On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

Abstract

Abstract. We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds 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 and the Navier-Stokes model) for the fluid velocity u suitably coupled with a convective Allen-Cahn equation for the order (phase) parameter φ. We give a unified analysis of the entire three-parameter family of twophase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α → 0 limit in α-models. Then, we also 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. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of [44]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.

Singular Limits of Voigt Models in Fluid Dynamics

“…Our proof of the existence of the global attractor slightly differs from the one in [33]. While here we obtain the optimal regularity in one step (also as in [21,Section 6]), the method of [33] requires a bootstrapping procedure which implies first the regularity in V α . Here, we also wanted to rephrase those results into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α.…”

“…We emphasize that all the results proven in this article are also valid in a more general setting, when Ω is a compact Riemannian manifold with or without boundary and in the presence of other boundary conditions. We refer the reader to [21,26] for more details. System (2.1)-(2.3) can be rewritten as an abstract evolution equation of the form…”

“…In recent years, many regularized equations have been proposed for the purpose of direct numerical simulations of turbulent incompressible flows modeled by the Navier-Stokes (NSE) equations [26] (for models of two-phase incompressible fluid flows, we refer the reader to [21] for further discussion and results). One such regularized model is the Navier-Stokes-Voigt (NSV) equations introduced by Oskolkov [41] as a model for the motion of a linear, viscoelastic, incompressible fluid.…”

“…While the first issue (a) has already been treated in [21,Section 6] to some extent as a special case of a general family of regularized models, in this contribution we also wanted to rephrase the results associated with the 3D NSV into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α > 0. One feature of this analysis is that we are able to obtain the optimal regularity of the global attractor in one step as opposed to employing a complicated bootstrapping procedure as in [33].…”

“…We emphasize that the former procedure has proved quite useful in the treatment of other regularized models for the 3D NSE (cf. [21,Section 6]). …”