Algorithm for solving the system of equations describing liquid crystal statics

I. V. Smolekho

Institute of Computational Modeling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russian Federation

Abstract: We present a mathematical model and algorithm for the numerical solution of the system of equations describing nematic liquid crystal statics. The model is derived from a simplified dynamic formulation within the acoustic approximation. The system includes two equations for pressure and shear stress describing translational motion; an equation for the rotation angle, whose right-hand side depends on shear stress (analogous to Hooke's law in elasticity); a heat conduction equation accounting for temperature distribution and the anisotropy caused by molecular orientation; and a system of determining equations for displacement, pressure, shear stress, temperature, and rotation angle.
The equations for pressure and shear stress satisfy the Cauchy–Riemann conditions, reducing the problem to a complex variable analysis. By further reducing it to a non-homogeneous singular integral equation, we applied the LU decomposition method for numerical solution. We used the Sokhotski–Plemelj theorem to impose boundary conditions. Based on this algorithm, we developed a MATLAB program and performed a series of test calculations. The results demonstrate the accuracy and efficiency of the proposed algorithm and implementation.

Keywords: liquid crystal, statics, complex analysis, Cauchy–Riemann conditions, Sokhotski–Plemelj theorem.