Theoretical analysis of Marangoni instability of an evaporating droplet by energy method

Abstract

Evaporation of a droplet is a process of heat and mass transports. When the droplet evaporates, heat is removed from the surface and the surface temperature is reduced thereby, resulting in a temperature gradient normal to the free surface. Such a situation is similar to that with a liquid layer heated from below. Thus, Marangoni instability as described by Pearson [J. Fluid Mech. 4 (1958) 489] may probably exist in an evaporating droplet, which is a very important and interesting phenomenon both from the academic and application points of view. In the present work, the energy method, which is appropriate for unsteady flows, is applied to investigate the stability of an evaporating droplet against disturbances of any amplitude. The results predicted by the present study possess similar trends with those acquired by the linear stability analysis. Both studies indicate that, as time proceeds, both the increase of the surface temperature reduction and the growth of the thermal boundary layer near the free surface are conducive to the onset of instability. The critical Marangoni number and wave number increase with the droplet initial temperature. For all the cases studied, the critical Marangoni numbers predicted by the energy method are smaller than those calculated by the linear stability analysis. However, the subcritical region between the Ma c's as calculated by these two methods is very narrow. This consistency indicates the validity of linear stability analysis as a first approach to the analysis of Marangoni instability of an evaporating droplet.