On a Cahn–Hilliard system with convection and dynamic boundary conditions

Abstract

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are in the unknowns ρ, the order parameter, and µ, the chemical potential. In the above equations, τ Ω is a nonnegative constant, f ′ is the derivative of a double-well potential f , and u is a given velocity field.Typical and significant examples of f are the so-called classical regular potential, the logarithmic double-well potential , and the double obstacle potential , which are given bywhere the constants in (1.3) and (1.4) satisfy c 1 > 1 and c 2 > 0, so that that f log and f 2obs are nonconvex. In cases like (1.4), one has to split f into a nondifferentiable convex part (the indicator function of [−1, 1] in the present example) and a smooth perturbation. Accordingly, one has to replace the derivative of the convex part by the subdifferential and interpret the second identity in (1.1) as a differential inclusion.As far as the conditions on the boundary Γ := ∂Ω are concerned, instead of the classical homogeneous Neumann boundary conditions, the dynamic boundary condition for both µ and ρ are considered, namely,

Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics

“…which can be seen as a local convective viscous Cahn-Hilliard equation with an additional source term in the definition of the chemical potential. It is well-known (see [26] for example) that such problem admits a unique weak solution (v, µ v ) with v ∈ H 1 (0, T ; L 2 (Ω)) ∩ L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) , µ v ∈ L 2 (0, T ; H 1 (Ω)) , satisfying (3.7)-(3.9) for example in the sense of distributions. Hence, the map Γ λ ε : L 2 (0, T ; H s (Ω)) → H 1 (0, T ; L 2 (Ω)) ∩ L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) associating to every w ∈ L 2 (0, T ; H s (Ω)) the solution v to (3.7)-(3.9) is well-defined.…”

“…Moreover, since u λ ε ∈ H 1 (0, T 0 ; L 2 (Ω)) ∩ L ∞ (0, T 0 ; H 1 (Ω)), then u λ ε is weakly continuous with values in T 0 : this allows us to obtain the pointwise regularity u λ ε (T 0 ) ∈ H 1 (Ω). Such regularity is then enough to extend the solution to the next subinterval [T 0 , 2T 0 ] (see [26]): using a standard patching argument in time allows to extend the solution to the whole interval [0, T ].…”

“…The interest in additional convective contributions is connected with applications in mixing and stirring of fluids, as well as in biological realizations of thin films via Langmuir-Blodgett transfer [6,42]. We mention in this direction the contributions [8,26,31,52] on the local Cahn-Hilliard with convection, [29,30,50] dealing with the nonlocal Cahn-Hilliard with local convection, and [32,41] on the nonlocal case with nonlocal convection. A nonlocal convective Cahn-Hilliard type system modelling phase-separation has been analyzed in [24,25].…”

Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model

“…Concerning the existence of solutions to (1) to (5) and (8) together with no-slip for v and a homogeneous Neumann (or no-flux) boundary condition for ϕ as well as with different assumptions on b and W , we refer to [22,[24][25][26]. For the boundary conditions considered here we are not aware of such results, but refer to [27] for the Cahn-Hilliard Navier-Stokes system with equal densities, to [28] for analytical results for the Cahn-Hilliard system with dynamic boundary conditions, and to [11] for a Cahn-Hilliard Navier-Stokes model with dynamical contact angle condition, but no-slip condition for the Navier-Stokes equation. Concerning sharp interface limits, we refer to [8] for the bulk model with homogeneous boundary conditions.…”

“…Here yc denotes the center of mass at time t = 2, vmax denotes the maximum rising velocity that appears at time tv max , and c min denotes the minimal circularity that appears at time tc min . See (26)- (28) or [32] for the definition of these values. As reference we choose the results from the 3rd group participating in [32] (ref.…”