OSMOSIS FOR NON-ELECTROLYTE SOLVENTS IN PERMEABLE PERIODIC POROUS MEDIA

The paper gives a rigorous description, based on mathematical homogenization theory, for flows of solvents with not charged solute particles under osmotic pressure for periodic porous media permeable for solute particles. The effective Darcy type equations for the flow under osmotic pressure distributed within the porous media are derived. The effective Darcy law contains an additional flux term representing the osmotic pressure. Coefficients in the effective homogenized equations are related to the values of the phenomenological coefficients in the Kedem-Katchalsky formulae (2).


1.
Introduction. The goal of the present paper is to give a rigorous description, based on mathematical homogenization theory, of flows of non-electrolyte solutions (that is electrically neutral solute particles) under osmotic pressure for periodic porous media permeable for solute particles.
Osmosis is historically the term for a phenomenon of spontaneous passage of water or other solvents through a membrane that is permeable to the solvent but is impermeable for solute particles. If a solution is separated by such a semipermeable membrane from the pure solvent, the pure solvent will move through the membrane making the solution at the other side of the membrane more dilute. This process can be stopped by applying external counter pressure that gives an idea of osmotic pressure.
Osmosis explains in particular how living cells as red blood cells or plant cells adapt their shape to the environment stress by changing concentration of solutes (sucrose in case of plants cells) inside them. This phenomenon was discovered by French experimental physicist Jean-Antoine Nollet in 1748 in natural membranes but was first studied in detail by a German plant physiologist Wilhelm Pfeffer only in 1877. The term osmose or osmosis was introduced by a British chemist, Thomas Graham in 1854.
The Dutch chemist van't Hoff showed in 1886 that, for dilute solutes the osmotic pressure varies with concentration and temperature similarly to an ideal gas. A classical formula by van't Hoff for osmotic pressure acting on solvent at the border of the membrane impermeable for solute particles reads [33]: where ρ is the concentration of solute particles, T is temperature and k is the Boltzmann constant. This relation led to practical methods for determining molecular weights of solutes.
Osmotic pressure plays important role in biological processes as transport in plants [19] and through cell membranes [21], [15] and also in several modern membrane technologies in particular for desalination and sustainable power generation [10], [38], [24].
There is a physical phenomenon called by chance similarly -electroosmosis. Electro-osmotic flows in micro channels are driven by external electric fields, acting on charged solute particles that initiate the solvent flow through the viscous interaction. This phenomenon was discovered by F.F. Reuss [28] in 1809. Electroosmosis of charged particles with corresponding electrokinetic models [25], [31] and osmosis of neutral particles have certain similarities from the mathematical point of view, but are rather different in physical nature.
In many situations porous membranes are not completely impermeable to solute particles, but depending on the size of pores, obstruct to some extend the passage of particles. The effect of osmotic pressure in this case is not concentrated on the surface of the membrane, but is distributed within the membrane's volume. A combination of several complicated phenomena define the joint transport of solute and solvent through the membrane in this case. The question about the nature of osmotic pressure in such intermediate regimes did not have a rigorous answer up to now.
Several phenomenological models based on general thermodynamical principles were suggested to extend formula (1) to the case when a porous membrane is partially permeable to neutral solute particles, as the Kedem-Katchalsky formulae J u = L p ∆p − L pD ∆p osm (2) J D = −L Dp ∆p + L D ∆p osm that connect fluxes J u and J D of solvent and of solute particles through the slab of a porous material with the value of the pressure drop ∆p in the solvent and the solute concentration jump ∆ρ [20], [21]. Here the phenomenological coefficients L p , L pD , L Dp , L D are called coefficients of filtration, osmotic transport, ultrafiltration and diffusion, respectively. The relation between the osmotic transport coefficient and the filtration coefficient is called membranes reflection coefficient. The goal of the present paper is to derive using mathematical homogenization theory a consistent macroscopic model for transport of solvents and neutral solutes in porous media that are permeable for solute particles. We consider as a microscopic model a system of equations of Nernst-Planck-Stokes type describing a slow flow of viscous fluid solvent together with the advection-diffusion of the solute particles through a periodic porous solid microstructure with period ε 1 under the effect of potential forces acting on the solute particles through a potential V concentrated along the surface of the porous structure. Such kind of models for flows under osmotic pressure were considered in the case of one dimensional flows in thin channels by Anderson and his coauthors [8] [7] and were developed also in [35], [17], [16]. They were applied to simple geometries in [37], [19], [36]. Neither rigorous mathematical analysis nor numerical analysis of these models in case of general geometry has been done up to now.
