet at 24 μs. If the same is taken as an example of 26 μs, the pressure of the liquid film surface at the arc length of 1400 μm is 0.55 × 10⁵ Pa higher than that at 1300 μm. It can be seen that the pressure gradient and velocity gradient on the left and right sides of the liquid inside the crater are very large during the formation process of the liquid crown. Due to this large gradient, the motion of the fluid is interrupted, thus forming the initial liquid crown.

Table 2. Data table for surface velocity and pressure of liquid film under non-charged condition.

Condition	Non-Charged
Arc Length (μm)	1300					1400
Time (μs)	22	24	26	28	30	22	24	26	28	30
Velocity (m/s)	0.18	0.66	1.65	2.95	4.09	2.73	7.43	10.93	11.46	9.91
Pressure (×105 Pa)	0.01	0.03	0.06	0.1	0.1	0.23	0.50	0.61	0.52	0.31

presents the specific values of the surface velocity and pressure of the liquid film at different time points when the wavelengths are set to 1300 and 1400 in the non-charged state.

After the liquid crown is formed in the initial stage, the liquid crown continues to extend upward under the action of inertial force. At the beginning of the extension, the wall thickness of the liquid crown is relatively uniform, and then the liquid crown gradually becomes thinner and forms fluctuations on the surface. The surface fluctuations are considered to be capillary waves. Under the action of capillary wave on the surface of liquid crown, the top liquid mass, liquid bridge, and liquid crown base are formed on the wall of liquid crown. The shape of left liquid crown at 240 μs is shown in Figure 12a, and the liquid bridge is formed at the surface trough. As shown in Figure 12b, the top liquid mass is mainly driven by the inertial force to pull the liquid crown upward, and the base of the liquid crown is mainly pulled by the liquid viscous resistance to pull the liquid crown back into the liquid film. At the same time, the surface tension of the liquid crown accelerates the fracture of the liquid crown at the liquid bridge position. The liquid mass at the top of the liquid crown contracted into a spherical droplet under surface tension after separation of the liquid bridge from the liquid crown, as shown in Figure 12c for 1000 μs, plotting a two-dimensional intercept through the center of the droplet, and extracting the volume fraction of the liquid phase in the baseline arc length, as shown in Figure 12d for the droplet region at a volume fraction of fluid 2 of 1, measuring the droplet with a diameter of about 36–38 μm.

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Analysis of droplet splashing and fragmentation behavior under charged conditions

Figure 13 shows the morphological changes of the liquid crown excited by the collision of charged droplet on the liquid film at initial velocities of 18, 18.5, 19, 20, and 25 m/s.

Comparing with the splash-breaking behavior of the charged droplet colliding with the liquid film at 18 m/s, no splash breaking occurs when the charged droplet collides with the liquid film at a critical speed of 18.5 m/s, and the corresponding critical Weber number was 1463.05. With the further increase of the initial velocity of the charged droplet, the liquid crown excited by the liquid film surface gradually becomes higher and thinner, which is similar to the collision behavior under non-charged conditions, including the initial stage of liquid crown formation, the development stage of liquid crown, and the liquid mass separation stage at the top of liquid crown. The droplet velocities of 18 m/s and 17 m/s are defined as the critical velocities for the splash crushing under the charged and non-charged conditions, and 19 m/s and 17 m/s are defined as the critical velocities for the splash crushing. By comparing the lower critical velocity, it can be found that due to the higher initial velocity and the effect of electric field, the crater formed at 30 μs is obviously deeper and wider, and the liquid crown formed at 200 μs is thinner and longer. In addition, it can be found that the liquid crown formed after charging has a certain bending. Comparing with the upper critical velocity, it can be found that under the charged condition, due to its higher initial velocity and greater inertial force, the liquid crown breaks earlier. Different to the non-charged condition, the top liquid mass does not move upward after separating from the top of the liquid crown, but tilts downward toward the liquid film, and contacts with the liquid film surface at 450 μs, and then fuses with the liquid film. Comparing the collision behavior under the two conditions when the velocity is 20 m/s, it can be found that the height of the liquid crown is lower and the liquid film is basically parallel under the charged condition, and the top liquid mass is separated from the liquid crown and integrated into the liquid film. Under the non-charged condition, the angle between the liquid crown and the liquid film is about 30 degrees at 150 μs, and the top liquid mass escapes from the computational boundary at 600 μs after being separated from the liquid crown. When the velocity is 25 m/s, the formation and development of the liquid crown under the two conditions are similar. The top liquid mass formed by the large inertial force escapes the calculation boundary, but the escape speed of the liquid mass under the charged condition is slower.

