The modified Kachanov method. Evaporation of multiple droplets

Continuum Mech. Thermodyn. (2026) 38:19 https://doi.org/10.1007/s00161-026-01456-6 O R I G I NA L A RT I C L E Ivan Argatov The modified Kachanov method. Evaporation of multiple droplets Received: 17 December 2025 / Accepted: 1 February 2026 / Published online: 12 February 2026 © The Author(s) 2026 Abstract Evaporation of multiple sessile droplets deposited on an impermeable flat substrate is considered in the diffusion-limited isothermal regime, with particular emphasis on determining the total quasi-stationary vapor fluxes from the droplet surfaces. The proposed approximate solution is based on Kachanov’s approximation for the vapor concentration field in the semi-infinite air domain, expressed as a linear combination of solutions to single-droplet problems, with the coefficients determined by imposing appropriate Bubnov– Galerkin orthogonality relations to enforce the boundary conditions on the droplet surfaces. A comparison of the proposed modified Kachanov method with other approximate approaches is presented. The case of multiple thin circular sessile droplets is examined in detail. Keywords Kachanov’s method · Multiple evaporation · Sessile droplets · Contact interaction · Asymptotic modeling 1 Introduction The interaction between multiple evaporating sessile droplets on a flat surface is of significant practical interest, and recent advances in both experimental and theoretical studies have been reviewed in [1]. Important analytical approximations for the competitive diffusion-limited evaporation of multiple sessile droplets were derived by Wray et al. [2] in the thin-droplet limit and by Masoud et al. [3] for spherical-cap droplets. Recent work [2,3] has clarified how neighboring droplets compete for vapor, leading to non-uniform evaporative fluxes. For thin sessile droplets, this interaction produces spatially varying “shielding,” reducing evaporation in regions where droplets are closest. On both the single- and multi-droplet levels, much analytical and semi-analytical work continues to rely on Maxwell’s classical quasi-steady solution for diffusion-limited mass transfer from a spherical droplet — adopted here as well — extended to incorporate interfacial effects [4]. Comprehensive reviews [5,6] show how such solutions are embedded in larger modelling frameworks, including discrete and quasi-discrete approaches, with extensions to transient and multidimensional internal transport. As it was shown earlier [7,8], the problem of evaporation in the diffusion-limited regime reduces to a harmonic potential problem: the solution of the Laplace equation in a parametrically time-dependent semiinfinite domain occupied by air with the Dirichlet boundary condition on the droplet surface, a mathematical problem formulation that (independently of its physical context) has been widely studied in the literature I. Argatov Department of Biomedical Science, Malmö University, Jan Waldenströms gata 25, Malmö SE-205 06, Sweden I. Argatov (B) Biofilms – Research Center for Biointerfaces, Malmö University, Per Albin Hanssons väg 35, Malmö SE-205 06, Sweden E-mail: 19 Page 2 of 14 I. Argatov x3 x3 S+j a) c(x) = csk c(x) = csj Pj ~ c∞ c(x) = S j0 Pk Sj ~ c∞ c(x) = c(x) = csk c(x) = csj Pj Sk Pk Sk+ Djk b) Fig. 1 a) Schematic of multiple droplets (side view) in a semi-infinite air domain; b) An infinite “air domain” obtained by applying a symmetry reflection for canonical domains. In particular, for a single spherical-cap sessile droplet, an exact analytical solution is available in different forms [9,10]. Analytical treatments of droplet–droplet interaction typically assume diffusion-limited evaporation in quiescent air and convective effects are negligible. Under these assumptions, superposition and multipole methods from potential theory apply, with droplets represented as Dirichlet boundary patches, yielding interaction coefficients or shielding factors as functions of droplet size and spacing [11]. In simple geometries, the resulting interaction-corrected evaporation rates can be expressed via asymptotic expansions for well-separated or nearly touching droplets [12]. It goes without saying that a general strategy for solving the problem of multiple-droplet evaporation is to reduce it to a sequence of single-droplet problems. However, it can be shown (see, e.g., [13]) that the straightforward application of an iterative method is accompanied by an error that grows with the number of droplets in the system, and the multiple-droplet problem prompts the development of special approaches that account for interaction and cooperative effects. Foldy [14] appears to have been the first to consider multiple scattering of scalar (e.g., acoustic) waves [15] and to introduce a self-consistent method that results in solving a system of simultaneous linear algebraic equations for the scattering amplitudes. The problem of multiple-droplet evaporation is, in this sense, analogous to multiple scattering by a cluster of sound-soft obstacles, particularly in the long-wavelength limit [16], and asymptotic modeling approaches developed in that context [17,18] can be directly applied to its analysis. To effectively approximate potentials for bodies containing clusters of small defects, the method of mesoscale asymptotic approximations was developed [19,20]. Holm [21] was probably the first to observe a strong interaction effect between contact spots on the constriction resistance of a cluster of electrical microcontacts, which markedly differs from the non-interaction approximation. This microcontact interaction effect was later studied in a number of publications [22–24] following the seminal paper by Greenwood [25]. The problem of multiple evaporation of thin sessile droplets is, to a certain degree, equivalent to the problem of multiple frictionless contacts or to the electrostatics problem for a system of infinitesimally thin disks located in a single plane. This analogy motivates the application of the Kachanov method [26] to its analysis, though this method has been developed for multiple contacts. Kachanov’s method was originally introduced in the context of crack interactions [27,28] and is based on self-consistency relations linking the average tractions acting on individual cracks. The method was later adapted for the approximate analysis of frictionless contact problems involving multiple punches indenting an elastic half-space [26]. More recently, it has been extended to multi-punch contact interface problems [29] and to multiple contact problems involving bonded punches [30]. Here, we develop a modified Kachanov method applied to the problem of multiple evaporation. 2 Problem of multiple evaporation We consider N liquid droplets on the flat surface, x 3 = 0, of an impermeable substrate, which are exposed to a quiescent air, x3 > 0 (outside the droplets), referred to a Cartesian coordinate system x = (x1 , x2 , x3 ). Let S 0j and S + j denote respectively the con (...truncated)