Abstract
We introduce a diffuse interface model describing the evolution of a mixture of twodifferent viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature onthe flow are taken into account. In the mathematical model,the evolution of the velocity $u$ is ruled by the Navier-Stokessystem with temperature-dependent viscosity, while the order parameter $\psi$ representingthe concentration of one of the components of the fluid is assumed to satisfy aconvective Cahn-Hilliard equation. The effects of the temperature are prescribed by asuitable form of the heat equation. However, due to quadratic forcing terms, this equationis replaced, in the weak formulation, by an equality representing energyconservation complemented with a differential inequality describing production of entropy.The main advantage of introducing this notion of solutionis that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulationis more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problemassociated to the weak formulation of the model is proved by deriving suitable a prioriestimates and showing weak sequential stability of families of approximating solutions.
Published Version
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