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arXiv · 1411.5401

Approximation of Smectic-A liquid crystals

Francisco Guillén-González·Giordano Tierra
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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 $ϕ$ such that ${\bf n}=\nabla ϕ$ 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 $({\bf u},p)$ 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 reaches an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.

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BibTeXRIS

Francisco Guillén-González, Giordano Tierra. 2014-11-19. Approximation of Smectic-A liquid crystals. https://doi.org/10.1016/j.cma.2015.03.015

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