\[appendix\_b\]. To be recalled and to be used for comparison with the simulation results, the preconditions and the perturbed quantities are detailed here. We prescribe the wall-parallel velocity $V_{B1}$, and employ standard lubrication assumption, i.e., $\varepsilon \ll 1$. We also suppose small capillary number ${\rm Ca}\ll 1$. As implied in (\[eq:def\_f\_lub\]), the deflection is $O({\rm Ca}\varepsilon^{-1/2}a)$, which has to be sufficiently smaller than the gap $\varepsilon a$ if the tilt angle of the near-wall interface from the plane wall is supposed to be small. Here we adopt an additional constraint $\delta\equiv\varepsilon^{-3/2}{\rm Ca}\ll 1$. For inner coordinates $(R,Z,\phi)$ (here $\varepsilon^{1/2}R=r$ and $\varepsilon^{-1}Z=z$ as illustrated in figure \[fig:schem\](b), (see e.g. [@gol1967; @one1967])), a parabolic profile $Z=H(R)\equiv 1+R^2/2$ represents the interface within the inner region $r\sim \varepsilon^{1/2}$, if the deformation is absent. Using $\varepsilon$, we write the velocity vector, the pressure, and the deflection $f^{\rm (Ca)}$ of the interface in an expansion form with respect to $\delta$ $$\begin{split}
u_r(r,\phi,z)=&
\hat{U}_r^{(0)}(R,Z)\cos\phi+
\delta\hat{U}_r^{(f)}(R,Z)
\\&
+\varepsilon\hat{U}_r^{(1)}(R,Z)\cos\phi
+\delta\hat{U}_r^{(f,2)}(R,Z)\cos2\phi
+O(\delta^2)+O(\varepsilon\delta)+...,
\label{eq:ur_lub}
\end{split}$$ $$\begin{split}
u_\phi(r,\phi,z)=&
\hat{U}_\phi^{(0)}(R,Z)\sin\phi\\
&+\varepsilon\hat{U}_\phi^{(1)}(R,Z)\sin\phi
+\delta\hat{U}_\phi^{(f,2)}(R,Z)\sin2\phi
+O(\delta^2)+O(\varepsilon\delta)+...,
\end{split}$$ $$\begin{split}
u_z(r,\phi,z)=&
\varepsilon^{1/2} \left(\hat{U}_z^{(0)}(R,Z)\cos\phi+
\delta\hat{U}_z^{(f)}(R,Z)\right.\\
&\left.+\varepsilon\hat{U}_z^{(1)}(R,Z)\cos\phi
+\delta\hat{U}_z^{(f,2)}(R,Z)\cos2\phi
+O(\delta^2)+O(\varepsilon\delta)+...\right),
\end{split}$$ $$\begin{split}
p(r,\phi,z)=&
\varepsilon^{-3/2} \left(\hat{P}^{(0)}(R)\cos\phi
+\delta \hat{P}^{(f)}(R)\right.\\
&\left.+\varepsilon\hat{P}^{(1)}(R,Z)\cos\phi
+\delta\hat{P}^{(f,2)}(R,Z)\cos2\phi
+O(\delta^2)+O(\varepsilon\delta)+...\right),
\label{eq:p0_lub}
\end{split}$$ $$f^{\rm (Ca)}(r,\phi)=
\varepsilon^{-1/2}
\left(
\hat{F}(R)\cos\phi
+O(\delta)+O(\varepsilon)
\right),
\label{eq:def_f_lub}$$ whose 