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On the development and generalizations of Allen–Cahn and Stefan equations within a thermodynamic framework

Abstract

Starting from a simplified framework of the theory of interacting continua in which the mass balance equations are considered for each constituent but the balance of linear momentum and the balance of energy are considered for the mixture as a whole, we provide a thermodynamic basis for models that include the Allen–Cahn and Stefan equations as particular cases. We neglect the mass flux due to diffusion associated with the components of the mixture but permit the possibility of mass conversion of the phases. As a consequence of the analysis, we are able to show that the reaction (source) term in the mass balance equation leads to the Laplace operator that appears in the Allen–Cahn model and that this term is not related to a diffusive process. This study is complementary to that by Heida et al. (Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 63, 145–169, 2012), where we neglected mass conversion of the species but considered mass diffusion effects and derived the constitutive equations for diffusive mass flux (the framework suitable for capturing other interface phenomena such as capillarity and for generalizing the Cahn–Hilliard and Lowengrub–Truskinovsky models).

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Correspondence to K. R. Rajagopal.

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This research has been partially performed during the stay of Martin Heida at the Charles University in Prague, the stay was supported by the Jindřich Nečas Center for Mathematical Modeling (the project LC06052 financed by MSMT).

Josef Málek’s contribution is supported by GACR 201/09/0917. K. R. Rajagopal thanks the National Science Foundation for its support.

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Heida, M., Málek, J. & Rajagopal, K.R. On the development and generalizations of Allen–Cahn and Stefan equations within a thermodynamic framework. Z. Angew. Math. Phys. 63, 759–776 (2012). https://doi.org/10.1007/s00033-011-0189-1

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Mathematic Subject Classification (1991)

  • Primary 76T
  • Secondary 74F
  • 76A

Keywords

  • Mass conversion
  • Diffusion
  • Mixture theory
  • Rate entropy production
  • Navier–Stokes–Fourier system