of the inverse distance $\kappa (=(1+\varepsilon)^{-1})$ and the clearance parameter $\varepsilon$, respectively. The inset of figure \[fig:migvel\_comp\_lub\] shows the scaled force (\[eq:fm\_lub\]). The simulation results are consistent with two asymptotic behaviors based on the lubrication theory for $\varepsilon\ll 1$ as well as the mirror image technique for $\varepsilon \gg 1$ (i.e., $\kappa\ll 1$). As seen from figure \[fig:migvel\_comp\_lub\], the theories provide the different exponents of the migration velocity scaling with respect to $\varepsilon$, namely, $V_{B3}/V_{B1}\propto {\rm Ca}\ \varepsilon ^{-2}$ for $\varepsilon\gg 1$ and $V_{B3}/V_{B1}\propto {\rm Ca}\ \varepsilon ^{-1}$ for $\varepsilon\ll 1$.

The puzzling finding in [@tak2009] that the theory of [@mag2003] accurately predicts the deformation but fails to predict quantitatively the deformation-induced migration velocity is explained from the fact that the ratio of the simulation result of $V_{B3}$ to the theoretical prediction is more sensitively dependent upon $\varepsilon$ than that of $\hat{f}_2^{({\rm Ca})}$ when the lubrication effect becomes relevant. For instance, for $\varepsilon=0.4$, $\varepsilon=0.2$ and $\varepsilon=0.1$, the simulation-to-theory ratios of $\hat{f}_2^{({\rm Ca})}$ are respectively $1.004$, $1.1$ and $1.3$ (figure \[fig:deformationr\_kap\]), while those of $V_{B3}$ are respectively $2.5$, $4.3$ and $7.3$ (figure \[fig:migvel\_comp\_exp\]). The lubrication effect is likely to compensate the large discrepancy between the migration velocities of the experiment and 