Maximum Deformation Ratio of Droplets of Water-Based Paint Impact on a Flat Surface

In this research, the maximum deformation ratio of water-based paint droplets impacting and spreading onto a flat solid surface was investigated numerically based on the Navier–Stokes equation coupled with the level set method. The effects of droplet size, impact velocity, and equilibrium contact angle are taken into account. The maximum deformation ratio increases as droplet size and impact velocity increase, and can scale as We1/4, where We is the Weber number, for the case of the effect of the droplet size. Finally, the effect of equilibrium contact angle is investigated, and the result shows that spreading radius decreases with the increase in equilibrium contact angle, whereas the height increases. When the dimensionless time t* < 0.3, there is a linear relationship between the dimensionless spreading radius and the dimensionless time to the 1/2 power. For the case of 80◦ ≤ θe ≤ 120◦, where θe is the equilibrium contact angle, the simulation result of the maximum deformation ratio follows the fitting result. The research on the maximum deformation ratio of water-based paint is useful for water-based paint applications in the automobile industry, as well as in the biomedical industry and the real estate industry. Please check all the part in the whole passage that highlighted in blue whether retains meaning before.


Introduction
The widespread application of the impact and spread of liquid droplets in such areas as agriculture, thermal spray, lab-on-a-chip, and coating [1][2][3], continues to receive attention.After impact onto a solid surface, there are four possible phases proposed by Rioboo et al. [4], i.e., the kinetic phase, the spreading phase, the relaxation phase, and the wetting/equilibrium phase.After some time, the droplet reaches a maximum spreading diameter.Due to the important fluid mechanics in these processes, both experiments and numerical simulations have been implemented in recent years to research droplets' dynamic behaviors after impact, which has been helpful in understanding droplet interactions with liquid and solid surfaces.
In terms of experimental research, great efforts have been made.Roisman et al. [5] experimentally and theoretically studied the impact of a single droplet onto a dry surface by implementing the inertial effect, surface tension, viscous, and wettability.Amirfazli et al. [6][7][8] conducted series experiments to study the dynamic behaviors of liquid droplets under different effects, e.g., the droplet size dependence on contact angles, the electric fields on contact angles, the surface tension of droplets for different materials, and the receding contact angles of droplets and the rebound time.Mao et al. [9] presented a rebound model of a droplet upon impact.Ukiwe and Kwok [10] reported that drop impact dynamics was influenced by impact energy of the droplet at impact, physical properties of the liquid droplets and solid surface tensions.Clanet et al. [11] studied the impact of liquid droplets of low viscosity on a super-hydrophobic surface, and the experimental results presented the maximal spreading ratio scaled as We 1/4 , where We is the Weber number.Based on energy balance, Park et al. [12] predicted the maximum deformation ratio at a low-impact velocity by experiment.
On the other hand, various numerical investigations have also been carried out.Bussmann et al. [13] presented a methodology to simulate the fingering and splashing of droplet impacts onto a solid surface.Merdasi et al. [14] investigated the deformation of two droplets within microfluidic T-junctions on a solid substrate by LBM (Please define).They reported that the deformation of the two droplets increases significantly with the increase in the relative velocity of the inlet flow, droplet size, and surface tension.The flow pattern in pipe flows had been simulated for drag reducing fluids using a low Reynolds number k-ε model by Dhotre et al. [15].
In this study, a numerical method was adopted to simulate the impact and spread of a water-based paint droplet onto a flat steel surface.The dynamics of the impact and spread were observed and used to establish guidelines for the maximum deformation ratio due to the droplet size, impact velocity, and equilibrium contact angle.The experimental results are compared with the prediction models.

Navier-Stokes Equations
The Navier-Stokes equations for the incompressible laminar two-phase flow are implemented, and its expression is written as: In the above equations, v, ρ, and η denote velocity, density, and dynamic viscosity, I is the 3 × 3 identity matrix, g is constant and gravitational acceleratio, p is pressure, and f stf is the surface tension force.The surface tension force only exists at an interface which separates the droplet and air.

