  We analyze the creeping flow generated by a spherical particle moving through
a viscous fluid with nematic directional order, in which momentum diffusivity
is anisotropic and which opposes resistance to bending. Specifically, we
provide closed-form analytical expressions for the response function, i.e. the
equivalent to Stokes's drag formula for nematic fluids. Particular attention is
given to the rotationally pseudo-isotropic condition defined by zero resistance
to bending, and to the strain pseudo-isotropic condition defined by isotropic
momentum diffusivity. We find the former to be consistent with the rheology of
biopolymer networks and the latter to be closer to the rheology of nematic
liquid crystals. These "pure" anisotropic conditions are used to benchmark
existing particle tracking microrheology methods that provide effective
directional viscosities by applying Stokes's drag law separately in different
directions. We find that the effective viscosity approach is phenomenologically
justified in rotationally isotropic fluids, although it leads to significant
errors in the estimated viscosity coefficients. On the other hand, the mere
concept of directional effective viscosities is found to be misleading in
fluids that oppose an appreciable resistance to bending. Finally, we observe
that anisotropic momentum diffusivity leads to asymmetric streamline patterns
displaying enhanced (reduced) streamline deflection in the directions of lower
(higher) diffusivity. The bending resistance of the fluid is found to modulate
the asymmetry of streamline deflection. In some cases, the combined effects of
both anisotropy mechanisms leads to streamline patterns that converge towards
the sphere.
