Numerical Approximation of Nematic Fluids

Friday, January 19, 2018 - 9:00am - 9:50am
Lind 305
Noel Walkington (Carnegie Mellon University)
This talk focuses on the issues that arise when modeling and
simulating fluids containing rod--like molecules (nematics). The
(average) orientation of these fluids is typically modeled by a unit
vector field which complicates both the analysis and numerical
solution of these equations. In particular,

i) The unit length constraint gives rise to topological
singularities. While singularities are observed ubiquitously in
liquid crystals, classical models assign infinite elastic
energy to these configurations.

ii) In a numerical context, imposing the unit length constraint can
lead to locking; that is, too few polynomial functions exist
which satisfy the constraint.

iii) The the unit length constraint is frequently relaxed to
accommodate the formation of singularities with finite energy or
isotropic regions. However, there is not definitive physical
principle to determine the evolution of the phases.

iv) The head--to--tail symmetry of the nematic molecules allows them
to form non--orientable direction fields and degree half
singularities, so director take values in real projective space.

v) Currently there is no St. Venant principle guaranteeing
that the flow and director configuration away from a singularity is
not sensitive to the fine structure of the core.

The development of numerical schemes in this context will be discussed.