We construct a Bogovskii-operator for a heterogeneous domain consisting of two bulk regions separated by a thin perforated layer with thickness and periodicity of order ɛ. In particular, we consider various types of boundary conditions at the outer boundary of the domain and determine the explicit dependence of the operator’s norm on ɛ. Finally, we apply this result to establish uniform a priori bounds for the pressure and then derive the effective problem for the corresponding microscopic Stokes model.

We study the homogenization of the compressible isentropic Navier-Stokes equations in periodically perforated domains where the size of the obstacles is of the same order as the distance between neighboring obstacles. Using the two-scale convergence method, which can be characterized via the unfolding operator, we derive the corresponding macroscopic model determined by Darcy's law. In particular, the macroscopic density satisfies the porous medium equation. The main challenge lies in identifying the pressure term in the limit. We overcome this by establishing the strong two-scale convergence of the densities, which is achieved by controlling the oscillation defect measure of the unfolded densities. A crucial contribution of our work is the development of a methodological framework applicable to more complex compressible fluid models. Furthermore, regarding conservative forces, we extend existing results from the literature to adiabatic constants γ> 9/5.

This investigation develops basic methods for the multi-scale analysis for problems in thin porous layers. More precisely, we provide tools for the homogenization in case of "tangentially" periodic structures and dimensional reduction letting the layer thickness tend to zero proportional to the scale parameter $\epsilon$. A crucial point is the identification of scale limits of functions $v_{\epsilon}$ in subsets of function spaces characterized by uniform a priori estimates with respect to $\epsilon$, arising for solutions of differential equations in heterogeneous media with thin layers, e.g., of a Navier-Stokes system, models in linear elasticity, or problems with fluid-structure interaction. Often in problems from continuum mechanics, in a first step, the symmetric gradients of arising vector fields can be controlled and Korn's inequality in porous layers is required to estimate the gradients, such that crucial constants do not depend on $\epsilon$. Controllable pore-filling extension are constructed and, thus, the analysis is reduced to a fixed basic domain. The proof of the required Korn-inequalities for porous thin layers, formulated with respect to $L^p$-spaces, is based on these constructions. Also, the investigation of compactness with respect to two-scale convergence and the characterization of the scale limits is strongly based on the extension theorem and the Korn-inequalities. To illustrate the range of applications of the developed analytic multiscale method a semi-linear elastic wave equation in a thin periodically perforated layer with an inhomogeneous Neumann boundary condition on the surface of the elastic substructure is treated and an homogenized, reduced system is derived.

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin epsilon-periodic perforated layers wit a thickness of order epsilon alpha with alpha is an element of (0, 1). Therefore the thickness of the layer is large compared with its porosity. The aim is the derivation of effective models for epsilon -> 0, when the thickness of the layer tends to zero. For the flow problem, we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer. The transport problem is given by reaction-diffusion-advection problem with advective flow governed by the fluid velocity from the Stokes model. Furthermore, we treat different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with the Darcy velocity depending only on the derivative of the Darcy pressure in the vertical direction. The effective equation for the transport problem is again one of the diffusion advection-type including homogenized coefficients, and with advective flow given by the Darcy velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where, in the case of high diffusion in the horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit, we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii operator for thin perforated domains with arbitrary boundary conditions on the top/bottom, and the strong two-scale convergence for the microscopic solution of the transport equation, which is necessary to pass to the limit in the advective term. This strong convergence is established by using a Kolmogorov-Simon compactness argument.

We consider the homogenization of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size epsilon. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer, which is assumed to be radially symmetric with a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection-reaction-diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for epsilon -> 0, based on epsilon-uniform a priori estimates. As a result of the passage to the limit, the processes on the macroscale are described by an advection-reaction-diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells that depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro-macro model. Homogenization results have been obtained recently for the case of reactive-diffusive transport coupled with microscopic domain evolution, but in the absence of advective transport. We extend these models by including the advective transport, which is driven by the Stokes equations in the a priori unknown evolving pore domain.

