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.
Highlights
We study a non-isothermal diffuse interface model for the flow of a mixture of two viscous incompressible Newtonian fluids of equal density in a bounded domain Ω ⊂ R3
Despite the large amount of mathematical literature on free boundary problems related to fluids with a classical sharp interface, most papers are confined to the case of flows without singularities in the interface and so far there is no satisfactory existence theory of weak solutions for a two-phase flow of two viscous, incompressible, immiscible fluids with a classical sharp interface
We describe the non-isothermal evolution of the fluid by means of a model that keeps its thermodynamical consistency in a wide temperature range
Summary
We study a non-isothermal diffuse interface model for the flow of a mixture of two viscous incompressible Newtonian fluids of equal density in a bounded domain Ω ⊂ R3. At least up to our knowledge, a non-isothermal model for two-phase fluids has been analyzed only in the reference [40], where a linearization of the internal energy balance is used in order to describe the evolution of the temperature. This permits the authors to get rid of the quadratic terms in the right hand side of (1.5) and of the coupling beween (1.4) and (1.5). The proof of this result occupies the remainder of the paper and is split into two steps: a-priori estimates, which are described in Section 4, and weak-sequential stability, which is proved in the last Section 5
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