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Journal of Non-Equilibrium Thermodynamics

Founded by Keller, Jürgen U.

Editor-in-Chief: Hoffmann, Karl Heinz

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Ed. by Michaelides, Efstathios E. / Rubi, J. Miguel

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Volume 43, Issue 2

Issues

A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids

Denis Anders
  • Computational Mechanics and Fluid Dynamics, Faculty of Computer Science and Engineering Science, TH Köln/University of Applied Sciences, Claudiusstraße 1, 50678 Cologne, Germany
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/ Kerstin WeinbergORCID iD: http://orcid.org/0000-0002-2213-8401
Published Online: 2018-03-15 | DOI: https://doi.org/10.1515/jnet-2017-0052

Abstract

The de-mixing properties of heterogeneous viscous fluids are determined by an interplay of diffusion, surface tension and a superposed velocity field. In this contribution a variational model of the decomposition, based on the Navier–Stokes equations for incompressible laminar flow and the extended Korteweg–Cahn–Hilliard equations, is formulated. An exemplary numerical simulation using C1-continuous finite elements demonstrates the capability of this model to compute phase decomposition and coarsening of the moving fluid.

Keywords: Cahn–Hilliard equation; multiphase flow; Korteweg stress; phase-field method

References

  • [1]

    D. G. Dalgleish, Adsorption of protein and the stability of emulsions, Trends Food Sci. Technol. 8 (1997), 1–6.CrossrefGoogle Scholar

  • [2]

    R. N. Grugel, T. A. Lograsso and A. Hellawell, The solidification of monotectic alloys - microstructures and phase spacings, Metall. Trans. A 15 (1984), 1003–1012.CrossrefGoogle Scholar

  • [3]

    D. Anders and K. Weinberg. An extended stochastic diffusion model for ternary mixtures, Mech. Mater. 56 (2013), 122–130.Web of ScienceCrossrefGoogle Scholar

  • [4]

    J. S. Higgins, J. E. G. Lipson and R. P. White, A simple approach to polymer mixture miscibility, Philos. Trans. R. Soc. A. 368 (2010), 1009–1025.CrossrefGoogle Scholar

  • [5]

    D. Lakehal, M. Meier and M. Fulgosi, Interface tracking towards the direct simulation of heat and mass transfer in multiphase flows, Int. J. Heat Fluid Flow 23 (2002), 242–257.CrossrefGoogle Scholar

  • [6]

    H. Terashima and G. Tryggvason, A front-tracking/ghost-fluid method for fluid interfaces in compressible flows, J. Comput. Phys. 228 (2009), no. 11, 4012–4037.CrossrefWeb of ScienceGoogle Scholar

  • [7]

    M. Rudman, Volume-tracking methods for interfacial flow calculations, Int. J. Numer. Methods Fluids 24 (1997), no. 7, 671–691.CrossrefGoogle Scholar

  • [8]

    L. Q. Chen, Phase-field models for microstructural evolution, Annu. Rev. Mater. Res. 32 (2002), 113–140.CrossrefGoogle Scholar

  • [9]

    D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30 (1998), 139–165.CrossrefGoogle Scholar

  • [10]

    M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6 (1996), no. 6, 815–831.CrossrefGoogle Scholar

  • [11]

    A. Morro, Phase-field models for fluid mixtures, Math. Comput. Model. 45 (2007), no. 9–10, 1042–1052.CrossrefWeb of ScienceGoogle Scholar

  • [12]

    P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10 (1942), 51–61.CrossrefGoogle Scholar

  • [13]

    M. L. Huggins, Theory of solutions of high polymers, J. Am. Chem. Soc. 64 (1942), 1712–1719.CrossrefGoogle Scholar

  • [14]

    J. W. Cahn, On spinodal decomposition in isotropic systems, J. Chem. Phys. 42 (1965), 93–99.CrossrefGoogle Scholar

  • [15]

    J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system I, J. Chem. Phys. 28 (1958), 258–267.CrossrefGoogle Scholar

  • [16]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49 (1977), no. 3, 435–479.CrossrefGoogle Scholar

  • [17]

    C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D 179 (2003), 211–228.CrossrefGoogle Scholar

  • [18]

    J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A 454 (1998), 2617–2654.CrossrefGoogle Scholar

  • [19]

    I. Szabo, Höhere Technische Mechanik, Springer-Verlag, Berlin, 1958.Google Scholar

  • [20]

    H. P. Langtangen, K.-A. Mardal and R. Winther, Numerical methods for incompressible viscous flow, Adv. Water Resour. 25 (2002), no. 8, 1125–1146.CrossrefGoogle Scholar

  • [21]

    D. Anders and K. Weinberg, A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation, Z. Angew. Math. Mech. 91 (2011), no. 8, 609–629.CrossrefWeb of ScienceGoogle Scholar

  • [22]

    D. Anders and K. Weinberg. Simulation of diffusion induced phase separation and coarsening in binary alloys. Comput. Mater. Sci., 50 (2011), no. 4, 1359–1364.Web of ScienceCrossrefGoogle Scholar

  • [23]

    D. Anders, K. Weinberg and R. Reichardt, Isogeometric analysis of thermal diffusion in binary blends, Int. J. Comput. Math. Sci. 52 (2012), no. 1, 182–188.CrossrefGoogle Scholar

  • [24]

    D. Anders, M. Dittmann, and K. Weinberg, A higher-order finite element approach to the Kuramoto–Sivashinsky equation, Z. Angew. Math. Mech. 92 (2012), no. 8, 599–607.CrossrefGoogle Scholar

  • [25]

    I. Müller, Die Kältefunktion eine universelle Funktion in der Thermodynamik viskoser wärmeleitender Flüssigleiten, Arch. Ration. Mech. Anal. 40 (1971), 1–36.CrossrefGoogle Scholar

  • [26]

    I. Müller, The coldness a universal function in thermoelastic bodies, Arch. Ration. Mech. Anal. 41 (1971), 319–332.Google Scholar

  • [27]

    I.-S.Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal. 46 (1972), 131–148.Google Scholar

  • [28]

    D. Anders and K. Weinberg. Thermophoresis in binary blends. Mech. Mater. 47 (2012), 33–50.CrossrefWeb of ScienceGoogle Scholar

  • [29]

    K. R. Rajagopal and A. R. Srinivasa, On thermomechanical restrictions of continua, Proc. R. Soc. Lond. A 460 (2004), 631–651.CrossrefGoogle Scholar

  • [30]

    M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys. 63 (2012), no. 1, 145–169.Web of ScienceCrossrefGoogle Scholar

  • [31]

    V. A. Cimmelli, D. Jou, T. Ruggeri and P. Ván, Entropy principle and recent results in non-equilibrium theories. Entropy 16 (2014), no. 3, 1756–1807.CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2017-10-06

Revised: 2018-02-15

Accepted: 2018-02-21

Published Online: 2018-03-15

Published in Print: 2018-04-25


Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 2, Pages 185–191, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2017-0052.

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