and the surface tension, respectively).'
author:
- 'KAZUYASU SUGIYAMA$^1$ and FUMIO TAKEMURA$^{2, 1}$'
title: On the lateral migration of a slightly deformed bubble rising near a vertical plane wall
---

Introduction
============

Recent technical progress in generating microbubbles (e.g. [@gar2006; @mak2006]), including potentials as actuator and sensor, has enhanced the range of applications, e.g. additives to reduce a turbulence friction ([@ser2005]), drug delivery capsules ([@sho2004]), and contrast agents ([@cor2001]). In many situations, a bubble encounters a boundary wall during its transport process, and a hydrodynamic interaction occurs as characterized by the inter-scale between the bubble and the wall. In practice, it is of primary importance that the bubble undergoes a repulsive or attractive force in the wall-normal direction, which causes a lateral migration ([@lea1980; @mag2003; @hib2007]) and determines the bubble distribution, when translating parallel to the wall. As the simplest model system, one may raise a phenomenon of a spherical bubble rising near a vertical infinite plane wall in a creeping (Stokes) flow. However, there is no mechanism to generate the lateral migration force, as kinematic reversibility is ensured by symmetry of the boundary and by linearity in the Stokes equation ([@lea1992], chapter 4). In fact, the migration force stems 