  We examine theoretically the spreading of a viscous liquid drop over a thin
film of uniform thickness, assuming the liquid's viscosity is regulated by the
concentration of a solute that is carried passively by the spreading flow. The
solute is assumed to be initially heterogeneous, having a spatial distribution
with prescribed statistical features. To examine how this variability
influences the drop's motion, we investigate spreading in a planar geometry
using lubrication theory, combining numerical simulations with asymptotic
analysis. We assume diffusion is sufficient to suppress solute concentration
gradients across but not along the film. The solute field beneath the bulk of
the drop is stretched by the spreading flow, such that the initial solute
concentration immediately behind the drop's effective contact lines has a
long-lived influence on the spreading rate. Over long periods, solute swept up
from the precursor film accumulates in a short region behind the contact line,
allowing patches of elevated viscosity within the precursor film to hinder
spreading. A low-order model provides explicit predictions of the variances in
spreading rate and drop location, which are validated against simulations.
