Approximation of Smectic-A liquid crystals

TL;DR

This paper develops energy-stable numerical schemes for a complex Smectic-A liquid crystal model involving coupled hydrodynamics and microscopic order parameters, and demonstrates their effectiveness through 2D simulations.

Contribution

It introduces the first numerical analysis for a coupled hydrodynamic and microscopic model of Smectic-A liquid crystals using energy-stable schemes.

Findings

Numerical schemes successfully simulate the evolution to equilibrium.

Second order finite differences and $C^0$-finite elements ensure stability.

Simulations illustrate the model's behavior in 2D domains.

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 $({\bf u},p)$ 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 ${\bf n}$. We start from the formulation given in \cite{E} by using the so-called layer variable $\phi$ such that ${\bf n}=\nabla \phi$ and the level sets of $\phi$ 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 $({\bf u},p)$ with a fourth order parabolic equation for $\phi$. 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 reaches an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.