Transport in Thin Gravity-driven Flow over a Curved Substrate

TL;DR

This paper analyzes steady gravity-driven flow of a thin viscous fluid over curved substrates with large-scale topographical variations, revealing complex particle transport behaviors including chaos and trapping phenomena.

Contribution

It develops a lubrication theory-based method to compute velocity fields on curved substrates and explores how substrate shape influences particle transport and chaos.

Findings

Flow exhibits trapped and untrapped particle trajectories.

Chaotic jumps occur between different particle motion states.

Transport properties are highly sensitive to substrate shape.

Abstract

We consider steady gravity-driven flow of a thin layer of viscous fluid over a curved substrate. The substrate has topographical variations (`bumps') on a large scale compared to the layer thickness. Using lubrication theory, we find the velocity field in generalized curvilinear coordinates. We correct the velocity field so as to satisfy kinematic constraints, which is essential to avoid particles escaping the fluid when computing their trajectories. We then investigate the particle transport properties of flows over substrates with translational symmetry, where chaotic motion is precluded. The existence of trapped and untrapped trajectories leads to complicated transport properties even for this simple case. For more general substrate shapes, the trajectories chaotically jump between trapped and untrapped motions.