Kinetic approach to the analysis of the flow around a levitating microdroplet

. Two-dimensional direct Monte Carlo simulation of the flow around an evaporating water microdroplet in the air atmosphere above a dry or liquid surface has been carried out. We consider a cylindrical droplet 1  m in diameter at a distance of 1  m above the surface. The temperature of the droplet is taken equal to 65°C and 80°C. A significant decrease in the air concentration between the droplet and the liquid film surface is observed. The features of water vapor flow separately from the droplet surface and from the liquid film surface are considered. It is shown that, under certain conditions, condensation on the lower side and evaporation on the upper side on the droplet can occur simultaneously.


Introduction
The * phenomenon of microdroplet levitation (droplet size ranging from ~1 to ~10 μm) near the surface of the heated liquid was first qualitatively described by V.Schaefer in 1971 [1].The mechanism of droplet formation is related to the upward movement of hot vapor-air mixture (Stefan flow) into an area of lower temperature where droplet condensation occurs.These droplets continue to grow through condensation and move downward under the force of gravity.At some point, the Stefan flow balances gravity, and the droplets ultimately levitate above the surface, often forming large ordered arrays.In 2003, A. Fedorets observed this phenomenon [2] when studying photo-induced thermocapillary flows, and it was named "droplet cluster" due to the use of a localized heat source with the diameter of 1 mm.Ordered arrays of levitating liquid microdroplets can be observed above various types of hot aqueous solutions, such as tea, coffee, water with detergents, tap water, boiled water, and distilled water [3].In [4], it was shown for the first time that microdroplet levitation and self-organization can occur not only above the liquid surface but also above a solid surface (underheated to the saturation temperature).It was found that the transition of microdroplets from a wet to a dry surface is accompanied by a significant change in the levitation height above the contact line [5].
An analytical model of microdroplet levitation above a dry surface was developed in [4], based on the representation of the droplet as a point evaporating source and using the method of images to estimate the flow velocity around the droplet.This model provides a good description of experimentally measured levitation heights for relatively small droplets.Taking into account droplet size and temperature heterogeneity allowed us to describe levitation height for larger droplets [6].To describe microdroplet levitation above the hot liquid, vapor flow from the liquid surface was added to the model [7].
In this work, for the first time, the direct simulation Monte Carlo (DSMC) method [8] is used to analyze the flow around a microdroplet.This method makes it possible to obtain detailed information about the processes in the flow region at the molecular level, taking into account the possible nonequilibrium of the particle velocity distribution function.It is known that the DSMC solution agrees well with the exact solution of the Boltzmann equation [9,10] at a much higher computational efficiency.
The flow near a micron-sized droplet at atmospheric pressure is characterized by the Knudsen number Kn = /D ~ 0.05 (here  ~ 0.05 m is the local mean free path, D = 1 m is the droplet diameter).Such a relatively large number justifies the application of the kinetic approach to solving this problem.Previously, the kinetic approach based on solution of the ellipsoidal statistical Bhatnagar-Gross-Krook (BGK) equation was used to solve related problems of determining the radiometric forces acting on a spherical particle in a rarefied gas [11,12].On the other hand, the DSMC method has been used to study thermophoretic force on a microdroplet [13,14] and to analyze the evaporation of droplets at the microscopic level [15].Also, this method is used for simulation of weak evaporation [16,17].

Modelling
We use the traditional DSMC scheme [8] with some elements of the "sophisticated DSMC" version [18,19].The physical domain is divided into computational cells with the same cell size x.The solution is advanced in discrete time steps.The modelling process is divided E3S Web of Conferences 459, 08006 (2023) https://doi.org/10.1051/e3sconf/202345908006XXXIX Siberian Thermophysical Seminar into two parts: first, the collisionless movement of particles in space and, second, the simulation of collisions between particles.The temporal evolution of the gas cloud is constructed by following the model particles.Each model particle represents FN real molecules (FN >> 1).The state of each particle is determined by its position in space x and the velocity vector v.The interparticle collisions are simulated in accordance with the "no-time counter" scheme.The number of pairs of particles selected for collisions in a cell per time step t is calculated as where NC is the number of particles in a cell,  is the total collision cross section, cr,max is the maximum relative velocity of two particles, V is the cell volume.
For each pair of particles, the relative velocity cr = |v1 -v2| is calculated and a random number 0 < Rf < 1 is generated.If the condition cr/cr,max > Rf is satisfied, then velocities of the selected particles change; otherwise, the particle velocities remain unchanged.The time step t must not exceed the average time between collisions, and the cell size x must not exceed the mean free path

