Drop impacting on the liquid surface with a liquid film

. The paper presents the obtained data about the conditions for the splashes when a drop impacts the surface of a liquid with a liquid film. Based on the results of numerical simulation, it is shown that the presence of a liquid film allow significantly expand the area of drop impact without the splashes. The results can be used to implement an active method to reduce drop entrainment, for example, in gas-liquid separators.


Introduction
Despite its small scale, the problem of drop impact on a liquid or solid surface is of great importance in many scientific and technical applications: aerosol cooling or painting, thermal spraying [1,2], liquid-gas separation [3], inkjet printing [4,5], forensic practice. Under certain conditions, as a result of a drop impacting on the surface, single or multiple secondary drops of smaller diameter are ejected into the external gaseous medium, which can cause significant problems. For example, in gas-liquid separators, the dynamic pressure of the purified gas reducing the separation efficiency [3] can carry these secondary drops with a lower Froude number (Fr) away. In inkjet printing, this reduces the accuracy of ink application. The mechanical properties of a thermally sprayed coating also strongly depend on the drop impact kinematics [1]. For this reason, the main emphasis in such problems is on revealing the criterion for the ejection of secondary drops into a gaseous medium, the formation of craters or so-called coronas.
The paper [6] presents an exhaustive analysis of the studies performed at that time with various problem statements about the impact of a liquid drop. All these statements can be classified according to the type of surface on which the drop falls: a solid wall, a liquid, and a liquid film of finite thickness on a solid surface. In terms of the dimensionless liquid thickness β (the ratio of the liquid thickness to the droplet diameter), the boundary between these three statements is determined by the values β=0.05-0.1 [7,8] and β=3 [9].
To date, for each above statement, based on experimental and numerical studies, sufficiently detailed maps of the drop impact regimes in the coordinates of dimensionless numerical similarity complexes have been obtained. These results have formed the basis of engineering guidelines for the design of technical products, in order to prevent the formation of splashes, it is sometimes necessary to reduce the diameter or velocity of the falling drop, which artificially underestimates the potential performance of the equipment.
To increase or decrease the threshold level of the splashing criterion, the method of changing the pressure of the gaseous medium can be used [10][11][12]. Obviously, this is only possible for hermetic systems, for example, the separator tank.
The consequences of the impact and so the conditions for the appearance of splashes, change significantly if we consider immiscible drop and target liquids, for example, oil and water. However, there are few such studies in the literature. Moreover, these studies focus more on the interfacial dynamics of immiscible fluids [13][14][15][16][17][18][19] than the issue of splashing secondary droplets into a gaseous medium.
The present paper, for the first time, considers the problem statement that describes the fall of a drop liquid "A" onto a film of finite thickness of an immiscible target liquid "B" with a lower density. The target liquid "B" is on the surface of the tank with the target liquid "A". Of course, this statement seems a little artificial, but it can be used as an active measure to combat the formation of secondary drops, increasing the values of the threshold similarity criteria numbers.