As shown in Figure 14, the initial velocity of the charged droplet is 18.5 m/s when it hits the liquid film at 26 μs. Figure 14a–d are the phase field variable distribution, pressure field distribution, velocity field distribution, and charge distribution, respectively.

Comparing with Figure 10, it can be found that due to the larger initial velocity of the charged droplet, the crater formed by the smaller distance between the droplet and the liquid film is wider. The spatial pressure distribution and velocity distribution under the two working conditions are roughly the same. The maximum pressure in the center and the maximum air flow velocity in the pressure layer during charged are 1.418 × 10⁵ Pa and 32.77 m/s, respectively. The analysis shows that the charged condition is similar to the non-charged condition. Under the action of the pressure difference on the inner wall of the crater, the gas drag force formed by the air escape in the pressure layer, and the viscous resistance of the liquid itself, the liquid protruding liquid film near the edge of the crater forms the initial shape of the liquid crown. Different to the non-charged condition, under the charged condition, negative charge accumulates on the upper surface of the droplet, and positive charge accumulates on the lower surface of the droplet, forming an electrostatic attraction between the upper and lower surfaces of the droplet, so the charged droplet is more flat. Therefore, even if the velocity difference is 1 m/s, the maximum pressure difference is only 80 Pa.

Figure 15 shows the velocity and pressure curves of the liquid film surface at a time of 22–30 μs under charged conditions. The arc length between 1300–1400 μm is also defined as the study fluid region. From the velocity curve of the liquid film surface in Figure 15a, it can be seen that the velocity on the left and right sides of the crater under charged conditions also shows a trend of increasing first and then decreasing with time, but the maximum gas flow velocity appears at 24 μs. At this time, the velocity of the liquid film surface at the arc length of 1400 μm is 12 m/s larger than that at the arc length of 1300 μm. From the surface pressure curve of the liquid film in Figure 15b, it can be seen that the pressure also shows a decreasing trend with time. The pressure is the largest under the droplet at 22 μs. If the same is taken as an example at 26 μs, the surface pressure of the liquid film at the arc length of 1400 μm is 0.4 × 10⁵ Pa larger than that at the arc length of 1300 μm.

Table 3. Data table for surface velocity and pressure of liquid film under charged condition.

Condition	Charged
Arc Length (μm)	1300					1400
Time (μs)	22	24	26	28	30	22	24	26	28	30
Velocity (m/s)	0.13	0.97	2.79	4.47	5.59	4.11	13.08	12.25	9.99	8.35
Pressure (×105 Pa)	0.01	0.04	0.11	0.13	0.08	0.38	1.03	0.73	0.33	0.15

presents the specific values of the surface velocity and pressure of the liquid film at different time points when the wavelengths are set to 1300 and 1400 in the charged state.

The fracture process of the liquid crown under charging conditions is shown in Figure 16. Figure 16a shows the liquid crown on the left at 190 μs. It can be seen that the liquid crown at this time also consists of three parts: the top liquid mass, the liquid bridge and the base of the liquid crown. Comparing with the uncharged condition, the liquid crown is thinner and lower, and the top liquid mass is also smaller. Figure 16b shows the charge distribution of the liquid crown and the liquid film. It can be seen that there is electrostatic attraction between the upper and lower parts of the liquid crown and between the lower part of the liquid crown and the liquid film, so the height of the liquid crown is lower than that under the non-charged condition. In addition, because the curvature at the top of the liquid mass is greater than that at other positions of the liquid crown, according to the characteristics of charge distribution, it can be seen that the top liquid mass gathers more charges, so the electrostatic attraction formed is greater, which is manifested by the bending of the liquid crown to the surface of the liquid film. Figure 16c shows the electro-viscous effect between two interfaces with different charges. It can be seen that there is charge transfer between two similar liquid films with different charges. The moving charge will drive the surrounding micro-liquid mass to move due to the electro-viscous effect. The electro-viscous effect promotes the flow of liquid in the liquid crown to the liquid film, so the wall thickness of the liquid crown formed under charged conditions is thinner. The top liquid mass is separated from the liquid crown to form a small droplet, but the droplet is ellipsoidal due to the charge on the surface of the droplet. When the Figure 16d is 450 μs, the droplet morphology is ellipsoidal. The droplet size along the long axis and the short axis of the droplet is measured as shown in Figure 16 (e) and (f), respectively. The long axis diameter of the droplet is about 38–40 μm, and the short axis diameter is about 13–15 μm.

