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We calculate the drag coefficient of a liquid domain in a flat fluid membranesurrounded by three-dimensional fluids on both sides. In the membrane, thetangential stress should be continuous across the domain perimeter, which makesthe velocity gradient discontinuous there unless the ratio of the membraneviscosity inside the domain to the one outside the domain equals unity. Thegradient of the velocity field in the three-dimensional fluids is continuous.This field, in the limit that the spatial point approaches the membrane, shouldagree with the velocity field of the membrane. Thus, unless the ratio of themembrane viscosities is unity, we need to assume some additional singularity atthe domain perimeter in solving the governing equations. In our result, thedrag coefficient is given in a series expansion with respect to a dimensionlessparameter, which equals zero when the ratio of the membrane viscosities isunity and approaches unity when the ratio tends to infinity. We derive therecurrence equations for the coefficients of the series. In the limit of theinfinite ratio, our numerical results agree with the previous results for thedisk.

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