Gravitational Interactions of Two Small Evaporating Drops

Abstract

The effect of evaporation on relative trajectories of two spherical drops sedimenting due to gravity in air is investigated. Theoretical analysis and numerical simulation of the interactions are used to obtain results. Several assumptions are made in the model. The drops remain small enough that the Reynolds number, which is linear in the density of the surrounding fluid, is negligible. However, the Stokes number, which is proportional to droplet mass and is a measure of drop inertia, can be much larger than one. Another restriction is that evaporation is dominated by diffusion and that convection of mass is insignificant. In analyzing evaporation when two drops are present, it is noted that the loss of mass is not the same at each point on a droplet surface. That is, evaporation is non-uniform in a spatial sense. In order to maintain the required spherical drop shape, three approaches, involving the isolated droplet and bispherical coordinate solutions, were taken to determine the mass flux due to evaporation and subsequently the drop position at each time step. For a pair of water drops with radii between 2 and 30 $\upmu$m, the following conclusions were obtained. In all three methods, evaporation leads to weaker inertial effects and stronger hydrodynamic effects. Most importantly, in comparing critical horizontal offsets, when both attractive molecular forces and Maxwell slip are considered, all three approaches to evaporation lead to similar results, making the choice of method nearly inconsequential.

1 Introduction

Droplet evaporation has long been the subject of scientific inquiry, as discussed in several review articles, such as Erbil (2012). Recently, it has been important in the investigation of airborne transmission of SARS-CoV-2 and other viruses (Dombrovsky et al. 2020; Li et al. 2020; Hasan et al. 2021; Kwak et al. 2023; Norvihoho et al. 2024; Tian et al. 2024). Initial work focused on one stationary drop (Maxwell 1890). The evaporation rate was derived under isothermal conditions, ignoring evaporative cooling and yielding Maxwell’s relation for the linear decrease in the square of the drop radius as a function of time (Kukkonen et al. 1989; Erbil 2012). For drops larger than a few micrometers in radius, this time-linear decrease in the surface area has been verified (Fuchs 1959; Erbil 2012). More sophisticated models have been developed, taking into account evaporative cooling and radiative effects (Barrett and Clement 1989; Kukkonen et al. 1989).

For very small drops, with radii on the order of the mean-free path, the theory was extended to take into account an additional shell around the drop from which evaporation actually begins (Bradley et al. 1946). In addition, the temperature and concentration jumps at the interface, particularly for Knudsen numbers $Kn \gtrapprox 0.05$ (Finneran et al. 2021), have been analyzed (Ben-Dor et al. 2003), as well as kinetic effects (Zhao and Nadal 2023). Of late, an alternative to the theory of Fuchs has also been suggested (Vlasov 2021).

When two stationary drops are interacting, the effect of the second drop on evaporation must be taken into account, as has been done in bispherical coordinates for evaporation (Morse and Feshbach 1953; Ramachandran and Kleinstreuer 1985) and combustion (Labowsky 1980; Umemura et al. 1981). These solutions assume that no change in the drop temperature occurs. Moreover, Stefan flow is incorporated in some research for drops (Zhao and Nadal 2023) and particles (Jayawickrama et al. 2023).

The evaporative effect of ventilation due to a single drop freely falling in the atmosphere has frequently been considered (Dolezel 1944; Ranz and Marshall 1952; Duguid and Stampfer 1971; Hughes and Stampfer 1971; Carstens 1979; Kleinstreuer and Ramachandran 1987; Kukkonen et al. 1989; Chavarría et al. 2022; Sezen and Gungor 2023) Most commonly, the drop is spherical, but deformation has been a topic of interest, as well (Petera and Weatherly 2001). Interaction of two evaporating drops has been studied at low Reynolds number under the action of a prescribed blowing rate (Oguz and Prosperetti 1989), at intermediate Reynolds numbers (Raju and Sirignano 1990), and in combustion (Chiang and Sirignano 1993; Yang et al. 2020) for tandem motion. For two, very small drops in, or close to, the noncontinuum regime, relative motion has been explored for evaporation and condensation (Grashchenkov et al. 1998; Grashchenkov 2002; Aleksandrov et al. 2008), including temperature and concentration jumps at the interfaces and curvature effects. Typically, the drop radii are less than 2 $\upmu$m in this work.

All of the previous surveys of interactions of two drops have been for movement along the axis of symmetry and have not examined motion transverse to the drops’ line of centers. That is, complete relative trajectories have not been calculated. As a final note, several researchers have looked at arrays of drops in combustion (Annamalai and Ryan 1992; Sirignano 2014) and evaporation (Devarakonda and Ray 2003; Cossali and Tonini 2020; Masoud et al. 2021). A good discussion of single, double and multiple evaporating drops is found in the literature review of Kékesi et al. (2019).

Herein, the simplest version of drop evaporation is studied to determine its effect on trajectories and coalescence. For water drops falling in air, with radii between 2 and 30 $\upmu$m, the Péclet number for mass transfer remains less than 0.15, as discussed below, so that diffusion dominates convection, which is ignored in the analysis. Combining evaporation with quasi-steady relative drop motion in gravity at low Reynolds numbers allows examination of its effects on interactions. Through the use of resistance functions (Zinchenko 1982; Rother et al. 2022), as employed here, such an investigation is possible. In this way, interactions of two falling drops in Stokes flow can be studied, taking into account drop inertia, hydrodynamic interactions, including the lubrication force, van der Waals forces and Maxwell slip. Complete relative trajectories, with motion both along and perpendicular to the drops’ line of centers, are determined.

Investigation of drops at finite Stokes numbers and low Reynolds numbers has been conducted for gravitational motion (Hocking 1959; Hocking and Jonas 1970; Hocking 1973; Davis 1984), in the context of raindrop growth. In previous work, where the drops were treated as solid particles (Hocking 1959; Davis 1984), due to the large drop-to-medium viscosity ratio, either van der Waals forces (Davis 1984) or Maxwell slip (Davis 1972; Hocking 1973) needed to be incorporated in the analysis for coalescence to occur. More recent raindrop analysis has accounted for droplet viscous effects, including internal circulation(Rother et al. 2022; Ababaei and Rosa 2023). For viscous drops, coalescence is possible without any additional considerations, since the drainage between the drops is increased by internal circulation (Zinchenko 1982; Davis et al 1989; Zhang and Davis 1991; Rother et al. 2022).