3.3. Behavior of Multi-Angle Droplet Colliding with Liquid Film

3.3.1. Multi-Angle Collision of Non-Charged Droplet on Liquid Film

In practical machining processes, the nozzle is typically oriented at a specific angle relative to the surface of the workpiece, so the droplet formed by the nozzle atomization collides with the liquid film on the surface of the workpiece at a certain angle. The nozzle angle is usually between 15° and 30°. The behavior of non-charged droplet with a particle size of 150 μm colliding with the liquid film at multiple angles is shown in Figure 17.

The horizontal velocity u_x and u_y of the droplet are set according to the initial position of the droplet, so that the velocity when the droplet contacts the liquid film is 7.5 m/s, and the angles are 15°, 20°, 25°, and 30°, respectively. When the incident angle of the droplet is 15 °, the contact area between the droplet and the liquid film is small, and the contact part between the droplet and the liquid film is rapidly pulled and broken under the action of inertial force. The droplet presents a comet-like shape. After the droplet is separated from the liquid film, it quickly moves away from the contact area and gradually returns to a spherical shape under the action of surface tension. As the incident angle increases, the contact area between the droplet and the liquid film increases, and the interaction between the droplet and the liquid film becomes more obvious. Under the action of inertial force and viscous force, the droplet and some liquid in the liquid film are pulled to form a unilateral liquid line. The larger the incident angle, the thicker and shorter the liquid line. When the angle is 20° and 25°, the liquid line finally breaks to form small droplets. When the angle is 30°, the droplet is captured by the liquid film and the liquid line does not break.

By comparing Figure 17 and Figure 8a, it can be seen that when the droplet vertically collides with the liquid film, the droplet is completely spread and absorbed into the liquid film, while the droplet collides with the liquid film at a certain incident angle to form a relatively slender unilateral night line, and the smaller the incident angle, the more likely the liquid line is to break and form floating droplets. The analysis shows that there is only normal velocity in the initial state of the vertical collision of the droplet on the liquid film, and there is only normal pressure and velocity in the initial stage of the interaction between the droplet and the liquid film. The formation of the liquid crown is due to the pressure difference. In this process, the droplet completely enters the liquid film and its kinetic energy consumption is large. The liquid in the liquid crown comes from the liquid film with almost zero initial kinetic energy, so the generated liquid crown has low kinetic energy and is not easy to splash and break. Figure 18 is the distribution of phase field variables, velocity field, and pressure field when a droplet with a particle size of 150 μm collides with the surface of the liquid film at an incident angle of 20°. It can be seen that the droplet and the liquid film are only partially contacted during the collision process, so only part of the droplet kinetic energy is exchanged with the liquid film, and the remaining kinetic energy keeps the droplet moving to the right. Different to the vertical collision behavior, the distribution of pressure field and velocity field is not symmetrical when the droplet collides with the liquid film at an angle. The pressure on the right side of the liquid film is significantly higher than that on the left side. Therefore, under the action of pressure difference, a unilateral liquid crown is formed on the surface of the liquid film, and then the bottom of the droplet is combined with the top of the liquid crown to form a liquid line. Under the combined action of inertial force and viscous force, the liquid line is continuously stretched and then broken. From the change of velocity field, it can also be seen that the flow field distribution also moves to the right as a whole. The liquid velocity on the right side of the liquid film is obviously larger than that on the left side. The velocity in the liquid film is the widest in the formation and development stage of the liquid line, and the velocity distribution in the liquid film gradually decreases after the liquid line breaks. In summary, the analysis shows that when the droplet is angled and the liquid film is collided, the contact area is smaller than the vertical collision, so the kinetic energy of the droplet is smaller, so the inertial force to keep it moving to the right is larger, and the length of the liquid line is also larger. In the competition between inertial force and viscous resistance, inertial force dominates, so the liquid line breaks and forms scattered droplets.

Analysis of Figure 17 reveals that as the angle of incidence increases, the contact area between the droplet and the liquid film also increases, leading to a greater loss of initial kinetic energy in the droplet. When the incident angle is 15°, 20°, and 25°, inertia dominates, causing the liquid line to break and form scattered droplets. The smaller the incident angle, the larger the diameter of the scattered droplets. When the incident angle is 30°, viscous resistance dominates, the liquid line does not break, and the droplet is captured and eventually merges into the liquid film. Section 3.2.2 (1) and this section agree with the research conducted by Yao et al. [12] in the introduction regarding non-charged droplets. The droplet velocity and collision angle do indeed have a significant impact on collision dynamics.

