# Optimal free-surface pumping by an undulating carpet

## TL;DR

This paper demonstrates how undulating boundaries can effectively pump thin liquids at free surfaces, revealing an optimal wave speed for maximum flow and a more energy-efficient mode of transport compared to rigid boundaries.

## Contribution

It introduces a novel fluid pumping mechanism driven by traveling waves on an undulating boundary near a free surface, with analytical predictions of optimal conditions.

## Key findings

- Flow rate depends non-monotonically on wave speed.
- Optimal wave speed maximizes pumping efficiency.
- Undulating boundaries cause less energy dissipation than rigid ones.

## Abstract

Examples of fluid flows driven by undulating boundaries are found in nature across many different length scales. Even though different driving mechanisms have evolved in distinct environments, they perform essentially the same function: directional transport of liquid. Nature-inspired strategies have been adopted in engineered devices to manipulate and direct flow. Here, we demonstrate how an undulating boundary generates large-scale pumping of a thin liquid near the liquid-air interface. Two dimensional traveling waves on the undulator, a canonical strategy to transport fluid at low Reynolds numbers, surprisingly lead to flow rates that depend non-monotonically on the wave speed. Through an asymptotic analysis of the thin-film equations that account for gravity and surface tension, we predict the observed optimal speed that maximizes pumping. Our findings reveal a novel mode of pumping with less energy dissipation near a free surface compared to a rigid boundary.