However, for viscous drops, both Maxwell slip and van der Waals forces do increase drop coalescence over the rate in their absence. Van der Waals forces (Hamaker 1937) have been added to spherical droplet analysis at low Reynolds numbers in many situations (Zhang and Davis 1991; Wang et al. 1994; Rother and Davis 2004; Rother 2009; Rother et al. 2022), and their effect is particularly important for smaller drops, e.g.  those with radii less than 10 $\upmu$m in the case of raindrops. Maxwell slip has been studied for solid spheres using asymptotic methods (Hocking 1973; Davis 1984) and more rigorously with bispherical coordinates (Davis 1972). The case of slip for viscous drops has also been examined asymptotically (Zinchenko 1981) and by bispherical coordinates (Grashchenkov 1996; Rother et al. 2022; Ababaei and Rosa 2023), and has a significant impact over a wider range of drop sizes than van der Waals forces.

While the topic is controversial, some justifications are listed below for using a purely diffusive model for evaporation and assuming low Reynolds number flow for water drops in air with drop radii between 2 and 30 $\upmu$m:

• The effect of ventilation (convection) on a single falling drop is less than 1% for drop radii less than 5 $\upmu$m, approximately 8% for a radius of 30 $\upmu$m and approximately 30% at a radius of 100 $\upmu$m. For drops between 3 and 9 $\upmu$m in radius, the rates of evaporation for a single falling drop are most accurately described by Maxwell’s simple theory (Duguid and Stampfer 1971; Kukkonen et al. 1989).

• The temperature correction for the evaporation rate of stationary water droplets in atmospheric conditions is less than 3%, although it can be much higher for other liquids (Kukkonen et al. 1989).

• The effect of ventilation on temperature depression is less than 1% for radii less than 50 $\upmu$m (Kukkonen et al. 1989).

• Similarly, for drops between 3 and 9 $\upmu$m in radius, the effect of the temperature decrease on evaporation rate for a single falling drop is typically less than 1.5% (Duguid and Stampfer 1971).

• Stokes Law to calculate drag is ‘valid’ up to a Reynolds number of Re = 2 (Dolezel 1944; Duguid and Stampfer 1971).

These arguments are not meant to be conclusive, but simply to illustrate that the assumptions are not unreasonable.

In the remainder of this paper, Sect. 2 contains the problem statement and its requisite constraints; Sect. 3 includes results for evaporation, mass flux and relative trajectories for a physical system of two water drops in air, as well as modified collision efficiencies; and Sect. 4 consists of concluding remarks.

2 Problem Statement

In Fig. 1 two drops are shown interacting due to gravity and have come into close approach. In addition, evaporation is occurring in the process. The physical properties are the same for both drops and are indicated by a subscript ‘d’, i.e., density $\rho _d$ and viscosity $\upmu _d$, while those for the external medium are designated with subscript ‘e’. The smaller and larger drops initially have drop radii $a_{1,0}$ and $a_{2,0}$ with size ratio k = $a_{1,0}/a_{2,0}$, but due to evaporation the values change with time, $a_1(t)$ and $a_2(t)$, as do the drop size ratios $k_1(t)$ = $a_1(t)/a_{2,0}$ and $k_2(t)$ = $a_2(t)/a_{2,0}$. The drop-to-medium viscosity ratio, however, $\hat{\mu }$ = $\mu _d/\mu _e$ remains constant.

A common goal in trajectory analysis is to determine a critical value of the initial horizontal offset $d_{\infty }$, at infinite vertical separation parallel to gravity, that defines the boundary between interactions leading to coalescence and those in which the drops pass by one another. That critical horizontal offset $d_{\infty }^*$ dictates the collision efficiency $E_{12}$, which is used in population dynamics to describe the evolution of the drop-size distribution in dilute dispersions:

$$\begin{aligned} E_{12} = \left( d_{\infty }^* \over a_{1,0} + a_{2,0} \right) ^2. \end{aligned}$$

(1)

In some cases, such as gravitational motion of spherical drops without attractive forces, a closed-form solution for the collision efficiency is possible (Zhang and Davis 1991). However, if non-linear effects, including van der Waals forces or small deformation, are present, multiple relative trajectories are required to narrow the critical horizontal offset to within a prescribed tolerance. In the absence of evaporation, a large but finite vertical separation parallel to gravity is chosen from which to begin calculations. A typical value is 1000$a_2$ or one thousand larger drop radii. At this initial vertical offset, the error is generally less than 0.5% compared to the closed-form solution at infinite vertical separation (Rother et al. 2022). One then prescribes a minimum and maximum value for the initial horizontal offset $d_{\infty }$, which should encompass the critical value. A trajectory starting from a subcritical horizontal offset will lead to coalescence, while a supercritical value will result in separation of the drops. A simple bisection algorithm can be used to ascertain an acceptably accurate critical horizontal offset $d_{\infty }^*$.

In the case of evaporating drops, it is not possible to have an infinitely long trajectory, so in many of the results below, the initial vertical separation is 500$a_{2,0}$, that is, five hundred larger drop radii. Such a modified collision efficiency or a critical horizontal offset $d_{\infty }^*$ can be useful in examining the relative strength of hydrodynamic, gravitational and van der Waals forces, as well as Maxwell slip.

Fig. 1

Definition sketch for two spherical, evaporating drops in gravitational motion. The initial smaller and larger drop radii are $a_{1,0}$ and $a_{2,0}$, respectively, and, as functions of time, are $a_1(t)$ and $a_2(t)$

The assumptions required in the model are listed in Table 1.

