Lattice Boltzmann simulation of wetting gradient accelerating droplets merging and shedding on a circumferential surface

A lattice Boltzmann method (LBM) based on Shan-Chen pseudo-potential model is used to investigate the process of droplets merging and shedding on a gradient wetting circular surface. The effects of wetting gradient, radius, and radius ratio on droplet merging and shedding were mainly explored. The results show that applying a wetting gradient on the circumferential surface can accelerate the process of droplet merging and shedding. The velocity of droplets merging and shedding increase with the increase of the wetting gradient. The more hydrophilic the original surface, the better the optimization effect of accelerating drainage is by applying a wetting gradient. The radius and radius ratio significantly affect droplets merging and shedding on the gradient wetting surface. The larger the radius and radius ratio, the shorter the droplets merging and shedding time.


Introduction
Film condensation and dropwise condensation are the main modes of condensation heat transfer in industrial production. The dropwise condensation heat transfer coefficient is obviously higher than that of film condensation because of its lower heat transfer resistance. For dropwise condensation, speeding up droplets' detachment rate outside the condenser tube can reduce thermal resistance and increase heat transfer efficiency.
Wetting gradient surfaces can drive droplets' directional movement and capture droplets. Many researchers have done a lot of experimental and numerical simulation studies in this area (Bai et al., 2018;Chowdhury et al., 2019;Hou et al., 2016;C. Liu et al., 2017;Q. Liu & Xu, 2016). Hirai et al. (2017) used silicon nanostructures to prepare wetting gradient surfaces, studied the upward movement process of droplets on inclined surfaces, and analyzed the driving force and resistance of droplets during movement. Lee et al. (2014) studied the transport process of droplets on a surface with micropillars structure and analyzed the droplets' behavior from the energy perspective. They pointed out that when the droplets move from the hydrophobic area to the hydrophilic area, the contact area increases and the surface free energy changes inversely. The reason is that the droplet's shape tends to minimize the total surface energy on all interfaces. Xu et al. (2019) used the CONTACT Ming Gao gaoming@usst.edu.cn VOF model to simulate the sliding motion of droplets on an equidistantly spaced hydrophilic and hydrophobic inclined surface. They found that when the contact angle difference is large, the droplets can be captured and collected by the surface. Some researchers have also studied the phenomenon of droplet aggregation and shedding. F. C. Wang et al., 2011 constructed a theoretical model of energy conservation to analyze the kinetic behavior of droplet coalescence, and analyzed the effects of surface energy and gravitational potential energy released during droplet coalescence on droplet shedding. H. Wang et al., 2020) simulated the condensation process on the surface of a two-dimensional inverted groove under the action of gravity and studied the nucleation, growth, merger, and shedding processes of droplets on surfaces with different wettability. They found two modes of droplet shedding: pure gravity-induced shedding and gravitycapillary force-induced shedding. Malekzadeh and Roohi (2015) used OpenFOAM software to investigate regimes in a two-dimensional T-junction microchannel geometry. They found that different droplet formation states can be obtained by changing the capillary number, contact angle, Reynolds number, and flow rate ratio. Therefore, according to the above studies, applying the wetting gradient on the tube surface is feasible to accelerate the droplets' movement and induce their coalescence and shedding.
In recent years, the lattice Boltzmann method (J. J. Huang et al., 2014;W. X. Huang & Sung, 2010;Montessori et al., 2020;X. Wang et al., , 2021 has received extensive attention from researchers. It is a mesoscopic numerical method. No matter from which point of view, LBM is different from the conventional flow simulation method based on the macroscopic Navier-Stokes (N-S) equation (Guo & Zheng, 2009). The lattice model retains the basic components of fluid physics while discarding irrelevant details as much as possible, allowing people to recover fluid dynamics equations in the macroscopic limit (Roberto & Succi, 1992) Due to the mesoscopic characteristics of the LBM, it can easily describe the interaction between different phases in multiphase flow, so it has a good application in multiphase flow (He et al., 2009). At present, in the field of multiphase flow, four models of multiphase flow are mainly proposed, for example, Shan-Chen pseudo-potential model (Shan & Chen, 1993), color model (Gunstensen et al., 1991), free energy model (M. R. Swift, 1995Swift, , 1996, and phase field model (He et al., 1998(He et al., , 1999. Among those models, Shan-Chen pseudo-potential model uses a pseudopotential to describe the interaction between particles more directly, so it has been widely used. Montessori et al. (2017) propose an entropic pseudo-potential approach with a variable vapor-liquid viscosity ratio. They present several applications of the entropic pseudo-potential approach: the impact of droplets with hydrophobic walls and head-on and off-axis collisions between droplets.
