Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall*

A deformation-induced lateral migration of a nearly spherical bubble rising near a vertical plane wall in a stagnant creeping liquid flow is numerically studied by means of a boundary-fitted finite-difference approach (Sugiyama \& Takemura (2010) J. Fluid Mech. accepted). The migration velocity is obtained using Lorentz's reciprocal theorem as a function of $\varepsilon$, corresponding to a ratio of a bubble-wall gap to the bubble radius. For $\varepsilon\gg 1$, the simulated migration velocities are consistent with an available analytical solution for the wide-gap case (Magnaudet {\it et al.} (2003) J. Fluid Mech. {\bf 476}, 115). With decreasing $\varepsilon$, the lift force is found to be more affected by the high-order deformation modes. The simulation and the lubrication analysis (Hodges {\it et al.} (2004) J. Fluid Mech. {\bf 512}, 95) 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 liquid viscosity, and the surface tension, respectively). However, the experimentally measured migration velocity is considerably higher by a factor of about 3 than the simulated one, implying that unexplored factors may be involved in the system.