Approximation of Smectic-A liquid crystals

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 (u, p) 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 n. We start from the formulation given in [E’97] by using the so-called layer variable φ such that n = ∇φ and the level sets of φ 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 (u, p) with a fourth order parabolic equation for φ. 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 C 0 - finite elements in space. Finally, numerical simulations are presented for 2D-domains, showing the evolution of the system until it reachs an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.