Table 1 Model restrictions for the larger drop radius $a_2$

In Table 1, $V_2^{(0)}$ is the Hadamard–Rybczynski velocity of the larger drop $a_{2,0}$:

$$\begin{aligned} V_2^{(0)} = {2 \over 9} \left( {\hat{\mu }+ 1 \over \hat{\mu }+ 2/3} \right) {|\rho _d - \rho _e| g \over \mu _e} a_{2,0}^2. \end{aligned}$$

(2)

The term $V_{12}^{(0)}$ = $|V_2^{(0)} - V_1^{(0)}|$ is the absolute value of the difference between the Hadamard-Rybczynski velocities of the larger and smaller drops. In addition, the mean-free path is $\lambda _M$ = 0.1 $\upmu$m (Davis 1984), the surface tension between water and air is $\sigma$, $D_{AB}$ is the mass transfer diffusivity between water and air, and $D_{12}^{(0)}$ is the Brownian diffusivity (Zhang and Davis 1991):

$$\begin{aligned} D_{12}^{(0)} = \left( \hat{\mu }+ 1 \over \hat{\mu }+ 2/3 \right) {k_B T (1 + k^{-1}) \over 6 \pi \mu _e a_{2,0}}, \end{aligned}$$

(3)

with Boltzmann’s constant $k_B$ = 1.380649 $\times$ 10$^{-16}$ erg/K.

Values for the dimensionless parameters governing the problem are presented in Table 2 for a system of water drops in air at 20 $^\textrm{o}$C. The relevant physical properties are $\rho _d$ = 1 g/cm$^3$, $\mu _d$ = 0.01 g/cm$\cdot$s, $\rho _e$ = 0.0013 g/cm$^3$, $\mu _e$ = 0.00017 g/cm$\cdot$s, $D_{AB}$ = 0.246 cm$^2$/s, and $\sigma$ = 72.75 dyn/cm (Davis 1984; Zhang and Davis 1991; Rother et al. 2022). For the Brownian Péclet number, the size ratio k = 0.5 was used.

Table 2 Dimensionless parameters as a function of the larger drop radius $a_2$

Before the trajectory equations are described, it is necessary to consider drop evaporation. For a single drop, with water vapor concentration $c_s$ = $P^{vap} M_W/RT$ at each drop surface $a_i$ and ambient concentration $c_{\infty }$ far from each drop, the time rate of change of the i$^{th}$ drop mass $dm_i/dt$ is given by Erbil (2012):

$$\begin{aligned} {dm_i \over dt} = -4 \pi a_i D_{AB} (c_s - c_{\infty }), \end{aligned}$$

(4)

which can be integrated to give the drop radius as a function of time in dimensionless form:

$$\begin{aligned} k_i^2 = \left( {a_{i,0} \over a_{2,0}} \right) ^2 - 2 {(c_s - c_{\infty })\over \rho _d} {1 \over Pe} t,~~i = 1,2, \end{aligned}$$

(5)

where Pe is the Péclet number defined in Table 1, and

$$\begin{aligned} k_i = \left( {a_{i} \over a_{2,0}} \right) ,~~i=1,2,~~~~~~~~~~ \left( {a_{i,0} \over a_{2,0}} \right) = {\left\{ \begin{array}{ll} k, &{} i = 1 \\ 1, &{} i = 2. \end{array}\right. } \end{aligned}$$

(6)

In the discussion above, the vapor pressure, molecular weight, ideal gas constant and absolute temperature are $P^{vap}$, $M_W$, R, and T, respectively. In addition, for the non-dimensionalization of Eq. (4), the length scale is $a_{2,0}$, the velocity scale is $V_2^{(0)}$, and the time scale is $a_{2,0}/V_2^{(0)}$.

To take into account evaporative cooling it is possible to employ an ad hoc method, similar to that of Hardy et al. (2021). A more rigorous technique using the energy balance is also feasible (Ranz and Marshall 1952; Lim et al. 2008) but more complicated, especially for groups of drops. When necessary, it is possible to use the drop temperature as a fitting parameter between experiment and theory in a slightly modified form of the Maxwell model (Hardy et al. 2021) to estimate the temperature depression of the droplet.

When the presence of both drops is accounted for, the water vapor concentration exterior to the drops can be found using bispherical coordinates (Morse and Feshbach 1953; Ramachandran and Kleinstreuer 1985):

$$\begin{aligned} c^*= & {} {c - c_{\infty } \over c_s - c_{\infty }} \nonumber \\= & {} \sqrt{2(\cosh \mu - \cos \eta )} \sum _{n = 0}^{\infty } {(A_n - B_n) P_n(\cos \eta ) \over \sinh [(n + 1/2)(\mu _1 - \mu _2)]}, \end{aligned}$$

(7)

where

$$\begin{aligned} A_n = \textrm{e}^{(n+1/2)(\mu _2-\mu )} \sinh [(n+1/2)\mu _1] \end{aligned}$$

(8)

and

$$\begin{aligned} B_n = \textrm{e}^{(n+1/2)(\mu -\mu _1)} \sinh [(n+1/2)\mu _2]. \end{aligned}$$

(9)

The variables $\eta$ and $\mu$ are orthogonal curvilinear coordinates defined in Appendix 1, and $\mu _1$ and $\mu _2$ are the transformed drop radii $a_1$ and $a_2$.

Unlike the case of an isolated drop, when two droplets are present, there is asymmetric loss of mass. However, the drops remain spherical, as required by the very small capillary numbers. As has long been recognized (Carstens 1979), this difficulty represents a failure in the theory. Suggested ways forward include using some form of the isolated drop approach, although the accuracy of such a method is suspect, and developing a more sophisticated bispherical coordinate solution to the complete problem. The latter idea has been applied in a variety of contexts (Carstens et al. 1970; Oguz and Prosperetti 1989; Grashchenkov 2002), with some interesting results. Tentative conclusions from this research are that effects due to the overlap of diffusion profiles between droplets are small for drops falling at different rates (Williams and Carstens 1971) and that interactions are significant only for pairs with nearly equal radii (Carstens et al. 1970). More recent studies on van der Waals forces and Maxwell slip (Davis 1984; Rother et al. 2022) indicate that their importance for closely sized drops may dominate any evaporative consequences on interactions. In fact, one hypothesis of the current work is that with the inclusion of attractive molecular forces and slip, which is the most physically realistic model, the technique for coupling momentum and mass transport, with some caveats, is largely irrelevant. This claim will be investigated in Sect. 3.2 below.