The Level Set Method
In order to track the interface between the two phases, the level set function of ϕ is introduced.The function ϕ is expressed as where γ is the parameter controlling the interface thickness, and α is the reinitialization parameter.Both of γ and α are set parameters.In air, ϕ = 0, whereas ϕ = 1 in water-based paint.In the interface of the two phase flows, ϕ = 0.5.The density and viscosity are controlled by In the above equations, subscript 1 denotes air and 2 denotes water-based paint.The volume fractions for air and water-based paint are determined by In the level set method, the surface tension force f stf is calculated by In the above equation, T is expressed by where Î is the identity matrix, σ is surface tension coefficient of droplet, and the interface normal n n , and the delta function δ are calculated by

Model Description
A scheme of the experimental unit for the water-based paint droplet impact and spread is presented in Figure 1.In Figure 1, the radius (r w ) and the height (H) of the cylindrical computational domains are 100 µm.The diameter of the droplet is 40 µm.Impact velocities in the range of 0.9-2.0m/s are applied.The equilibrium contact angles are in the range of 40 • -120 • , and the equilibrium contact angle is expressed as θ e .In the numerical simulation, the wall boundary is the wetted wall.The velocity component normal to the wall is set to zero and is determined by A frictional boundary force ( f stf ) is added, and its expression is written by In the above equation, β is the slip length.where Î is the identity matrix, σ is surface tension coefficient of droplet, and the interface normal n n , and the delta function δ are calculated by

Model Description
A scheme of the experimental unit for the water-based paint droplet impact and spread is presented in Figure 1.In Figure 1, the radius (rw) and the height (H) of the cylindrical computational domains are 100 μm.The diameter of the droplet is 40 μm.Impact velocities in the range of 0.9-2.0m/s are applied.The equilibrium contact angles are in the range of 40-120°, and the equilibrium contact angle is expressed as θe.In the numerical simulation, the wall boundary is the wetted wall.The velocity component normal to the wall is set to zero and is determined by A frictional boundary force ( f f ) is added, and its expression is written by In the above equation, β is the slip length.As shown in Figure 2, local mesh refinement is adopted, and the maximum mesh size is 2.9 μm in the mesh refinement region, whereas the coarse mesh is adopted in other regions.A conservative form has been used, which results in exact conservation of the mass.Some parameters for the material properties in the simulation are listed in Table 1.
In addition, several relevant numbers are defined as As shown in Figure 2, local mesh refinement is adopted, and the maximum mesh size is 2.9 µm in the mesh refinement region, whereas the coarse mesh is adopted in other regions.A conservative form has been used, which results in exact conservation of the mass.Some parameters for the material properties in the simulation are listed in Table 1.
In addition, several relevant numbers are defined as In the above definitions, v 0 is the initial impact velocity, r max is the maximum spreading radius of droplet, D 0 is the initial diameter of droplet, r(t) is equal to half of the diameter of the droplet with variation in time, We is the Weber number, Re is the Reynolds number, ξ max is the maximum deformation ratio, t * d is the dimensionless time, and r * d is the dimensionless spreading radius.The spreading radius r(t) and dynamic contact angle (θ) are defined in Figure 3.

Numerical Validation
The Navier-Stokes equation has been successfully applied for the simulation of the impact and spread of liquid droplets onto solid surfaces [16,17].To further verify the numerical validation, the simulations of the impact and spread of droplets of water-based paint onto a flat surface have been implemented for the cases of θe = 40°, 50°, 80° and 95°.Impact velocity 0 v is 1.5 m/s.The droplet diameter 0 D is 40 μm. Figure 4 presents the shape of the droplets in the final equilibrium state.The measurement software Digimizer was adopted to measure the contact angle in Figure 4, and the measurement results show that the real contact angle θ is equal to 38.4°, 48.7°, 82.9° and 92.9°, respectively.Therefore, the results agree well with the theoretical solutions.

Numerical Validation
The Navier-Stokes equation has been successfully applied for the simulation of the impact and spread of liquid droplets onto solid surfaces [16,17].To further verify the numerical validation, the simulations of the impact and spread of droplets of water-based paint onto a flat surface have been implemented for the cases of θe = 40°, 50°, 80° and 95°.Impact velocity 0 v is 1.5 m/s.The droplet diameter 0 D is 40 μm. Figure 4 presents the shape of the droplets in the final equilibrium state.The measurement software Digimizer was adopted to measure the contact angle in Figure 4, and the measurement results show that the real contact angle θ is equal to 38.4°, 48.7°, 82.9° and 92.9°, respectively.Therefore, the results agree well with the theoretical solutions.

