  Dynamics at low Reynolds numbers experiences recent revival in the fields of
biophysics and active matter. While in bulk isotropic fluids it is exhaustively
studied, this is less so in anisotropic fluids and in confined situations.
Here, we combine the latter two by studying the rotation of a disk-like
inclusion in a uniaxially anisotropy, globally oriented, incompressible
two-dimensional fluid film. In terms of a perturbative expansion in parameters
that quantify anisotropies in viscosity and in additional linear friction with
a supporting substrate or other type of confinement, we derive analytical
expressions for the resulting hydrodynamic flow and pressure fields as well as
for the resistance and mobility coefficients of the rotating disk. It turns out
that, in contrast to translational motion, the solutions remain well-behaved
also in the absence of the additional linear friction. Comparison with results
from finite-element simulations show very good agreement with those from our
analytical calculations. Besides applications to describe technological
systems, for instance, in the area of microfluidics and thin cells of aligned
nematic liquid crystals, our solutions are important for quantitative
theoretical approaches to fluid membranes and thin films in general featuring a
preferred direction.