At this point, the issue then becomes determining how the predicted asymmetry affects the shape of the drops and the gap between them. Three approaches are taken here to try to handle the problem:

• 1. Use the isolated spherical drop result, Eq. (5). Referred to below as ‘single sphere.’

• 2. Use the bispherical coordinate result, Eq. (7), to update the drop volumes but fit the result to a sphere at the current location. Only the trajectory equations are allowed to determine drop positions. Referred to below as ‘fixed center.’

• 3. Use the bispherical coordinate result to update both the drop volumes and position in the fitting, so that both evaporation and the trajectory equations determine the drop positions. Referred to below as ‘moving center.’

None of these approaches is necessarily correct, but they are three logical alternatives that will be used for the purpose of comparison.

To determine the trajectory equations, it is necessary to begin with a force balance on each drop:

$$\begin{aligned} m_i a_i = \sum F_i = F_{g,i} + F_{h,i} + F_{mol,i}, \end{aligned}$$

(10)

where $m_i$ and $a_i$ are the mass and acceleration of the $i^{th}$ drop; $F_{g,i}$ consists of the weight and buoyant force; $F_{h,i}$ is the hydrodynamic force, including the presence of other drop, and the lubrication force in close approach; and $F_{mol,i}$ is the attractive van der Waals force, retarded or unretarded. The expression for $F_{h,i}$ will be in terms of resistance functions (Zinchenko 1982), and when Maxwell slip is present, the resistance functions will be modified for their inclusion (Rother et al. 2022). The terms on the left side of Eq. (10) are required for the case of significant droplet inertia, as measured by the Stokes number. The hydrodynamic forces remain linear in the drop velocities, but the sum of the forces is no longer zero (Davis 1984).

In dimensionless form, the resulting equations for motion of two interacting drops become

$$\begin{aligned}{} & {} k_1^2 St {d\mathbf{V_1} \over dt} = k_1^2 \left( {\hat{\mu }+ 2/3 \over \hat{\mu }+ 1} \right) \textbf{g} \nonumber \\{} & {} \quad - \left[ \Lambda _{11}(\mathbf{V_1 - V_2 })^{\parallel } + \Lambda _{12} \mathbf{V_2}^{\parallel } + T_{11}(\mathbf{V_1 - V_2})^{\bot } + T_{12} \mathbf{V_2}^{\bot } \right] , \end{aligned}$$

(11)

$$\begin{aligned}{} & {} k_2^2 St {d\mathbf{V_2} \over dt} = k_2^2 \left( {\hat{\mu }+ 2/3 \over \hat{\mu }+ 1} \right) \textbf{g} \nonumber \\{} & {} \quad - \left[ \Lambda _{21}(\mathbf{V_2 - V_1})^{\parallel } + \Lambda _{22} \mathbf{V_2}^{\parallel } + T_{21}(\mathbf{V_2 - V_1})^{\bot } + T_{22} \mathbf{V_2}^{\bot } \right] . \end{aligned}$$

(12)

The quantities $\Lambda _{ij}$ and $T_{ij}$ are the resistance functions, which are discussed in more detail elsewhere (Zinchenko 1982; Rother et al. 2022), while the superscript symbols $\parallel$ and $\bot$ mark components of velocity parallel and perpendicular to the drops’ line of centers. The resistance values depend on k, $\hat{\mu }$ and s and are identical when the size ratio is replaced by 1/k. The number of terms required for convergence in the governing series goes to infinity as the drop separation becomes vanishingly small. It should be mentioned that $k_1$ and $k_2$ are functions of time due to evaporation, as discussed after Fig. 1.

Two other physical phenomena remain to be considered: Maxwell slip and van der Waals forces. As discussed elsewhere (Grashchenkov, 1996; Rother et al., 2022), slip is accounted for by modifications to the resistance functions $\Lambda _{ij}$ and $T_{ij}$ in Eqs. (11) and (12). Although the drops are large enough that Knudsen effects are not taken into account, when the gap becomes very small, noncontinuum behavior is possible, as observed in previous work (Davis 1984; Rother et al. 2022).

Attractive molecular forces have proven to be important for drops less than 10 $\upmu$m in radius (Davis 1984; Rogers and Davis 1990; Rother et al. 2022). Expressions for $F_{mol,i}$ in Eq. (10) can be found in Appendix 2. Calculations for unretarded van der Waals forces are shown in Sect. 3, where retarded molecular forces (Rother et al. 2022) include a weakened attraction due to the electromagnetic consequences of the finite speed of light. In subsequent figures, the interparticle force parameter $Q_{12}$ (Zhang and Davis 1991) represents the relative strength of the buoyant force to the attractive molecular force:

$$\begin{aligned} Q_{12} = {2\pi \over 3}k(1-k^2){|\rho _d-\rho _e|g a_2^4 \over A_H}, \end{aligned}$$

(13)

where $A_H$ is the Hamaker constant (Hamaker 1937). In this way, the smaller the value of $Q_{12}$, the greater the strength of van der Waals attraction. The Hamaker constant for the water-air system is 5 $\times$ 10$^{-13}$ erg (Davis 1984; Zhang and Davis 1991). For water drops in air at 20 $^\textrm{o}$C with a drop size ratio of k = 0.5, typical values of the interparticle force parameter are $Q_{12}$ = 2.46 at $a_2$ = 2 $\upmu$m, $Q_{12}$ = 1540 at $a_2$ = 10 $\upmu$m, and $Q_{12}$ = 9.62 $\times$ 10$^5$ at $a_2$ = 50 $\upmu$m.

3 Results

3.1 Evaporation and Mass Flux

Fig. 2

Comparison of dimensionless water vapor concentration using the isolated drop and bispherical coordinate results at a center-to-center separation of $r/a_2$ = 2.25. The size ratio is $k = a_1/a_2 = 0.5$

In Fig. 2a, b, the dimensionless water vapor concentration $c^*$ external to the drops at 20 $^\textrm{o}$C is shown, by contour plot and quantitative graphs, respectively. In both figures, the water vapor concentration far from the drops is $c_{\infty }$ = 0. The smaller drop is more affected by the presence of the second drop, as is consistent with previous work, and in the region between the two liquid spheres, evaporation is significantly reduced. The isolated drop result for $c^*$ from Eq. (5) is accurate for the larger drop, away from the gap, in comparison to calculations including both drops from Eq. (7). In Fig. 2b in the gap between the drops, the intersection of the isolated sphere concentration curves indicates the overlap of the drops’ diffusive regions.