Splash kinematics of drop impact onto liquid surface
In classical statement, the liquid drop impact onto liquid surface can proceed by to four scenarios: complete or partial rebound from the liquid [20][21][22][23], a drop after impact remains floating on the surface, penetration into the liquid without splashing and with the formation of splashes. Figure 1 shows the boundary of the last two regimes.
In the case of splashing, immediately after the impact, a sheet cylindrical jet (crown splash) expanding in diameter rises above the liquid surface, which can collapse forming a bubble. According to [12], as a rule, the crown splash consists of the target liquid; later papers [15][16][17] present more detailed studies on the phase composition of crown splashes depending on the problem statement. From the edge of this crown due to Rayleigh-Plateau instability [14,24], small secondary droplets can be ejected into the gaseous medium. The liquid surface is strongly deformed, and a hemispherical gas crater is formed in it, the radius of which can be an order greater than the radius of the drop. Further, when the crater begins to close, a jet of liquid is ejected vertically upwards from the center of the crater; when broken, this jet can be a source of splashes. Because of this loss in stability, the jet is often referred to as the Rayleigh jet or the Worthington jet, after one of its first explorers. Thus, when a drop impacts, there are two sources of splashes: the crown splash and the Rayleigh jet.
The process of drop impact is due to the action of viscous forces, inertia, surface tension, gravitation, which are determined by the problem parameters: R is the drop radius, U is the drop velocity at the moment of impact, ρ is the density of the continuous medium, σ is the coefficient of surface tension between continuous media, μ is coefficient of dynamic viscosity of a continuum. Usually, three dimensionless numerical complexes express all these parameters: Onesorge (Oh), Weber (We) and Reynolds (Re) (1). Re .
Oh R U R We UR (1) In addition, there are also the Froude (Fr) or Bond (Bo) numbers, which include the action of gravitational forces from the gravitational acceleration g (2). Nevertheless, to reduce the analyzed similarity numbers, it is more convenient to replace the action of gravitational forces with the resulting finite velocity U of the drop before impact. However, the drop falling in a gaseous medium can lead to internal circulation of the liquid in the drop, change and fluctuation of its shape, which may also be important, but often neglected [ (2) However, one of the first criteria for the formation of secondary droplets for water was implemented in the coordinates of Re and Fr numbers in [25]. Figure 1 presents this criterion for convenience in the coordinates of three other similarity numbers -Re, Oh, and We. Maps with larger intervals by similarity number can be found in more recent papers such as [24]. According to these two above papers, the minimum values of the We number for the appearance of secondary drops correspond to the Re≈1800…2000; the present paper use the map [25], which is also more detailed in scale. One should note the We number is determined both by the droplet diameter and by the radius in such papers, so the critical values of the We number can differ by a factor of two. Fig. 1. Similarity numbers for splash and coalescence zones for water drops impacting on water pools [25].  Figure 2 shows the geometry of the computational scheme of our problem statement, which is in many respects similar to the computational schemes for numerical simulation of droplet impact [19,26]. A spherical water drop of R=2.8 mm radius with U velocity falls in an incompressible air medium at atmospheric pressure in the field of gravity onto a film of silicone oil of h thickness; under film there is water. According to [24,25], the secondary drops with a gradual increase in the We number will first be formed due to the breakup of the axisymmetric Rayleigh jet. For this reason, to search for the We critical values of the Rayleigh jet breakup, the problem can be solved in an axisymmetric formulation. The linear dimensions of the domains of water and air were fixed and chosen based on the absence of the effect of boundary conditions on the considered time interval of the process [19,26]. The initial drop velocity, oil film thickness, oil density and kinematic viscosity, surface tension coefficient were variable and determined the values of the analyzed hydrodynamic similarity numbers We, Re, Oh, and Fr (Table 1). At the initial moment of time, the water drop was at a distance of 0.5 mm from the liquid surface. No-slip condition was used on the lower wall and side walls, and the condition of constant static atmospheric pressure was used on the upper boundary.

Results
All regime parameters of the calculations can be traced in Table 1. At the first stage, the kinematic viscosity of the silicone oil was chosen νo=50 cSt. By varying the β value, further, the critical value of the β at which the ejection of the secondary droplet occurred after the impact was obtained. The fact of the ejection was determined on the basis of the analysis of the storyboard of the phase fields, the graphic files of which were recorded by the Fluent , Figures 3 and 4. After that, the other values for the viscosity of silicone oil from table 1 were considered. Drawing the results of calculations on the regime maps made it possible to reveal clear dividing lines on two maps (νo/νw; β) and (Re; β) (regime map (Oh; β) is equal to (νo/νw; β)), Figure 5.

Conclusions
The present paper shows the obtained data about the conditions for splashes when a drop impacts the surface of a liquid with a liquid film. Artificial deposition of a liquid film on the surface of a liquid allow significantly expand the range of similarity numbers when a drop impacts without the secondary splashes. For example, the Reynolds number of a drop can be increased by almost an order. Therefore, in the presence of a liquid film, the product of the drop diameter and its velocity can be increased by the same time. When β>3, as expected, the liquid film loses its "film properties" and becomes a new target liquid. The results can be used to implement an active method to reduce drop entrainment, for example, in gas-liquid separators.