Relaxation models of phase transition flows

Helluy, Philippe; Seguin, Nicolas

doi:10.1051/m2an:2006015

Cited by 48 publications

(59 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then we compute the equilibrium of two phases satisfying the perfect gas law. The results are in agreement with those available in [Ja01] or [HS06]. Our algorithm is also applied to the van der Waals EOS: applying twice the Legendre transform is equivalent to a convexification process.…”

Section: Introductionsupporting

confidence: 87%

“…Then, we investigate the thermodynamics theory of mixtures. First, we consider an immiscible mixture of two phases of a same pure body as in [HS06].…”

Section: Introductionmentioning

confidence: 99%

“…As it is noticed in [HS06], the equilibrium energy can be related to classical operations in convex analysis that are the inf-convolution and the Legendre transform. These two operations are closely linked: the inf-convolution is transformed into an addition by the Legendre transform.…”

Section: Introductionmentioning

confidence: 99%

“…Although it is usual to consider the system through its entropy function [HS06], we decide here to describe it by its internal energy E. The fundamental equation of a simple system is then of the form…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Pressure Laws and Fast Legendre Transform

Helluy

Mathis

2011

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

Abstract. In this paper we investigate algorithms based on the Fast Legendre Transform (FLT) in order to compute tabulated Equation Of State (EOS) for fluids with phase transition. The equation of state of a binary mixture is given by an energy minimization principle. According to the miscible or immiscible nature of the mixture, the energy of the system is either a convex envelope or an inf-convolution of the energies of the two phases. Because these operations are closely linked to Legendre Transform, it is possible to construct O(N ) algorithms that compute efficiently these operations, where N is the number of points in the table. In addition, it appears that the natural mathematical tool for studying mixture thermodynamics in the Legendre space is the max-plus algebra theory.

show abstract

Section: Introductionsupporting

confidence: 87%

“…Then, we investigate the thermodynamics theory of mixtures. First, we consider an immiscible mixture of two phases of a same pure body as in [HS06].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pressure Laws and Fast Legendre Transform

Helluy

Mathis

2011

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

show abstract

“…Recall that, for the fully conservative coupling, several distinct entropy criteria have been proposed, each selecting a distinct weak solution in agreement with the physical context. (See [14] for a review and [41,30,7]). …”

Section: Introductionmentioning

confidence: 99%

Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws

Boutin¹,

Coquel²,

LeFloch³

2015

Math. Comp.

View full text Add to dashboard Cite

39 pagesInternational audienceThis series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well–balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem

show abstract

Derivation of a two‐phase flow model accounting for surface tension

Mathis

2024

Z Angew Math Mech

View full text Add to dashboard Cite

This paper presents the derivation of a two‐phase flow model that incorporates surface tension effects using Hamilton's principle of stationary action. The Lagrangian functional, which defines the action consists of kinetic energy–accounting for interface characteristics–and potential energy. A key feature of the model is the assumption that the interface separating the two phases possesses its own internal energy, which satisfies a Gibbs form that includes both surface tension and interfacial area. Consequently, surface tension is considered in both the kinetic and potential energy terms that define the Lagrangian functional. By applying the stationary action principle, a set of partial differential equations (PDEs) governing the dynamics of the two‐phase flow is derived. This includes evolution equations for the volume fraction and interfacial area, incorporating mechanical relaxation terms. The final model is proven to be well‐posed, demonstrating hyperbolicity and satisfying Lax entropy conditions.

show abstract

Relaxation models of phase transition flows

Cited by 48 publications

References 22 publications

Pressure Laws and Fast Legendre Transform

Pressure Laws and Fast Legendre Transform

Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws

Derivation of a two‐phase flow model accounting for surface tension

Contact Info

Product

Resources

About