Related mathematical problems for Nernst-Planck-Poisson and Nernst-Planck-Poisson-Stokes systems for non-stationary electrokinetic models were considered in papers [25], [30], [31], [6], [5]. In [25] the homogenization problem for periodic micro-structures for the stationary Nernst-Planck-Poisson-Stokes system is considered, and formal asymptotic expansions for solutions are constructed. Rigorous justification of convergence to homogenized solution is given for the non-stationary Nernst-Planck-Poisson-Stokes system in [31], [6]. Similar results for non-ideal transport when finite size of ions is taken into account, were obtained in [5]. A number of works on electro-osmosis in porous media is available in physical literature, see for example [12], [9], [29].
The main results of the present paper are following. Introduction and mathematical analysis of a new model for the microscopic picture of osmotic flow for non-electrolyte (not charged) solute transport at the pore level. Derivation using mathematical homogenization theory of new effective Darcy's type equations for the flow under osmotic pressure distributed within the porous media. The new formula (5.3) for the distribution of osmotic pressure inside the porous media gives a quantitative answer about the nature of the osmotic transport. Coefficients in the derived homogenized equations relate values of the phenomenological coefficients in (2) with properties of the osmotic flow at the pore level.
The present paper deals with the stationary transport of neutral solute particles where the potential of forces acting on the particles is given and can grow infinitely for points approaching the boundary. This leads to possible degeneracy of the diffusion equation in the vicinity of the boundary and to corresponding complications in mathematical analysis. In this respect the considered model is mathematically more complicated than models for electro-osmosis where the potential satisfies the Poisson equation and is regular. One of the new features of the studied problem is the choice of boundary conditions for the flow equations describing a flow through a reservoir with prescribed pressure drop between the inflow and outflow parts of the boundary.
We consider in the present paper an N -dimensional porous structure with N = 2, 3, that fills an open domain Ω surrounded by solid lateral walls Γ 0 and by flat inflow and outflow boundaries S 1 and S 2 in two planes orthogonal to one of the coordinate axes. It is assumed that Γ 0 is a Lipschitz continuous surface.
Through this paper we suppose that the boundary of the porous structure is a Lipschitz continuous and periodic surface. The periodicity cell is denoted by Y. Without loss of generality we suppose that Y = [0, 1) N . We denote by Y F an open set on Y and assume that it is Lipschitz and its periodic extension to R N is a connected set. In what follows we refer to Y F as the fluid part of the porous medium. Y S = Y \Y F denotes the solid part of the structure in Y . The scaled periodicity cell is denoted by Y ε . Cells including the structure match exactly the outflow and inflow boundaries S 1 and S 2 of Ω. Ω ε denotes the fluid part of the domain Ω together with the porous structure, and ∂Ω ε is its boundary. Γ ε is the solid part of the boundary ∂Ω ε of the flow domain including the structure boundary and the solid boundary Γ 0 of Ω. The inflow and outflow parts of ∂Ω ε are denoted by S ε 1 and S ε 2 . We denote by C ε the union of scaled periodicity cells that are completely included into the domain Ω: The fluid solvent is described by the Stokes equations for velocity u ε and pressure p ε with external forces coming from friction between the particles and the fluid. We impose non-slip boundary conditions for the velocity u ε in the Stokes equations on the solid boundary Γ ε and impose boundary conditions on the inflow and outflow boundaries S ε 1 and S ε 2 for pressure p ε as constant values P 1 and P 2 , and for tangential component of velocity as u ε,τ = 0.
