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

A one-sided model of the thermal Marangoni instability owing to evaporation into an 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 and 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 using Biot numbers. 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 wave-number 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.

Figure 1

Sketches of the horizontal-layer (a) and spherical-droplet (b) systems.

Figure 2

Temperature reference profiles in the liquid in the horizontal-layer case at different dimensionless times for $H$= 2 [(a) and (b)], $H$= 11 [(c) and (d)], and $H$= 101 [(e) and (f)] for the fixed-temperature [(a), (c), and (e)] and the zero-flux conditions at the bottom [(b), (d), and (f)].

Figure 3

Temperature reference profiles in the liquid at different dimensionless times in the spherical-droplet case.

Figure 4

Dimensional [(a), (c), and (e)] and dimensionless [(b), (d), and (f)] time versus the system size at the critical condition for a layer with FTB [(a) and (b)], a layer with TIB [(c) and (d)] (both for $H=2$, 11, and 101), and a droplet [(e) and (f)], comparing the results using the wave-number-dependent Biot number (solid lines) with those using just a constant Biot number ${\text{Bi}}_0$ (dashed lines) as typically the case in the literature (e.g., see Refs. [15, 17, 18, 27]).

Figure 5

Critical wave numbers versus the layer thickness in the FTB (a) and TIB (b) cases (both for $H=2$, 11, and 101), comparing the results using the wave-number-dependent Biot number (solid lines) with those using just a constant Biot number ${\text{Bi}}_0$ (dashed lines). The direction of time along the curves is symbolically indicated by the $t$ arrows

Figure 6

The Pearson-like marginal curves for steady reference states (linear temperature profiles) of the layer with FTB. The same arrangements as in Fig. 4 are explored.