Formulation of the problem
The problem is solved in a two-dimensional planar formulation.A rectangular area of 5 by 5 µm is set, divided into square cells.The cell size is x = 0.025 m.
We consider a mixture of model gases with molecular masses of ma = 29 u and mw = 18 u, which correspond to molecules of air and water.The internal degrees of freedom of the considered model particles are not taken into account in the calculation.The model of hard spheres is used.
For simplicity, we assume the same collision cross section for air and water molecules with a hard sphere diameter of d = 4.27 Å, which generally corresponds to known literature data [8,20,21].For such a molecular model, the mean free path of molecules at atmospheric pressure p = 10 5 Pa and the temperature T = 20ºC is  = ( ) where k is the Boltzmann constant.
Initially the area is filled with model molecules of air.In the center of the region there is a cylindrical drop with a diameter of 1 μm, from the surface of which model particles of water evaporate.
The lower boundary of the calculation area corresponds to either a solid or a liquid surface.The case of a solid surface corresponds to the levitation of a microdroplet over a dry surface.In this case diffuse reflection of all particles from the solid surface takes place.
The case of the liquid surface corresponds to the droplet levitation over a thin liquid film.Figure 1 shows a typical scheme of the computational domain for this case.In this case, air particles are reflected from the surface, while water particles are absorbed, and water particles evaporate with parameters corresponding to the liquid temperature.At the top of the area we set the absorption of all particles and the inlet flow of air particles with parameters corresponding to zero velocity and ambient temperature.At the left and right boundaries there is specular reflection for air particles and absorption of water particles.Also, a specular surface can be specified on the left and right boundaries, which corresponds to modeling a chain of identical microdroplets at the same height above the surface.This formulation of the boundary conditions qualitatively corresponds to the experimentally observed monolayer of droplets located at the same height above the surface with the same distance between the droplets [7].where  is the evaporation/condensation coefficient, pS is the saturated vapor pressure at the surface temperature TS, nS = pS/(kTS).For the velocities of evaporating particles, a half-Maxwellian distribution function is given where u, v, w are the components of the velocity vector.The temperature of the droplet and the liquid film is set as TS = 80°C with the corresponding pressure of saturated water vapor on the surface pS = 0.5 atm.The ambient temperature of air molecules at the boundaries of the computational domain is set to 20°C.The evaporation / condensation coefficient for water molecules is set to  = 0.1 [22].
Three types of particles are considered separately: (1) air molecules, (2) water molecules evaporated from the surface of the microdroplet, and (3) water molecules evaporated from the liquid surface below.An analysis of the movement of water vapor particles of various origins (from the droplet or from the liquid film) allows one to obtain a more detailed understanding of the processes taking place around the droplet.

