On the lateral migration of a slightly deformed bubble rising near a vertical plane wall

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 short clearance $c$ between the bubble interface and the wall. Motivated by the fact that numerically and experimentally measured migration velocities are considerably higher than the velocity estimated by the available analytical solution using the Fax\'{e}n mirror image technique for $a/(a+c)\ll 1$ (here $a$ is the bubble radius), when the clearance parameter $\varepsilon(= c/a)$ is comparable to or smaller than unity, the numerical analysis based on the boundary-fitted finite-difference approach solving the Stokes equation is performed to complement the experiment. The migration velocity is found to be more affected by the high-order deformation modes with decreasing $\varepsilon$. The numerical simulations are compared with a theoretical migration velocity obtained from a lubrication study of a nearly spherical drop, which describes the role of the squeezing flow within the bubble-wall gap. The numerical and lubrication analyses consistently demonstrate that when $\varepsilon\leq 1$, the lubrication effect makes the migration velocity asymptotically $\mu V_{B1}^2/(25\varepsilon \gamma)$ (here, $V_{B1}$, $\mu$, and $\gamma$ denote the rising velocity, the dynamic viscosity of liquid, and the surface tension, respectively).