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In this paper, we propose a mathematical model of phase separation for a quasi-incompressible binary mixture where the spinodal decomposition is induced by an heat flux governed by the Cattaneo–Maxwell equation. As usual, the phase separation is considered in the framework of phase-field modeling so that the transition is described by an additional field, the mixture concentration c. The evolution of the mixture concentration is described by the Cahn–Hilliard equation, and in our model, it is coupled with the Navier–Stokes equation. Since thermal effect is included, the whole set of evolution equations is set up for the velocity, the mixture concentration, the temperature and the heat flux. The model is compatible with thermodynamics and a maximum theorem holds

Phase separation in quasi-incompressible fluids: Cahn–Hilliard model in the Cattaneo–Maxwell framework