Droplet levitation over non-isothermal surface of evaporating liquid layer

Experimental studies of heated liquid layers show that large arrays of microscopic levitating droplets can be observed near the layer surface [1]. These droplets form spontaneously above the layer as a result of re-condensation of vapor produced by the evaporation from the liquid surface. In order for levitation to occur, there must be a repulsive force from the liquid layer to compensate for the weight of the droplet. Understanding this force is important for describing everyday phenomena, such as the formation of a layer of mist over a cup of coffee [2] and for a number of important applications, such as containerless biochemical reactors and droplet deposition in human airways [3‐6]. For the latter case, inhaled droplets of respiratory fluids potentially carrying pathogens and droplets of medications administered via an inhaler are both of interest. The interaction force between the droplet and evaporating aqueous layer of mucus covering the airway wall affects how far the droplets can travel through the respiratory system, which in turn determines the probability of infection and the success of the therapy in the two respective situations. In most numerical models of processes in the human respiratory system, the repulsive interaction between the droplets and the walls of the airways is not accounted for.

Simple diffusion models of evaporating/condensing droplets at moderate temperatures with the assumption of a quiescent gas phase [7, 8] cannot describe experimental data on levitation. It is now well understood that the force balancing the weight of the droplet is essentially the Stokes drag force due to a viscous air flow around the droplet [9, 10]. This flow, although affected by the droplet, is not induced by its presence. In fact, the flow originates from phase change at the liquid layer surface. An early study by Stefan [11] begins to define this convective mass transfer, later named Stefan flow. The mechanism of the flow generation can be understood as follows. During evaporation from a heated liquid surface, there is a gradient of vapor concentration established to transport vapor away from the surface. If one assumes that the total density of the gas phase is constant, there must be a gradient of the other (non-vapor) molecules in air (nitrogen, oxygen, etc.) in the opposite direction. From this assumption, there would be a diffusion of these dry air molecules toward the liquid layer surface. As these molecules do not penetrate the surface, there must be an induced flow to transport them away from the surface. As the evaporation intensity increases, so does the magnitude of this flow.

Mathematical models for a droplet levitating over a heated liquid layer, accounting for Stefan flow, have been developed by Zaitsev et al. [10, 12]. They first considered the condensing droplets as point sinks [10] and obtained a simple formula for levitation height as a function of droplet size but required the use of adjustable parameters to match the experimental data. Their follow-up model [12] added complexity by considering a droplet of finite size. A constant condensation rate was set along the surface of the droplet, despite a strong gradient of temperature around the droplet [13]. Due to this simplification, the rates of phase change on both surfaces could not be determined, and instead had to be filled in by experimental results or treated as fitting parameters.

The sizes of droplets of interest for the present study are well below the capillary length of the liquid (about 2.7 mm for water), suggesting small Bond numbers and the surface of the droplet being dominated by capillary forces. Thus, the droplets are nearly spherical. This vastly simplifies the mathematical modeling of the droplet, including the steady diffusion equation for vapor distribution around the droplet [7] and heat transfer inside and outside of the droplet. All of these could be solved using separation of variables in bipolar coordinates [14, 15]. Furthermore, an analytical solution for Stokes flow around a droplet which is interacting with a flat surface has been found using a similar approach [16]. The Stokes flow equations are reformulated into a single equation for the Stokes stream function. This equation is then solved analytically using the method of separation of variables. This solution has been combined with models of heat transfer and diffusion by Zhang and Gogos [14] in the context of an evaporating droplet near a hot dry surface. Physical parameters for this study were motivated by the Leidenfrost phenomenon [17, 18], so the rate of phase change was high. Although the geometry and governing equations of this study are similar to the problem discussed in the present work, the boundary conditions are different, most notably due to the flat surface under the droplet in our study being a liquid rather than flat dry solid. Several recent studies investigated Leidenfrost-type levitation over liquid surfaces [19‐22]. While many ideas from these studies, most notably the approach to coupling between heat transfer solutions in different phases [20, 22], provided inspiration for our work, it is important to note that we are motivated by levitation experiments in which the droplet sizes are much smaller [9, 10], leading to different values of the relevant nondimensional parameters.

