From kinetic to fluid models of liquid crystals by the moment method

Degond, Pierre; Frouvelle, Amic; Liu, Jianguo

doi:10.3934/krm.2021047

Cited by 2 publications

(1 citation statement)

References 64 publications

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“…By the Hilbert expansion, Li and Wang [27] rigorously justified the connection between the inertial Qian-Sheng theory and the full Ericksen-Leslie theory. Based on the moment method, Degond, Frouvelle and Liu [28] presented the convergence of the Doi-Onsager theory to the Ericksen-Leslie theory. These works were mainly concerned with the hydrodynamic theory of nematic phases.…”

Section: Tensor Theorymentioning

confidence: 99%

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

Feng

2022

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The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were derived from those in the molecular model. The energy dissipation law was inherited from the tensor model. Some special cases for the model were also discussed.

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Section: Tensor Theorymentioning

confidence: 99%

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

Feng

2022

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From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

Jiang,

Luo,

Zhang

2024

Math. Models Methods Appl. Sci.

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We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

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From kinetic to fluid models of liquid crystals by the moment method

Cited by 2 publications

References 64 publications

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

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