Homogenization of Evolutionary Incompressible Navier–Stokes System in Perforated Domains

Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

“…We emphasize that the Darcy's law in [LY23,All90b] is exactly the same as (1.10) whereas the Darcy's law in [Tar80,Mik91] differs quantitatively, in terms of a different resistance tensor R per which is obtained analogously as R from (1.5) but by solving the Stokes equations in the torus instead of the whole space. The reason for this difference is that in the case α = 1 the particle diameter is comparable to the interparticle distance.…”

“…Formally they are obtained by setting γ = 0 in (1.6) and taking the limit ε → 0. The critical regime, α = 3, leading to the Navier-Stokes-Brinkman equations, has been considered by Feireisl, Nec ˇasová and Namlyeyeva [FNN16], whereas the subcritical case α > 3 and the supercritical case α ∈ (1, 3) has been treated recently by Lu and Yang [LY23].…”

“…These functions w ε from [All90a, All90b] (which go back to corresponding functions in [Tar80] and similar functions for the Poisson equations used by Cioranescu and Murat in [CM82]) have been used with some modifications in many related works, see e.g. [GH19,LY23]. However, w ε truncates on an ε-neighborhood around the particles, and therefore we could only use them directly in the present context provided the Reynolds number on the εlengthscale is small.…”

Homogenization of the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in a perforated pipe

“…In many of such studies, of particular relevance is the possibility of calculating the velocity net fluxes from prescribed pressure drops, or alternatively, of finding the pressure drops that originate these net fluxes [27]. From a strictly mathematical point of view, starting from the pioneering works of Marchenko & Khruslov [38, Chapter IV], Sánchez-Palencia [19,45] and Tartar [48], passing through the benchmark contributions of Allaire [1][2][3] and Conca [12][13][14], homogenization methods in the context of viscous incompressible fluid flows (in stationary regime, see also [34]) have attracted the interest of several authors that adapted and expanded such techniques to models involving unsteady incompressible [22,37,41], viscous compressible [20,36,39,43] or heat-conducting [21,23,35,44] fluid flows, among many others (the list of references presented is far from being exhaustive). Concerning the boundary conditions imposed on the velocity field, while Navier boundary conditions on the surface of the perforations have been treated in [3,12], the no-slip boundary condition is typically assumed on the external boundary, that is, the velocity field is set to be zero on the boundary of the domain containing the perforations.…”