Drag coefficient of a circular inclusion in a near-critical binary fluid membrane

We calculate the drag coefficient of a circular liquid domain, which is placed in a flat fluid membrane composed of a binary fluid mixture lying in the homogeneous phase near the demixing critical point. Assuming a sufficiently small correlation length, we regard the domain dynamics as independent of the critical fluctuation and use the Gaussian free-energy functional for the mixture. Because of the near-criticality, the preferential attraction between the domain component and one of the mixture components generates a significant composition gradient outside the domain, which can affect the drag coefficient. We first consider a domain having the same membrane viscosity as the domain exterior. The drag coefficient is expanded with respect to a dimensionless strength of the preferential attraction. It is numerically shown that the magnitude of the expansion coefficient markedly decreases as the order of the strength increases and that the first-order term of the series usually gives a good approximation for practical material constants. The effect of the preferential attraction is shown to be able to become significantly large in practice. We secondly consider cases where the membrane viscosities of the domain interior and exterior are different. The first-order term of the expansion series decreases to approach zero as the domain viscosity increases to infinity. This agrees with previous numerical results showing that the hydrodynamics makes the effect of the preferential attraction negligibly small for a rigid disk.