A recent study by Davis et al. [23] considered a spherical droplet levitating over a heated liquid surface with the assumption that the temperature in the liquid layer was constant. The heat transfer in and around the droplet, as well as the vapor concentration around the droplet, was solved for. Using these results, the Stokes stream function around the droplet was calculated, which described the moist air flow induced by the phase change. The stream function was used to find the force acting on the droplet from the viscous flow around it. Balancing this force with the weight of the droplet led to predictions of levitation height.

A common feature of the previous theoretical studies of droplet levitation, including [23], is the assumption of uniform temperature at the top surface of the evaporating liquid layer. It is clear that such an assumption cannot be strictly valid, especially when the droplet is very close to the surface of the layer. Note that in levitation experiments, the liquid in the layer is the same as in the droplet, typically water, and therefore has the same thermal conductivity. The key objective of the present study is to overcome the limitation of the constant evaporating liquid surface temperature assumption by considering heat transfer in the layer. Since the method based on separation of variables in bipolar coordinates does not work for the flat layer geometry, we propose a hybrid approach in which the analytical solution in the domain above the layer surface is coupled to straightforward finite-difference numerical solution for heat transfer in the liquid layer. The remarkably fast convergence of the analytical solution allows us to obtain accurate results with just a few terms in the truncated series expansions, thus dramatically reducing the computational efforts needed to obtain a complete description of the heat and mass transfer in the system that includes both the droplet and the liquid layer.

The geometric configuration considered in the present study is illustrated in Fig. 1a. A spherical droplet is levitating over a surface of an evaporating liquid layer. The top of the liquid layer is assumed to correspond to \(z=0\) in the cylindrical coordinates shown in the sketch. The temperature in the droplet and the air around it is coupled with the temperature in the liquid layer. Beneath the liquid layer, there is a heating element held at constant temperature. We begin by considering the distributions of vapor concentration \(c^*\), temperature in the air around the droplet \(T^*_a\), temperature in the droplet \(T^*_l\), and temperature in the liquid layer \(T^*_f\) (where ‘f’ stands for ‘film’). We nondimensionalize the temperatures, usingwhere \(k=a,l,f\), and \(T_H\), the temperature at the surface of the heating element, is assumed constant. We then proceed to nondimensionalize the vapor concentration,where \(c_{\textrm{sat}}\) is the equilibrium vapor saturation concentration at the flat liquid-air interface when no droplet is present but the heater is on. The air velocity scale is set by Stefan flow originating at the liquid-air boundary [8] and can be expressed as \(U=DMc_{\textrm{sat}}/(\rho R_0)\), where D is the vapor diffusivity, M is the molar mass of vapor, \(\rho \) is total moist air density, and \(R_0\) is the characteristic radius of the droplet also used as the length scale for all length variables.

The temperature and vapor concentration follow the advection–diffusion equations, which after being nondimensionalized, take the formHere, D/Dt denotes the material derivative, t is the time scaled by \(R_0/U\), and the Péclet numbers are defined,where \(\kappa _k\) denotes the thermal diffusivities of the corresponding fluids. In experiments, Péclet numbers are typically found to be small [10, 12]. This follows from the small droplet size (\(\sim 10\) \(\upmu \)m) and relatively slow flow speed (\(< 10^{-2}\) m/s). From these observations, we can take \(\text{ Pe}_k,\text{ Pe}_m \rightarrow 0.\) This leads to the conclusion that \(c,T_a,T_f,\) and \(T_l\) all satisfy Laplace’s equations,For the heat transfer and vapor diffusion problems, we will re-formulate them in bipolar coordinates and then proceed to solve these problems using separation of variables. Suppose h is the nondimensional height at which the droplet levitates and \(\alpha = \cosh ^{-1} h\). The bipolar coordinates \(\xi \)   and   \(\eta ,\) illustrated in Fig. 1b, are defined byThe quantities \(c,T_a,\) and \(T_l\) satisfy several boundary conditions at \(\xi = \alpha .\) The first is the fact that temperature is continuous across the surface of the droplet, orWe also assume that the concentration and the temperature are related through the factor \(\gamma \), where \(\rho _{v}^e = Mc_{\textrm{sat}}\), \(\rho _{vT}^e\) is the rate of change of the equilibrium saturation vapor density with temperature, \(T^i\) is the reference temperature which corresponds to layer surface temperature when no droplet is present. Equation (9) represents a linearization of a more general nonlinear relation between the equilibrium concentration and temperature and is typically accurate for temperature variations of up to a few degrees K, as verified by direct comparison with the experimental data in Sefiane et al. [24].

