Self-similar cuspidal formation by runaway thermocapillary forces in thin liquid films

Abstract

Many physical systems give rise to dynamical behavior leading to cuspidal shapes which represent a singularity of the governing equation. The cusp tip often exhibits self-similarity as well, indicative of scaling symmetry invariant in time up to a change of scale. Cuspidal shapes even occur in liquid systems when the driving force for fluid elongation is sufficiently strong to overcome leveling by capillarity. In almost all cases reported in the literature, however, the moving interface is assumed to be shear-free and the operable forces orient exclusively in the direction normal to the advancing boundary. Here we focus on a system in which a slender liquid film is exposed to large thermocapillary stresses, a system previously shown to undergo a linear instability resembling microlens arrays. We demonstrate by analytic and numerical means how in the nonlinear regime runaway thermocapillary forces induce cuspidal formations terminated by a conical tip whose slope is given by an analytic relation. On a fundamental level, this finding broadens our understanding of known categories of flows that can generate cuspidal forms. More practically, the system examined here introduces a potentially novel lithographic method for one-step non-contact fabrication of cuspidal microarrays.