Special Session 120: Mixtures: Modeling, analysis and computing

Finite element methods for electroneutral multicomponent convection-diffusion

Aaron Baier-Reinio University of Oxford England Co-Author(s):

Abstract: We present novel high-order finite element methods for the coupled Navier--Stokes and Onsager--Stefan--Maxwell equations, which model bulk momentum transport, multicomponent cross-diffusion and electrical effects within mixtures. We discuss some salient properties of our numerical schemes and discuss challenges that arise when discretising the governing equations. Our approach naturally extends to handle local electroneutrality when the species carry electrical charge, making our methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the flow in a Hull cell of a cosolvent electrolyte mixture used in lithium-ion batteries.

Structure-preserving approximation for the quasi-incompressible Maxwell-Stefan-Navier-Stokes system

Aaron Brunk Johannes-Gutenberg University Mainz Germany Co-Author(s):    Ansgar Juengel, Maria Lukacova-Medvidova

Abstract: In this talk, we present a conforming finite element method for the quasi-incompressible Navier--Stokes--Maxwell--Stefan system. The model couples the Navier--Stokes equations with a quasi-incompressibility constraint and a cross-diffusion Maxwell--Stefan system describing multicomponent mass transport. \noindent Our scheme combines a mixed explicit--implicit time discretization with a spatially conforming finite element approximation, ensuring the preservation of partial masses, strict enforcement of the quasi-incompressibility condition, and dissipation of a discrete analogue of the total energy. Furthermore, we present an a priori error analysis of the fully discrete scheme based on the relative entropy method. The proof leverages the structure-preserving properties of the discretization, including discrete energy dissipation and the control of the cross-diffusion terms. Numerical experiments in two space dimensions complement the analysis. They illustrate the predicted convergence rates, confirm the robustness of the error estimate, and demonstrate the correct qualitative behaviour of multicomponent flows, including mass conservation and energy decay. The results show that the proposed scheme provides a reliable and physically consistent computational framework for the simulation of quasi-incompressible multicomponent fluid mixtures. If time permits we will discuss extensions to the non-isothermal case. This is joint work with Ansgar J\ungel and Maria Lukacova-Medvidova

Stability of the rest state for some models of viscous chemically reacting fluids

Miroslav Bul\\'{\\i}\\v{c}ek Charles University Czech Rep Co-Author(s):

Abstract: We analyse a model for heat-conducting, chemically reacting fluid mixtures with cross-effects and discuss the stability of the rest state. Our main goal is to provide a rigorous stability result at the level of weak solutions. The main idea is to follow the relative energy and entropy framework to construct a suitable Lyapunov-like functional, specifically designed for the considered fluid system.

Multicomponent flow in domains with moving boundary: discretizations and numerical simulations

Lenka Ko\\v{s}\\'{a}rkov\\'{a} Charles University Czech Rep Co-Author(s):    Jaroslav Hron, Patrick Farrell

Abstract: We consider a multicomponent flow in which diffusion is described either by Fick`s law or by the Stefan-Maxwell equations, where the latter one also allows to capture cross-diffusive effects. We account for the motion of the domain and reformulate the equations in the Arbitrary Lagrangian-Eulerian framework. We then discuss various discretizations of the governing equations under different boundary conditions and present numerical simulations of the system in some simplified settings.

Chemically reacting mixtures: asymptotic stability of steady solutions

Tobi\\'{a}\\v{s} Krupa Charles University Czech Rep Co-Author(s):    Miroslav Bul\`{i}\v{c}ek

Abstract: In many applications, we are interested in mixtures in which chemical reactions occur. There is a well-known theory of mixtures that can capture this effect in its equations. However, the terms describing the interchange of mass between the constituents are often quite complicated and nonlinear. One important particularly studied aspect is the equilibrium state, especially in cases involving multiple independent chemical reactions. In such cases, we study the existence, uniqueness, and, most importantly, the stability of the steady solution to the coupled system of equations for the velocity field and the concentrations of the constituents equipped with various boundary conditions. The analysis is performed within the framework of mixture models, where the mixture moves collectively together at a single barycentric velocity, while the partial fluxes of the constituents are modeled at the constitutive level.

