Drag coefficient of a liquid domain with distinct viscosity in a fluid membrane

Abstract

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

Drag Coefficient of a Circular Inclusion in a Near-Critical Binary Fluid Membrane

“…With the aid of numerical integrations, Eq. (3.9) is calculated for −1 < κ < 1 and some values of ν o in Tani and Fujitani [26], where the calculation results in the limit of κ → 1− are shown to agree with the results for a rigid disk in Saffman & Delbrück's model [12][13][14]. Figure 2 shows numerical results of Eq.…”

“…We introduce the Fourier transforms with respect to θ, e.g., Fujitani [26], we can use the incompressibility conditions to delete one of the two undetermined functions of ζ, and thus have only to consider one undetermined function of ζ. We write A(ζ) for this function.…”

“…It is helpful in calculating the integration over a semi-infinite interval numerically to find how the integrand behaves for its large variable. As described in Appendix C of Tani and Fujitani [26], part of MJ 0 has a peculiar logarithmic dependence for its large variable, although [NMJ 0 ](R) is continuous at R = 1. Let us defineB (0) n as (0) n − β (0) n J 0 to calculate MJ 0 separately.…”

“…Some researchers measured the diffusion coefficient of a circular liquid domain, and analyzed the results by using the theoretical result for a rigid disk [23] or by assuming that the membrane viscosities are the same in the domain interior and exterior [24]. The drag coefficient of a circular liquid domain with a distinct membrane viscosity in a flat fluid membrane has recently been calculated [25,26]. Multicomponent membranes near the demixing critical point were also studied experimentally.…”

“…26),(2.32), and (2.35) determine c 1 , c 2 , andÂ, and then γ with the aid of Eqs. (2.18) and(2.22).…”