Linear and nonlinear aspects of Marangoni and evaporative instabilities

Abstract

Surface-tension gradient driven instability or Marangoni instability and phase-change driven evaporative instability are two classical examples of interfacial instability in fluid layers subject to a vertical temperature gradient. We review the physics of these two problems, comparing their behavior when the interfaces are taken to be either non-deformable or deformable. First, a full linear stability analysis is carried out. Conditions for neutral stability as well as linear growth rates are computed. In both problems, when the interface is non-deformable, a finite short-wavelength instability is seen beyond a critical threshold temperature difference. The presence of a deformable interface renders both problems unstable to long-wave instability with significant deformation of the interface. In this case, both problems become unstable for any non-zero value of imposed temperature difference. The velocity and temperature perturbation profiles at the onset of instability is shown to be different and as a result the differing role of thermal convection in the two problems is explained. Next, we look at the nonlinear evolution of the long-wave instability in these problems. Using the method of weighted residual integral boundary layer, nonlinear evolution equations retaining thermal inertia are derived to track the interface position. The approach to interfacial rupture is shown to be different in both problems. Evaporation results in sharp troughs due to an increased rate of evaporative mass flux near the wall, while the Marangoni problem results in a cascade of buckling events due to a symmetric drainage about the troughs. The presence of thermal inertia in evaporation is destabilizing and is shown to result in a shorter rupture time, while it results in lower linear growth rates in Marangoni instability and therefore a larger rupture time.