Locomotion on a lubricating fluid with spatial viscosity variations

TL;DR

This paper derives a formula for crawler locomotion speed on a fluid with spatially varying viscosity, revealing how viscosity interfaces and gradients affect movement efficiency.

Contribution

It introduces a novel analytical approach to quantify how spatial viscosity variations influence locomotion, including explicit formulas and analysis of different environmental scenarios.

Findings

Speed decreases at viscosity interfaces, especially when entering more viscous layers.

Transverse waves slow down the crawler on viscosity gradients, while longitudinal waves are less affected.

Environmental interactions significantly impact locomotion efficiency and behavior.

Abstract

We studied locomotion of a crawler on a thin Newtonian fluid film whose viscosity varied spatially. We first derived a general locomotion velocity formula with fluid viscosity variations via the lubrication theory. For further analysis, the surface of the crawler was described by a combination of transverse and longitudinal travelling waves and we analysed the time-averaged locomotion behaviours 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 time-scale analysis, we derived 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 corrections are of a higher order compared with the uniform viscosity case. As an application of the derived locomotion velocity formula, we also analysed the impacts of a substrate topography to the average speed. Our analysis illustrates the fundamental importance of interactions between a locomotor and its environment, and separating the time scale behind the locomotion.