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Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg–Landau/Cahn–Hilliard-type equations (Schmuck et al 2012 Proc. R. Soc. A 468 3705–24), we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large Péclet number. As for classical convection–diffusion problems, we obtain systematically diffusion–dispersion relations (including Taylor–Aris-dispersion). Our results also provide a convenient computational framework to macroscopically track interfaces in porous media. In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells. [ABSTRACT FROM AUTHOR]

We study a diffuse interface model of the two-component Hele–Shaw flow. This is an advective Cahn–Hilliard equation for the relative concentration , where the incompressible velocity field is determined by the Darcy’s law depending on the Korteweg force . Here μ is the derivative of a nonlocal non-convex free energy characterized by a logarithmic potential. The system is subject to no-flux boundary conditions for and μ along with an initial condition for . First of all, we prove that the corresponding problem is globally well posed with respect to a natural notion of weak solution. Also, we establish the existence of global strong solutions. In dimension two, we show the validity of the so-called instantaneous separation property. This means that any solution, which is not a pure phase initially, stays away from the pure phases, uniformly with respect to the initial energy and total mass. Finally we prove the existence of the global attractor for the corresponding dynamical system as well as the convergence to a single equilibrium of any weak solution in the two-dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins–Daruud et al in Hawkins–Daruud et al (2011 Int. J. Numer. Math. Biomed. Eng. 28 3–24). The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side of the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. [ABSTRACT FROM AUTHOR]

We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng. 30 726–54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn–Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity satisfies , where ν is the outer normal to the boundary of the domain. [ABSTRACT FROM AUTHOR]

We propose a two-dimensional phase field model for solid state dewetting where the surface energy is weakly anisotropic. The evolution is described by the Cahn–Hilliard equation with a bi-quadratic degenerate mobility together with a bulk free energy based on a double-well potential and a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. We show that in contrast to the frequently used quadratic degenerate mobility, the resulting sharp interface model for the bi-quatratic mobility is consistent with the pure surface diffusion model. In addition, we show that natural boundary conditions at the substrate obtained from the first variation of the total free energy including contributions at the substrate imply a contact angle condition in the sharp-interface limit which recovers the Young–Herring equation in the anisotropic and Young’s equation in the isotropic case, as well as a balance of fluxes at the contact line (or contact point). [ABSTRACT FROM AUTHOR]

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier–Stokes system and a Cahn–Hilliard equation. For the Cahn–Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the -truncation method we prove existence of weak solutions for a power-law exponent , d  =  2, 3. [ABSTRACT FROM AUTHOR]

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier–Stokes system coupled with a convective non-local Cahn–Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of a global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective non-local Cahn–Hilliard equation with degenerate mobility and singular potential in dimension three. [ABSTRACT FROM AUTHOR]