  The study of viscous thin film flow has led to the development of highly
nonlinear partial differential equations that model how the evolution of the
film height is affected by different forces. We investigate a model of
interaction between surface tension and the thermocapillary Marangoni effect,
with a particular focus on the long-time limit. In this limit, the model
predicts the creation of an infinite cascade of successively smaller satellite
droplets near points where the film thickness vanishes. Motivated by recent
progress on the analysis of discrete self-similarity in thin film equations, we
compute solutions in a space- and time-rescaled coordinate system. Using this
rescaled system we observe the dynamics much further in time than has
previously been achieved. The observed behaviour is close to, but distinct
from, previous observations of discretely self-similar thin film flows, in that
the rescaled system does not settle down to a periodic solution, but instead
has aspects that continue to evolve monotonically in scaled time. This
discovery suggests there are as-yet unexplored ways in which discrete
self-similarity may be exhibited.