X. Wang and Chen (2019) used the Shan-Chen pseudo-potential model to simulate the directional migration and merging behavior of droplets on gradientwetted surfaces. They analyzed the influence of droplet size and wetting gradient on the droplets' dynamic behavior on the gradient-wetted surface. B. Huang et al. (2018) used Shan-Chen pseudo-potential model to simulate the condensation process of droplets and found that nanostructures based on microstructure design could further improve the condensation efficiency of droplets. X. L. Liu and Cheng (2015) used the Shan-Chen pseudo-potential model to simulate the process of droplets merging and bouncing on a super-hydrophobic surface. Yuan et al. (2021) used a high-density-ratio pseudopotential LBM with tune-able surface tension to investigate the dynamics of two droplets impacting a liquid film. They found that the energy loss during the impact process and the velocity discontinuity in the liquid film are the two key factors affecting the stability and evolution process of the crown. Jiang et al. (2021) used the lattice Boltzmann methods to simulate the air film flow in the narrow clearance of a GLJB. They investigated the changing rules of the air film pressure, thickness, and velocity distributions under different bearing speeds, eccentricity ratios and slip conditions.
In the field of condensation heat transfer, the heat transfer effect of dropwise condensation is much stronger than that of film condensation. In order to maintain the dropwise condensation, it is necessary to remove the condensate in time. The traditional way mostly relies on gravity to remove the droplets, and the wetting gradient surface can provide a new idea for accelerating the drainage process. At present, there are many kinds of research on directional droplet movement on a horizontal surface with a wetting gradient, but there are few studies on the non-horizontal surface. There are many studies on merging double droplets, but most researchers focus on uniformly wetted surfaces. In this paper, a wettinggradient surface is introduced to study the process of accelerating the merging and shedding of two droplets on the circumferential surface, the influence of radius and radius ratio on the shedding process of two droplets is investigated, and the velocity field of droplets sliding downward and shedding process is analyzed. In the simulation of this paper, applying a wetting gradient on the circumferential surface is a major difficulty encountered in this study. Finally, we divide the circumferential surface into two parts with left and right symmetry, and each part is divided into 12 sections with continuously changing wettability to form a wetting gradient.

Calculation model
Lattice Boltzmann method (LBM) is a mesoscopic simulation method between macroscopic and microscopic scales. Compared with other numerical simulation methods, it has the advantages of clear physical meaning, easy programming, simple boundary condition setting, and good parallel performance (He et al., 2009). We simulate the fluid flow process through the collision and migration of fluid particles' distribution function on the grid points.
D2Q9 model (Qian et al., 1992) is adopted in this paper, and its discrete velocity calculation is defined by: 6,7,8 (1) Where i is the particle's movement direction.
A single relaxation time collision operator is used. The fluid particles' evolution equation can be defined as follows: Where t is time step, it usually takes 1. τ is the dimensionless relaxation time, and the relationship between τ and kinematic viscosity ν is ν = c 2 s (τ − 0.5) t. f i and f eq i are the velocity distribution function and the particle's equilibrium state velocity distribution function. f eq i is given by: In the formula, ρ and u are the macroscopic fluid particle's density and velocity at the lattice point x at time t, and the calculation equations are as follows: c s is the sound speed of lattice, c s = √ RT, RT = 1/3, ω i is the weight coefficient, which represents the probability of fluid particles in all directions. i = 0, ω i = 4/9; i = 1, 2, 3, 4, ω i = 1/9; and i = 5, 6, 7, 8, ω i = 1/36. This paper uses the Shan-Chen pseudo-potential model to calculate the particles' interaction force, which mainly includes two parts, one is the fluid particles' interaction force F int , and the other is the interaction force between fluid particles and solid particles F ads .
