Linear stability of thermocapillary flow in a droplet attached to a hot or cold substrate

Abstract

The symmetry-breaking instability of the steady axisymmetric thermocapillary flow in a nonvolatile sessile droplet on a hot or cold substrate under zero gravity conditions is investigated by linear stability analysis. The dependence of the critical Marangoni number on the contact angle and the Prandtl number is computed, and the structures of the most dangerous perturbation modes are analyzed. For a small contact angle and a high-Prandtl-number droplet with Pr = 16.36 (corresponding to 1 cSt silicone oil) on a hot wall, we find classical Marangoni instabilities near the center of the droplet. In contrast, no instability is observed for small-contact-angle droplets on a cold wall. For large contact angles with α > 90° either on a hot or a cold wall, the first instability of the basic axisymmetric toroidal vortex is inertial for low Prandtl numbers and of hydrothermal-wave type for high Prandtl numbers.