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  • Open Access

Locomotion on a lubricating fluid with spatial viscosity variations

Takahiro Kanazawa1,2,3,* and Kenta Ishimoto4,†

  • *Contact author: kt@g.ecc.u-tokyo.ac.jp
  • †Contact author: kenta.ishimoto@math.kyoto-u.ac.jp
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Phys. Rev. Fluids 10, 103102 – Published 21 October, 2025

DOI: https://doi.org/10.1103/yrzq-k45v

Abstract

We study locomotion of a model crawler corresponding to a deforming upper boundary of finite length above a thin Newtonian fluid film whose viscosity varies spatially. We first derive a general locomotion velocity formula with fluid viscosity variations via the lubrication theory. For further analysis, the surface of the crawler is described by a combination of transverse and longitudinal traveling waves and we find that under a uniform viscosity a transverse wave results in a retrograde crawler, while a longitudinal wave leads to a direct crawler. We then analyze the time-averaged locomotion behaviors under two scenarios: (i) a sharp viscosity interface and (ii) a linear viscosity gradient. Using the asymptotic expansions of small surface deformations and the method of multiple timescale analysis, we derive an explicit form of the average velocity that captures nonlinear, accumulative interactions between the crawler and the spatially varying environment. (i) In the case of a viscosity interface, the time-averaged speed of the crawler is always slower than that in the uniform viscosity, for both the transverse and longitudinal wave cases. Notably, the speed reduction is most significant when the crawler's front enters a more viscous layer and the crawler's rear exits from the same layer. (ii) In the case of a viscosity gradient, the crawler's speed becomes slower for the transverse wave, while for the longitudinal wave, the locomotion speed does not change significantly. Our analysis illustrates the fundamental importance of interactions between a locomotor and its environment, and separating the timescale behind the locomotion.

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Phys. Rev. Fluids 10, 103102– Published 21 October, 2025
  • Received 11 March 2025
  • Accepted 21 July 2025
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