A projection‐based time‐splitting algorithm for approximating nematic liquid crystal flows with stretching

Abstract

A numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time-splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection-based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal-order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

A Finite Element Approximation for Nematic Liquid Crystal Flow with Stretching Effect Based on Nonincremental Pressure-Correction Method

“…Moreover, we give the evolution of the director field and velocity field of a rotating flow. Clearly, the annihilation time is much smaller than that obtained in reference [ 11 ]. Finally, the convergence of the numerical solution in time and space is analyzed.…”

“…Additionally, parameter is a constant that determines the geometry of the molecule. For instance, when , the molecules are rod-shaped, spherical, and disk-shaped [ 10 , 11 , 18 ], respectively.…”

“…We consider the initial conditions of the nematic liquid crystal with stretching effect ( 1 )–( 3 ) are where . This example was also used to research nematic liquid crystal in [ 5 , 6 , 10 , 11 , 13 , 22 ]. In this experiment, we choose computational domain , time step size , and use a grid in the computation.…”

“…Ref. [ 11 ] used finite element method to obtain spatial discretization and the time-splitting method based on a nonincremental velocity correction projection scheme [ 8 ] to decouple variables. The literature in [ 12 ] used a spectral method to study this kind of problem.…”

A Finite Element Algorithm for Nematic Liquid Crystal Flow Based on the Gauge-Uzawa Method

“…A fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, has been shown in [12]. Besides, Cabrales et al [6,7] have presented a projection-based time-splitting algorithm, where the velocity and pressure are computed by using a projection-based algorithm and the director is computed jointly to an auxiliary vari-able. An accurate and efficient Legendre-Galerkin method which can preserve energy law in discrete form has been designed [35].…”