The solute concentration ρ ε satisfies the advection diffusion equation with drift force defined in terms of the potential V ε with support concentrated along the solid boundaries. V is a periodic function on Y, and we denote the scaled potential by V ε (x) = V x ε . We apply zero normal flux boundary condition for the solute concentration ρ ε on the solid boundary Γ ε and the Dirichlet boundary conditions for ρ ε on inflow and outflow boundaries S ε 1 and S ε 2 defined as S ε i = S i ∩ Ω ε , i = 1, 2. We consider a boundary value problem for the system of PDEs consisting of the Stokes equations for velocity u ε and pressure p ε of the solvent with the osmotic force ρ ε ∇V ε and the advection-diffusion equation with advection velocity u ε and drift term div (κρ ε ∇V ε ) .
The strong formulation of the boundary value problem reads: for the Stokes equations and for the advection-diffusion equation. Here µ is viscosity, λ is diffusion constant, κ is the mobility of solute particles, θ 2 ≥ 0 is a constant, and β ε (x) = exp(− κ λ V ε (x)); n is the exterior normal on ∂Ω ε .
The weak formulation of problem (5)-(6) and conditions for well posedness of this problem are given in Sections 2 and 3.
We notice that according to the Einstein-Smoluchowski relation [14], [34] λ κ = kT where T is absolute temperature and k is the Boltzmann constant, and van't Hoffs formula (1) for osmotic pressure can in our notations be rewritten as To illuminate the effects of osmosis in the Stokes equation we observe that and rewrite the equation (5a) as with the expression λ κ ∇ρ ε = ∇p ε,osm for the osmotic pressure (8) included explicitly.
We formulate here also a boundary value problem for pressure that follows from (5) Only the difference δP = P 1 − P 2 between pressure values at the inflow and outflow boundaries S 1 and S 2 has physical meaning. We will control only δP and will normalize pressure by the condition Ωε p ε dx = 0 (12) The main result of the work is deriving a limit macroscopic system consisting of an effective diffusion equation (95) and a Darcy type equation (96) with additional flux representing the effect of the osmotic pressure distributed within the structure.
In the case of a flat membrane, the corresponding effective matrices B D and B osm (90) in (96), are related to the filtration and osmotic transport coefficients L p and L pD in the Kedem-Katchalsky formula (2). The paper is organized as follows. In Section 2 we provide the problem setup and obtain apriori estimates in weighted Sobolev spaces for solutions of the studied system. In the second part of this section we use contraction arguments to justify the well posedness of the system under consideration.
In Section 3 we pass to the two-scale limit in the advection-diffusion equation with potential forces. Here we use two-scale convergence in the variable spaces approach [39]. Section 4 is devoted to the homogenization of velocity and pressure satisfying the Stokes system with osmotic forces originated in potential forces acting on the solute and in the density gradient of the solute.
The goal of Section 5 is to derive the macroscopic Darcy's law with osmotic pressure distributed within the porous structure. The result is obtained by excluding the fast variable from the two-scaled effective system of equations.
Finally, in the Appendix we adapt results on the Friedrichs and Poincare type inequalities from [22] and [27] to the weighted Sobolev spaces specific for our problems. Also we provide nontrivial examples of potentials and corresponding weights such that the desired Friedrichs and Poincare inequalities hold true.