Numerical Validation
The Navier-Stokes equation has been successfully applied for the simulation of the impact and spread of liquid droplets onto solid surfaces [16,17].To further verify the numerical validation, the simulations of the impact and spread of droplets of water-based paint onto a flat surface have been implemented for the cases of θ e = 40 • , 50 • , 80 • and 95 • .Impact velocity v 0 is 1.5 m/s.The droplet diameter D 0 is 40 µm.Figure 4 presents the shape of the droplets in the final equilibrium state.The measurement software Digimizer was adopted to measure the contact angle in Figure 4, and the measurement results show that the real contact angle θ is equal to 38.4 • , 48.7 • , 82.9 • and 92.9 • , respectively.Therefore, the results agree well with the theoretical solutions.Based on the research of Vadillo et al. [18], the contact diameter is determined by

The Effect of Droplet Size
In this section, the effect of volume on droplet spread is investigated.Here, initial impact velocity ( 0 v ) is 1.5 m/s, the equilibrium contact angle (θe) is 60°, and the initial droplet diameters ( 0 D ) are 20, 30, 40, 50, and 60 μm. Figure 6 shows the variation in the maximum deformation ratio ( max

 ) versus
We 1/4 for five different volumes of water-based paint droplets.As we can see, the larger-sized droplets have greater spreading due to the greater inertia.Based on the experimental data shown in Figure 6, the max  associated with We 1/4 is obtained in the form of In good agreement with the experimental data, where the numerical coefficient is equal to 0.97.Therefore, we can obtain The above equation agrees with the result of Clanet et al. [11].Based on the research of Vadillo et al. [18], the contact diameter is determined by 1/3 (17) In the above equation, e is the equilibrium contact angle, D 0 is 40 µm, and D s , e is the spreading diameter of the droplet in the equilibrium stage.Figure 5   Based on the research of Vadillo et al. [18], the contact diameter is determined by In the above equation, θe is the equilibrium contact angle, 0 D is 40 μm, and Ds,e is the spreading diameter of the droplet in the equilibrium stage.

The Effect of Droplet Size
In this section, the effect of volume on droplet spread is investigated.Here, initial impact velocity ( 0 v ) is 1.5 m/s, the equilibrium contact angle (θe) is 60°, and the initial droplet diameters ( 0 D ) are 20, 30, 40, 50, and 60 μm. Figure 6 shows the variation in the maximum deformation ratio ( max

 ) versus
We 1/4 for five different volumes of water-based paint droplets.As we can see, the larger-sized droplets have greater spreading due to the greater inertia.Based on the experimental data shown in Figure 6, the max  associated with We 1/4 is obtained in the form of In good agreement with the experimental data, where the numerical coefficient is equal to 0.97.Therefore, we can obtain The above equation agrees with the result of Clanet et al. [11].

The Effect of Droplet Size
In this section, the effect of volume on droplet spread is investigated.Here, initial impact velocity (v 0 ) is 1.5 m/s, the equilibrium contact angle (θ e ) is 60 • , and the initial droplet diameters (D 0 ) are 20, 30, 40, 50, and 60 µm.Figure 6 shows the variation in the maximum deformation ratio (ξ max ) versus We 1/4 for five different volumes of water-based paint droplets.As we can see, the larger-sized droplets have greater spreading due to the greater inertia.Based on the experimental data shown in Figure 6, the ξ max associated with We 1/4 is obtained in the form of ξ max = 0.55 • We 1/4 + 0.89 (18) In good agreement with the experimental data, where the numerical coefficient is equal to 0.97.Therefore, we can obtain The above equation agrees with the result of Clanet et al. [11].

The Effect of Impact Velocity
This section investigates the effect of impact velocity on the maximum deformation ratio of droplets.In this set of experiments, the equilibrium contact angle (θe) is 60°, and the initial droplet diameter ( 0 D ) is 40 μm.In the above equation, the value of R-Square is 0.996, and a good agreement has been achieved.

