  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.
