  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.
