Regularized family of models for incompressible Cahn–Hilliard two-phase flows. (June 2015)

• Record Type: Journal Article
• Authors: Gal, Ciprian G.; Medjo, T. Tachim
• Is Part Of: Nonlinear analysis. Volume 23(2015)
• Page Start: 94
• Page End: 122
• Publication Date: 2015-06
• Subjects: Regularized Navier–Stokes -- Navier–Stokes-α -- Simplified Bardina -- Navier–Stokes–Voight -- Cahn–Hilliard equations -- Exponential attractor; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2014.11.005
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We consider a general family of regularized models for incompressible two-phase flows based on the Cahn–Hilliard formulation in n -dimensional compact Riemannian manifolds (with or without boundary) for n = 2, 3 . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- α -like model, the Leray- α model, the Modified Leray- α model, the Simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes model, and many others) for the fluid velocity u suitably coupled with a convective Cahn–Hilliard equation for the order (phase) parameter ϕ . We give a unified analysis of the entire three-parameter family of two-phase models. We first establish existence, stability and regularity results. Then, we show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory ( u, ϕ ) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.