Drag Coefficient of a Liquid Domain in a Fluid Membrane

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane, which is surrounded by three-dimensional fluids, by using the Stokes approximation. Taking into account the difference between the membrane viscosities inside and outside the domain, we obtain an integral equation involving a parameter, which is defined as unity minus the ratio of the latter viscosity to the former. We solve the equation up to the linear order with respect to the parameter to calculate the drag coefficient numerically. Its derivative with respect to the domain viscosity, evaluated when there is no difference between the membrane viscosities, is shown to almost vanish.