F int is as follows: Where G is the fluid particles' interaction coefficient and ψ is the interaction potential, which is given by: Where, G = −120,ψ 0 = 4, ρ 0 = 200 according to references (Sukop & Thorne, 2007). The expression of fluid-solid force F ads is given as follows: Where G ads is the interaction coefficient between fluid particles and solid wall surface. s(x + e i t) represents the switching function. When the lattice point is a fluid particle, s(x + e i t) = 0; when the lattice point is a solid particle, s(x + e i t) = 1.
Under the action of a force, the fluid velocity will change. The true fluid's velocity is the average value of the velocity before and after the particle collision, expressed as: Where F is the total force, which mainly includes the fluid particles' interaction force F int , the interaction force F ads between fluid and solid particles, and the gravity F g on fluid particles. The equation for F is as follows: Where the calculation formula of F g is given by: Where ρ(x) is the fluid particle density, ρ v is the vapor phase density, and g is the gravitational acceleration.

Verification of laplace law
In this paper, the Young-Laplace law is used to verify the correctness of the above model. Young-Laplace law shows that for a steady droplet, the droplet's inside and outside pressure difference P has a linear relationship with the reciprocal of radius, which satisfies the following equation: The calculation area is taken as NX×NY = 200 × 200, with periodic boundaries around. The saturated liquid and vapor phase densities are set to ρ l = 524.39 and ρ v = 85.7, respectively. In the beginning, place a droplet in the center of the calculation area. The different pressure differences are obtained by changing the size of the radius. We choose radius r = 15, 20, 25, 30, 35, 40. The simulation results are shown in Figure 1, which conforms to the Young-Laplace law.

Verification of contact angle
The contact angle represents the wettability of the liquid on the solid wall. In Shan-Chen pseudo-potential model, the contact angle can be adjusted by changing the value G ads . In this paper, the calculation region is NX×NY = 200 × 200, the upper and lower boundaries adopt bounce back boundaries, and the left and right boundaries adopt periodic boundaries. As shown in Figure 2, the contact angle θ has a linear relationship with the fluid-solid interaction coefficient G ads , which is consistent with the reference (Sukop & Thorne, 2007).

Numerical validation
The droplet movement problem on the horizontally wetted gradient surface is similar to the problem studied in this paper. In order to verify the reliability of the algorithm in this paper, the droplet movement velocity and theoretical prediction equation on the horizontally wetted gradient surface are verified in this paper. In the theoretical model, (Raphaël, 1988) assumed that no deformation occurred during droplet movement, and obtained the theoretical equation of droplet movement velocity, namely: Where γ is the surface tension of the droplet; θ d is the dynamic contact angle, 2 cos θ d = cos θ a + cos θ r ; θ a is the equilibrium forward contact angle; θ r is the  equilibrium receding contact angle; l is a parameter that measures the ratio of macroscopic scale to molecular scale, and ν is the droplet viscosity. Figure 3 is a schematic diagram of a droplet in the horizontal wetting gradient movement. The surface on the right side is more hydrophilic. The calculation area is NX×NY = 100 × 150, the initial radius of the droplet is 15, the upper and lower boundaries adopt the standard bounce back format, and the left and right boundaries adopt periodic boundary conditions. Comparing the simulation results with the theoretical prediction results of equation (12), as shown in Figure 4, the trend is in good agreement, but the specific values have deviated. This is because the theoretical equation assumes that the droplet does not deform when it moves, whereas in reality they do. There will be deformation during the movement, and the simulation results are exactly in line with the actual situation. The deformation of the droplet needs to consume a certain amount of energy, so the movement speed of the droplet obtained by the simulation is lower than the theoretical prediction value.

