  Capillary microflows of liquid crystal phases are central to material,
biological and bio-inspired systems. Despite their fundamental and applied
significance, a detailed understanding of the stationary behaviour of nematic
liquid crystals (NLC-s) in cylindrical capillaries is still lacking. Here,
using numerical simulations based on the continuum theory of Leslie, Ericksen
and Parodi, we investigate stationary NLC flows within cylindrical capillaries
possessing homeotropic (normal) and uniform planar anchoring conditions. By
considering the material parameters of the flow-aligning NLC, 5CB, we report
that instead of the expected, unique director field monotonically approaching
the alignment angle over corresponding Ericksen numbers (dimensionless number
capturing viscous v/s elastic effects), a second solution emerges below a
threshold flow rate (or applied pressure gradient). We demonstrate that the
onset of the second solution, a nematodynamic bifurcation yielding
energetically degenerate director tilts at the threshold pressure gradient, can
be controlled by the surface anchoring and the flow driving mechanism
(pressure-driven or volume-driven). For homeotropic surface anchoring, this
alternate director field orients against the alignment angle in the vicinity of
the capillary center; while in the uniform planar case, the alternate director
field extends throughout the capillary volume, leading to reduction of the flow
speed with increasing pressure gradients. While the practical realization and
utilization of such nematodynamic bifurcations still await systematic
exploration, signatures of the emergent rheology have been reported previously
within microfluidic environments, under both homeotropic (Sengupta et al.,
Phys. Rev. Lett. 110, 048303, 2013) and planar anchoring conditions (Sengupta,
Int. J. Mol. Sci. 14, 22826, 2013).
