  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. Cusp formation can 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 \textit{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 these surface forces undergo
self-similar runaway behavior leading to cusp formation with a conical tip
whose slope can be prescribed from the analytic relation derived. On a
fundamental level, this finding broadens our understanding of known categories
of flows capable of cusp formation. More practically, the system geometry
proposed offers a potentially novel lithographic method for one-step
non-contact fabrication of cuspidal microarrays.

Fluid Dynamics