Grid independence verification
The motion of double droplets on the horizontally wetted gradient surface is similar to the problem studied in this paper. Therefore, grid independence verification is carried out for the merging process of two droplets on the horizontally wetted gradient surface, as shown in Figure 5. We selected grid numbers 200lu×100lu, 300lu×150lu and 400lu×200lu for calculation. The calculation results are shown in Figure 6. Under three kinds of grid numbers, the variation range of double droplet merge time is small, which satisfies the grid independence. In this study, 500lu×500lu grids were selected for calculation, so the calculation result was independent of the number of grids. on both sides. The wettability of the circumferential surface is symmetrically distributed and increases from top to bottom. So the upper wettability is the worst, and the lower wettability is the strongest.

Bottom of the circumference is hydrophilic
At the initial moment, a circular solid with a radius of 100 units is set in the center of the calculation domain, and droplets with a radius of 20 units are placed symmetrically on the left and right sides of the circular solid. After running 5,000 steps, the droplets reach a stable state, and then the wetting gradient is applied on the circumference solid surface, and the gravity is applied to the droplets. Figure 8 shows the coalescence and shedding process of droplets on wetted surfaces with different gradients when the bottom of the circumference is hydrophilic (θ b = 63.4°). The contact angles θ t at the top of the tube are 63.4°, 83.5°, 96.8°, 106.1°, and 115.4°, respectively. The bottom contact angles θ b are 63.4°, and the wetting gradients θ are 0°, 1.7°, 2.8°, 3.6°, and 4.3°, respectively. Initially, the wettability of the surface is different. Under the combined action of unbalanced tension and gravity, the droplets will slide downward along the circumferential surface. As the droplets move to the bottom, the two droplets will merge into a large one. After the combination, a certain amount of surface energy and gravitational potential energy will be released, prompting the droplets to move downward and finally fall off from the circumferential surface. During the downward sliding process, the shape of the droplets changes continuously, which is due to the combined effect of gravity, non-equilibrium tension, and viscous resistance on the droplet (Gong & Cheng, 2012). Due to the small contact Angle at the bottom of the circumference, the surface is hydrophilic and has a large adhesion force to the droplets. The combined droplets cannot fall off from the surface as a whole but break into two parts, one part falls off, and the other part sticks to the circumference surface. Zhang et al. (2020) also found this phenomenon. With the increase of the wetting gradient, the time required for droplets merging and shedding decreases. Because the larger the wetting gradient is, the larger the non-equilibrium tension will be in the downward sliding process of the droplets, and the greater the acceleration will be in the downward sliding process.
To further analyze the effects of the wetting gradient on the droplets merging and shedding process, several cases with different contact angles at the bottom of the circumferential surface are also selected for further simulation analysis. The bottom contact angles θ b are 48.4°, 55.1°, 63.4°, 67.1°, 71.7°, 81.2°, 90.6°and 98.7°, respectively. Figure 9 and Figure 10 show the changes in droplet merging time and shedding time with the wetting gradient. Merging time is defined as the time of the two droplets to slide down from the beginning just to touch each other at the bottom of the circumference. Shedding time refers to the time from merging to dripping off the surface. As shown in Figure 9 , applying a wetting gradient can accelerate the coalescence of droplets on the circumferential surface compared with θ = 0 • (i.e. uniform wetting surface). For example, when the bottom contact angle θ b = 48.4°and the wetting gradient θ = 4.3°, the merging time is only 64.8% of the original. This means that the wetting gradient greatly reduces the droplet merging time. This is because droplets moving on gradient-wetted surfaces are subjected not only to gravity but also to non-equilibrium tension, which together pushes the droplets downward. The larger the wetting gradient, the more obvious the acceleration effect on the droplet merger. When the bottom contact angle θ b is 48.4°, the drop shedding time on the surface with the wetting gradient is significantly reduced compared with that on the uniform wetting surface. For example, when the wetting gradient θ is 4.3°, the shedding time is only 57% of the original. This may be because the uniform wetting surface is relatively hydrophilic at this time, and the adhesion force to the droplet is relatively large. After  the wetting gradient is applied, the droplet motion can be accelerated, and the droplet obtains more initial kinetic energy to overcome the adhesion force of the surface before merging, thus promoting drop shedding.
