Nonstationary models for liquid crystals: A fresh mathematical perspective

Abstract

In this article we discuss nonstationary models for inhomogeneous liquid crystals driven out of equilibrium by flow. Emphasis is put on those models which are used in the mathematics as well as in the physics literature, the overall goal being to illustrate the mathematical progress on popular models which physicists often just solve numerically. Our discussion includes the Doi-Hess model for the orientational distribution function, the Q-tensor model and the Ericksen-Leslie model which focuses on the director dynamics. We survey particularly the mathematical issues (such as existence of solutions) and linkages between these models. Moreover, we introduce the new concept of relative energies measuring the distance between solutions of equation systems with nonconvex energy functionals and discuss possible applications of this concept for future studies.Nonstationary models for liquid crystals: A fresh mathematical perspective physicists, chemists, material scientists, and even (applied) mathematicians. This interest is recently also triggered by the important role of liquid-crystal physics in the fields of biophysics (e.g., for the structure of the cytoskeleton or the movement between actin and myosin, see Ahmadi et al. [6]), in active matter Ref.[7] and in astrophysics (emergence of topological defects). Many of these contexts involve physical situations outside thermal equilibrium, where the material properties generally depend on time. The purpose of the present article is to summarize modeling approaches for such nonstationary (out-of-equilibrium) liquid crystals from both, a mathematical and a physical perspective.Clearly, the presence of orientational degrees of freedom makes the theoretical description of liquid crystals more complex than that of ordinary (atomic) fluids. This holds for microscopic ("bottom-up") approaches such as classical density functional theory (see Ref. [8,9]), but also for coarse-grained approaches such as phase-field crystal modeling (see Ref.[10]) and for mesoscopic (continuum) approaches involving appropriate order parameter fields (see Ref. [11]) or even macroscopic variables, such as a stress tensor (see Sec. 4). Such mesoscopic approaches have become particularly popular for the description of liquid crystals under flow, a situation of major relevance for many applications (see Ref. [12]). Mathematically, continuum approaches for nonstationary liquid crystals involve typically nonlinear coupled (partial) differential equations. While physicists just tend to solve these equations numerically and explore the emerging physical behavior, there are many open problems from the mathematical (and numerical) side concerning, e.g., the existence and uniqueness of solutions. From the physical side, this poses the danger of overseeing important dynamical features, while from the mathematical side, there is a certain risk to concentrate on too simplistic (or even unphysical) models.It is in this spirit that we here aim at giving an overview over some of the most relevant nonstationary mode...

Maximally dissipative solutions for incompressible fluid dynamics

“…The proposed solution concept is very general and may be applied to various kinds of problems, and we want to introduce the concept here for a general system, but the main idea is to apply it to the Navier-Stokes and Euler equations. But it can be applied in the sense of Definition 2.2 (below) to other systems featuring the relative energy inequality like systems in complex fluids like nematic liquid crystals [10], models in phase transition [23] or [22], or more generally GENERIC systems [14].…”

Existence of weak solutions to the Ericksen‐Leslie model for a general class of free energies

“…For a broader overview of results concerning the analysis of liquid crystal models, we refer to Lin and Liu, 24 Lin and Wang, 25 and to Emmrich et al 26 Recently, also the existence of measure-valued solutions has been established for more general models by the second author (see the work of Lasarzik 27 ) and results on the weak-strong uniqueness of weak solutions (see the work of Emmrich and Lasarzik 28 ) and of measure-valued solutions (see the work of Lasarzik 29 ) have been proved.…”

Flow and Nematic Director Profiles in a Microfluidic Channel: The Interplay of Nematic Material Constants and Backflow

“…Since L 2 1, the velocity profile is not influenced by the discontinuities and is still given by Equation (26).…”

“…In recent years, there are rigorous existence and regularity results for the Beris-Edwards framework too [11,14,23] and numerical simulations for microfluidic set-ups in [24,25]. The various dynamical theories of nematic liquid crystals and the key results are surveyed in [26] and in [27]; the authors rigorously derive the Ericksen-Leslie equations from the Beris-Edwards model. In [28], the authors use a lattice-Boltzmann algorithm to study nematodynamics in a microfluidic channel, in the Beris-Edwards framework, with both Dirichlet and mixed boundary conditions on the channel walls.…”