Fig. 3

Dimensionless mass flux is shown at the gap ($\eta$ = $\pi$) and away from the gap ($\eta$ = 0) for separations between 10$^{-5}$ and 100 larger drop radii. The size ratio is $k = a_1/a_2 = 0.5$. The isolated drop flux (dashed lines) from Eq. (14) is $-2$ for the smaller drop and $-1$ for the larger drop. See Fig. 10 for definition of the bispherical coordinate $\eta$

Fig. 4

Dimensionless water vapor flux at the drops’ surfaces at center-to-center separations of $r/a_2$ = 7.2, 2.25 and 1.501, from top to bottom, respectively, determined by Eq. (15). The size ratio is $k = a_1/a_2 = 0.5$

The dimensionless evaporative flux from the drops $-\textbf{n} \cdot \nabla c / [(c_s - c_{\infty }) / a_2]$ appears in Figs. 3 and 4, where again $c_{\infty }$ = 0. For the case of isolated drops, the flux is

$$\begin{aligned}{} & {} -\textbf{n} \cdot D_{AB} \nabla c / [D_{AB} (c_s - c_{\infty }) / a_2] \nonumber \\{} & {} \quad = -\textbf{n} \cdot \nabla c / [(c_s - c_{\infty }) / a_2] = -{1 \over k_i} ( \hat{\textbf{r}} \cdot \hat{\textbf{r}}) = -{1 \over k_i}, ~~ i = 1,2, \end{aligned}$$

(14)

where $\textbf{n}$ is the outward-pointing unit normal vector, $\hat{\textbf{r}}$ is the unit position vector from the center of each sphere, and the mass diffusivity $D_{AB}$ is assumed a constant. For the bispherical coordinate result in Eq. (7), the dimensionless flux can be expressed as

$$\begin{aligned}{} & {} -\textbf{n} \cdot \nabla c / [(c_s - c_{\infty }) / a_2] = - {(\cosh \mu - \cos \eta ) \over c_{\ell }} {\partial c^* \over \partial \eta } (\mathbf{e_{\eta }} \cdot \mathbf{e_{\eta }})\\{} & {} \quad = {\sinh \mu \over \sqrt{2(\cosh \mu - \cos \eta )}} \sum _{n = 0}^{\infty } {(A_n - B_n) P_n(\cos \eta ) \over \sinh [(n + 1/2)(\mu _1 - \mu _2)]} \end{aligned}$$

$$\begin{aligned} - \sqrt{2(\cosh \mu - \cos \eta )} \sum _{n = 0}^{\infty } {(n + 1/2) (A_n + B_n) P_n(\cos \eta ) \over \sinh [(n + 1/2)(\mu _1 - \mu _2)]}, \end{aligned}$$

(15)

where $\textbf{n}$ is again the outward-pointing unit normal vector, $\mathbf{e_{\eta }}$ is the unit vector in the bispherical coordinate $\eta$-direction, which is perpendicular to the surface of each sphere (see Fig. 10), and the length $c_{\ell }$ is defined in Eq. (22). The coefficients $A_n$ and $B_n$ are provided in Eqs. (8) and (9).

Fig. 5

Diagram illustrating integration of mass flux $j_A$ over the surface of a drop from Eq. (18) to find the total rate at which mass is lost $\dot{m}$ by evaporation

Once the mass flux is known from Eq. (15), the rate at which mass is lost from each drop $\dot{m}$ is found from

$$\begin{aligned} \dot{m} = {dm \over dt} = - \iint _S {D_{AB} (c_s - c_{\infty }) \over a_{2,0}} \nabla c^* \cdot \textbf{n}~dS = -\iint _S j_A dS, \end{aligned}$$

(16)

where S is the surface area of the drop. Integrating over the top half of the drop, Eq. (16) can be simplified as

$$\begin{aligned} \dot{m} = - \int _{-a_i}^{+a_i} (2 \pi r)j_Ads, \end{aligned}$$

(17)

where $j_A$ is the water vapor flux with SI units of kg/m$^2 \cdot$s, and s is the arc length. In discretized form, due to the axisymmetry of evaporation about the drops’ line of centers, the time rate of change in mass becomes

$$\begin{aligned}{} & {} \dot{m} \approx - \sum _{i=1}^{n} 2 \pi {(y_{i+1}+y_i) \over 2} {(j_{A,i+1}+j_{A,i}) \over 2}\nonumber \\{} & {} \quad \times \sqrt{(y_{i+1}-y_{i})^2+(x_{i+1}-x_{i})^2} \end{aligned}$$

(18)

An illustration of the numerical approach is shown in Fig. 5.

This numerical technique is employed for the rate of mass loss, because it has the advantage of being able to handle any axisymmetric profile and is useful for implementation of the moving center method. It can also be generalized to an arbitrary functional dependence, while not being numerically prohibitive. The method was checked with results for isolated drops and then with the bispherical formulation. Convergence to less than 0.2% was obtained for critical horizontal offsets at an initial vertical separation of 500 larger drop radii with n = 100 at each time step.

3.2 Relative Trajectories

Fig. 6

Relative trajectories for two water drops in air at 20 $^\textrm{o}$C at $a_2$ = 30 $\upmu$m, k = 0.7, and St =45.45 without van der Waals forces or Maxwell slip. The initial horizontal separation is $\Delta x_0/a_{2,0}$ = 1, with an initial vertical separation of $\Delta z_0/a_{2,0}$ = 500, parallel to gravity. In the top interaction, the droplets are not evaporating and coalesce. In the bottom two trajectories, the isolated-drop evaporation rate from Eq. (5) is employed, with the water vapor concentration far from the droplets decreasing from $c_{\infty }$ = 0.5$c_s$ to 0. The uniform mass flux leaving the drops is indicated by the colored contours. The larger drop is shown only at closest approach

In Fig. 6 two-drop interactions are shown using the isolated drop model in Eq. (5) for water droplets in air at 20 $^\textrm{o}$C. In the upper frame, comparison with non-evaporating drops is made. As the water vapor concentration far from the drops decreases in the lower two frames of the figure, the gap between the drops increases due to weakening of drop inertia and strengthening of hydrodynamic forces as the droplets become smaller. The dimensionless flux obeys Eq. (14), which is proportional to the reciprocal of the dimensionless drop ratio $1 / k_i$. We note that the symbol $\Delta$ is used to represent the difference between values in the caption to Fig. 6 and elsewhere throughout.