3.3.2. Multi-Angle Collision of Charged Droplet on Liquid Film

The behavior of the charged droplet colliding with the liquid film at multiple angles is shown in Figure 19. The droplet size is also set to 150 μm, the collision velocity is 7.5 m/s, the incident angles are 15°, 20°, 25°, and 30°, and the voltage is 20 kV. It can be seen that the action mechanism of the droplet and the liquid film under the charged state at different incident angles is roughly the same, but there are also some differences. When the incident angles are 15° and 20°, the liquid line breaks to form scattered droplets. When the angles are 25° and 30°, the droplet is captured by the liquid film and gradually merges into the liquid film. The small droplets formed by the splashing and fragmentation of the charged droplet can settle to the surface of the liquid film again and merge with the liquid film quickly. In addition, it can be seen that the liquid film itself recovers faster under charged conditions. Analysis shows that after the droplet is charged, there is a surface charge on the surface of the droplet. The electrostatic force and electro-viscous effect between different charges lead to the difference in collision behavior under the two conditions.

As shown in Figure 20a,b, when the incident angle is 20° and the incident time is 600 μs, the droplet has been separated from the top of the liquid line. At this time, there is a negative charge on the upper surface of the droplet and the surface of the liquid film, and there is a positive charge on the lower surface of the droplet. Under the action of electrostatic attraction, the droplet moves to the surface of the liquid film, so the secondary collision adsorption of the scattered droplet occurs.

As shown in Figure 20c,d, the liquid line did not break when the incident angle was 25° at 600 μs. At this time, there was a positive charge distributed below the liquid line, and a negative charge distributed above the liquid line and on the liquid film surface. The positive charge will move to the direction of the liquid film for electrical neutralization. The movement of the electro-viscous effect charge will drive the surrounding micro-liquid mass to move together, which promotes the fusion of the liquid and the liquid film in the liquid line. In summary, it is considered that when the charged droplet collides with the liquid film at a certain angle, the droplet is more easily captured by the liquid film. Even if the liquid line breaks to form a scattered droplet, the droplet will undergo secondary collision adsorption under the electrostatic force.

4. Conclusions

This paper primarily integrates the actual conditions of droplets in the practical working conditions of electro-spraying, conducts numerical simulation calculations of the dynamic process of droplet collides with liquid films, tracks the rebound, spreading, and splashing fragmentation behavior during the droplet collision process, and analyzes the influence mechanism of the external electric field on the behavior of a droplet colliding with liquid films.

(1)

Based on the assumptions that the fluid is incompressible, the liquid viscosity and surface tension do not change during the collision process, and the gas–liquid two-phase has no phase transition behavior during the collision process; combined with the actual situation of droplets in the practical working conditions of electro-spraying, the two-phase flow field continuity equation, momentum conservation equation, and electric field control equation are used to construct a dynamic numerical calculation model of a charged droplet colliding with liquid film.

(2)

The phase field variable is used to track the diffusion behavior between the two-phase interfaces, and the droplet collision behavior is calculated and analyzed by changing the initial velocity of the droplet. Research indicates that as the initial velocity of the droplet increases, the process of the droplet colliding with the liquid film sequentially exhibits three behaviors: rebound, spreading, and splashing fragmentation. A comparative analysis was conducted on the similarities and differences in the collision behavior of charged droplets and non-charged droplets.

(3)

By tracking the changes in the velocity field, pressure field, and spatial charge density during the process of the droplet colliding with the liquid film, the study analyzed the kinetic mechanism of the formation of the liquid crown during the collision process, revealing the kinetic mechanism of the rebound, spreading, and splashing fragmentation behavior of the droplets. The study calculated the critical velocity and corresponding critical Weber number for the spreading and splashing fragmentation behavior of droplets under vertical collision conditions. The results showed that due to the electro-viscous effect in the process of charged droplets colliding with the liquid film, the critical velocity of its splashing fragmentation was increased, concluding that the external electric field is conducive to the fusion of droplets and liquid films.

(4)

The collision behaviors of charged droplet and non-charged droplet at impact angles of 15°, 20°, 25°, and 30°were compared and analyzed, and the dynamic mechanism of forming unilateral liquid line under charged and non-charged conditions was compared and analyzed. The research results indicate that as the incidence angle decreases, the tangential velocity of the droplet continuously increases, making it more prone to splashing and fragmentation. Compared to non-charged droplets, the critical angle at which charged droplets undergo fragmentation and splashing is smaller, and the settling speed of the droplets formed by splashing and fragmentation is faster.