Existence analysis of an evolutionary compressible fluid model for heat-conducting and chemically reacting mixtures

Milan Pokorny Charles University Czech Rep Co-Author(s):    Miroslav Bulicek, Ansgar Juengel, Nicola Zamponi

Abstract: The existence of large-data weak solutions to the evolutionary compressible Navier--Stokes--Fourier system for chemically reacting fluid mixtures is proved. General free energies are considered satisfying some structural assumptions, with a pressure containing a $\gamma$-power law. The model is thermodynamically consistent and contains the Maxwell--Stefan cross-diffusion equations in the Fick--Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the pressure in $L^p$ with $p>1$, the effective viscous flux identity, and uniform bounds related to Feireisl`s oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

Modeling metamaterials by second-order rate-type constitutive relations between only the macroscopic stress and strain

Vit Prusa Charles University Czech Rep Co-Author(s):

Abstract: We discuss a novel thermodynamically based approach for constructing effective rate-type constitutive relations describing finite deformations of \emph{metamaterials}. The effective constitutive relations are formulated as \emph{second-order} in time rate-type Eulerian constitutive relations between only the Cauchy stress tensor, the Hencky strain tensor, and objective time derivatives thereof. In particular, there is no need to introduce additional quantities or concepts such as micro-level deformation, micromorphic continua, enriched continua, or elastic solids with frequency dependent material properties. The linearisation of the proposed fully nonlinear (finite deformations) constitutive relations leads, in Fourier space, to the same dispersion relations as those commonly used in metamaterial theories based on the concepts of frequency dependent density and/or stiffness. From this perspective the proposed constitutive relations reproduce the behaviour predicted by the frequency dependent density and/or stiffness models, but yet they work with constant --- that is motion independent --- material properties. The behaviour predicted by the linearised models is documented through numerical experiments exploiting a recently proposed dispersion-relation preserving discretisation. Finally, we argue that the proposed fully nonlinear (finite deformations) second-order in time rate-type constitutive relations do not fall into the traditional classes of models for elastic solids (hyperelastic solids/Green elastic solids, first-order in time hypoelastic solids), and that the proposed constitutive relations embody a new class of constitutive relations characterising elastic solids.

Homogenization Limits for Compressible Liquid-Vapour Flow with Phase Transition

Christian Rohde University of Stuttgart Germany Co-Author(s):    Jens Keim, Florian Wendt

Abstract: In the lecture we consider mathematical models for the evolution of two-phase fluids in free flow and porous media settings addressing diffuse-interface approaches. A special focus will be on Navier-Stokes-Korteweg (NSK) systems that describe on a detailed scale compressible liquid-vapour fluids which account for phase transition. We present rigorous results for homogenization in periodic porous media that lead to a Cahn-Hilliard-type evolution equation on the Darcy scale. Furthermore we discuss the upscaling of NSK motion for highy oscillating initial data on the corresponding Baer-Nunziato sacle. The latter limit results in a mixed type continuum-mechanical/kinetic evolution system.

Convergent numerical methods for viscoelastic fluid models

Dennis Trautwein University of Regensburg Germany Co-Author(s):    Endre Suli

Abstract: In this talk, we review a class of energy-stable numerical methods for the viscoelastic Oldroyd-B and Giesekus models. The model couples the incompressible Navier--Stokes equations with an evolution equation for an additional stress tensor accounting for elastic effects. This coupled evolution equation models transport and nonlinear relaxation effects and is usually stated in terms of the elastic deformation gradient, the conformation tensor, or the log-conformation approach. In the existing literature, numerical schemes for such models often suffer from accuracy limitations and convergence problems, usually due to the lack of rigorous existence results or inherent limitations of the discretization. The core of this presentation introduces a novel convergence result. We prove the (subsequence) convergence of a proposed numerical method to a large-data global weak solution of the Giesekus model in two dimensions. Crucially, this result is achieved without the use of artificial cut-offs or regularization in the limit system, providing a constructive alternative to the existence proof by Bul\`{i}\v{c}ek et al.~(Nonlinearity, 2022). Finally, we demonstrate the robustness of the method through numerical experiments, including convergence rate studies and typical benchmark problems.