Fig. 7

Relative trajectories for two water drops in air at 20 $^\textrm{o}$C for $a_2$ = 30 $\upmu$m, k = 0.9, St =45.45, and $c_s$ = 0. Two initial horizontal separations are shown: $\Delta x_0/a_{2,0}$ = 0.2242 (solid lines) and 0.1629 (dashed lines), with an initial vertical separation of $\Delta z_0/a_{2,0}$ = 500, parallel to gravity. The evaporation rate taking into account the presence of the second drop from Eq. (15) is employed, with the fixed center approach for droplet position. In the insets, the drop positions and evaporative flux are indicated just before coalescence for (1) $\Delta x_0/a_{2,0}$ = 0.1629 including Maxwell slip, but not van der Waals forces and (2) $\Delta x_0/a_{2,0}$ = 0.2242 including unretarded van der Waals forces, with $Q_{12}$ = 5.68 $\times$ 10$^4$, but not slip

The effects of Maxwell slip and unretarded van der Waals forces are considered in Fig. 7. Slip is a linear effect, so that the relative trajectories remain symmetric about $\theta$ = $\pi /2$ at low Stokes numbers, and the drops come into contact at $\theta _0$-values less than $\pi /2$, as seen in inset $\fbox {1}$. However, unretarded van der Waals forces are non-linear, generally become important at small separations, and can lead to coalescence at angles greater than $\pi /2$ (See inset $\fbox {2}$). Although the drops are relatively large, with $a_{2,0}$ = 30 $\upmu$m, van der Waals still have a significant impact. This result occurs, because the original size ratio is close to 1, allowing more time in close approach, and the drop radii have reduced by nearly half at the point of apparent contact due to evaporation, making molecular forces stronger.

Fig. 8

Comparison of gap and smaller drop size as a function of time, including unretarded van der Waals forces and Maxwell slip, for the case of non-evaporating drops, drops evaporating according to the isolated drop model, and drops evaporating according to the bispherical coordinate model, with fixed and moving centers. Insets are shown at closest approach for each method. The relevant parameters are k = 0.7, $a_{2,0}$ = 20 $\upmu$m, St = 13.47, $Q_{12}$ = 2.34 $\times$ 10$^4$, $c_{\infty }$ = 0.5 $c_s$, $\Delta x_0/a_{2,0}$ = 0.23 and $\Delta z_0/a_{2,0}$ = 500. Other physical parameters are for water and air at 20 $^\textrm{o}$C

Four types of relative trajectories are shown in Fig. 8: (1) no evaporation, (2) isolated drop evaporation, (3) fixed center, and (4) moving center, the latter two approaches to evaporation using the bispherical coordinate solution accounting for the presence of both drops in Eq. (7). Both unretarded van der Waals forces and Maxwell slip are considered in the calculations. Evaporation of isolated droplets leads to the greatest reduction in size and the largest separation. As mentioned with respect to Fig. 6, evaporation causes weaker inertial effects and greater hydrodynamic consequences.

When the presence of the second drop is included, mass loss between the droplets decreases (Fig. 2a, b). In the case of the fixed center method, where the position is determined by the hydrodynamics, placing the spherical drop at the point calculated from the trajectory equations, (11) and (12), is equivalent to increasing the gap between droplets. On the other hand, for the moving center technique, the gap is readjusted according to the evaporation at the leading and trailing edges of the drops. As a result, the trajectory for moving center evaporation shows a smaller droplet separation than the fixed center case. In fact, for the parameters used in Fig. 8, coalescence takes place in the latter instance, but not in the former.

Concerning the moving center method, an ad hoc approach was used to take into account the unequal evaporation across both drops. The amount of mass lost at the leading and trailing edges of each drop along the line of centers was calculated. For example, for drops in close approach, the leading edge toward the gap experiences little to no loss of mass, while the trailing edge away from the gap does lose some mass, as seen in Fig. 3. The changes in the radius at both drops edges are then found and the difference between the values used to adjust the drop positions back toward one another. One advantage to this approach is that it does not change the drop positions if there is no evaporation or if the evaporation is uniform, as in the single drop method. In the moving center method, the drop positions are never relocated beyond the original leading edge of the drop, so that the gap does not decrease due to evaporation.

Fig. 9

The effect of microphysics on the critical horizontal offset at an initial vertical separation of 500 larger drop radii. Curves are for both unretarded van der Waals forces and Maxwell slip (dashed-dotted lines), van der Waals forces and no slip (dashed lines), and no van der Waals forces and no slip (solid lines). Critical offsets are presented for no evaporation and two water vapor concentrations far from the drops, $c_{\infty }$ = 0.5$c_s$ and $c_{\infty }$ = 0. Results are also shown for the three evaporation models described in the text, but in most cases the models are indistinguishable from one another. (See discussion.) The physical parameters are for water and air at 20 $^\textrm{o}$C

With respect to the complete trajectories, with relative motion parallel and perpendicular to the drops’ line of centers, the possibility of the water vapor concentration being affected by the flow in the gap and asymmetric redistribution due to the changing gravitational orientation requires comment. A similar problem concerning interactions of slightly deformable drops (Rother et al. 1997) was handled with the same assumptions used here for evaporation, namely, that asymmetric effects and pumping flow in the gap have minimal consequences (Zinchenko and Davis 2005). The flow is at low Reynolds numbers with little to no convection. In addition, the drops have viscosity ratios which are normally order of magnitude 10 or higher, so that the interfaces are not highly mobile. And, finally, it is expected that van der Waals forces and slip will dominate when the drops are in close approach.

