Importance of wave-number dependence of Biot numbers in one-sided models of evaporative Marangoni instability: Horizontal layer and spherical droplet

Abstract

A one-sided model of the thermal Marangoni instability owing to evaporation into inert gas is developed. Two configurations are studied in parallel: a horizontal liquid layer and a spherical droplet. With the dynamic gas properties being admittedly negligible, one-sided approaches typically hinge upon quantifying heat/mass transfer through the gas phase by means of transfer coefficients (like in the Newton's cooling law), which in dimensionless terms eventually corresponds to the Biot number. Quite a typical arrangement encountered in the literature is a constant Biot number, the same for perturbations of different wavelengths and maybe even the same as for the reference state. In the present work, we underscore the relevance of accounting for its wavenumber dependence, which is especially the case in the evaporative context with relatively large values of the resulting effective Biot number. We illustrate the effect in the framework of the Marangoni instability thresholds. As a concrete example, we consider HFE-7100 (a standard refrigerant) for the liquid and air for the inert gas. * H.Machrafi@ulg.ac.be 2

Self-assembly by multi-drop evaporation of carbon-nanotube droplets on a polycarbonate substrate

“…The Marangoni instability in the droplet is rather strong for the dimensions used in this work (25 µl) [22] and we can expect that at these dimensions the motion in the droplet will be triggered as soon as the droplet is deposited. This implies that the timescale of droplet spreading, , (we have deposited the droplet in such a way as to avoid splashing, so that only the spreading phenomenon is of importance here) should be in the same order of magnitude as or smaller than that of thermocapillarity, (Marangoni effect).…”

Sensitivity of lateral heat transfer on the convection onset in a transient Rayleigh-Bénard-Marangoni flow

“…In that case, a generalized heat transfer coecient accounting for both the convective heat transfer and the thermal evaporation ux is introduced at the liquid/gas interface. In this scope, Machra et al [14] developed a one-sided model to study the thermal Marangoni instability produced by a liquid evaporation in an inert gas. The use of a linear stability analysis of the transient base state using the frozen time approach showed the conditions for which the introduction of a generalized heat transfer coecient, function of the perturbation wavenumber, is particularly relevant at the interface.…”

Effect of including a gas layer on the gel formation process during the drying of a polymer solution

“…We will also neglect the Stefan flow in the gas, but not at the liquid-gas interface. This is a reasonable assumption in case the solvent vapor content is low [25], which can safely be assumed when the solvent considered is water, whose saturation pressure is low with respect to the gas pressure. In situations for which the concentration of the vapor in the gas is high due to a high vapor pressure (e.g.…”

“…In situations for which the concentration of the vapor in the gas is high due to a high vapor pressure (e.g. HFE -7100 [25]), the Stefan flow should of course be added in the description Finally, we will not describe hydrodynamic instabilities and convection is thus not taken into account. In this context, and because the system is horizontally uniform, a one-dimensional description is proposed.…”

“…setting the term ℎ / ≡ 0 in Eqs. (24) and (25), changes the results of Fig. 4 by less than 0.1 % (in fact, the results with ℎ / = 0 were used as initial guess for the numerical procedure in bvp4c when ℎ / ≠ 0).…”