Rotation of an immersed cylinder sliding near a thin elastic coating

It is well known that an object translating parallel to a soft wall produces viscous stresses and a pressure field that deform the wall, which, in turn, results in a lift force on the object. Recent experiments on cylinders sliding near a soft incline under gravity confirmed previously developed theoretical arguments, but also reported an unexplained rotation of the cylinder at steady state (Saintyves et al. \emph{PNAS} 113(21), 2016). Here, we use the Lorentz reciprocal theorem to calculate the angular velocity of an infinite cylinder sliding near a soft incline, in the lubrication limit. Our results show that the softness-induced angular velocity of the cylinder is quadratic in the deformation of the elastic layer. This implies that a cylinder sliding parallel to a soft wall without rotation experiences an elastohydrodynamic torque that is proportional to the cube of the sliding speed. We compare the theoretical predictions of the rotation speed with experimental measurements. We then develop scaling and symmetry arguments that are generally applicable to hydrodynamically mediated interactions between soft systems, such as those in biological and geophysical settings.

众所周知，一个平行于软壁平移的物体产生粘性应力以及使壁变形的压力场，而这反过来又导致物体受到一个升力。最近在重力作用下圆柱体在软斜面附近滑动的实验证实了先前提出的理论观点，但也报道了圆柱体在稳态下出现无法解释的旋转（塞因蒂夫等人，《美国国家科学院院刊》113卷21期，2016年）。在这里，我们利用洛伦兹互易定理来计算在润滑极限下一个无限长圆柱体在软斜面附近滑动时的角速度。我们的结果表明，圆柱体由软度引起的角速度与弹性层的变形呈二次方关系。这意味着一个平行于软壁无旋转滑动的圆柱体受到一个与滑动速度的立方成正比的弹性流体动力扭矩。我们将转速的理论预测与实验测量结果进行了比较。然后我们提出了比例缩放和对称性的论点，这些论点通常适用于软系统之间由流体动力学介导的相互作用，比如生物和地球物理环境中的那些相互作用。