  A liquid drop spreading over a thin heterogeneous precursor film (such as an
inhaled droplet on the mucus-lined wall of a lung airway) will experience
perturbations in shape and location as its advancing contact line encounters
regions of low or high film viscosity. Prior work on spatially one-dimensional
spreading over a precursor film having a random viscosity field [Xu & Jensen
2016, Proc. Roy. Soc. A 472, 20160270] has demonstrated how viscosity
fluctuations are swept into a narrow region behind the contact line, where they
can impact drop dynamics. Here we investigate two-dimensional drops, seeking to
understand the relationship between the statistical properties of the precursor
film and those of the spreading drop. Assuming the precursor film is much
thinner than the drop and viscosity fluctuations are weak, we use asymptotic
methods to derive explicit predictions for the mean and variance of drop area
and the drop's lateral drift. For larger film variability, we use Gaussian
process emulation to estimate the variance of outcomes from a restricted set of
simulations. Stochastic drift of the droplet is predicted to be greatest when
the initial drop diameter is comparable to the correlation length of viscosity
fluctuations.
