Variational formulation of the constitutive relaxation at finite rate of the mechanical state in multiphase flows with the diffuse-interface method
Abstract
Pressure relaxation is a fundamental mechanism in compressible multiphase flows and heterogeneous materials, where local pressure disequilibria arise from unresolved microscale processes such as wave scattering, pore deformation, and interfacial dynamics. At the macroscopic scale, these phenomena manifest as dissipative volume redistribution mechanisms that progressively drive the system toward mechanical equilibrium. Despite their importance, pressure relaxation closures are commonly introduced through phenomenological source terms whose thermodynamic consistency and physical interpretation remain largely heuristic. This work proposes a variational framework for finite-rate pressure relaxation in compressible multiphase flows based on the theory of Generalized Standard Materials. The relaxation process is formulated as a constitutive evolution law governed by the competition between a stored energy and a convex dissipation pseudo-potential depending on the rates of volume fractions. The resulting evolution equations are obtained from the stationarity conditions of an incremental variational principle, providing a systematic derivation of relaxation closures while ensuring non-negative entropy production and thermodynamic consistency by construction. The framework is developed for both phase-wise and interaction-based dissipation potentials. A central result is that the interfacial pressure emerges naturally as a Lagrange multiplier associated with the saturation constraint on volume fractions, rather than being prescribed through ad hoc averaging or impedance-weighting procedures. This yields a rigorous thermodynamic interpretation of interfacial pressure and clarifies its role in energy exchanges between phases. The variational structure further induces symmetric tangent operators for the constitutive update problem, enabling efficient and robust monolithic Newton solution strategies. The proposed formulation is discretized within a Godunovtype finite-volume framework and applied to representative multiphase flow configurations. The results demonstrate the accuracy, robustness, and thermodynamic consistency of the approach, and highlight its potential as a unified constitutive framework for dissipative processes in compressible multiphase and heterogeneous media.
Related
- Reactive flash for ideal multiphase mixtures: Unified formulation and efficient computation
- A Gaussian Process State Space Formulation for Materials Constitutive Modeling: Application to Quasi-brittle Materials
- Variational formulation for the dynamics of soft matter including inertia
- Fixed-Point Formulation and Parallel Averaged Operator Algorithms for Generalized Variational Inclusion Problems
- On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation
BibTeX
@article{heuze2026variational,
title = {{Variational formulation of the constitutive relaxation at finite rate of the mechanical state in multiphase flows with the diffuse-interface method}},
author = {Thomas Heuzé and Nicolas Favrie},
journal = {Computer Methods in Applied Mechanics and Engineering},
year = {2026},
month = sep,
doi = {10.1016/j.cma.2026.119417},
url = {https://hal.science/hal-05752436v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Variational formulation of the constitutive relaxation at finite rate of the mechanical state in multiphase flows with the diffuse-interface method
AU - Thomas Heuzé
AU - Nicolas Favrie
PY - 2026
DA - 2026/09/15
JO - Computer Methods in Applied Mechanics and Engineering
DO - 10.1016/j.cma.2026.119417
UR - https://hal.science/hal-05752436v1
AB - Pressure relaxation is a fundamental mechanism in compressible multiphase flows and heterogeneous materials, where local pressure disequilibria arise from unresolved microscale processes such as wave scattering, pore deformation, and interfacial dynamics. At the macroscopic scale, these phenomena manifest as dissipative volume redistribution mechanisms that progressively drive the system toward mechanical equilibrium. Despite their importance, pressure relaxation closures are commonly introduced through phenomenological source terms whose thermodynamic consistency and physical interpretation remain largely heuristic. This work proposes a variational framework for finite-rate pressure relaxation in compressible multiphase flows based on the theory of Generalized Standard Materials. The relaxation process is formulated as a constitutive evolution law governed by the competition between a stored energy and a convex dissipation pseudo-potential depending on the rates of volume fractions. The resulting evolution equations are obtained from the stationarity conditions of an incremental variational principle, providing a systematic derivation of relaxation closures while ensuring non-negative entropy production and thermodynamic consistency by construction. The framework is developed for both phase-wise and interaction-based dissipation potentials. A central result is that the interfacial pressure emerges naturally as a Lagrange multiplier associated with the saturation constraint on volume fractions, rather than being prescribed through ad hoc averaging or impedance-weighting procedures. This yields a rigorous thermodynamic interpretation of interfacial pressure and clarifies its role in energy exchanges between phases. The variational structure further induces symmetric tangent operators for the constitutive update problem, enabling efficient and robust monolithic Newton solution strategies. The proposed formulation is discretized within a Godunovtype finite-volume framework and applied to representative multiphase flow configurations. The results demonstrate the accuracy, robustness, and thermodynamic consistency of the approach, and highlight its potential as a unified constitutive framework for dissipative processes in compressible multiphase and heterogeneous media.
SN - 0045-7825
SN - 1879-2138
ER -