Results and discussion
Figure 2 shows the flow field for the 1 µm droplet above a dry solid surface.There is a flow of water vapor from the droplet with a corresponding displacement of air and a decrease of its concentration.The forming flow is reflected from the solid surface, which leads to a relative increase in the concentration of water vapor in the region between the droplet and the surface.Figure 3 shows the similar flow field for a single droplet above a liquid film.Since in this case evaporation takes place not only from the droplet, but also from the liquid surface from below, the total flow of water vapor is directed upwards and to the sides (left and right) from the droplet.One can see a significant increase in the concentration of water vapor under the drop (Fig. 3a).
Figure 4 shows concentration distribution and streamlines separately for water vapor evaporating from the droplet surface and from the liquid film surface.In this case, absorption of water molecules takes place on the surface of the liquid film, so that a significant part of the water molecule from the droplet surface absorbs on the lower surface (Fig. 4a).At the same time, the flow of particles from the lower boundary absorbs on the surface of the droplet (Fig. 4b).Depending on the ratio of particle fluxes from the droplet and from the liquid film surface, the total flux of water particles from the droplet surface can be either positive (then the droplet evaporates and decreases in size) or negative (and then the droplet grows in size).In our case, 37% of the particles evaporated from the surface of the droplet come back and absorbs, i.e. the back flux is b = 0.37 0, where 0 is the total flux of evaporated molecules from the droplet surface.In addition, we have the flux of absorbed particles, initially evaporated from the liquid film surface, d = 0.29 0.Thus, the total flow from the droplet is positive ( = 0 -b -d = 0.34 0) and the droplet decreases in size over time.It should be noted that the situation can dramatically change if the droplet temperature is lower than the liquid film temperature.In this case, the vapor flow from the droplet decreases, and the balance can become negative, i.e. the droplet will increase in size over time.This is indeed usually the case in experiment, when droplets above a liquid film increase in size until they fall down under the action of gravity.Figure 5 shows the water vapor flow around the droplet for the case of the water vapor pressure on the droplet surface 0.25 atm, which corresponds to a temperature of 65°C.The back flux of particles evaporating from the droplet is practically the same, b = 0.39 0.However, the relative flux of absorbed particles from the liquid film surface increases, d = 0.63 0.As a result, the total flow from the droplet is negative ( = 0 -b -d = -0.020) and the droplet increases in size over time.Note that in this case, the condensation process dominates from below the droplet, where the droplet grows, while the evaporation process dominates from above, where the droplet decreases.This is clearly seen from the streamlines of water vapor directed from below towards the droplet and from above away from the droplet (Fig. 5), in contrast to the streamlines directed away from the droplet only as the droplet decreases (Fig. 3a).The above results refer to the single droplet regime.If we consider a chain of droplets, then the results some change.Figure 6 shows the flow field around the droplet for this case.Here, specular surfaces are specified on the left and right boundaries, which correspond to modeling a chain of identical droplets.In this case, in contrast to Figure 3a, the flow is directed only upwards.However, the flux of condensing water particles to the surface only slightly increases, which indicates the evaporation of the droplet for this case.

Conclusions
Direct Monte Carlo simulation of the flow around an evaporating water microdroplet above a dry or liquid surface has been carried out.The problem is solved in a two-dimensional planar formulation.We consider a cylindrical drop 1 m in diameter at a distance of 1 m above the surface.It is shown that there is a significant decrease in the air concentration near the droplet and the liquid lower surface.It is shown that for the same temperature of the droplet and the liquid surface, there is a decrease in the size of the droplet as a result of evaporation.If the droplet temperature is less than the surface temperature, the droplet grows due to condensation.

Fig. 1 .
Fig. 1.Scheme of the computational domain for a pair of microdroplets over an evaporating thin liquid film.The flux of water particles from the evaporation surface of the microdroplet or the liquid surface is determined by expression

Fig. 2 .
Fig. 2. Density distribution and streamlines for water vapor (a) and for air (b) for a droplet above a dry surface for the droplet temperature 80°C with the saturated vapor pressure 0.5 atm.

Fig. 3 .
Fig. 3. Density distribution and streamlines for water vapor (a) and air (b) for a droplet above a liquid film for the droplet temperature 80°C with the saturated vapor pressure 0.5 atm.

Fig. 4 .
Fig. 4. Density distribution and streamlines for water vapor evaporating from the surface of the droplet (a) and from the surface of a liquid film (b) for the droplet temperature 80°C with the saturated vapor pressure 0.5 atm.

Fig. 5 .
Fig. 5. Density distribution and streamlines for water vapor for a droplet above a liquid film for the droplet temperature 65°C with the saturated vapor pressure 0.25 atm.

Fig. 6 .
Fig. 6.Density distribution and streamlines for water vapor for a chain of droplets above a liquid film (specular boundary conditions at the left and right border) for the droplet temperature 80°C with the saturated vapor pressure 0.5 atm.