Conservation of energy at the liquid-air interface, \(\xi = \alpha \), leads to the following equation,where we introduced the ratio of thermal conductivities of the two phases, \(k=k_a/k_l\), and the nondimensional form of the latent heat,with \(\mathcal {L}\) being the dimensional latent heat. To provide more insight into the meaning of L, we observe that it can be represented as the product of the ratio of sensible-to-latent heat, i.e., the inverse Jakob number (Ja), the density ratio, the scaled interfacial temperature, and the inverse Lewis number (Le), \( L= \frac{ \rho _{v}^e T^i}{ \rho \text {Le}\text {Ja} }\). The Jakob number is defined by the temperature drop from the heating element to the liquid layer surface. In experiments, the value of L can be increased by decreasing the heater temperature.

Vapor concentration near the liquid layer surface, \(\xi =0\), can be related to temperature using the same approach as for the droplet surface,The liquid layer thickness d (scaled by \(R_0\)) is constant and assumed spatially uniform, so that the condition of fixed heater temperature (\(T_f=0\)) is applied at \(z=-d\). The conditions of continuity of temperature and energy balance at the layer surface ensure coupling between the temperature fields in the liquid film and the moist air above it and are discussed in detail below in Sect. 3.2.

Finally, the boundary condition in air far away from the droplet has to be formulated. Let us first discuss the temperature profiles in the moist air (\(T_{0a}\)) and in the liquid film (\(T_{0f}\)) when no droplet is present. Solving the one-dimensional steady heat conduction problem in the liquid film immediately givesFor the air domain above the liquid surface, \(z\ge 0\), the appropriate length scale \(L_a\) is many orders of magnitude larger than \(R_0\), so convective heat transfer can no longer be assumed negligible and the solution would have to be obtained from detailed numerical simulations of coupled heat transfer and air flow. Instead of conducting such simulations, we take advantage of the condition \(L_a \gg R_0\) to argue that locally, in the region where levitating droplets are found, the global solution for temperature can be approximated by a linear function, \(T_{0a} = T^i -Gz\). The scaled temperature gradient G is treated as a parameter of the model. When a droplet is introduced, we assume that this local profile remains unperturbed far away from the droplet, leading to the far-field condition \(T_a \rightarrow T^i - Gz\). A similar argument can be made for the concentration profile, with the corresponding far-field gradient denoted by \(G_c\).

For the developments that follow, it is convenient to re-define the nondimensional temperatures according toNote that both \(\tilde{T}_a\) and \(\tilde{T}_f\) decay to zero far away from the droplet.

Let us start by considering the limit of negligible phase change, \(L=0\), for a droplet of radius \(R_0=10 \,\upmu \textrm{m}\). The results presented here and below are for the computational domain size of \(l=10\), corresponding to \(100 \,\upmu \textrm{m}\) in dimensional terms. Values of nondimensional parameters listed in the caption of Fig. 3 are motivated by experimental studies of levitating water droplets [10]. The spatial distribution in Fig. 3a, corresponding to \(L=0\), shows that the layer temperature directly under the droplet is slightly reduced as compared to the surface values away from it. To provide better resolution of temperature gradients, only immediate vicinity of the droplet is shown; details of the solution in the layer are discussed below. The local cooling effect is expected since a droplet has much higher thermal conductivity than air and thus provides, locally, an efficient path for heat loss from the layer, as long as no phase change is considered. In dimensional terms, the layer cooling effect is on the order of 0.1 K, while the vertical temperature change seen in Fig. 3a is about 1 K. As the scaled measure of latent heat, L, is increased, so that phase change starts playing a more prominent role in the overall energy balance, the layer cooling effect is gradually diminished and eventually reversed, as seen in Fig. 3b. This could be explained by the geometric effect of the droplet: as more vapor is accumulated in the gap between the droplet and layer surface, evaporation intensity is reduced and so is evaporative cooling of the surface.

