Numerical solution of the Ericksen–Leslie dynamic equations for two-dimensional nematic liquid crystal flows

A finite difference method for solving nematic liquid crystal flows under the effect of a magnetic field is developed. The dynamic equations of nematic liquid crystals, given by the Ericksen–Leslie dynamic theory, are employed. These are expressed in terms of primitive variables and solved employing the ideas behind the GENSMAC methodology (Tomé and McKee, 1994; Tomé et al., 2002) [38] and [41]. These equations are nonlinear partial differential equations consisting of the mass conservation equation and the balance laws of linear and angular momentum. By employing fully developed flow assumptions an analytic solution for steady 2D-channel flow is found. The resulting numerical technique was then, in part, validated by comparing numerical solutions against this analytic solution. Convergence results are presented. To demonstrate the capabilities of the numerical method, the flow of a nematic liquid crystal through various complex geometries are then simulated. Results are obtained for L-shaped channels and planar 4:1 contraction for several values of Reynolds and Ericksen numbers.