In this paper, we investigate an effective model for reactive transport in elasticallydeformable perforated media. This model was derived by formal asymptotic expansions in [J. Knoch,M. Gahn, M. Neuss-Radu, and N. Neu{\ss},Transp. Porous Media, 147 (2023), pp. 93--123], startingfrom a microscopic model consisting of a linear elasticity problem on a fixed domain, i.e., in theLagrangian framework, and a problem for reactive transport on the current deformed domain, i.e.,in the Eulerian framework. The effective model is of micro-macro type and features strong nonlinearcouplings. Here, we prove global existence in time and uniqueness for the effective micro-macro modelunder a smallness assumption for the data of the macroscopic elasticity subproblem. Moreover, weshow numerically the convergence of microscopic solutions towards the solution of the effective modelwhen the scale parameter\varepsilon >0 becomes smaller and smaller, and also compute the approximationerror. The numerical justification of the formally derived effective micro-macro model is particularlyimportant, as rigorous analytical convergence proofs or error estimates are not available so far.Finally, we compare the effective micro-macro model with alternative, simpler effective descriptionsof transport in elastic perforated media.

This paper presents a rigorous derivation of an effective model for fluid flow through a thin elastic porous membrane separating two fluid bulk domains. The microscopic setting involves a periodically structured porous membrane composed of a solid phase and fluid-filled pores, with thickness and periodicity of order ε, small compared to the size of the bulk regions. The microscopic model is governed by a coupled fluid-structure interaction system: instationary Stokes equations for the fluid and linear elasticity for the solid, with two distinct scalings of the elastic stress tensor yielding different effective behaviors. Using two-scale convergence techniques adapted to thin domains and oscillatory microstructures, the membrane is reduced to an effective interface across which transmission conditions are derived. The resulting macroscopic model couples the bulk fluid domains via effective interface laws of Navier-slip-type including the dynamic displacement. The character of this coupling depends critically on the choice of the scaling in the elastic stress tensor, leading to either a membrane equation or a Kirchhoff-Love plate equation for the effective displacement. The resulting interface conditions naturally admit mass exchange between the adjacent fluid regions. In the analytical framework, a new two-scale compactness theorem for the symmetric gradient is established, underpinning the passage to the limit in the coupled system. Moreover, cell problem techniques are employed systematically to construct admissible test functions and to rigorously extract the effective macroscopic coefficients.

We study incompressible fluid flow through a thin poro elastic layer and rigorously derive a macroscopic model when the thickness of the layer tends to zero. Within the layer, we assume a periodic structure, and both the periodicity and the thickness of the layer are of order epsilon, which is small compared to the length of the layer. The fluid flow is described by quasi-static Stokes equations, and for the elastic solid, we consider linear elasticity equations, and both are coupled via continuity of the velocities and the normal stresses. The aim is to pass to the limit epsilon -> 0 in the weak microscopic formulation by using multiscale techniques adapted to the simultaneous homogenization and dimension reduction in continuum mechanics. The macroscopic limit model is given by a coupled Biot plate system consisting of a generalized Darcy law coupled to a Kirchhoff-- Love-type plate equation including the Darcy pressure.

In this paper, we study the asymptotic behavior of solutions to the compressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the full-slip boundary conditions, we show that in the asymptotic limit the fluid sticks completely to the boundary, provided the oscillations are non-degenerate, meaning not oriented in a single direction.

We consider a coupled model for fluid flow and transport in a domain consisting of two bulk regions separated by a thin porous layer. The thickness of the layer is of order ε , and the microscopic structure of the layer is periodic in the tangential direction also with period ε . The fluid flow is described by an instationary Stokes system, properly scaled in the fluid part of the thin layer. The evolution of the solute concentrations is described by a reaction–diffusion–advection equation in the fluid part of the domain and a diffusion equation (allowing different scaling in the diffusion coefficients) in the solid part of the layer. At the microscopic fluid–solid interface inside the layer, nonlinear reactions take place. This system is rigorously homogenized in the limit ε→ 0 , based on weak and strong (two-scale) compactness results for the solutions. These are based on new embedding inequalities for thin perforated layers including coupling to bulk domains. In the limit, effective interface laws for flow and transport are derived at the interface separating the two bulk regions. These interface laws enable effective mass transport through the membrane, which is also an important feature from an application point of view.

