n}
\left(
\frac{\partial{\bm U}}{\partial n}\!\cdot\!{\bm n}-{\bm U}\!\cdot\!\nabla_s
\right)
-{\bm u}\!\cdot\!\biggl(
\frac{\partial {\bm \Sigma}}{\partial n}\!\cdot\!{\bm n}
-{\bm \Sigma}\!\cdot\!\nabla_s
\biggr)
\\&
-\left\{
{\bm n}\cdot{\bm\Sigma}\cdot{\bm n}
-3x_1\langle x_1 {\bm n}\cdot{\bm\Sigma}\cdot{\bm n}\rangle_{S_B}
\right\}{\bm u}\cdot\nabla_s.
\end{split}$$ The migration velocity $V_{B3}={\rm Ca}V_{B3}^{({\rm Ca})}$ is expressed as $$\frac{V_{B3}^{({\rm Ca})}}{V_{B1}}=\frac{F_{M}^{({\rm Ca})}}{F_{DC}},
\label{eq:vb3ca}$$ where $$F_{DC}=\oint_{S_B}\!\!\!{\rm d}^2{\bm x}\ {\bm e}_3\cdot{\bm\sigma}\cdot{\bm n}
\label{eq:fdc}$$ denotes the drag force acting on the bubble translating perpendicular to the plane wall.

Simulation method
-----------------

The basic equations are numerically solved by the second-order finite-difference method discretized on the bipolar coordinates $(\xi,\eta)$ grid, which is boundary-fitted on both the bubble surface and the plane wall (see figure \[fig:schem\_bipolar\](a) in Appendix \[appendix\_a\]). We take care of the mass and momentum conservations in a discretized form. For technical detail on the discretization, see Appendix \[appendix\_a\]. The number of grid points is $N_\xi\times N_\eta = 200 \times 200$, and the grid is non-uniform and refined near the wall and the bubble surface. The computational procedure is based on a Simplified-Marker-And-Cell method ([@ams1970]) with a first-order Eulerian implicit time marching scheme. Such an unsteady scheme enables us to check whether the computation converges to the fully developed state through temporal changes in the budgets of the momentum and kinetic-energy transports. 