Homogenization of the evolutionary Navier–Stokes system

Abstract

We study the homogenization problem for the evolutionary Navier-Stokes system under the critical size of obstacles. Convergence towards the limit system of Brinkman's type is shown under very mild assumptions concerning the shape of the obstacles and their mutual distance.

The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier–Stokes system

“…We refer to [22] for a number of real world applications.The problem is relatively well understood in the framework of stationary, viscous fluid flows represented by the the standard Stokes and/or Navier-Stokes system of equations. Allaire [3], [4] (see also earlier results by Tartar [23]) identified three different scenarios for the case of periodically distributed holes:• the supercritical size of holes for which the asymptotic limit is Darcy's law;• the critical size giving rise to Brinkman's law;• the subcritical size of holes has no influence on the motion in the asymptotic limit -the limit problem coincides with the original one.Related results for the evolutionary (time-dependent) incompressible Navire-Stokes system were obtained by Mikelić [18] and, more recently, in [12].Considerably less is known in the case of compressible fluids. Masmoudi [17] identified rigorously the porous medium equation (Darcy's law) as a homogenization limit for the evolutionary barotropic (compressible) Navier-Stokes system in the case where the diameter of the holes is comparable to their mutual distance, which is a subcase of the supercritical case, similar results for the full Navier-Stokes-Fourier system were obtained in [13].In [9], we considered the compressible (isentropic) stationary Navier-Stokes system in the subcritical regime, where the spatial domain is perforated by a periodic lattice of holes of subcritical size.…”

Homogenization of the compressible Navier–Stokes equations in domains with very tiny holes

“…Allaire showed that when 1 ≤ α < 3 corresponding to the case of large holes, the limit fluid behavior is governed by the classical Darcy's law; when α > 3 corresponding to the case of tiny holes, the equations do not change in the homogenization process and the limit problem is determined by the same system of Stokes or Navier-Stokes equations; when α = 3 corresponding to the case of critical size of holes, in the limit there yields the Brinkman's law-a damping term is added to the original system, which looks like a combination of the original Stokes or Navier-Stokes equations and the Darcy's law. Related results for the evolutionary (time-dependent) incompressible Navier-Stokes system were obtained by Mikelić [20] and, more recently, by Feireisl, Namlyeyeva and Nečasová [11]. We note that the holes are assumed to be periodically distributed in Allaire's results, while in [11] more general distribution of holes was considered.…”

“…Related results for the evolutionary (time-dependent) incompressible Navier-Stokes system were obtained by Mikelić [20] and, more recently, by Feireisl, Namlyeyeva and Nečasová [11]. We note that the holes are assumed to be periodically distributed in Allaire's results, while in [11] more general distribution of holes was considered.…”

On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

“…Our method is directly inspired from the one introduced in [10], the main difference being the appearance of the so-called Stokes' resistance matrix that we introduce below. These matrices are related to the 'Stokes' capacity" used in [7] where the case of fixed particles is considered. Finally let us also mention the recent paper [9] which relies on the method of reflections, the proceeding paper [15] in which the case of different shapes is explored but only for scalar equations with constant boundary conditions and a rather formal presentation, the paper [11] for the identification of a dilute regime of sedimentation and the recent papers [5,6,13] dealing with the case of a dilute cloud of fixed particles for which the limit system is the Stokes equations in the full space without any Brinkman force.…”