  We consider the effect of droplet geometry on the early-stages of coffee ring
formation during the evaporation of a thin droplet with an arbitrary simple,
smooth, pinned contact line. We perform a systematic matched asymptotic
analysis of the small-capillary number, large-solutal Peclet number limit for
two evaporative models: a kinetic model, in which the evaporative flux is
constant across the droplet, and a diffusive model, in which the evaporative
flux is singular at the contact line. For both evaporative models, solute is
transported to the contact line by a capillary flow while, local to the contact
line, solute diffusion counters advection. The resulting interplay leads to the
formation of the nascent coffee ring. By exploiting a coordinate system
embedded in the contact line, we solve explicitly the local leading-order
problem, deriving a similarity profile (in the form of a gamma distribution)
that describes the coffee ring profile in its early stages. Notably, the ring
characteristics change due to the concomitant asymmetry in the shape of the
droplet free surface, the evaporative flux (in the diffusive evaporative
regime) and the mass flux into the contact line. We utilize the asymptotic
model to determine the effects of geometry on the growth of the coffee ring for
a droplet with an elliptical contact set. Our results offer mechanistic insight
into the effect of contact-line curvature on the development of the coffee-ring
from deposition up to jamming of the solute; moreover our model predicts when
finite concentration effects become relevant.
