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In the current work we propose an efficient and accurate numerical approach to the problem of electrical characterization of Nematic Liquid Crystal (N-LC) microstructures under the influence of low-frequency AC electric fields. N-LCs are anisotropic and their electrical properties are determined by the directors' tilt angles which in turn depend on the applied electric field. Therefore, the problem is governed by a coupled system of two-dimensional PDEs: A Poisson equation with variable coefficients for the electric potential and a highly nonlinear second-order equation for the tilt angle of the directors. Both equations are solved using finite-difference schemes with relaxation and the results are found to be in good agreement with the literature. Various 2-D geometries are considered and it is shown that a low DC voltage is sufficient to tune the average refractive index of the N-LC structures under consideration.

In the current work, the nonlinear problem of electromagnetic wave propagation in a Nematic Liquid Crystal (NLC) cell is solved numerically. The LC is sandwiched between two glass layers of finite thickness and a linearly polarized beam is obliquely incident to the cell. The dielectric properties of N-LCs depend on the tilt angle of the directors. When the excitation beam enters the cell, and providing the incident intensity is above the Freedericksz threshold, the directors reorient themselves changing the LC's relative permittivity tensor. In turn, this affects beam propagation throughout the crystal. The electromagnetic field is modeled by the time-harmonic Maxwell equations whereas the director field is governed by a nonlinear ordinary differential equation (ODE). Our solution method is iterative, consistently taking into account this interaction between the excitation beam and the director field. The Maxwell equations are solved employing the Mode-Matching Technique (MMT). The solution of the nonlinear differential equation for the director field is obtained with the aid of a finite difference (FD) scheme.

In the current work, we develop a numerical method suitable for treating the problem of nonlinear two‐dimensional flows in rectangular domains. For the spatial approximation we employ the Fourier‐Galerkin approach. More specifically, our basis functions are products of trigonometric and Beam functions. This choice means that the solutions automatically satisfy the boundary and periodic conditions in the x and y directions respectively.The accuracy of the method is assessed by applying it to model problems which admit exact analytical solutions. The numerical and analytic solutions are found to be in good agreement. The convergence rate of the spectral coefficients is found to be fifth‐order algebraic in the x‐direction and y‐direction, confirming the efficiency and speed of our technique.

In this work, we investigate the 2D flow in a rectangular cavity subject to both vertical and horizontal temperature gradients. The linearized model is studied and the effect of thermal relaxation, as described by the Maxwell‐Cattaneo law of heat conduction is examined. To this end, a spectral numerical model is created based on a Galerkin expansion. The basis is the Cartesian product of systems of beam functions and trigonometric functions.The natural modes of the system are derived for both the Fourier and non‐Fourier models. The results are compared to earlier works for the plain Fourier law. Our computations show that for the same set of parameters, the Maxwell‐Cattaneo law yields modes which are quantitatively different from the Fourier. It is found that the real parts of the eigenvalues increase with the Straughan number Sg, which quantifies the non‐Fourier effects. This confirms the destabilizing effect of the MC‐law on the convective flow.

We study the effect of the Maxwell‐Cattaneo law of heat conduction (MCHC) on the 1D flow in a vertical slot subject to both vertical and horizontal temperature gradients. The gravitational acceleration is allowed to oscillate, which provides an opportunity to investigate the quantitative contribution of thermal inertia as epitomized by MCHC. The addition of the time derivative in MCHC increases the order of the system. We use a spectral expansion with Rayleigh’s beam functions as the basis set, which is especially suited to fourth order boundary value problems (BVP). We show that the time derivative (relaxation of the thermal flux) has a dissipative nature and leads to the appearance of purely real negative eigenvalues. Yet it also increases the absolute value of the imaginary part and decreases the absolute value of the real part of the complex eigenvalues. Thus, the system has a somewhat more oscillatory behavior than the one based on Fourier’s heat conduction law (FHC).

In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonUnear problems. We make use of the formidas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10^'^ for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10^'. The foeam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former