ALEXEI HEINTZ AND ANDREY PIATNITSKI
2. Weak formulation of the problems and a priori estimates.
2.1. Apriori estimates for the advection diffusion equation with drift by osmotic forces. We derive in this section a week formulation of problem (6) in terms of weighted spaces L 2 (Ω ε , β ε ), W 1 2 (Ω ε , β ε ) with scalar products and weight The space L p (Ω ε , β ε ) is defined by the norm Typical potentials V ε (x) in our problems are nonnegative and bounded on compact subsets of Ω ε , and are rising, may be infinitely, for points tending to the solid part Γ ε of the boundary of Ω ε . By using the formula: with β −1 ε = 1/(β ε ), the following symmetrization of the advection diffusion equation with potential forces is achieved. We multiply the advection-diffusion equation (6a) by an arbitrary function of the form ψβ −1 ε ∈ W 1 2 (Ω ε , β ε ) and integrate the resulting relation by parts using (18). Boundary conditions imply that after integration by parts the sum of all fluxes on the solid boundary Γ ε is zero.
The connectedness of Ω ε and positivity of β ε on Ω ε implies that the measure β ε dx is ergodic in the sense of Zhikov [39]. By other words the equality Ωε |∇f | 2 β ε dx = 0 implies that f is constant almost everywhere with respect to the measure β ε dx. Potentials V ε (x) appearing in the problems of interest are natural to interpret as ). If the potential V ε (x) goes to infinity when x approaches the solid boundary Γ ε , that can naturally happen in applications, the weight β ε (x) degenerates at Γ ε .
We provide in the Appendix a number of sufficient conditions for the Friedrichs inequality and the Poincare inequality in weighted Sobolev spaces from [27]. We also give examples of potentials V ε (d Γ (x)) such that these conditions are satisfied . For later analysis of the coupled advection-diffusion and Stokes equations we consider first two auxiliary problems for the equation (19): one with homogeneous boundary conditions on S ε 1 ∪ S ε 2 with a given right hand side, and another one with inhomogeneous boundary conditions and zero right hand side.
The first problem in weak form reads: satisfies the integral relation: and the boundary conditions for an arbitrary function ψβ −1 ε ∈ W 1 2 (Ω ε , β ε ) that satisfies boundary conditions on the inflow and outflow parts S ε 1 and S ε 2 of the boundary. Here G n stands for the normal component of the vector function G. For the coupled system of advectiondiffusion and Stokes equations we will substitute G with G = 1 λ ρ ε β −1 ε u ε . The Friedrichs inequality implies that problem (22) is coercive and the solution operator . Namely the following bound holds: Notice that according to (20) the constant C R1 does not depend on ε.
The second auxiliary problem in weak form reads: find a ε β −1 ε ∈ W 1 2 (Ω ε , β ε ) such that a ε satisfies the integral relation: and the boundary conditions for an arbitrary function ψβ −1 In order to construct a solution of this problem, we first introduce a function for an arbitrary function ψβ −1 and the boundary conditions Combining an energy estimate following from (26) with ψβ −1 ε = g ε β −1 ε and the weighted Friedrichs inequality (20) we obtain by means of the Lax -Milgram lemma the existence and uniqueness of solutions to (26).
The corresponding solution operator R 2 (θ 2 ) = a ε β −1 ε is bounded and satisfies the estimate: 2.2. Weak formulation and apriori estimates for the Stokes equation. We introduce the space of smooth solenoidal vector valued functions equal to zero on the solid part Γ ε of the boundary, having zero tangential component ϕ τ on the inflow and outflow parts of the boundary S ε 1 ∪ S ε 2 , and possibly non zero normal component ϕ n on S ε 1 ∪ S ε 2 . We will also use the space and the space A weak formulation of the Stokes boundary value problem with given constant pressure p ε = P i and tangential velocity u ε,τ = 0 on the inflow and outflow boundaries S ε i , i = 1, 2, is formulated following ideas in [13] and [18]. We find a function u ε ∈ J # 1 (Ω ε ) that for arbitrary ϕ ∈ J # 1 (Ω ε ) satisfies the integral relation: To derive the weak formulation (33) from the strong one we multiply the Stokes equation (10) by a solenoidal test function ϕ ∈ D # (Ω ε ) and integrate by parts taking into account boundary conditions: u ε = ϕ = 0 on the solid boundary Γ ε ; p ε = P i and u ε,τ = ϕ τ = 0 on the inflow and outflow boundaries S ε i , i = 1, 2. This yields We observe that in the case of dimension N = 3, for two orthogonal tangential directions τ 1 and Therefore ∂uε,n ∂n ≡ 0 on S ε 1 ∪ S ε 2 and it leads to the following simplification: Similar formula evidently holds for dimension N = 2. Together with the relation following from the constraint div(ϕ) = 0 and from the boundary conditions (6c), (5d) for ρ ε , p ε it implies the equation (33).