To explore the optimization effect of the wetting gradient on droplet merging and shedding time on different hydrophilic and hydrophobic surfaces, we define the dimensionless time of droplet moving as the time when the wetting gradient is applied compared to the time required when the wetting gradient θ is 0. So when the wetting gradient θ is 0, the value of dimensionless moving time is 1. Figure 11 shows the variation of dimensionless moving time with wetting gradients. As shown in the figure, the slope of the curve reaches its maximum value when the bottom contact angle is 48.4°. It shows that applying a wetting gradient on the more hydrophilic surface has a better optimization effect on droplets merging and shedding. This is because the more hydrophilic the surface, the greater the adhesion force to the droplets, and the less likely they are to fall off from the surface. After applying the wetting gradient, the droplet movement can be accelerated, so that the droplet has more kinetic energy before merging, promoting its fall off from the surface. This paper defined the contact line length between the bottom of the two-dimensional droplets and the wall as the wetting length. Figure 12 shows the variation of wetting length on the circumferential surface with time under different wetting gradients. The movement of droplets on a gradient wetting surface can be divided into four stages: spreading, sliding, merging, and shedding. Since the droplet's initial contact angle before dropping is greater than the contact angle at the dropping position, the droplets will spread on the gradient surface at the beginning, and then the droplets begins to slide downward along the circumferential surface due to the combined action of gravity and surface tension. Show as in Figure 12, when the wetting gradient is small (e.g. θ ranges from 0°to 2.8°), the wetting length of the droplet decreases in the sliding process before merging. At this time, gravity is the dominant factor, and the droplets are dragged downward by gravity, so the wetting length gradually decreases. When the wetting gradient is large (e.g. θ = 3.6°and 4.3°), the wetting length of the droplet first decreases and then increases during the sliding stage. At this time, the unbalanced tension is the dominant factor, and the spreading ability of the droplets is gradually enhanced, which can offset the decrease in the wetting length caused by gravity. In the merging stage, when the two droplets merge, the wetting length increases instantaneously. After merging, the droplets begin to fall off, and the wetting length decreases gradually under the action of gravity. When the droplets fall off, the wetting length also increases sharply (See the rising part at the end of the curves.), because the bottom of the circumference is relatively hydrophilic, and the droplets cannot fall off completely, so a part of the droplet remains on the circumferential surface. The drop breaks and bounces back causing the wetting length to rise suddenly.
As shown in Figure 13, we give the velocity field of droplet motion when the wetting gradient θ is 0°, 2.8°, and 4.3°, respectively. Compared with θ = 0 • (i.e. uniform wetting surface), when θ = 2.8 • and θ = 4.3 • , the internal velocity vector is larger and denser in the droplet sliding process and has greater kinetic energy before merging. The greater the wetting gradient, the greater the velocity vector inside the droplet. This is because as the wetting gradient increases, the nonequilibrium tension generated by the difference in wettability obtained during the droplets' motion also increases, and thus the obtained acceleration is also greater. After merging, the unbalanced tension caused by wettability disappears (Since the wetting gradient is symmetrically distributed on both sides of the tube, the wettability is uniform at the bottom). At this time, the droplet is mainly affected by gravity, and the internal vector begins to change from horizontal to vertical and is symmetrically distributed. Since the bottom of the circumference is relatively hydrophilic, the adhesion force of the circumference surface to the droplet is large. As the droplet detaches, it forms a long neck due to the hydrophilic properties of the bottom. When the gravity is greater than the adhesion force, the droplet breaks into two parts from the neck. One part of the velocity vector is downward,   and it is separated from the circumference surface. The other part of the velocity vector is upward, and finally, it adheres to the circumference surface. After the droplet separates from the circumferential surface and loses the adhesion force, the velocity vector of the tail increases instantly and accelerates downward movement. Figure 14 shows the effect of the wetting gradient on droplet coalescence and shedding when the bottom of the circumference is hydrophobic. Since the bottom of the circumference is relatively hydrophobic, the adhesion force of