High-friction limits of Euler flows for multicomponent systems

Huo, Xiaokai; Jüngel, Ansgar; Tzavaras, Athanasios E.

doi:10.1088/1361-6544/ab12a6

Cited by 17 publications

(36 citation statements)

References 32 publications

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“…3. Without a porous medium contribution, a different (high friction) scaling for multicomponent flow is considered in Huo et al 23 For the limit → 0, the authors show the convergence to a hyperbolic zeroth-order system with a parabolic first-order correction system which is of Maxwell-Stefan type similar to (30).…”

Section: Compressible Multicomponent Flow In Porous Mediamentioning

confidence: 99%

“…Remark For the single‐component case

n = 1

, the system reduces to the porous media Equation . If no porous medium is present, that is,

M_{i} = 0

, the system in this framework corresponds for perfect gas laws to the following version of the Maxwell–Stefan equations formulated for the molar concentrations

c_{i}

often seen in the literature, eg, in Jüngel and Stelzer:

\begin{matrix} right \partial_{t} c_{i} + div J_{i} & left = 0, & right \\ right \nabla c_{i} & left = - \sum_{j = 1, j \neq i}^{n} \frac{c_{j} J_{i} - c_{i} J_{j}}{D_{i j}} . \end{matrix}

Here,

D_{i j} = \frac{R}{c {scriptM}_{i} {scriptM}_{j} λ_{i j}}

, with the ideal gas constant

R

, total molar concentration

c = \sum_{i = 1}^{n} c_{i}

, and molar masses

{scriptM}_{i}

. Without a porous medium contribution, a different (high friction) scaling for multicomponent flow is considered in Huo et al For the limit

ε \to 0

, the authors show the convergence to a hyperbolic zeroth‐order system with a parabolic first‐order correction system which is of Maxwell‐Stefan type similar to . …”

Section: Compressible Flow In Porous Mediamentioning

confidence: 99%

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Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Ostrowski

Rohde

2020

Math Meth Appl Sci

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We introduce a Darcy‐scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross‐diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell‐Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy‐scale model tend in a certain long‐time regime to solutions of a parabolic limit system.

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Section: Compressible Multicomponent Flow In Porous Mediamentioning

confidence: 99%

“…Remark For the single‐component case

n = 1

, the system reduces to the porous media Equation . If no porous medium is present, that is,

M_{i} = 0

, the system in this framework corresponds for perfect gas laws to the following version of the Maxwell–Stefan equations formulated for the molar concentrations

c_{i}

often seen in the literature, eg, in Jüngel and Stelzer:

\begin{matrix} right \partial_{t} c_{i} + div J_{i} & left = 0, & right \\ right \nabla c_{i} & left = - \sum_{j = 1, j \neq i}^{n} \frac{c_{j} J_{i} - c_{i} J_{j}}{D_{i j}} . \end{matrix}

Here,

D_{i j} = \frac{R}{c {scriptM}_{i} {scriptM}_{j} λ_{i j}}

, with the ideal gas constant

R

, total molar concentration

c = \sum_{i = 1}^{n} c_{i}

, and molar masses

{scriptM}_{i}

. Without a porous medium contribution, a different (high friction) scaling for multicomponent flow is considered in Huo et al For the limit

ε \to 0

, the authors show the convergence to a hyperbolic zeroth‐order system with a parabolic first‐order correction system which is of Maxwell‐Stefan type similar to . …”

Section: Compressible Flow In Porous Mediamentioning

confidence: 99%

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Ostrowski

Rohde

2020

Math Meth Appl Sci

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“…The relative energy method is used here to perform this limiting process for strong solutions of (1) in several space dimensions. This approach was successful for the relaxation limit in single-species fluid models [15,3], as well as for certain (weakly coupled through friction) multicomponent systems [9]. The relative energy method provides an efficient mathematical mechanism for stability analysis and establishing limiting processes; see [5] for early developments, [2,15,16] and references therein for applications to diffusive relaxation.…”

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confidence: 99%

The relaxation limit of bipolar fluid models

Alves¹,

Tzavaras²

2022

DCDS

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<p style='text-indent:20px;'>This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid models, and it is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.</p>

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“…The system (1.1)--(1.3) can be obtained as the high-friction limit of the multicomponent Euler equations [13]:…”

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confidence: 99%

“…It was proved in [13] that, when the total momentum is zero, the system (1.8) converges to (1.1)--(1.3) in the high-friction limit \varepsi \rightar 0. Moreover, (1.1)--(1.3) can be regarded as a gradient flow for F (\rho ).…”

mentioning

confidence: 99%

An Energy Stable and Positivity-Preserving Scheme for the Maxwell--Stefan Diffusion System

Huo¹,

Liu²,

Tzavaras³

et al. 2021

SIAM J. Numer. Anal.

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We develop a new finite difference scheme for the Maxwell--Stefan diffusion system. The scheme is conservative, energy-stable, and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

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High-friction limits of Euler flows for multicomponent systems

Cited by 17 publications

References 32 publications

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

The relaxation limit of bipolar fluid models

An Energy Stable and Positivity-Preserving Scheme for the Maxwell--Stefan Diffusion System

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