# Rotation of a low-Reynolds-number watermill: theory and simulations

## TL;DR

This study combines theory and simulations to analyze the hydrodynamics of a flow-driven micro-scale watermill, revealing how its rotation depends on the number of rods, with hydrodynamic interactions playing a crucial role.

## Contribution

It introduces a combined theoretical and numerical approach to understand the hydrodynamics of a low-Reynolds-number watermill, highlighting the importance of hydrodynamic interactions for accurate predictions.

## Key findings

- Rotational velocity is largely independent of the number of rods for N ≥ 4.
- Hydrodynamic interactions significantly affect the watermill's rotation and are essential for accurate modeling.
- Theoretical predictions align with simulations when hydrodynamic interactions are included.

## Abstract

Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven "watermill" at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally-spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill's instantaneous rotational velocity as a function of its rod number $N$, position and orientation. When $N \geq 4$, the RF theory predicts that the watermill's rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinity order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on $N$ and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.