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Drag coefficient of a liquid domain in a fluid membrane with the membrane viscosities being different across the domain perimeter

We calculate the drag coefficient of a liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. In the membrane, the tangential stress should be continuous across the domain perimeter, which makes the velocity gradient discontinuous there unless the ratio of the membrane viscosity inside the domain to the one outside the domain equals unity. The gradient of the velocity field in the three-dimensional fluids is continuous. This field, in the limit that the spatial point approaches the membrane, should agree with the velocity field of the membrane. Thus, unless the ratio of the membrane viscosities is unity, we need to assume some additional singularity at the domain perimeter in solving the governing equations. In our result, the drag coefficient is given in a series expansion with respect to a dimensionless parameter, which equals zero when the ratio of the membrane viscosities is unity and approaches unity when the ratio tends to infinity. We derive the recurrence equations for the coefficients of the series. In the limit of the infinite ratio, our numerical results agree with the previous results for the disk.