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On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids
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 temperaturedependent viscosity, while the order parameter Phi 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.
Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids
We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [12] where existence of weak solutions was proved in three space dimensions. Here, we aim at studying the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [12]. Moreover, we can admit slightly more general conditions on some material coefficients of the system.
Strong stationarity conditions for the optimal control of a Cahn--Hilliard--Navier--Stokes system
This paper is concerned with the distributed optimal control of a time-discrete Cahn-Hilliard-Navier-Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier-Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.
A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system
This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn-Hilliard-Navier-Stokes system with variable densities. The free energy density associated to the Cahn-Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier-Stokes equation. A dual-weighted residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.