Soft-lubrication effect on the lateral migration of a slightly deformed bubble rising near a vertical plane wall

TL;DR

This study investigates how a slightly deformed bubble's lateral migration near a vertical wall is affected by soft-lubrication effects, combining numerical, theoretical, and experimental approaches to understand the migration velocity in small clearance conditions.

Contribution

It introduces a combined numerical and theoretical analysis of bubble migration considering soft-lubrication effects, extending previous models to small clearance scenarios.

Findings

Migration velocity scales with Capillary number and inverse of clearance parameter.

Numerical analysis complements experimental data for small clearance cases.

Theoretical analysis based on soft-lubrication explains the migration behavior.

Abstract

Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a small clearance $c$ between the bubble interface and the wall. Motivated by the fact that experimentally measured migration velocity (Takemura et al. (2002, J. Fluid Mech. {\bf 461}, 277)) is higher than the velocity estimated by the available analytical solution (Magnaudet et al. (2003, J. Fluid Mech. {\bf 476}, 115)) using the Fax\'{e}n mirror image technique for $\kappa(=a/(a+c))\ll 1$ (here $a$ is the bubble radius), when the clearance parameter $\epsilon(=c/a)$ is comparable to or smaller than unit, the numerical analysis based on the boundary-fitted finite-difference approach by solving the Stokes equation is performed to complement the experiment. To improve the understandings of a role of the squeezing flow within the bubble-wall gap, the theoretical analysis based on a soft-lubrication approach (Skotheim & Mahadevan (2004, Phys. Rev. Lett. {\bf 92}, 245509)) is also performed. The present analyses demonstrate the migration velocity scales $\propto{\rm Ca}\ \epsilon^{-1}V_{B1}$ (here, $V_{B1}$ and ${\rm Ca}$ denote the rising velocity and the capillary number, respectively) in the limit of $\epsilon\to 0$.