higher modal contributions for $n\geq 4$ become no longer disregarded. Moreover, the fact that all the shown modal contributions decay as $\kappa\rightarrow 1$ indicates that the further higher-order contributions become considerable, and the regular perturbation approach with respect to $\kappa$ is no longer effective.

Comparison with small deformation theory in the lubrication limit {#sec:abe_lub}
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Figures \[fig:scale\_deform\], \[fig:deformationr\_kap\] and \[fig:modal\] imply that for small $\varepsilon$, the bubble deformation is preferentially enhanced within the narrow bubble-wall gap, and then its squeezing effect promotes the bubble migration. To shed more light on the role of the hydrodynamics in the gap, comparisons with a small deformation theory in the lubrication limit will be made. It should be noticed that [@hod2004] have performed a lubrication study for a nearly spherical drop near a tilted plane in so-called ‘slipping’ regime, and derived the deformation-induced normal force. One can also access a relevant physical picture in theoretical studies on the lift force on an elastic body induced by its deformation ([@sek1993; @sko2004; @sko2005; @urz2007]). Following the spirit of the lubrication theory, we evaluate the migration velocity in the limit of $\varepsilon\rightarrow 0$.

The basic equations for the lubrication analysis and the solutions are shown in Appendix 