  The diffusion coefficient of a circular shaped inclusion in a liquid membrane
is investigated by taking into account the interaction between membranes and
bulk solvents of arbitrary thickness. As illustrative examples, the diffusion
coefficients of two types of inclusions - a circular domain composed of fluid
with the same viscosity as the host membrane and that of a polymer chain
embedded in the membrane are studied.The diffusion coefficients are expressed
in terms of the hydrodynamic screening lengths which vary according to the
solvent thickness. When the membrane fluid is dragged by the solvent of finite
thickness, via stick boundary conditions, multiple hydrodynamic screening
lengths together with the weight factors to the diffusion coefficients are
obtained from the dispersion relation. The condition for which the diffusion
coefficients can be approximated by the expression including only a single
hydrodynamic screening length are also shown.