We consider the homogenisation of a coupled Stokes flow and advection–reaction–diffusion problem in a perforated domain with an evolving microstructure of size ε. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer having a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection–reaction–diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for ε→ 0, based on ε-uniform a priori estimates. As a result of the limit passage, the processes on the macroscale are described by an advection–reaction–diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells, which depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro–macro model. For pure reactive–diffusive transport coupled with microscopic domain evolution but without advective transport, homogenisation results have recently been presented. We extend these models by advective transport which is driven by the Stokes equation in the a priori unknown evolving pore domain.

In this paper we present the homogenization for nonlinear viscoelastic second-grade non-simple perforated materials at large strain in the quasistatic setting. The reference domain Ω _ε is periodically perforated and is depending on the scaling parameter ε which describes the ratio between the size of the whole domain and the small periodic perforations. The mechanical energy depends on the gradient and also the second gradient of the deformation, and also respects positivity of the determinant of the deformation gradient. For the viscous stress we assume dynamic frame indifference and it is therefore depending on the rate of the Cauchy-stress tensor. For the derivation of the homogenized model for ε→ 0 we use the method of two-scale convergence. For this uniform a priori estimates with respect to ε are necessary. The most crucial part is to estimate the rate of the deformation gradient. Due to the time-dependent frame indifference of the viscous term, we only get coercivity with respect to the rate of the Cauchy-stress tensor. To overcome this problem we derive a Korn inequality for non-constant coefficients on the perforated domain. The crucial point is to verify that the constant in this inequality, which is usually depending on the domain, can be chosen independently of the parameter ε . Further, we construct an extension operator for second order Sobolev spaces on perforated domains with operator norm independent of ε .

We investigate a reaction-diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk-domains are given by a reaction-diffusion equation for the traces of the bulk-solutions, including nonlinear reaction-kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenization results for surface diffusion. To cope with the nonlinear terms we derive strong two-scale results.

We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: 1) dimensional reduction of the layer and 2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter $\epsilon>0,$ which is related to the thickness of the layer as well as the size of the periodic structure. By letting $\epsilon$ tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit velocity and limit pressure satisfy a two pressure Stokes model, from which a Darcy law for thin layers can be derived. Due to non-standard boundary conditions, some additional terms appear in Darcy's law.

In this paper, we derive an effective model for transport processes in periodically perforated elastic media, taking into account, e.g., cyclic elastic deformations as they occur in lung tissue due to respiratory movement. The underlying microscopic problem couples the deformation of the domain with a diffusion process within a mixed Lagrangian/Eulerian formulation. After a transformation of the diffusion problem onto the fixed domain, we use the formal method of two-scale asymptotic expansion to derive the upscaled model, which is nonlinearly coupled through effective coefficients. The effective model is implemented and validated using an application-inspired model problem. Numerical solutions for both, cell problems and macroscopic equations, are investigated and interpreted. We use simulations to qualitatively determine the effect of the deformation on the transport process.

In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. In the initial state, the microscopic geometry is a periodically perforated domain, each perforation being a spherical solid grains. A small parameter ϵ is characterizing both the distance between two neighboring grains, and the radii of the grains. For each grain, the radius depends on the unknown (the solute concentration) at its surface. Therefore, the radii of the grains change in time and are model unknowns, so the model involves free boundaries at the micro scale. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method, we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to ϵ. Finally, letting ϵ→0, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.

In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order ϵ, where ϵ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to an interface Σ in the limit ϵ→0 using two-scale convergence. The effective model consists of the Stokes equation coupled to a time-dependent plate equation on the interface Σ including homogenized elasticity coefficients carrying information about the micro structure of the layer. In the zeroth-order approximation we obtain continuity of the velocities at the interface, where only a vertical movement occurs and the tangential components vanish. The tangential movement in the solid is of order ϵ and given as a Kirchhoff-Love displacement. Additionally, we derive higher-order correctors for the fluid in the thin layer.

Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order \begin{document}$ \epsilon $\end{document}, where \begin{document}$ \epsilon $\end{document} is much smaller than the size of the bulk domains. For the singular limit \begin{document}$ \epsilon \to 0 $\end{document}, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard a priori estimates are not enough to pass to the limit \begin{document}$ \epsilon \to 0 $\end{document} in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).