The Effect of Impact Velocity
This section investigates the effect of impact velocity on the maximum deformation ratio of droplets.In this set of experiments, the equilibrium contact angle (θ e ) is 60 • , and the initial droplet diameter (D 0 ) is 40 µm.

The Effect of Impact Velocity
This section investigates the effect of impact velocity on the maximum deformation ratio of droplets.In this set of experiments, the equilibrium contact angle (θe) is 60°, and the initial droplet diameter ( 0 D ) is 40 μm.In the above equation, the value of R-Square is 0.996, and a good agreement has been achieved.According to Akao et al. [19] study, the maximum deformation ratio can be expressed as Senda et al. [20] proposed another prediction model on the maximum deformation ratio, and its expression is

The Effect of Equilibrium Contact Angle
To research the influence of the initial contact angle on the maximum deformation ratio, the initial droplet diameter ( 0 D ) was set to 40 μm, and the initial impact velocity ( 0 v ) to 1.5 m/s. Figure 9 shows the variation in the spreading radius (r(t)) and the center height (h(t)) with the increase in time (t) under equilibrium contact angle (θe = 40°, 60° and 120°).Figure 6 presents the variation in spreading radius and height of droplet with the time.As shown in Figure 6, the spreading radius of droplets decreases with the increase in equilibrium contact angle, whereas the height of droplet increases with the increase in equilibrium contact angle.By comparing the results for θe = 40°, 60° and 120° in Figure 9a, it has been proven that the spreading radius decreases with the increase in θe.For the case of θe = 40° and 60°, the spreading radius coincides in one line, and the spreading velocity is faster than the case of θe = 120° when time t ≤ 0.9 ms in Figure 9a.As shown in Figure 10, the fitting curve for the case of θe = 40° was generated to show

The Effect of Equilibrium Contact Angle
research the influence of the initial contact angle on the maximum deformation ratio, the initial droplet diameter (D 0 ) was set to 40 µm, and the initial impact velocity (v 0 ) to 1.5 m/s. Figure 9 shows the variation in the spreading radius (r(t)) and the center height (h(t)) with the increase in time (t) under equilibrium contact angle (θ e = 40 • , 60 • and 120 • ). Figure 6 presents the variation in spreading radius and height of droplet with the time.As shown in Figure 6, the spreading radius of droplets decreases with the increase in equilibrium contact angle, whereas the height of droplet increases with the increase in equilibrium contact angle.By comparing the results for θ e = 40 • , 60 • and 120 • in Figure 9a, it has been proven that the spreading radius decreases with the increase in θ e .For the case of θ e = 40 • and 60 • , the spreading radius coincides in one line, and the spreading velocity is faster than the case of θ e = 120 • when time t ≤ 0.9 ms in Figure 9a.As shown in Figure 10, the fitting curve for the case of θ e = 40 • was generated to show r * d = 0.321 The above equation appears to be accurate when t * < 0.3.Similar expressions were obtained by Rioboo et al. [4] and Gupta et al. [21].

