# Sliding drops - ensemble statistics from single drop bifurcations

## TL;DR

This paper investigates the complex dynamics of sliding drops on inclined surfaces, revealing how merging and splitting processes influence the steady-state size distribution through numerical simulations and a statistical model.

## Contribution

It introduces a combined approach of numerical simulations and bifurcation analysis to understand ensemble drop dynamics and develops a Smoluchowski-type model for these processes.

## Key findings

- Drop ensembles exhibit a balance of coalescence and breakup.
- The size distribution reaches a stationary state.
- The statistical model agrees well with direct simulations.

## Abstract

Ensembles of interacting drops that slide down an inclined plate show a dramatically different coarsening behavior as compared to drops on a horizontal plate: As drops of different size slide at different velocities, frequent collisions result in fast coalescence. However, above a certain size individual sliding drops are unstable and break up into smaller drops. Therefore, the long-time dynamics of a large drop ensemble is governed by a balance of merging and splitting. We employ a long-wave film height evolution equation and determine the dynamics of the drop size distribution towards a stationary state from direct numerical simulations on large domains. The main features of the distribution are then related to the bifurcation diagram of individual drops obtained by numerical path continuation. The gained knowledge allows us to develop a Smoluchowski-type statistical model for the ensemble dynamics that well compares to full direct simulations.