the bubble migration. The inset shows that for small $\kappa$, the contribution of the mode $n=2$ is proportional to $\kappa^2$, while that of the mode $n=3$ to $\kappa^5$, whose exponent is not trivially proven but may be predictable extending the regular perturbation to the higher-order. The difference in the exponent ensures that the relative contribution of the mode $n=3$ to $n=2$ becomes more significant with $\kappa$. It should be noticed that although we confirmed that the migration force contribution $F_{M, 2}^{({\rm Ca})}$ of the mode $n=2$ increases as $\kappa\rightarrow 1$ (i.e., $\varepsilon\rightarrow 0$), the velocity contribution $V_{B3, 2}^{({\rm Ca})}$ reduces as shown in figure \[fig:modal\](a). It is because the slope of the $n=2$ migration force, $-{\rm d}\log F_{M, 2}^{({\rm Ca})}/{\rm d}\log \varepsilon$, in a logarithmic plot is more gentle than that of the drag force, $-{\rm d}\log F_{DC}/{\rm d}\log \varepsilon\rightarrow 1$ in the denominator of (\[eq:vb3nca\_fm\]) as $\varepsilon\rightarrow 0$. Figure \[fig:modal\](b) shows the modal contribution $V_{B3,n}^{({\rm Ca})}$ compared with the migration velocity $V_{B3}^{({\rm Ca})}$. The contribution of the mode $n=2$ monotonically decreases with $\kappa$. The contribution of the mode $n=3$ increases with $\kappa$ in the range of $\kappa< \sim 0.8$, while decreases in the greater $\kappa$. It is because the 