Abstrakt
In this paper, we present a generic energy-stable numerical scheme for a Smectic-A liquid crystal model. This model involves the hydrodynamic velocity-pressure variables (u, p) and the order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal given by the unitary vector n.
Our starting point is the model presented by E in [6] using the so-called layer variable phi such that n = del phi, where is a strongly non-linear parabolic system is considered coupling velocity and pressure of the Navier-Stokes equations (u, p) with a fourth order parabolic equation for phi. We give a reformulation as a mixed second order problem which let us to define 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 model.