The Effect of Equilibrium Contact Angle
To research the influence of the initial contact angle on the maximum deformation ratio, the initial droplet diameter ( 0 D ) was set to 40 μm, and the initial impact velocity ( 0 v ) to 1.5 m/s. Figure 9 shows the variation in the spreading radius (r(t)) and the center height (h(t)) with the increase in time (t) under equilibrium contact angle (θe = 40°, 60° and 120°).Figure 6 presents the variation in spreading radius and height of droplet with the time.As shown in Figure 6, the spreading radius of droplets decreases with the increase in equilibrium contact angle, whereas the height of droplet increases with the increase in equilibrium contact angle.By comparing the results for θe = 40°, 60° and 120° in Figure 9a, it has been proven that the spreading radius decreases with the increase in θe.For the case of θe = 40° and 60°, the spreading radius coincides in one line, and the spreading velocity is faster than the case of θe = 120° when time t ≤ 0.9 ms in Figure 9a.As shown in Figure 10, the fitting curve for the case of θe = 40° was generated to show  Based on the research of Pasandideh-Fard et al. [22], the max  can be followed by In the above equation, θa is the advancing contact angle.Under different equilibrium contact angles, θa is different, and the actual value of θa in the simulation is obtained by Digimizer.Figure 11 presents the variation in max  versus the set equilibrium contact angle.As shown in Figure 11, max  decreases with the increase in equilibrium contact angle.The fitting line in Figure 11 is the best fit of the results of the prediction model obtained by Pasandideh-Fard et al. [19], and the relation between max  and θe can be described as The R-Square is equal to 0.997 in the fitting.Simulation results are compared with the expression of Equation (24) as shown in Figure 11, and the results agree well with Equation (24) for 80° ≤ θe ≤ 120°.Based on the research of Pasandideh-Fard et al. [22], the max  can be followed by In the above equation, θa is the advancing contact angle.Under different equilibrium contact angles, θa is different, and the actual value of θa in the simulation is obtained by Digimizer.Figure 11 presents the variation in max  versus the set equilibrium contact angle.As shown in Figure 11, max  decreases with the increase in equilibrium contact angle.The fitting line in Figure 11 is the best fit of the results of the prediction model obtained by Pasandideh-Fard et al. [19], and the relation between max  and θe can be described as The R-Square is equal to 0.997 in the fitting.Simulation results are compared with the expression of Equation (24) as shown in Figure 11, and the results agree well with Equation (24) for 80° ≤ θe ≤ 120°.Based on the research of Pasandideh-Fard et al. [22], the ξ max can be followed by In the above equation, θ a is the advancing contact angle.Under different equilibrium contact angles, θ a is different, and the actual value of θ a in the simulation is obtained by Digimizer.Figure 11 presents the variation in ξ max versus the set equilibrium contact angle.As shown in Figure 11, ξ max decreases with the increase in equilibrium contact angle.The fitting line in Figure 11 is the best fit of the results of the prediction model obtained by Pasandideh-Fard et al. [19], and the relation between ξ max and θ e can be described as

Conclusions
In summary, the maximum deformation of water-based paint droplets was studied by numerical simulation based on the Navier-Stokes equations coupled with the level set method.Here, the effects of droplet size, impact velocity, and initial contact angle on the maximum deformation ratio of the water-based paint droplet were investigated.By the variation in the above three parameters, the maximum deformation ratio would change.As droplet size increased, the maximum deformation ratio of droplet scaled as We 1/4 .As impact velocity increased, the maximum deformation ratio increased, and there was a linear relation between the maximum deformation ratio and We.Finally, the effect of equilibrium contact angle was studied, and the results showed a relation between the maximum deformation ratio and equilibrium contact angle.

Conclusions
In summary, the maximum deformation of water-based paint droplets was studied by numerical simulation based on the Navier-Stokes equations coupled with the level set method.Here, the effects of droplet size, impact velocity, and initial contact angle on the maximum deformation ratio of the water-based paint droplet were investigated.By the variation in the above three parameters, the maximum deformation ratio would change.As droplet size increased, the maximum deformation ratio of droplet scaled as We 1/4 .As impact velocity increased, the maximum deformation ratio increased, and there was a linear relation between the maximum deformation ratio and We.Finally, the effect of equilibrium contact angle was studied, and the results showed a relation between the maximum deformation ratio and equilibrium contact angle.

Figure 1 .
Figure 1.Schematic domain of the experiment for the water-based paint droplet impact and spread.(a) 2D; (b) 3D.

Figure 1 .
Figure 1.Schematic domain of the experiment for the water-based paint droplet impact and spread.(a) 2D; (b) 3D.
definitions, 0 v is the initial impact velocity, max r is the maximum spreading radius of droplet, 0 D is the initial diameter of droplet, ( ) r t is equal to half of the diameter of the droplet with variation in time, We is the Weber number, Re is the Reynolds number, max  is the maximum deformation ratio, * d t is the dimensionless time, and * d r is the dimensionless spreading radius.The spreading radius ( ) r t and dynamic contact angle (θ) are defined in Figure 3.

Figure 2 .Table 1 .
Figure 2. A typical triangular mesh of the domains.

Figure 3 .
Figure 3. Sketch of a droplet of water-based paint on the solid surface after impact.

Figure 2 .Table 1 .
Figure 2. A typical triangular mesh of the domains.

