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June  2015, 35(6): 2497-2522. doi: 10.3934/dcds.2015.35.2497

## On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids

 1 Dipartimento di Matematica ed Informatica “U. Dini”, viale Morgagni 67/a, I-50134 Firenze 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin 3 Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia

Received  January 2014 Revised  April 2014 Published  December 2014

We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity $u$ is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter $\psi$ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a priori estimates and showing weak sequential stability of families of approximating solutions.
Citation: Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497
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