the surface to the droplet is relatively small. At this time, gravity is the dominant factor. Under the influence of gravity, the wetting length of the droplet on the surface is small. As the two droplets move to the bottom of the circumference, the deformation is more severe due to gravity, the two droplets' heads first contact and then merge. While in Figure 8, when the bottom of the circumference is hydrophilic, the feet of the droplets contact first and then merge. In Figure 14, when the bottom is hydrophobic, the gravity can counterbalance the adhesion force, and the droplet detaches from the surface and does not break as shown in Figure 8. On the hydrophobic surface, the droplet shedding time decreases with the increase of wetting gradient. Figure 15 shows the process of droplet merging and shedding on the gradient wetting surface under different radii. In each case, the droplet diameters are the same on both sides of the circumference surface. In Figure 15, the droplet radius are 20, 21, 22, 23, and 24, respectively. The larger the radius, the greater the effect of gravity on the droplet. When the two droplets move to the bottom of the circumference, the deformation is more intense. Due to the strong wettability at the bottom of the circumference surface, a slender neck will be formed when the droplet falls off, and then it will break off at the neck under the action of gravity. With the increase of the radius, the effect of gravity on the droplet also increases. During the downward shedding process, the confrontation between the adhesion force and the gravitational force is stronger, so the length of the neck formed when the droplet falls off will be longer. Figure 16 shows the changes of droplet merging and shedding time with radius. With the increased radius, the time required for droplets merging and shedding on the gradient wetting surface are both decreases. The larger the droplet radius, the greater the driving force and the faster the droplets slide down the surface. After the  droplets merges, the droplets with a larger radius are also subjected to a greater downward driving force, which can overcome the circumferential surface adhesion and thus fall off faster. Figure 17 shows the variation of the droplet wetting length on the gradient wetting surface with time under different radii. Figure 17 and Figure 12 have similar curve trends. In the spreading stage, the larger the radius, the larger the initial contact area between the droplet and the surface, and therefore, the longer the wetting length. In the sliding stage, the droplet is greatly affected by the downward drag of gravity, the surface tension cannot be counterbalanced with gravity, and the wetting length decreases gradually. The larger the radius, the more severe the droplet wetting length decreases. In the merging stage, the wetting length suddenly increases as the two droplets merge into a large one and then decreases gradually. In the shedding stage, part of the droplet will stick to the surface because the droplet cannot be completely shed, so the wetting length increases suddenly. Figure 18 shows the process of merging and shedding two droplets on the gradient wetting surface at different radius ratios. The radius ratios are 1, 1.05, 1.1, 1.15, and 1.2, respectively. As shown in Figure 18, the large droplets reach the bottom of the circumference faster than small droplets in the downward sliding process. The larger the droplet radius, the greater the influence of gravity, and the degree of deformation is more significant when the droplet moves downward along the circumference surface than the droplet with a smaller radius. At different radius ratios, the shapes of droplets are different when they merge and fall off. When the radius ratio is equal to 1, the droplets move symmetrically to the bottom and eventually fall off from the center of the circumferential surface. When the radius ratio is greater than 1, since the larger droplet travels to the bottom of the circle before the smaller one, the two droplets are no longer symmetrical when they are combined, and the combined droplets swing left and right, and the final shedding position shifts to the left. The larger the radius ratio, the faster the large droplet moves to the bottom of the circle, the more distance it crosses the center of the bottom, and the more the droplet is shifted to the left. Figure 19 shows the changes in droplet merging and shedding time at different radius ratios. It can be seen that when the radius ratio increases, the time required for shedding decreases, while the time required for merging first decreases and then increases. This is because the large droplet is deformed more strongly by gravity. When reaching the bottom of the circumference surface, the large droplet wetting length is shorter due to gravity, and the small droplet needs to travel a longer distance to coalesce with the large one. As shown in Figure 19, the larger the radius