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

We introduce a di\ufb00use interface model describing the evolution of a mixture of two di\ufb00erent viscous incompressible \ufb02uids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the \ufb02ow 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 \u3d5 representing the concentration of one of the components of the \ufb02uid is assumed to satisfy a convective Cahn-Hilliard equation. The e\ufb00ects 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 di\ufb00erential 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 solution