To get a broader perspective, critical horizontal offsets $d_{\infty }^*/a_{2,0}$ at an initial vertical offset $\Delta z_{0}/a_{2,0}$ parallel to gravity of 500 larger drop radii are displayed in Fig. 9. Evaporative effects on the hydrodynamics are important, particularly when slip and attractive molecular forces are absent. The evaporative model matters in close approach from a microscopic viewpoint, as seen in Figs. 7 and 8. For more global parameters, like critical offset, the difference in the results among the three approaches to evaporation is relatively small when both van der Waals forces and Maxwell slip are present. This issue is discussed in greater detail below concerning Tables 3 and 4.

In Fig. 9a, the critical horizontal offset will approach the Smoluchowski limit as the larger drop size increases. That is, $\lim _{a_{2,0} \rightarrow \infty } \left( d_{\infty }^*/a_{2,0} \right) = 1+k$, or 1.7 for the parameters in the figure. For the case of no van der Waals forces and no slip, results are shown only for the single sphere model to avoid congesting the image. However, the greatest difference in the model results (not shown, typically more than 10% between the single sphere and moving center approaches) occurs without van der Waals forces and slip, because of the importance of hydrodynamic forces.

Some difference is visible with van der Waals forces but no slip (dashed lines) in Fig. 9a, b when $c_{\infty }$ = 0. The upper dashed line marks bispherical, fixed center results, while the lower dashed line is for the single sphere technique. For both attractive molecular forces and Maxwell slip, the curves (dashed-dotted lines) are nearly indistinguishable for all three evaporative approaches with $c_{\infty }$ = 0 and 0.5 $c_s$ in Fig. 9a. This trend of virtually equal values of the critical horizontal offset $d^*_{\infty }/a_{2,0}$ holds as a function of k in Fig. 9b, as well, so long as the calculations encompass the two coalescence-enhancing effects together, viz., slip and van der Waals forces. We note that inclusion of both microphysical phenomena is likely the most accurate model.

The assertion that the evaporation model used in conjunction with the hydrodynamics is mostly unimportant remains to be discussed. To this end, Tables 3 and 4 are presented. The critical horizontal offsets for the three evaporation techniques considered here are shown at several values of the drop size ratio k and larger drop radius $a_{2,0}$, when both van der Waals forces and Maxwell slip are included in the analysis, with an initial horizontal offset parallel to gravity of 500 larger drop radii. The data are with the same parameters as those used in Fig. 9, and the subscripts ‘ss,’ ‘fc,’ and ‘mc’ indicate the single sphere, fixed center and moving center approaches.

It should be noted that the attractive molecular forces are unretarded in both the figures and the tables. This choice was made to keep the figures from being too overcrowded. However, retarded van der Waals forces, which are perhaps the most realistic, have been used in similar calculations (Rother et al. 2022). The results for retarded attractive molecular forces alone fall between those for unretarded van der Waals forces alone and those for slip alone. The conclusions drawn from Tables 3 and 4 are unchanged by the substitution of retarded for unretarded molecular forces.

Table 3 Dimensionless critical horizontal offsets as a function of the drop size ratio k for water drops in air at 20 $^\textrm{o}$C with a larger drop radius of $a_2$ = 30 $\upmu$m

Table 4 Dimensionless critical horizontal offsets as a function of the larger drop radius $a_{2,0}$ for water drops in air at 20 $^\textrm{o}$C with a drop size ratio of k = 0.7

The maximum relative difference in Table 3 is less than 10% for all but one point and is less than 3% for the vast majority. Similar behavior is observed in Table 4, with two of the points having a maximum relative difference greater than 13%, but most showing a relative error of less than 3.6%. The largest discrepancy between the methods occurs at the smallest and largest size ratios in Table 3 and the smallest drop radii in Table 4. There is no corresponding trend at larger drop sizes in Table 4, such as 40 $\upmu$m, because drop inertia dominates the interactions, with a Stokes number greater than 100. (See Table 2.) In fact, the horizontal offsets are nearly identical at larger drop radii. Additionally, the relative differences are also smaller for $c_{\infty }$ = 0.5$c_s$ than for $c_{\infty }$ = 0, an outcome which arises since the evaporation rate is smaller when the driving force is smaller.

The differences at small size ratios and drop radii take place at parameter values close to points where the critical offset becomes zero and are partly an artifact of the stopping criterion used in the calculations. Generally, any relative trajectory ended if the smaller drop radius dropped below 2 $\upmu$m, since Knudsen and Brownian effects can become significant. However, if the final drop size is increased to 3 or 4 $\upmu$m, for example, the critical offsets at small size ratios and drop sizes decrease, sometimes to zero.

At size ratios above 0.9, the variation in offsets depends on the amount of time the drops spend in close approach. When evaporating drops are large enough that drop inertia can dominate, for example, $a_{2,0}$ $\approx$ 30 $\upmu$m, the critical horizontal offset can go to zero if the size ratio is too close to one, due to the complete vanishing of the smaller droplet. For smaller drops of radii 10 or 20 $\upmu$m, the critical horizontal offset can increase as the size ratio approaches unity due to the relative strength of van der Waals forces. It is in this region, at k $\approx$ 0.9, but before any complete evaporation of the smaller drop, that the method of coupling mass and momentum transfer is most important. A variety of physical phenomena have been predicted in bispherical coordinate analysis due to diffusio-phoretic and thermo-phoretic effects, such as drop motion toward or away from one another (Williams and Carstens 1971; Oguz and Prosperetti 1989; Grashchenkov 2002). The analysis here does not take into account these possibilities; however, their influence could be significant at this point. Nonetheless, van der Waals forces and droplet inertia may overwhelm the consequences of diffusive interactions for small or large drops, respectively.

These results indicate that with the most true-to-life physics, when van der Waals forces and slip are present, values for the critical horizontal offset are relatively insensitive to the evaporative model. The limitations on this conclusion include (1) the transition from zero to non-zero offsets at small size ratios and small drop sizes, and (2) size ratios of 0.9 or so, where the rate of evaporation and the relative importance of attractive van der Waals forces, droplet inertia and any diffusio-phoretic or thermo-phoretic effects matter most. As a quantitative guideline, if the modified collision efficiency $E_{12}^M$ is greater than 0.011, the maximum relative error among the methods is less than 10%, where

$$\begin{aligned} E_{12}^M = \left( d_{\infty }^*|_{(\Delta z_0/a_{2,0}=500)} \over a_{1,0} + a_{2,0} \right) ^2 = \left( d_{\infty }^*|_{(\Delta z_0/a_{2,0}=500)} / a_{2,0} \over 1 + k \right) ^2 \end{aligned}$$

(19)

is the collision efficiency from Eq. (1) evaluated with the critical horizontal offset at a finite initial vertical separation of 500 larger drop radii. Similarly, the larger the modified collision efficiency, that is, the closer it is to one, the smaller the relative error.

