Analytical modeling of the evaporation of sessile drop linear arrays

An analytical model for predicting the competitive evaporation of two and three sessile drops is proposed, based on an analytical solution, in terms of Mehler functions, of the steady species and energy conservation equations for the gaseous phase. The assessment through a comparison with accurate numerical solutions of the species conservation equations is reported in order to quantify the accuracy of the analytical solution. The model is validated against three available sets of experiments on two and three sessile drops on a line array. The decrease of the evaporation rate caused by the vicinity of sessile drops is reported in terms of a screening coefficient given by a relatively simple analytical expression. The influence of wall wettability on the evaporation of pairs of sessile drops is analyzed, and a parameter is proposed to quantify the effect of geometry in a unified way.

Figure 1

Schematic of a sessile drop: (left) hydrophilic substrate and (right) hydrophobic substrate.

Figure 2

Schematic of a pair of sessile drops: geometric parameters.

Figure 3

Average values of the difference between the values of ${\mathrm{\Phi }}_{12}$ on one of the drop surfaces and the correct boundary condition, as a function of drop interdistance ${\lambda }_2=\frac{x_0}{R_d}$, for different contact angles.

Figure 4

Schematic of a triplet of sessile drops: three drops on a line configuration.

Figure 5

Inner boundaries of the domain for the solution of the Laplace equation over all the space; $\mathrm{\Omega }$ is a generic closed surface surrounding one of the inner boundaries.

Figure 6

Surface nondimensional flux as function of angular position, $\alpha$ (see Fig. 1), from numerical (line) and analytical (symbols) solutions; hydrophilic substrate with ${\theta }_{\mathrm{c}}={45}^{\circ }$.

Figure 7

Screening coefficient as function of nondimensional distance ${\lambda }_2$ for three drop configurations as predicted by the analytical model (line) and by the numerical calculation (symbols).

Figure 8

Value of the ratio $V/V_0$ as reported in Fig. 4 of [23], for the nondimensional time interval $\tau \in (0-0.6)$.

Figure 9

Comparison between the screening factors $\gamma$ as obtained from the experimental data reported in [23] and those calculated by the proposed model.

Figure 10

Transient profiles of drop nondimensional volume for a single drop and seven values of drop interdistance configurations, from [21].

Figure 11

Experimental values of $V/V_0={(R_d/R_{d,0})}^3$ versus time, as obtained from the data reported in Fig. 4 of [24], for a single drop (solid circle) and for the central drop of a triplet (open squares), for four different confinements (see [24] for details).

Figure 12

Values of the screening coefficient ${\gamma }_c$ for the central drop of a triplet of sessile drops, as obtained from the data reported in [21] (solid circles) and [24] (open squares) and calculated through the analytical approach (lines).

Figure 13

Screening factor $\gamma$ predicted by the analytical model as a function of (a) ${\lambda }_2=x_0/R_d$ and (b) ${\lambda }_1=x_0/R_c$, for different values of the contact angle.

Figure 14

(a) Values of the screening factor $\gamma$ as a function of the contact angle, for different values of ${\lambda }_3=x_0/R_{eq}$. (b) Values of the screening factor $\gamma$ as a function of the nondimensional drop distance ${\lambda }_3=x_0/R_{eq}$, for different values of the contact angle.

Figure 15

Values of the screening factor $\gamma$ as a function of the nondimensional drop distance ${\lambda }_1=x_0/R_c$, for three values of the contact angle (${30}^{\circ }$, ${20}^{\circ }$, ${10}^{\circ }$) and the thin film prediction from [32].