Figure 4a provides a more detailed study of the effect of the droplet on the temperature of the liquid layer surface. Profiles of the temperature at \(z=0\) are shown for different values of the scaled phase change rate, illustrating the regimes when the surface is locally cooled due to the droplet (lower L) and when it is locally heated (higher L). The heating effect shows a saturation trend, suggesting that increasing L above 0.2 is not likely to increase the maximum temperature much further. The two regimes are separated by a critical value of the parameter L such that the layer surface is isothermal and the overall temperature distribution is qualitatively similar to the one shown in Fig. 3b. We found the critical value to be \(L^*=0.048\) and observed it to be independent of the levitation height and the values of the external gradients, \(G_c\) and G. In order to explain this rather puzzling observation, we propose the following argument. Consider a function \(\mathcal{F} \equiv c - 1 - \gamma (T_a - T^i)\) defined on the domain corresponding to the gas phase, \(\Omega _g\). By construction, this function satisfies Laplace’s equation and, according to Eqs. (9), (12), is zero everywhere on the boundary, \(\partial \Omega _g\). Therefore, based on the well-known property of the solutions of Laplace’s equation, the function has to be zero everywhere in \(\Omega _g\). Differentiation of \(\mathcal{F}\) with respect to \(\xi \) then leads to the conditionApplying this condition at the droplet surface and combining it with Eq. (10) leads toThus, the thermal field is not affected by the presence of the interface as long asWhen this condition is satisfied, the droplet presence, regardless of its distance from the layer, does not change the layer surface temperature since the droplet surface has no effect on the global temperature field. We verified that the previously found critical value of \(L^*\) satisfies Eq. (31).

In order to better understand how phase change affects the layer temperature, we plot scaled evaporative flux profiles at \(z=0\), defined by \(\hat{J}=-G_c^{-1} \partial c/\partial z\), for three different values of L near \(L^*\), see Fig. 4b. The result provides quantitative justification to the explanation of the effect of phase change on layer temperature proposed above. Indeed, increasing L is seen to reduce the local evaporative flux, as vapor is accumulated in the gap between the droplet and the layer. Thus, the evaporative cooling is weaker and the local layer temperature under the droplet increases.

Changes in the value of L also affect the temperature of the droplet, as shown in Fig. 3; let us discuss this in more detail. Figure 4c shows droplet surface temperature as a function of the polar angle \(\theta \), for different L. While the temperature is nearly uniform at \(L=0\), even very small non-zero values of the scaled latent heat lead to strongly non-uniform temperature profiles, with bottom of the droplet (\(\theta =\pi \)) being hotter. The temperature non-uniformity increases with increase in the latent heat, but tends to saturate at L near 0.2. In experiments, the values for L usually range between \(L=0.03\) and \(L=0.2\). The values for the dimensional parameters used in the estimates of L are found based on Marrero and Mason [25] and Tsilingiris [26].

The spatial non-uniformity of the flat surface temperature seen in Fig. 4a is caused by the effect of heat conduction in the layer, but the local depression in the evaporative flux is also observed when the layer surface temperature is fixed, as in Davis et al. [23]. In Fig. 4d, the evaporative flux profile at \(L=0.2\) is compared to the profile found from their model at the same experimental conditions, corresponding to slightly different values of nondimensional parameters, as these parameters were based on the dimensional liquid-air interfacial temperature rather than \(T_H\). The comparison shows that taking into account heat conduction in the liquid layer leads to stronger evaporation under these conditions. This observation can be explained by the increase of the layer temperature under the droplet at \(L=0.2\), seen in Fig. 4a, leading to higher local equilibrium vapor concentration (according to Eq. (12)) and thus suggesting a stronger concentration gradient.

