标题： 在具有空间粘度变化的润滑流体上的运动 显示英文标题

标题： Locomotion on a lubricating fluid with spatial viscosity variations

摘要： 我们研究了在粘度空间变化的薄牛顿流体膜上爬行器的运动。 我们首先通过润滑理论推导了一个考虑流体粘度变化的通用运动速度公式。 为了进一步分析，爬行器的表面由横向和纵向行波的组合描述，并我们在两种情况下分析了时间平均的运动行为：(i) 粘度界面分明和 (ii) 线性粘度梯度。 利用小表面变形的渐近展开和多重时间尺度分析方法，我们推导出了一个显式形式的平均速度，该速度捕捉了爬行器与空间变化环境之间的非线性、累积相互作用。 (i) 在粘度界面的情况下，无论是横向波还是纵向波的情况，爬行器的时间平均速度始终比均匀粘度情况下的速度慢。 值得注意的是，当爬行器的前端进入更粘的层而爬行器的后端离开同一层时，速度下降最为显著。 (ii) 在粘度梯度的情况下，对于横向波，爬行器的速度变慢，而对于纵向波，修正项相对于均匀粘度情况来说是更高阶的。 作为推导出的运动速度公式的应用，我们还分析了基底形貌对平均速度的影响。 我们的分析说明了运动器与其环境之间相互作用的基本重要性，并区分了运动背后的时标。 显示英文摘要

摘要： 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.