Global Solutions for Two-Phase Complex Fluids with Quadratic Anchoring in Soft Matter Physics
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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Acknowledgments.
The authors wish to thank Francesco Ballarin, Andrea Cianchi, and Cristiana De Filippis for fruitful discussions, and the two reviewers for helpful comments that improved the quality of our paper.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 6057 - 6120
DOI: 10.1137/23M1608902
ISSN (online): 1095-7154
Copyright
© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 12 October 2023
Accepted: 24 May 2024
Published online: 4 September 2024
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Funding Information
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM/GNAMPA): CUP E53C23001670001
Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR): CUP I57G22000700001
Funding: The first author was supported by the European Research Council (ERC), under the European Union’s Horizon 2020 research and innovation program, through the project ERC VAREG - Variational Approach to the Regularity of the Free Boundaries (grant agreement 853404) and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM) through the INdAM-GNAMPA project 2024 CUP E53C23001670001. The first author also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001. The second author was supported by the MUR grant Dipartimento di Eccellenza 2023-2027 of Dipartimento di Matematica, Politecnico di Milano, and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale per l’Alta Matematica (INdAM).
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