The plots in Fig. 4 suggest that droplet-induced perturbations in the surface temperature and flux are localized in the radial direction, rapidly decaying to almost zero when \(r=5\), which allows us to restrict the plots to that value rather than the maximum value of \(r=10\). An even faster decay to zero in the vertical direction is seen in the two-dimensional solution plot in Fig. 5, showing \(\hat{T}_f\) with the linear gradient subtracted out to better illustrate the temperature correction magnitude. The domain dimensions in both directions for the finite-difference formulation are \(l=10\), but the region of non-zero temperature gradients is clearly localized near the upper left corner of the domain. It is therefore not surprising that very little change in the solution is seen when the value of l is varied, as long as it is above \(\sim 5\), as illustrated by the data in Table 1. Thus, we conclude that the choice of \(l=10\) provides a good approximation to experimental configurations in which the layer thickness is typically two orders of magnitude larger than \(R_0\), while the size of the heating element which defines the horizontal length scale for the evaporation zone is three orders of magnitude larger than the droplet size [10, 27].

A comparison of the two parts of Fig. 3 shows that the temperature distributions inside the droplet can also be drastically different depending on the intensity of the phase change. We note that the small conductivity ratio, \(k=0.04\), makes it natural to assume that the droplet should be approximately isothermal, as seen, e.g., in Fisher and Golovin [15] and in our Fig. 3a above. However, Fig. 3b clearly shows that there can be strong temperature gradients inside the droplet. We believe that an explanation for this phenomenon is related to the relatively large value of the external concentration gradient, \(G_c\). When vapor concentration decreases rapidly with z around the droplet, favorable conditions are created for evaporation near the top. Local evaporative cooling at the top enhances vertical temperature gradient inside the droplet. This argument suggests that reducing the value of \(G_c\) should lead to more uniform temperature inside the droplet, which is indeed observed in our solutions. For example, taking \(G_c=10^{-3}\) leads to the plot shown in Fig. 6. Even though all other parameters are the same as in Fig. 3b, there is almost no temperature gradient inside the droplet. In contrast to the situation illustrated in Fig. 3a, droplet temperature is significantly higher than that of moist air surrounding it. We note that qualitatively similar observations of almost uniform elevated temperature inside the droplet were made by Davis et al. [23] at comparable \(G_c\), suggesting that the effect of droplet being heated through by heat exchange with the liquid layer below it is not sensitive to the details of the heat conduction model in the layer.

The dependence of surface tension on temperature can lead to fluid motion known as Marangoni flow [28, 29]. Our conclusions on the direction of temperature gradient at the layer surface suggest the different effects Marangoni stresses can have on levitation. It is well known that such stresses can lead to gas being pushed into the gap between the droplet and the liquid layer, as seen in, e.g., studies of flotation [30]. This could happen in our configuration and thus support levitation, but only for relatively low scaled latent heat, when the layer is locally cooled. Marangoni stresses tend to oppose levitation when evaporation effects are stronger and the shear stress pushes air out of the gap between the droplet and the layer. It is also important to note that in our case the effect of Marangoni stresses is much weaker than for larger droplets in flotation studies. Following Geri et al. [30], we define the Marangoni number bywhere \(\textrm{d}\sigma /\textrm{d}T\) is the derivative of surface tension with respect to temperature, \(\mu _l\) and \(\kappa _l\) are the dynamic viscosity and thermal diffusivity of liquid, respectively, and \(\Delta T\) is the characteristic temperature difference. In our configuration, using \(R_0=10 \,\upmu \)m, \(\Delta T \sim 0.01\) K, and the physical properties of water, we find Ma \(\sim 0.2\), an indication that thermocapillarity is a secondary factor, especially since in experiments with water it tends to be weaker than predicted theoretically. Note that in flotation problems, typical values of the Marangoni number are \(10^2 - 10^3\) [30].

The studies of diffusion and heat transfer described above allow us to accurately determine the rate of evaporation at the liquid layer surface and thus set the stage for describing the flow of moist air around the droplet. Viscous stresses associated with this flow are widely recognized as the key factor in droplet levitation [9, 27]. Using the same scales as in Sect. 2 above, the momentum and continuity equations for the flow of moist air around the droplet take the formThe Reynolds number is defined bywhere \(\mu _a\) is the dynamic viscosity of the air, and the effective pressure is scaled by \(\mu _a U/R_0\).

