Dynamics and steady state of squirmer motion in liquid crystal

TL;DR

This paper rigorously proves the existence of steady states and finite-time solutions for a nonlinear PDE system modeling microswimmer motion in liquid crystals, and derives a reduced model for collective behavior.

Contribution

It provides the first rigorous mathematical analysis of squirmer dynamics in liquid crystals, including steady state existence and a reduced collective model.

Findings

Existence of steady states for the coupled PDE system.

Finite-time existence of solutions to the time-dependent problem.

Derivation of a reduced model for collective swimming behavior.

Abstract

We analyze a nonlinear PDE system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer's motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer's surface. The liquid crystal is described using the well-established Beris-Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem. Finally, using a two-scale asymptotic expansion we derive a reduced model for the collective swimming of squirmers as they reach their steady state orientation and speed.