scaling relations are suitable to all equations in Appendix \[appendix\_b\]. It should be noted that the terms with the superscript $(0)$, $(1)$ or $(f,2)$ in (\[eq:ur\_lub\])–(\[eq:p0\_lub\]) are proportional to $\cos\phi$, $\sin\phi$, $\cos 2\phi$ or $\sin 2\phi$, and thus provide no wall-normal force. Here we make clear the physical domain of validity of the condition $\delta\ll 1$ when the rising velocity $V_{B1}$, which is needed to evaluate ${\rm Ca}(=\mu V_{B1}/\gamma)$, is unknown. Following the analysis by [@one1967], we write the drag force $F_D$ acting on the spherical bubble translating parallel to the wall in a form $F_{D}=6\pi\mu a V_{B1}(A\log \varepsilon+B)$, where $A$ and $B$ are independent of $\varepsilon$. Considering the free-slip boundary condition on the bubble surface, we can analytically find $A=-1/5$. From our numerical data, we approximately estimate $B=0.6$. Hence, from the force balance $F_D=4\pi \rho a^3g/3$, for $\varepsilon\ll 1$, we evaluate the rising velocity $$V_{B1}=
\frac{1}{(-\log\varepsilon+3)}\frac{10\rho a^2 g}{9\mu},$$ and obtain the following relation for the bubble radius $a$ to satisfy the condition $\delta\ll 1$. $$a \ll \sqrt{\frac{9\gamma}{10\rho g}}\varepsilon^{3/4}(-\log \varepsilon+3)^{1/2}.$$

To assure us of the appearance of the lubrication effect, figure \[fig:scale\_pres\] shows the profiles of the pressure $\hat{P}^{(0)}(=3R/(5H^2))$ and deflection $-\hat{F}(=3\log H/(5R))$ in the lubrication limit (see (\[eq:p\_lub\]) and 