Let us consider the limit when the Reynolds number is small, as is the case in experiments with microscale levitating water droplets at moderate evaporation rates [10]. The approach to an analytical solution of the resulting Stokes flow equations is described in detail in Davis et al. [23] and involves writing the Stokes stream function \(\psi \) as an infinite series of the formwhere \(C_{n+1}^{-1/2}\) are the Gegenbauer polynomials. The coefficients of our expansion are written in the formexcept for \(W_0\) and \(W_{-1}\) which simplify according toHere and below, we use \(n_-=n - 1/2\), \(n_+=n+3/2\). The gas flow velocity near the droplet surface, as well as the liquid layer surface, is equal to the velocity of the Stefan flow induced by the phase change. Thus, in bipolar coordinates we can write the conditionsat the droplet surface, andat the liquid layer surface. As our system has a large liquid-to-gas viscosity ratio, and based on the fact that interface immobilization has been observed on a microscopic scale, we impose no-slip conditions on both the droplet surface, and the liquid layer surface,Direct experimental evidence of water-air interface immobilization can be found in the pioneering studies of Manor et al. [31, 32]; subsequent studies provided additional discussion of the topic [33, 34].

Equations (40) and (41) suggest that the flow velocity at boundaries is defined by evaporation/condensation flux profiles. Since evaporation from the liquid layer was discussed above, let us now focus on phase change at the droplet surface. We found that for a range of realistic parameter values, condensation takes place near the bottom of the droplet, while evaporation occurs near the top. Scaled condensation flux (with negative values corresponding to evaporation) is shown in Fig. 7a. Comparison with the predictions from a previous study of Davis et al. [23] shows stronger condensation when the effects of heat conduction in the layer are accounted for, as long as L is above the critical value of \(L^*\). This observation can be explained by substantial increase in the local vapor concentration near the layer surface due to increase in temperature, a solution feature already used in interpreting Fig. 4d. Vapor concentration at the droplet surface is almost the same as in the models without layer heat conduction, so the difference and therefore the concentration gradient increase and condensation is promoted. The top half of the droplet is mostly characterized by evaporation and, being further away from the liquid layer, is only weakly affected by heat conduction there.

We can now solve for the coefficients \({\bar{A}}, {\bar{B}}, {\bar{A}}_n, {\bar{B}}_n, {\bar{C}}_n, \) and \({\bar{D}}_n\) using the stream function boundary conditions formulated above, as described in the Appendix. Typical flow streamlines based on this solution are shown in Fig. 7b. In contrast to classical pattern of flow around a sphere without phase change, here we observe some of the streamlines originating at the liquid layer surface terminate at the droplet, a pattern made possible by condensation there. The density of streamlines indicates that flow is weaker under the droplet, as expected due to reduced evaporation intensity there compared to the flat layer surface far away from the droplet. Evaporation at the top of the droplet leads to streamlines originating there.

The upward force acting on the droplet can now be found in terms of the solution for the flow using the well-known formulas [35, 36]. The procedure for balancing this force with the weight of the droplet, described in Davis et al. [23], leads to the ability to express levitation height h in terms of the scaled droplet radius, defined bywhere \(\rho _l\) is liquid density and g is the acceleration of gravity. Typical results for levitation height predicted by our model are shown in Fig. 8a, solid line. Comparison with the dashed line found from the solution of Davis et al. [23] shows that the levitation height is increased when the heat conduction in the layer is accounted for. The origin of this effect can be understood by examining Fig. 4d which clearly points to stronger evaporation from the liquid layer and thus stronger force acting on the droplet, which in turn translates into increase of the levitation height. However, this conclusion is only valid when the temperature under the droplet is higher than the layer surface temperature away from it, i.e., for sufficiently large L. The effect of the scaled latent heat L on the levitation height is illustrated in Fig. 8b for a droplet of fixed size (\(\hat{R}=2\)). The value of h increases as phase change rate is increased since the upward force is directly related to the intensity of phase change. The increase is initially nearly linear, but slows down for higher values of L, suggesting a limitation on how much the levitation height can be increased by changing the heater temperature.

