On a Model for Mixture Flows: Derivation, Dissipation and Stability Properties

Goudon, Thierry; Vasseur, Alexis

doi:10.1007/s00205-015-0925-3

Cited by 7 publications

(5 citation statements)

References 51 publications

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“…Definition 2 (Global weak solution in terms of u) The couple ( , u) is called a weak solution to system (5) and (14) if the following regularity properties are satisfied…”

Section: The Case Of ϕ and µ Related By (7)mentioning

confidence: 99%

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Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

Bresch

Giovangigli

Zatorska

2015

Journal de Mathématiques Pures et Appliquées

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In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics with periodic boundary conditions. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. We introduce a new mathematical entropy which clearly shows existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, Comm. Partial Diff. Eqs. 1987] to the global weak solutions framework. Second, we present the ghost effect system studied by [C.D. Levermore, W. Sun, K. Trivisa, SIAM J. Math. Anal. 2012] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.

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“…Definition 2 (Global weak solution in terms of u) The couple ( , u) is called a weak solution to system (5) and (14) if the following regularity properties are satisfied…”

Section: The Case Of ϕ and µ Related By (7)mentioning

confidence: 99%

“…Depending on the nonlinearity ϕ, such systems are used to model various phenomena like motion of mixtures and avalanches, salt and pollutant spreading or combustion. In the recent paper [14] a more complex system is derived to model a flow of mixture in the multi-dimensional setting and investigated in the one-dimensional domain.…”

Section: Introductionmentioning

confidence: 99%

Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

Bresch

Giovangigli

Zatorska

2015

Journal de Mathématiques Pures et Appliquées

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“…Imposing a specific relation between the viscosity µ and the diffusion coefficient κ further estimates (roughly speaking that control ∇ x ρ) can be derived which are useful to justify the existence of solutions [11,51,52]. In [15,34], we reinterpret the derivation of the KS equations by means of hydrodynamic regimes, starting from a coupled fluid-kinetic system. This viewpoint allows us to derive a new model which can be interpreted as a generalized KS system and which has remarkable dissipation properties.…”

Section: Modelling Of Mixtures: the Kazhikhov-smagulov Systemmentioning

confidence: 99%

Simulations of non homogeneous viscous flows with incompressibility constraints

Calgaro

Creusé

Goudon

et al. 2017

Mathematics and Computers in Simulation

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This presentation is an overview on the development of numerical methods for the simulation of non homogeneous flows with incompressibility constraints. We are particularly interested in systems of partial differential equations describing certain mixture flows, like the Kazhikhov-Smagulov system which can be used to model powder-snow avalanches. It turns out that the Incompressible Navier-Stokes system with variable density is a relevant step towards the treatment of such models, and it allows us to bring out some interesting numerical difficulties. We should handle equations of different types, roughly speaking transport and diffusion equations. We present two strategies based on time-splitting. The former relies on a hybrid approach, coupling finite volume and finite element methods. The latter extends discrete duality finite volume schemes for such non homogeneous flows. The methods are confronted to exact solutions and to the simulation of Rayleigh-Taylor instabilities.

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“…[6] derived versions of (1.4) without θ 2 -nonlinearity by specific choice of µ and θ. Cook-Dimotakis [7] investigated (1.2) with the constant viscosity and (1.3) to understand Rayleigh-Taylor instability between miscible fluids. Goudon-Vasseur [10] re-analyzed Kazhikhov-Smagulov type modeling of mixture flows from a more microscopic viewpoint to propose a generalized model.…”

Section: Introductionmentioning

confidence: 99%

Existence of global weak solutions of inhomogeneous incompressible Navier-Stokes equations with mass diffusion

Kacedan¹,

Soga²

2023

Preprint

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This paper proves existence of a global weak solution of the inhomogeneous (i.e., non-constant density) incompressible Navier-Stokes system with mass diffusion. The system is well-known as the Kazhikhov-Smagulov model. The major novelty of the paper is to deal with the Kazhikhov-Smagulov model possessing the nonconstant viscosity without any simplification of higher order nonlinearity. Any global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of the Aubin-Lions-Simon type.

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On a Model for Mixture Flows: Derivation, Dissipation and Stability Properties

Cited by 7 publications

References 51 publications

Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

Simulations of non homogeneous viscous flows with incompressibility constraints

Existence of global weak solutions of inhomogeneous incompressible Navier-Stokes equations with mass diffusion

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