Weakly nonlinear study of Marangoni instabilities in an evaporating liquid layer

Abstract

We propose a theoretical study of Marangoni driven convection in an evaporating liquid layer surmounted by an inert gas-vapor mixture. After reduction of the full two layer problem to a one-sided model we use a Galerkin-Eckhaus method leading to a finite set of amplitude equations for the weakly nonlinear analysis of the problem. We analyse the stability of the roll, square and hexagonal patterns emerging above the linear stability threshold for a water-air and for an ethanol-air system.

Bénard instabilities in a binary-liquid layer evaporating into an inert gas

“…It also occurs in nature when for instance a salty lake dries out due to the evaporation of water, leaving behind structures on the soil. Several theoretical works have already been published on evaporation-driven Bénard instability of a one-component liquid layer [9][10][11][12][13][14][15], with the liquid evaporating in either an inert gas [9][10][11][12][13] or in its own vapor [6,[14][15]. To our knowledge, the studies of two-component systems has been rather limited in this context.…”

“…We shall also choose the mass fraction imposed at the bottom of the system as the typical concentration , , introduced in (1), (11) and (20). Similarly, the concentration imposed at the top of the system is used as the typical concentration , introduced in (2).…”

“…For instance, this is in the sense that the material properties of the liquid have all been fixed at (section 2), whereas at least some of them are rather sensitive to : these include the diffusion coefficient [24], the Soret coefficient (11) and even -cf. …”

On the use of Galerkin-Eckhaus method to study the nonlinear regime of Marangoni-Bénard instabilities in an evaporating liquid layer

“…Above the linear stability threshold, a weakly nonlinear stability analysis has been carried out by Dondlinger et al [11] and Margerit et al [12] for the nondeformable configuration, and by Sultan et al [13] for the additional case of a thin film within the lubrification theory framework. All these nonlinear studies assume that the gas layer is passive, i.e., that the evaporation rate remains small enough for convection in the gas to be negligible, and it was verified that the gas conductivity can also be neglected in general, because the ratio of the gas to liquid conductivities is small: these hypotheses define the so-called 1.5-sided model, which was introduced in [12], and consists of the liquid layer equations plus the needed diffusion equation for the vapor in the gas to account for evaporation.…”

“…This method has already been considered in [11,12], but with the restriction that only the evolution of the critical modes was considered. For this reason, the solu-tions were asymptotically correct only close to the threshold and the stability diagrams presented in these papers are only qualitative results.…”