Experimental measurements of levitation height using optical methods have been conducted by Zaitsev et al. [10, 12]. Table 2 shows the physical parameters corresponding to their experiment conducted at \(63\,^\circ \textrm{C}\) and the calculated values of the nondimensional parameters. Figure 9 presents comparison between theory and experiment. While experimental data are also available at higher heater temperatures, equation (10) would need to be replaced by a nonlinear equation discussed in e.g., [14] in order to describe it; we do not consider such a model in the present work. Since the value of L in the experiment we consider is relatively low, the temperature under the droplet is not expected to increase due to heat conduction in the layer and thus the correction to the levitation height is negative rather than positive. The origin of slight under-prediction of the experimental data by both theories remains unexplained.

The calculation of the force acting on the droplet in the present section is based on the standard continuum description of the flow, valid when the Knudsen number, the ratio of the mean free path of the gas molecules to \(R_0\), is small. Levitation condition is determined from the balance between the droplet weight and the force due to Stefan flow, a theoretical approach confirmed by previous comparisons with experiments [9, 12, 23]. This approximation may appear inconsistent in the limit of small Péclet numbers, suggesting that viscous stresses, which depend linearly on the characteristic flow velocity \(U \sim |\nabla \rho _v|\), should be much smaller than the diffusion contributions to stress, just as advective flow is weaker than diffusion mass transfer. However, a closer examination based on the theory reviewed by, e.g., Mills [38], shows that this is not true, as the diffusion contributions to the stress tensor are quadratic in \(|\nabla \rho _v|\) and thus decrease rapidly as the gradient is decreased, which also corresponds to \(\text{ Pe }\rightarrow 0\).

The theoretical description of levitation of a microscale droplet over the surface of a liquid layer requires detailed knowledge of temperature and vapor concentration around the droplet. Both of these are coupled to heat transfer inside the droplet and in the layer under it. While the former was investigated by Davis et al. [23], the latter has not been considered in the previous studies. We develop a comprehensive model of heat transfer and vapor diffusion in the system by coupling analytical solutions for temperature and concentration above the layer surface to numerical finite-difference-based solution below it. We identify a key nondimensional parameter, the scaled latent heat of phase change L, that characterizes heat exchange between the droplet and the layer. For values of L below a critical value defined by a simple analytical relation, Eq. (31), the droplet locally cools the layer, with effect being stronger when the distance between the two is reduced. However, above the critical value, the layer temperature is locally increased as a result of heat exchange with the droplet. In this regime, comparison with the model of Davis et al. [23] shows stronger evaporation rate at the layer surface under the droplet, a rather counterintuitive result explained by the increase of the local vapor concentration there.

The temperature distribution inside the droplet also tends to be sensitive to the rate of phase change. Nearly uniform temperature is seen for small L, but gradients can develop when the evaporation rate is increased and a significant local concentration gradient \(G_c\) is present. However, when \(G_c\) is reduced, the droplet temperature is again nearly uniform and significantly higher than the temperature of air around the droplet.

The modeling approach used in the present study relies in part on separation of variables in bipolar coordinates, an analytical technique which allows one to obtain the solutions for heat/mass transfer and flow equations in terms of rapidly convergent series. Even though the geometry of the flat liquid layer does not allow one to separate variables in bipolar coordinates, the advantages of the remarkably fast convergence of the series are fully preserved in the hybrid approach we propose, with the series solutions coupled to the standard finite-difference method applied in the liquid layer. This approach could be applied to other situations when analytical solutions are available, but only for part of the domain. Of particular interest would be exploring this for the coupled fluid flow equations in two different fluid phases, a topic beyond the scope of the present article.

The results of our study have several implications for applications such as interaction of respiratory droplets with walls of airways. In particular, temperature of the droplet approaching the wall just before deposition can be either higher or lower than the temperature of the surroundings, depending on the evaporation rate. The temperature affects the phase change rate (and thus the droplet size) as well as the force of repulsive interaction with the airway and therefore the location of droplet deposition.