Viscous friction acting on a solid disk falling in confined fluid: lessons for the scaling analysis

We fill a viscous liquid in a vertically stood cell of millimeter thickness, called the Hele-Shaw cell, and insert a disk in the liquid whose thickness is smaller than the cell thickness. The disk starts falling in the liquid due to gravity with opposed by viscous friction. We focus on the case in which lubricating films formed in the gap between the cell surface and the disk surface are thinner than the disk thickness. As a result, we find an apparent scaling regime for the falling velocity of a disk, in which the thickness of the lubricating film characterizes the dynamics. We further show that the apparent scaling regime is explained simply as a result of competition of two scaling regimes, elucidating physics of the viscous friction to make the present study relevant to fundamental issues and applications in various fields such as microfluidics, bioconvection, and active matter. The simple scenario for explaining an apparent scaling law demonstrated in the present study would be useful in diverse fields.