A multiwell phase-field model for arbitrarily strong total-spreading case

Karel Tuma Faculty Of Mathematics And Physics, Charles University Czech Rep Co-Author(s):    J. Kozl\`{i}k, O.Sou\v{c}ek, J. Dobrza\`{n}ski, S. Stupkiewicz

Abstract: Motivated by the $\beta$--$\omega$ transformation in titanium alloys, we study the total-spreading regime in which two $\omega$ variants can not form a direct interface and must remain separated by the parent $\beta$ phase. Building on the classical multiwell framework and the consistency requirements discussed by Boyer and Lapuerta (2006), we show that the standard model works only in a limited parameter range and may fail under strong chemical driving force. To overcome this limitation, we propose a modified multiwell phase-field model with an additional interfacial-energy term that penalizes mixed $\omega$--$\omega$ states and enforces separation of $\omega$ variants by the $\beta$ phase. At the same time, the model preserves the desired behavior of purely two-phase $\beta$--$\omega$ states. Numerical examples in one spatial dimension show proper separation of four $\omega$ variants by the $\beta$ phase and monotone decay of the total free energy. Two- and three-dimensional simulations further show the formation of thin $\beta$ layers at contacts between variants and the subsequent coarsening of the microstructure. The model provides a natural starting point for future extensions with a more realistic chemical background.

Bulk-Surface Coupling in Brain Tissue: A Multiscale Mixture Model for Amyloid-beta

Lucie Wintrova Mathematical Institute of Charles University Czech Rep Co-Author(s):    Marie Rognes, Miroslav Bulicek

Abstract: Transport and clearance of amyloid-$\beta$ in the brain tissue involve coupled diffusion-reaction processes in the interstitial bulk and receptor-mediated interactions on cellular interfaces, leading naturally to a bulk-surface system. These mechanisms operate across strongly separated spatial scales: microscopic cellular geometries determine local transport behavior, whereas clinically relevant observations concern macroscopic tissue domains. We formulate a microscale class I mixture model posed on a periodically perforated domain, incorporating bulk diffusion, nonlinear surface reactions, and interfacial exchange. Using two-scale convergence techniques, we rigorously derive an effective macroscopic model that retains explicit dependence on the underlying microstructure through effective transport and reaction coefficients. The analysis establishes a systematic link between microscopic structure and macroscopic transport laws, yielding a model suitable for large-scale simulation.

Multiphase cross-diffusion models for tissue structures

Sara Xhahysa TU Wien Austria Co-Author(s):    Ansgar J\ungel, Cordula Reisch

Abstract: Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell--Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak--strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

Richards flow in porous media with cross diffusion

Nicola Zamponi University of Augsburg Germany Co-Author(s):    Esther S. Daus, Josipa-Pina Mili\v{s}i\`c

Abstract: We study a non-isothermal simplified two-phase air-water flow in porous media, where the water phase is modelled as a mixture of N chemical components. The governing system consists of mass conservation equations for each component, an energy balance equation, and a capillary pressure relation. The model is thermodynamically consistent and incorporates cross-diffusion effects arising from multicomponent interactions. Our main result establishes the sequential stability of weak variational entropy solutions. The analysis relies on a priori estimates derived from the entropy balance and the total energy balance, together with a dynamic capillary pressure law. Compactness arguments are carried out using the Div-Curl lemma, which allows us to pass to the limit in the nonlinear terms.

Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential.

Margherita Zanella Politecnico di Milano Italy Co-Author(s):    Andrea di Primio, Luca Scarpa

Abstract: In this talk we present the study of the long-time behaviour of a stochastic Allen-Cahn-Navier-Stokes system. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically relevant Flory-Huggins logarithmic potential. We first show existence of ergodic invariant measures. Secondly, we prove that if the noise acting in the Navier-Stokes equation is nondegenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure which is asymptotically stable with respect to a suitable Wasserstein metric. The talk is based on a joint work with A. Di Primio and L. Scarpa.

On the low Mach number limit for compressible two-fluid system

Ewelina Zatorska University of Warwick England Co-Author(s):    Yang Li, Nilasis Chaudhuri

Abstract: In this talk I will focus on two results justifying the low Mach Number limit for the compressible two-fluid system in the regime of strong and weak solutions. In the framework of local-in-time strong solutions, we prove that, for well-prepared initial data, solutions to the rescaled compressible two-fluid system exist on a time interval independent of the Mach number and converge to the solution of the incompressible Navier--Stokes equations as the Mach number tends to zero. In the framework of weak solutions, our recently developed method of relative entropies, can be applied to identify the inhomogeneous incompressible Navier--Stokes equations as the limiting system when the Mach number goes to zero, even in the ill-prepared data case. The proof is based on the relative entropy method and the dispersive estimates of the acoustic waves.