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

TL;DR

This paper investigates the viscous friction on a falling disk in a confined fluid, revealing an apparent scaling regime governed by lubricating film thickness, which enhances understanding of low Reynolds number flows in microfluidics and related fields.

Contribution

It introduces a simple explanation for the apparent scaling law of a falling disk's velocity in confined viscous flow based on competing scaling regimes.

Findings

Identification of an apparent scaling regime for falling velocity

Demonstration that film thickness governs the dynamics

Relevance to microfluidics and low Reynolds number physics

Abstract

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 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 the physics of the viscous friction. The present study is thus relevant to fundamental issues and applications in various fields in which small-scale physics in the flow at low Reynolds numbers is essential, 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, considering that the generality and strength of scaling analysis in science and that simple arguments usually lead to a few different scaling laws for a given problem.