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

Published online by Cambridge University Press:  15 June 2018

Lailai Zhu Open the ORCID record for Lailai Zhu [Opens in a new window]  and
Howard A. Stone
Lailai Zhu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Stockholm, SE-10044, Sweden
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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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 $N\geqslant 4$, the RF theory predicts that the watermill’s rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinite 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.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: MEMS/NEMS Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 57 - 75
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Copyright
© 2018 Cambridge University Press 

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