# American Institute of Mathematical Sciences

January  2014, 19(1): 73-88. doi: 10.3934/dcdsb.2014.19.73

## Phase transition and separation in compressible Cahn-Hilliard fluids

 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy 2 DICATAM, Università di Brescia, Via Valotti 9, 25133 Brescia, Italy 3 DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova, Italy

Received  October 2012 Revised  July 2013 Published  December 2013

The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are compressible fluids thus improving the model of a previous approach. Moreover a possible saturation effect for the concentration of a constituent is made explicit. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed to occur. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. Shear-induced mixing and demixing are examined. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
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