Approximation of Smectic-A liquid crystals
arXiv:1411.5401 · doi:10.1016/j.cma.2015.03.015
Abstract
In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector . We start from the formulation given in \cite{E} by using the so-called layer variable such that and the level sets of describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations with a fourth order parabolic equation for . We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and -finite elements in space. Finally, numerical simulations are presented for -domains, showing the evolution of the system until it reaches an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.
19 pages