(\[eq:f\_lub\]), respectively), which are compared with the simulation results of the scaled interfacial pressure $\varepsilon^{3/2} P/\cos\phi$ and the scaled deflection $\varepsilon^{1/2}\hat{f}^{({\rm Ca})}$ near the wall as a function of $\varepsilon^{-1/2}\theta$. For sufficiently small $\varepsilon$, the simulation data of the scaled pressure collapse onto the curve (\[eq:p\_lub\]) as shown in figure \[fig:scale\_pres\](a). Therefore, the scaled deflection profile reasonably approaches the lubrication solution (\[eq:f\_lub\]) with decreasing the bubble-wall gap as shown in figure \[fig:scale\_pres\](b). As plotted as the dashed-dotted curve in figure \[fig:scale\_deform\], the deflection based on (\[eq:def\_f\_lub\]) and (\[eq:f\_lub\]) is consistent with the measured and simulated deflections in the wall neighbour ($\theta\sim 0$). Hence, the discrepancy between the deflections of the narrow-gap experiment and the wide-gap theory ([@mag2003]) is attributable to the lubrication effect.

From [@bar1968], the drag force acting on the bubble translating perpendicular to the plane wall is $F_{DC}\rightarrow 3\pi \mu /(2\varepsilon)$ as $\varepsilon\rightarrow 0$. Substituting this relation and (\[eq:fm\_lub\]) (i.e., $F_M$ in the lubrication limit) into (\[eq:vb3ca\]), we obtain the asymptotic solution of the migration velocity $$\frac{V_{B3}}{V_{B1}}=\frac{F_M}{F_{DC}}
\rightarrow 
\frac{{\rm Ca}}{25}\varepsilon^{-1}
\ \ \ {\rm as}\ \ \ 
\varepsilon\rightarrow 0.
\label{eq:vb3_lub}$$

To make comparisons with the asymptotic solutions, figures \[fig:migvel\_comp\_exp\] and \[fig:migvel\_comp\_lub\] show the scaled migration velocity as functions 