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.

Introduction

Liquid crystals (LCs) are anisotropic fluids in which the orientation of the rigid molecules they comprise are affected by the history of the flow; all the more so since most of the properties of the LC will greatly depend on the fluctuations of the unit vector field (the so-called directors $\text{d}$) which represents the different average orientation of the particles. Furthermore, the actual coupling between flow and particle orientation plays in both directions, according to rules not yet ascertained. Typically, LC molecules (like the extensively studied pentyl-cyano-biphenyl, or 5CB molecule) possess moderately high aspect ratios, and liquid crystalline polymers (LCPs) have rigid units that can behave as rodlike particles, with extremely high aspect ratios.

Along with the Leslie–Ericksen theory [1], [2], [3] and the Landau-de Gennes model (see, e.g. [4]), the Hess–Doi model ([5], [6]) has been widely employed to simulate complex flows of LCPs. In the Hess–Doi model, the LCP is envisioned as an ensemble of particles with infinite aspect ratio (rigid rods) that undergo Brownian motion and interact via a self-consistent mean-field potential. However, the complexity of the problem demands further simplifications, like reducing the dimensions of the configuration space or employing simplified expressions for the rotary diffusivity and the mean-field potential.

Using the Hess–Doi model, remarkable insight has been gained on the behavior of LCPs in pure shear homogeneous flows [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], Poiseuille flow (see, e.g. [19], [20]) and pressure-driven flows [21]; however, complex flows have received comparatively little attention, possibly due to the computational cost of accurately reflecting the historical effects and the flow modification induced by the so-altered particle orientations. An exception is the work by Feng and Leal [22], who investigated the start-up of LCPs according to the Hess–Doi model in an eccentric cylinder geometry.

In order to avoid a direct simulation of the distribution function appearing in the Hess–Doi kinetic model, it is a common approach to combine the macroscopic and the statistical models in some form of closure approximation [9], [10], [11], [16], [18]. Since closure approximations can severely alter the time evolution of the alignment tensor, a wrong choice could lead to inconsistencies with the kinematic model they were based on [23], or produce macroscopic results qualitatively different from those predicted by the distribution function. In this respect, the recent effort by Kröger and co-workers [24] should definitely help in the election of a statistically consistent closure approximation. An investigation of appropriate closure approximations and their possible qualitative impact on the results in the eccentric cylinder geometry can be found in [25], [26]. As an alternative, hybrid methods propose a simultaneous resolution of both the macroscopic equations (Navier–Stokes equations with a source term due to the polymer) and the diffusion equations for the particle orientations.

Thus, the two main families of methods used nowadays attempt either to solve the diffusion equation in configuration space (Fokker–Planck), or to couple discretization techniques with a direct solution of the associated stochastic differential equation for a large ensemble of trajectories. The direct solution of the Fokker–Planck equation is arguably superior in terms of convergence for small numbers of degrees of freedom in homogeneous fields, if the proper trial functions are chosen; spherical harmonics have been traditionally employed [7], but recently, Suen et al. [27] presented the wavelet expansion as a sound alternative. As to the second class of methods, explored in [28], [29], [30], they are still rather competitive when complex flows and precise, rheological information come into scene (see, e.g. the review by Keunings on micro–macro methods for viscoelastic fluids for a more detailed exposition [31]).

Following this latter approach, we intend to devise an efficient and accurate micro–macro method to simulate complex flows of LCPs, with a focus on the solution of the fully coupled problem, and not so much about how the LCP reacts to given velocity fields. The purpose is threefold: (1) to introduce a general method for the simulation of LCPs, assessing its capabilities and drawbacks in the context of complex LCPs flows; (2) to address the discrepancies or concordances between the values predicted by this method and other solution methods such as closure approximations or the spherical harmonics technique for the rheological variables of interest, namely, the scalar order parameter S or the director vector $\text{d}$; and (3) to study the effect of the LCP over a macroscopic, quasi-Newtonian Stokes flow.

The present paper is organized as follows. Section 1 serves as an introduction to the topic of LCPs and their simulation. The mathematical description of the problem is exposed in Section 2. Section 3 provides numerical details on the algorithm, including a succinct description of the micro–macro method used. In Section 4 we show results for the eccentric cylinder geometry, comparing them with those of previous works and discussing the discrepancies that might arise. Finally, Section 5 summarizes the main objectives and conclusions of this work.