As a final, parenthetical remark, to my knowledge, no experiments have been performed with a goal of determining relative trajectories for two evaporating drops falling due to gravity. However, some experiments have taken into account the effect of gravity and ventilation on the evaporation rate for a linear chain of separated drops. An important feature of the experiments in Hardy et al. (2021) is that the rate of evaporation for falling drops with radii of 17.5 and 22 $\upmu$m can be explained in terms of diffusion only, without including convective effects. This result helps confirm one assumption made in Sects. 1 and 2 above, concerning the decrease in drop size as a function of time for droplets with radii between 2 and 30 $\upmu$m.

4 Concluding Remarks

Three approaches to evaporation and their influence on hydrodynamics have been taken. Extension of the isolated sphere equations to a pair of drops leads to the largest gap between droplets, followed by fixed center and moving center techniques, which employ the bispherical coordinate solution for two liquid particles. All three methods lead to increased hydrodynamic effects and weaker inertial impact due to evaporation. When van der Waals forces and Maxwell slip are included in the analysis, the influence of the evaporative method on the outcome, as measured by critical horizontal offsets at finite vertical separations, is significantly reduced. That is, any of the three approaches can be used with a typical percent relative difference in critical offsets of less than 10%, often less 1%.

The largest discrepancy among the methods occurs for smaller drops and smaller size ratios, where the critical offset is rapidly decreasing toward zero, and at larger size ratios. The most important deviations, not at least partly ascribable to the stopping criterion for each trajectory, take place at drop size ratios close to one, i.e., k $\approx$ 0.9 or larger. In this region of the parameter space, diffusio-phoretic or thermo-phoretic effects could become prominent. It is worth noting, however, that van der Waals forces and droplet inertia may mask or partly mask their relevance. Although no direct observations of relative trajectories of two small evaporating droplets falling due to gravity can be found in the literature, recent experiments (Hardy et al. 2021) verify the assumption that evaporation is governed by diffusion for drops with radii between 2 and 30 $\upmu$m.

Appendices

Appendix 1. Bispherical coordinates

Fig. 10

Definition sketch of the bispherical coordinate system

Some discussion of the bispherical coordinate system, used to determine the evaporation rate of two drops, is made here. One of the advantages of this set of coordinates is that the Laplace equation is separable for a system of two unequally sized spheres, with each sphere described by a constant value of $\mu$ (see Fig. 10). Bispherical or bipolar coordinates have thus been employed to find solutions for two-sphere problems (Morse and Feshbach 1953; Rother 2009; Happel and Brenner 2012; Gilbert and Giacomin 2019).

As described in Happel and Brenner (2012), the bispherical coordinates $(\eta , \mu , \theta )$ are related to the cylindrical coordinates $(\rho , \theta , z)$ by

$$\begin{aligned}{} & {} z = {c_{\ell } \sinh \mu \over \cosh \mu - \cos \eta }~~~, \quad \rho = \sqrt{x^2+y^2}\nonumber \\{} & {} \quad = {c_{\ell } \sin \eta \over \cosh \mu - \cos \eta }~~~, \theta = \theta . \end{aligned}$$

(20)

If two drops of radii $a_1$ and $a_2$, with size ratio k = $a_1/a_2$, are separated by a center-to-center distance r, then the dimensionless distance s is

$$\begin{aligned} s = {2r \over a_1 + a_2}. \end{aligned}$$

(21)

The length $c_{\ell }$, at which the drop poles are located, determining the bispherical coordinates is then

$$\begin{aligned} c_{\ell } = {\sqrt{1 - q^2} \over q } a_1, \qquad q = {2k \lambda \over (k + 1) + \lambda ^2(k-1)}, \qquad \lambda = 2/s. \end{aligned}$$

(22)

The radii $\textrm{a}_i$ are thus transformed as $\textrm{a}_i = c_{\ell } \, \textrm{cosech} \, |\mu _i|$ for i = 1 or 2, and $\mu _i$ is a constant ($\mu _1 > 0$ and $\mu _2 < 0$). As illustrated in Fig. 10, the center-to-center distance r can be converted back using

$$\begin{aligned} r = d_1 + d_2 = c_{\ell }~\coth |\mu _1| + c_{\ell }~\coth |\mu _2|. \end{aligned}$$

(23)

In Fig. 10, as drawn, 0 < $c_2$ < $c_1$ < $\pi$.

The resistance functions, without and with slip, were determined using bispherical coordinates for motion along the line of centers and multipole techniques for transverse motion. More discussion can be found in Rother et al. (2022).

Appendix 2. Van der Waals forces

For unretarded van der Waals forces, the molecular force on the $i^{th}$ drop, as shown in Eq. (10), is

$$\begin{aligned} F_{mol,i} = \pm {k(1-k^2) \over 2} \left( {\hat{\mu }+ 2/3 \over \hat{\mu }+ 1} \right) {1 \over Q_{12}} {d \Phi _{12,nret} \over dr} {\hat{\textbf{r}}}, \end{aligned}$$

(24)

where

$$\begin{aligned}{} & {} {d \Phi _{12,nret} \over dr} = {kr \over (r-(k+1)^2)}+ {kr \over (r-(k-1)^2)} \nonumber \\{} & {} \quad - {2kr \over (r-(k+1)^2)(r-(k-1)^2)} \end{aligned}$$

(25)

and $Q_{12}$ is defined in Eq. (13). Please note that in Eqs. (24) and (25) the center-to-center distance has already been made dimensionless by the initial larger drop radius $a_{2,0}$.

Similar equations hold for retarded molecular forces. More details and discussion of retarded van der Waals forces can be found in Rother et al. (2022).