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A note on hyperbolic relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows.

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Title: A note on hyperbolic relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows.
Authors: Keim, Jens1 (AUTHOR) jens.keim@iag.uni-stuttgart.de, Konan, Hasel-Cicek2 (AUTHOR) Hasel-Cicek.Konan@mathematik.uni-stuttgart.de, Rohde, Christian2 (AUTHOR) christian.rohde@mathematik.uni-stuttgart.de
Source: ESAIM: Proceedings & Surveys. 2025, Vol. 78, p188-212. 25p.
Subjects: Two-phase flow, Incompressible flow, Numerical analysis, Hyperbolic differential equations, Hyperbolic functions, Interface dynamics
Abstract: We consider the two-phase dynamics of two incompressible and immiscible fluids. As a mathematical model we rely on the Navier-Stokes-Cahn-Hilliard system that belongs to the class of diffuse-interface models. Solutions of the Navier-Stokes-Cahn-Hilliard system exhibit strong non-local effects due to the velocity divergence constraint and the fourth-order Cahn-Hilliard operator. We suggest a new first-order approximative system for the inviscid sub-system. It relies on the artificialcompressibility ansatz for the Navier-Stokes equations, a friction-type approximation for the Cahn- Hilliard equation and a relaxation of a third-order capillarity term. We show under reasonable assumptions that the first-order operator within the approximative system is hyperbolic; precisely we prove for the spatially one-dimensional case that it is equipped with an entropy-entropy flux pair with convex (mathematical) entropy. For specific states we present a numerical characteristic analysis. Thanks to the hyperbolicity of the system, we can employ all standard numerical methods from the field of hyperbolic conservation laws. We conclude the paper with preliminary numerical results in one spatial dimension. [ABSTRACT FROM AUTHOR]
Copyright of ESAIM: Proceedings & Surveys is the property of EDP Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: A note on hyperbolic relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows.
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  Data: <searchLink fieldCode="AR" term="%22Keim%2C+Jens%22">Keim, Jens</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> jens.keim@iag.uni-stuttgart.de</i><br /><searchLink fieldCode="AR" term="%22Konan%2C+Hasel-Cicek%22">Konan, Hasel-Cicek</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> Hasel-Cicek.Konan@mathematik.uni-stuttgart.de</i><br /><searchLink fieldCode="AR" term="%22Rohde%2C+Christian%22">Rohde, Christian</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> christian.rohde@mathematik.uni-stuttgart.de</i>
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  Data: <searchLink fieldCode="JN" term="%22ESAIM%3A+Proceedings+%26+Surveys%22">ESAIM: Proceedings & Surveys</searchLink>. 2025, Vol. 78, p188-212. 25p.
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  Data: <searchLink fieldCode="DE" term="%22Two-phase+flow%22">Two-phase flow</searchLink><br /><searchLink fieldCode="DE" term="%22Incompressible+flow%22">Incompressible flow</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Hyperbolic+differential+equations%22">Hyperbolic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Hyperbolic+functions%22">Hyperbolic functions</searchLink><br /><searchLink fieldCode="DE" term="%22Interface+dynamics%22">Interface dynamics</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We consider the two-phase dynamics of two incompressible and immiscible fluids. As a mathematical model we rely on the Navier-Stokes-Cahn-Hilliard system that belongs to the class of diffuse-interface models. Solutions of the Navier-Stokes-Cahn-Hilliard system exhibit strong non-local effects due to the velocity divergence constraint and the fourth-order Cahn-Hilliard operator. We suggest a new first-order approximative system for the inviscid sub-system. It relies on the artificialcompressibility ansatz for the Navier-Stokes equations, a friction-type approximation for the Cahn- Hilliard equation and a relaxation of a third-order capillarity term. We show under reasonable assumptions that the first-order operator within the approximative system is hyperbolic; precisely we prove for the spatially one-dimensional case that it is equipped with an entropy-entropy flux pair with convex (mathematical) entropy. For specific states we present a numerical characteristic analysis. Thanks to the hyperbolicity of the system, we can employ all standard numerical methods from the field of hyperbolic conservation laws. We conclude the paper with preliminary numerical results in one spatial dimension. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of ESAIM: Proceedings & Surveys is the property of EDP Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1051/proc/202578188
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      – Code: eng
        Text: English
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        PageCount: 25
        StartPage: 188
    Subjects:
      – SubjectFull: Two-phase flow
        Type: general
      – SubjectFull: Incompressible flow
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Hyperbolic differential equations
        Type: general
      – SubjectFull: Hyperbolic functions
        Type: general
      – SubjectFull: Interface dynamics
        Type: general
    Titles:
      – TitleFull: A note on hyperbolic relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows.
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            NameFull: Keim, Jens
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            NameFull: Konan, Hasel-Cicek
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            NameFull: Rohde, Christian
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            – D: 01
              M: 02
              Text: 2025
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              Y: 2025
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