Nonlinear Cross-Diffusion with Size Exclusion

Burger, Martin; Francesco, Marco Di; Pietschmann, Jan‐Frederik; Schlake, Bärbel

doi:10.1137/100783674

Cited by 121 publications

(269 citation statements)

References 33 publications

(41 reference statements)

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“…Here, we have introduced the dimensionless spatial coordinate r/ℓ → r and time τ = νt, and ∂ τ stands for the partial time derivative. Note that equations of the form (2), as well as their generalizations for two-and multicomponent systems, also appear in the problems of nonlinear crossdiffusion with size exclusion [50], diffusion in monolayers of reagents on the surface of a catalyst [51] and serve as a model of fast ionic conductors [37].…”

Section: Modelmentioning

confidence: 99%

Effects of gas interparticle interaction on dissipative wake-mediated forces

Kliushnychenko

Lukyanets

2017

Phys. Rev. E

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The effect of concentration-dependent switching of the non-equilibrium depletion interaction between obstacles in a gas flow of interacting Brownian particles is presented. When increasing bath fraction exceeds half-filling, the wake-mediated interaction between obstacles switches from effective attraction to repulsion or vice-versa, depending on the mutual alignment of obstacles with respect to the gas flow. It is shown that for an ensemble of small and widely separated obstacles the dissipative interaction takes the form of induced dipole-dipole interaction governed by an anisotropic screened Coulomb-like potential. This allows one to give a qualitative picture of the interaction between obstacles and explain switching effect as a result of changes of anisotropy direction. The non-linear blockade effect is shown to be essential near closely located obstacles, that manifests itself in additional screening of the gas flow and generation of a pronounced step-like profile of gas density distribution. It is established that behavior of the magnitude of dissipative effective interaction is, generally, non-monotonic in relation to both the bath fraction and the external driving field. It has characteristic peaks corresponding to the situation when the common density "coat" formed around the obstacles is most pronounced. The possibility of the dissipative pairing effect is briefly discussed. All the results are obtained within the classical lattice-gas model.

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Section: Modelmentioning

confidence: 99%

Effects of gas interparticle interaction on dissipative wake-mediated forces

Kliushnychenko

Lukyanets

2017

Phys. Rev. E

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“…The prevention of overcrowding has already been considered in several papers, either in the parabolic-elliptic case [Burger et al 2008; or the parabolic-parabolic case [Di Francesco and Rosado 2008] (with diffusiondominated large-time asymptotics) and [Burger et al 2010] (where, additionally, the case of several species and cross-diffusion was taken in to account). In these papers the emphasis was put on asymptotic behaviors, with a discussion of the possible asymptotic states and behaviors depending on nonlinearities in [Burger et al 2006] and a study of plateau-like quasistationary solutions and their motion in [Burger et al 2008].…”

Section: Motivationsmentioning

confidence: 99%

Untitled

2015

Math. Mech. Compl. Sys.

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In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results, and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated with such solutions.Dynamical stability in the neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows us to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter, with all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss the qualitative properties of the solutions using theoretical methods and numerical computations.

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“…Equations (12) and (13) together with Equation (11) fully describe our model and define a set of two fourth-order partial differential equations. Let us analyse the derived model.…”

Section: Modelmentioning

confidence: 99%

“…Let us analyse the derived model. If we assume constant diffusion coefficients, D 1 = D 2 = D, the evolution of ϕ in Equation (12) can be compared to the Cahn-Hilliard model [10]. A standard Cahn-Hilliard model minimizes the interfacial energy F I , by a gradient flow with constant mobility.…”

Section: Modelmentioning

confidence: 99%

“…It is worth noting that due to this asymmetry, separated domains of material do not coarsen, but voids in the material do. The first term on the right-hand side of Equation (12) shows that unequal diffusion coefficients connected with a concentration gradient lead to material flux. This is again a direct result of our modelling assumption, as every concentration flux is connected to a mass flux of a single component.…”

Section: Modelmentioning

confidence: 99%

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The interplay of morphological and compositional evolution in crystal growth: a phase-field model

Backofen

Bergamaschini

Voigt

2014

Philosophical Magazine

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We present a closed and consistent model for growth phenomena of binary systems. Starting with the free energy of mixtures, a mean-field description of the material is derived that accounts for morphological as well as compositional evolution, which are strongly coupled to each other. The derived phase-field model describes surface effects, e.g. surface diffusion and anisotropic surface tension, as well as codiffusion in the evolving bulk phase. The model leads to a set of nonlinear partial differential equations of high order, which are solved simultaneously using adaptive finite elements and used to analyse the interplay of growth, surface and compositional dynamics. The external driving force is a mass flux from a fluid or vapour phase which allows to control the growth process.

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Nonlinear Cross-Diffusion with Size Exclusion

Cited by 121 publications

References 33 publications

Effects of gas interparticle interaction on dissipative wake-mediated forces

Effects of gas interparticle interaction on dissipative wake-mediated forces

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The interplay of morphological and compositional evolution in crystal growth: a phase-field model

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