Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis

Abstract

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wavenumber and time. The method is applied to transient Rayleigh-Bénard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tensiondriven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

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

“…The viscosity thresholds were obtained for a wide range of uid layer thicknesses and these results turned out to be in qualitative agreement with the experimental data by Toussaint et al [19]. The linear stability analysis by Doumenc et al [7], performed with the non normal approach,…”

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

“…This would still contrast with the approach used elsewhere [17], [29], corresponding in our terms to Bi ≡ Bi 0,amb , even for the perturbations.…”

“…Thus, the problem for the liquid temperature reference profile finally reduces to Eq. (12), with the interface conditions (4), (11), (13) and (14), bottom/center conditions (15) or (16), and the initial condition (17). Note that it has actually been reduced to a one-sided problem, owing to the large value of diffusivities in the gas phase (as discussed in section II).…”

“…Indeed, an Ansatz similar to (18)- (20) is often encountered in the literature (e.g. [17], [29]), in the case when mass/heat exchange with the gas phase is modeled by means of Newton's law of cooling. However, the problem is that this same constant Biot number subsequently appears in the analysis of perturbations as well, whereas our goal in the present paper is to show that the Biot number for perturbations should actually be different and wavenumber-dependent.…”

“…This kind of approach to describe the gas influence has been used quite often in the literature, both for horizontal layers [13], [14], [15] and for spherical droplets [16]. In the case of evaporation (into ambient air), one arrives again at a single constant effective thermal Biot number, now dependent on both mass and heat transfer coefficients in the gas [17]. On the other hand, no heuristic transfer coefficients and Biot numbers are adopted in the studies [18], [19] for evaporating spherical droplets.…”