Drop spreading and drifting on a spatially heterogeneous film: capturing variability with asymptotics and emulation

TL;DR

This paper studies how liquid drops spread and drift on heterogeneous thin films, using asymptotic analysis and emulation to predict variability in drop behavior caused by spatial viscosity fluctuations.

Contribution

It extends previous one-dimensional models to two dimensions, deriving explicit predictions for drop area and drift variance under weak heterogeneity and employing emulation for larger variability.

Findings

Drop drift is maximized when initial drop size matches viscosity correlation length.

Explicit formulas for mean and variance of drop area are derived.

Gaussian process emulation effectively estimates outcome variability.

Abstract

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.