Global Solutions for Two-Phase Complex Fluids with Quadratic Anchoring in Soft Matter Physics

TL;DR

This paper establishes the first mathematical results on the global dynamics of two-phase complex fluids with quadratic interfacial anchoring, including existence, uniqueness, and regularity of solutions, supported by numerical simulations.

Contribution

It provides the first rigorous analysis of global solutions for two-phase complex fluids with interfacial quadratic anchoring in soft matter physics.

Findings

Existence of global weak solutions in 2D and 3D.

Uniqueness and regularity of solutions in 2D.

Numerical simulations illustrating polarization and anchoring effects.

Abstract

We study a diffuse interface model describing the complex rheology and the interfacial dynamics during phase separation in a polar liquid-crystalline emulsion. More precisely, the physical systems comprises a two-phase mixture consisting in a polar liquid crystal immersed in a Newtonian fluid. Such composite material is a paradigmatic example of complex fluids arising in Soft Matter which exhibits multiscale interplay. Beyond the Ginzburg-Landau and Frank elastic energies for the concentration and the polarization, the free energy of the system is characterized by a quadratic anchoring term which tunes the orientation of the polarization at the interface. This leads to several quasi-linear nonlinear couplings in the resulting system describing the macroscopic dynamics. In this work, we establish the first mathematical results concerning the global dynamics of two-phase complex fluids with interfacial anchoring mechanism. First, we determine a set of sufficient conditions on the parameters of the system and the initial conditions which guarantee the existence of global weak solutions in two and three dimensions. Secondly, we show that weak solutions are unique and globally regular in the two dimensional case. Finally, we complement our analysis with some numerical simulations to display polarization and interfacial anchoring.