Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

On a novel approach for modeling liquid crystalline flows

Pages: 359 – 378

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n2.a4

Author

Stefan Metzger (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)

Abstract

In this paper, we derive a new model for the description of liquid crystalline flows. While microscopic Doi type models suffer from the high dimensionality of the underlying product space, the more macroscopic Ericksen–Leslie-type models describe only the long-time behavior of the flow and are valid only close to equilibrium. By applying an energetic variational approach, we derive a new macroscopic model which shall provide an improved description far from equilibrium. The novelty of our approach lies in the way the energy is minimized. Distinguishing between the velocities of particles and fluid allows us to define the energy dissipation not in terms of chemical potentials but in terms of friction induced by the discrepancies in the considered velocities. We conclude this publication by establishing the existence of weak solutions to the newly derived model.

Keywords

nematic flow, Ericksen–Leslie, Navier–Stokes, energetic variational approach, existence of weak solutions, non-Newtonian fluids

2010 Mathematics Subject Classification

35Q35, 76A05, 76D03, 76D05

This work was supported by the NSF through grant number NSF-DMS 1759536.

Received 2 May 2019

Accepted 5 October 2019

Published 11 May 2020