Hydrodynamics of a disk in a thin film of weakly nematic fluid subject to linear friction

To make progress towards the development of a theory on the motion of inclusions in thin structured films and membranes, we here consider as an initial step a circular disk in a two-dimensional, uniaxially anisotropic fluid layer. We assume overdamped dynamics, incompressibility of the fluid, and global alignment of the axis of anisotropy. Motion within this layer is affected by additional linear friction with the environment, for instance, a supporting substrate. We investigate the induced flows in the fluid when the disk is translated parallel or perpendicular to the direction of anisotropy. Moreover, expressions for corresponding mobilities and resistance coefficients of the disk are derived. Our results are obtained within the framework of a perturbative expansion in the parameters that quantify the anisotropy of the fluid. Good agreement is found for moderate anisotropy when compared to associated results from finite-element simulations. At pronounced anisotropy, the induced flow fields are still predicted qualitatively correctly by the perturbative theory, although quantitative deviations arise. We hope to stimulate with our investigations corresponding experimental analyses, for example, concerning fluid flows in anisotropic thin films on uniaxially rubbed supporting substrates.