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Colloidal Homogenisation for the Hydrodynamics of Nematic Liquid Crystals

Francesco De Anna, Anja Schloemerkemper, Arghir Zarnescu

Abstract

This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg-Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles, which impose no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenisation problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenisation process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system's energy. We address this challenge with a novel variation of the Aubin-Lions lemma, specifically adapted for homogenisation problems.

Colloidal Homogenisation for the Hydrodynamics of Nematic Liquid Crystals

Abstract

This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg-Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles, which impose no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenisation problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenisation process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system's energy. We address this challenge with a novel variation of the Aubin-Lions lemma, specifically adapted for homogenisation problems.
Paper Structure (11 sections, 11 theorems, 146 equations)

This paper contains 11 sections, 11 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. An overview of the analysis of static LC colloids
  3. Homogenisation of Stokes and Navier-Stokes equations
  4. Challenges of homogenisation for complex fluids and our contributions
  5. Main result
  6. A preliminary toolbox of homogenisation
  7. Aubin-Lions Lemma Revisited: a Homogenisation Extension
  8. The extension of the pressure
  9. The Darcy law
  10. Homogenisation of the director equation
  11. A compactness-like result

Key Result

Theorem 2.1

Consider a family of initial data $d_{\varepsilon, \rm in}$ as in in-cdt with $H^1(\Omega_\varepsilon)$-norms uniformly bounded in $\varepsilon>0$. Assume that the extensions $\tilde{d}_{\varepsilon, \rm in}$ weakly-$\ast$ converges to a vector function $\theta\,d_{\rm in}$ in $L^\infty(\Omega)$, wi Define the two-dimensional vector field $G \in L^2((0,T)\times \Omega)$ with components Then the f

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 13 more