ratio, the shorter the time required for droplet shedding. This is because the larger the radius ratio, the greater the surface energy and gravitational potential energy released after merging, and the droplet can gain more kinetic energy to fall off the circumferential surface. Figure 20 shows the velocity field of droplets at different radius ratios. In the droplet's downward sliding process, the velocity vector inside the large droplet is larger and denser than that inside the small one, so the large droplet reaches the bottom faster than the small droplet. When merging, the velocity vector inside the small droplet is larger than that inside the large one. This is because the large droplet has uniform wettability at the position, and the driving force generated by the wetting gradient disappears. At the merging moment, the large droplet is mainly affected by gravity, and the velocity vector changes from horizontal to downward, while the small droplet is still in the non-uniform wetting region. Inside the small droplet, gravity and nonequilibrium tension jointly provide the driving force for forwarding movement, and the small droplet is the main force of merging at this time. After merging, the internal velocity vector gradually changes its direction and finally uniformly downwards. The velocity vector on the left is larger than that on the right. When the droplet falls off, due to the strong wettability of the bottom, the droplet breaks off from the neck, and the velocity vector direction in the two broken parts is different. The smaller part is due to the large adhesion force on the circumference surface and finally adheres to the circumference surface.

Conclusions
In this paper, the Shan-Chen pseudo-potential model is used to simulate the process of droplets merging and shedding on the gradient wetting circumferential surface. The effects of wetting gradient, radius size, and radius ratio on the process of droplets merging and shedding are studied. The variation of wetting length with time and the velocity field distribution in the process of droplets movement are analyzed. According to the simulation results, the following conclusions are obtained: (1) The wetting characteristics at the bottom of the circumferential surface affect the droplet shedding state. When the bottom is hydrophobic, the droplets merge and fall off the circumference surface as a whole. When the bottom is hydrophilic, the droplet breaks from the neck, most of the droplet falls off, and a small part adheres to the bottom surface. The more hydrophobic the surface, the more conducive to droplet shedding. (2) The wetting gradient applied on the circumferential surface can accelerate the droplet merging and shedding process. The stronger the hydrophilicity of the surface, the better the effect of the wetting gradient applied on accelerating liquid drainage. In the case of the same contact angle at the bottom, the higher the wetting gradient, the faster the droplets merge and fall off. Therefore, rapid droplet movement and coalescence can be achieved by changing the surface wetting gradient outside the condensation tube.
(3) The size of the radius has a significant effect on the droplets coalescence and shedding process. The larger the radius, the easier for droplets to detach from the circumferential surface, and the time required for coalescence and detachment decreases. It indicates that smaller droplets are less likely to fall off from the surface. Therefore, a wettability gradient can be applied on the surface to induce smaller droplets to merge and fall off. (4) With the increase of radius ratio, the droplet merging time first decreases and then increases. The overall shedding time decreases with the increase in radius ratio. When the radius ratio is greater than 1, the droplet shedding point shifts to the left. (5) In practical industrial applications, we can prepare hydrophobic surfaces with wetting gradients to accelerate drainage.
There are still some deficiencies in our study: (1) In this paper, the study of droplets' movement on the wetted gradient surface is based on a smooth surface, while the actual solid surface is rough. Therefore, the influence of surface roughness on droplets' movement process needs to be considered to make the simulation results closer to the actual situation.
(2) The simulation in this paper is based on the isothermal model, which does not involve the temperature change in the simulation process. Still, the actual droplet movement is often accompanied by heat transfer, condensation, and evaporation. Therefore, temperature changes should be considered in future studies to investigate whether temperature changes affect the conclusions.
(3) Due to the limitation of computing resources, the model in this paper is based on two-dimensional simulation, which has low accuracy in computing results. Therefore, future studies should consider three-dimensional simulation to improve the accuracy.