On a scale-by-scale basis we consider the equilibrium and nonequilibrium behaviour of films and droplets on horizontal and inclined homogeneous substrates. First, we employ classical density functional theory (DFT) [1] to determine binding potentials and Derjaguin (disjoining) pressures that encode the adsorption and wetting behaviour of liquids at solid substrates including the possible layered packing of molecules at the substrate [2]. These pressures are incorporated in a mesoscopic gradient dynamics (hydrodynamic long-wave) model to study the spreading of individual (terraced) drops on both, an adsorption (or precursor) layer and completely dry substrates. To achieve this, the hydrodynamic long-wave model is modified in such a way that for very thin layers a diffusion equation is recovered [3].

Next, we study the dynamics of individual sliding drops on an incline [4]. In particular, we employ continuation techniques to analyse sliding drops and their transformations in dependence of the driving force. We show that a number of shape transitions occur at saddle-node bifurcations. Further there is a global bifurcation that results in dynamic states where a main sliding droplet emits small satellite droplets at its rear (pearling instability) that subsequently coalesce with the next droplet. These pearling states show the period-doubling route to chaos [5].

The single-drop results are then related to direct numerical simulations on a large domain that examine the interaction of many sliding drops.  The ongoing merging and pearling behavior results in a stationary distribution of drop sizes, whose shape depends on the substrate inclination and the overall liquid volume.  We illustrate that aspects of the steady long-time drop size distribution may be deduced from the bifurcation diagrams for individual drops. In the final coarse graining step we use the single-drop diagram to construct a statistical model for the time evolution of the drop size distribution and show that it captures the main features of the full scale simulations.

[1] A.P. Hughes, U. Thiele, and A.J. Archer, Am. J. Phys. 82, 1119-1129 (2014).
[2] A.P. Hughes, U. Thiele and A.J. Archer, J. Chem. Phys. 142, 074702 (2015) and preprint at http://arxiv.org/abs/1611.06957.
[3] H. Yin, D.N. Sibley, U. Thiele  and A.J. Archer, preprint at http://arxiv.org/abs/1611.00390.
[4] T. Podgorski, J.-M. Flesselles and L. Limat, Phys. Rev. Lett. 87, 036102 (2001).
[5] S. Engelnkemper, M. Wilczek, S. V. Gurevich and U. Thiele, Phys. Rev. Fluids 1, 073901 (2016).