We confirmed that the relative errors in the migration velocity $V_{B3}$ and the drag forces for the perpendicular and parallel bubble motions to those obtained on the base meshes $N_\xi\times N_\eta = 200 \times 200$ for various $\varepsilon$ decrease with increasing the size $R_{\rm max}$ and the resolutions $N_\xi$ and $N_\eta$. From the convergence behaviors, we deduce to obtain the migration velocities $V_{B3}$ with errors of much less than $1\%$ using the present base meshes, which are accurate enough for the subsequent discussion.

Migration velocity
------------------

As shown in the inset of figure \[fig:migvel\_comp\_exp\], the ratio of the simulation to the analytical migration velocity becomes close to unity as $\kappa (=(1+\varepsilon)^{-1})$ approaches zero, indicating the simulation result is consistent with the analytical solution ([@mag2003]) for small $\kappa (< \sim 0.3)$. Such a consistency between the different approaches may deny the erroneous prediction of the migration velocity in [@mag2003], of which the possibility was pointed out by [@wan2006]. By contrast, the simulated migration velocity for the bubble closer to the wall with the clearance shorter than the bubble radius (i.e., for $\kappa > \sim 0.5$ presumably beyond the theoretical precondition $\kappa\ll 1$) is considerably higher than the analytical solution. Although the 