Stochastic semi-Lagrangian micro–macro calculations of liquid crystalline solutions in complex flows

Abstract

A general method for the simulation of complex flows of liquid crystalline polymers (LCPs) using a stochastic semi-Lagrangian micro–macro method is introduced. The macroscopic part uses a spatial-temporal second order accurate semi-Lagrangian algorithm, where ideas from the finite element and natural element methods are mixed in order to compute average quantities. The microscopic part employs a stochastic interpretation of the Doi–Hess LCP model, which is discretized with a second order Richardson extrapolated Euler–Maruyama scheme. The new method is validated and tested using the benchmark problem of flow between rotating eccentric cylinders. In a decoupled analysis, a discussion on the sensibility of the scalar order parameter to the macroscopic flow is offered. For the coupled situation, the proposed method predicts disclinations at certain regions of the geometry, as well as an accentuated abatement of the flow as the strength of the micro–macro interaction increases. Further examples are provided at different Peclet and concentration numbers to gain insight on the behavior of complex flows of LCPs in the eccentric cylinder geometry. The generality and robustness of the method, as well as its accurate prediction of LCP behavior under complex flows are main features of the implementation.