Figure 3 .
Figure 3. Sketch of a droplet of water-based paint on the solid surface after impact.

Figure 3 .
Figure 3. Sketch of a droplet of water-based paint on the solid surface after impact.

Figure 4 .
Figure 4. Shape of droplets of water-based paint with different equilibrium contact angles.
the above equation, θe is the equilibrium contact angle, 0 D is 40 μm, and Ds,e is the spreading diameter of the droplet in the equilibrium stage.Figure5presents the simulation results and the prediction model of Vadillo et al. θe is in the range of 40°-95° in Figure5.In Figure5, the value of Ds,e is equal to the final spreading diameter of the droplet for the study of the effect of the equilibrium contact angle.As shown in Figure5, the simulation result is in accordance with the prediction model of Vadillo et al.Therefore, the Navier-Stokes equation coupled with the level set method can be used to simulate the dynamics of droplets after the impact onto the solid surface.

Figure 5 .
Figure 5.Comparison of the simulation result and the prediction model of Vadillo et al.

Figure 4 .
Figure 4. Shape of droplets of water-based paint with different equilibrium contact angles.
presents the simulation results and the prediction model of Vadillo et al. θ e is in the range of 40 • -95 • in Figure 5.In Figure 5, the value of D s , e is equal to the final spreading diameter of the droplet for the study of the effect of the equilibrium contact angle.As shown in Figure 5, the simulation result is in accordance with the prediction model of Vadillo et al.Therefore, the Navier-Stokes equation coupled with the level set method can be used to simulate the dynamics of droplets after the impact onto the solid surface.Coatings 2017, 7, 81 5 of 10

Figure 4 .
Figure 4. Shape of droplets of water-based paint with different equilibrium contact angles.

Figure 5
presents the simulation results and the prediction model of Vadillo et al. θe is in the range of 40°-95° in Figure5.In Figure5, the value of Ds,e is equal to the final spreading diameter of the droplet for the study of the effect of the equilibrium contact angle.As shown in Figure5, the simulation result is in accordance with the prediction model of Vadillo et al.Therefore, the Navier-Stokes equation coupled with the level set method can be used to simulate the dynamics of droplets after the impact onto the solid surface.

Figure 5 .
Figure 5.Comparison of the simulation result and the prediction model of Vadillo et al.

Figure 5 .
Figure 5.Comparison of the simulation result and the prediction model of Vadillo et al.

Figure 6 .
Figure 6.Maximum deformation ratio as a function of We 1/4 under different diameters.

Figure 7
presents the results for different impact velocity of droplet.With the increase in impact velocity, max  increases.Based on the experimental result, we can obtain

Figure 7 .Figure 8
Figure 7. Maximum deformation ratio as a function of We under different impacting velocities.

Figure 6 .
Figure 6.Maximum deformation ratio as a function of We 1/4 under different diameters.

Figure 7 10 Figure 6 .
Figure 6.Maximum deformation ratio as a function of We 1/4 under different diameters.

Figure 7
presents the results for different impact velocity of droplet.With the increase in impact velocity, max  increases.Based on the experimental result, we can obtain

Figure 7 .Figure 8
Figure 7. Maximum deformation ratio as a function of We under different impacting velocities.

Figure 7 .
Figure 7. Maximum deformation ratio as a function of We under different impacting velocities.

Figure 8 10 Figure 8 .
Figure 8 presents the experimental results and the results based on the prediction models from Akao et al. and Senda et al.In Figure 8, we can see the maximum deformation ratio increases due to the greater impact velocity, and the prediction model of Akao et al. is similar to Equation (20).Coatings 2017, 7, 81 7 of 10

Figure 8 .
Figure 8.Comparison of the maximum deformation ratio by the experimental result and the prediction models.

Figure 8 .
Figure 8.Comparison of the maximum deformation ratio by the experimental result and the prediction models.

Figure 10 .
Figure 10.Variation in dimensionless spreading radius * d r versus * t .

Figure 9 .Figure 9 .
Figure 9. Variation in (a) spreading radius and (b) height versus the time.

Figure 10 .
Figure 10.Variation in dimensionless spreading radius * d r versus * t .