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

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-09-04 DOI:10.1137/23m1608902

Giulia Bevilacqua, Andrea Giorgini

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引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6057-6120, October 2024.
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 of 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. Second, 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.

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软物质物理学中具有二次锚定的两相复杂流体的全局解决方案

SIAM 数学分析期刊》，第 56 卷第 5 期，第 6057-6120 页，2024 年 10 月。摘要。我们研究了描述极性液晶乳液相分离过程中复杂流变学和界面动力学的扩散界面模型。更确切地说，该物理系统由浸入牛顿流体中的极性液晶两相混合物组成。这种复合材料是软物质中出现的复杂流体的典型例子，表现出多尺度的相互作用。除了浓度和极化的金兹堡-朗道弹性能和弗兰克弹性能之外，该系统的自由能还具有二次锚定项的特征，可调整界面上极化的方向。这导致在描述宏观动力学的结果系统中产生了几个准线性非线性耦合。在这项研究中，我们首次建立了关于具有界面锚定机制的两相复合流体全局动力学的数学结果。首先，我们确定了一组关于系统参数和初始条件的充分条件，这些条件保证了二维和三维全局弱解的存在。其次，我们证明了弱解在二维情况下是唯一和全局规则的。最后，我们用一些数值模拟来补充我们的分析，以显示极化和界面锚定。

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.