) and the restriction of β ε (x) on S ε i is a periodic function with period ε on cells of dimension N − 1. There is an auxiliary function with the constant C independent of ε. We reformulate the equation (33) by substracting Π ε (x) from the pressure p ε . Find function u ε ∈ J # 1 (Ω ε ), that for arbitrary ϕ ∈ J # 1 (Ω ε ), satisfies the integral relation: similar to (33) but with zero boundary terms on S ε i . For a fixed ρ ε β −1 ε ∈ W 1 2 (Ω ε , β ε ) and for Π ε ∈ W 1 2 (Ω ε ) this equation and the equivalent equation (33) have a unique solution in J # 1 (Ω ε ) by the Lax Milgram Lemma since the linear functional . This argument is classical, see [23], [13], [18]. We consider corresponding estimates in more detail later.
We deal in this section with estimating solutions of the Stokes equation. This estimate is crucial for the homogenization analysis.
We consider first a general form of the Stokes equations with zero boundary terms on S ε 1 and S ε 2 : Consider this integral relation for ϕ = u ε : The scaling argument for Friedrichs inequality on the periodicity cell implies and after one more similar argument an a priory estimate for the [L 2 (Ω ε )] N norm of u ε follows: Therefore the solution operator with the potential Π ε representing as above, the effect of the hydrostatic pressure drop P 1 − P 2 between S ε 1 and S ε 2 together with the classical osmotic pressure λ κ ρ ε , has a solution operator S 2 satisfying estimates Remark. Notice that if the density ρ ε of the solute has a constant value r 2 on S ε 2 and is zero on S ε 1 , the last estimates depend just on a simple balance between the hydrostatic pressure drop P 1 − P 2 and the osmotic pressure p osm = λ κ r 2 : 3. Abstract contraction argument for quadratic non-linearity and apriori estimates for the coupled system. We consider now the following joint system of equations for flow and advection-diffusion. Structure interacts with the solute through the potential V ε and by that acts on the solvent. The Stokes equations and the advection-diffusion equation are coupled here through the first order terms.
The joint system in weak form reads: with velocity u ε ∈ J # 1 (Ω ε ), arbitrary ϕ ∈ J # 1 (Ω ε ), scaled concentration ρ ε β −1 ε (x) = 0 for x ∈ S ε 1 . We reformulate this system of equations in abstract form using notations for solution operators of the decoupled auxiliary equations considered above: Formally we can write down a non-linear operator equation for ρ ε only: and want to show that for small ε the nonlinear operator in the right hand side of (45) is a contraction in W 1 2 (Ω ε , β ε ). To reach this goal we estimate first R 1 ρ ε β −1 ε u ε and ρ ε β −1 ε u ε that appear in the weak form of the advection-diffusion equation. The Hölder inequality, the estimate β ε ≤ 1 and the Sobolev imbedding theorems W We point out that despite the fact that Ω ε depends on ε, the last estimates are uniform with respect to ε → 0 because the porous structure boundary Γ ε is Lipschitz on the periodicity cell Y and therefore admits a uniformly bounded extension within the same Sobolevs class. The interpolation inequality together with the earlier estimates (41) for the Stokes solution operators implies that Using estimates (49) and (47), for the operators we obtain the following inequalities : and similarly The last two estimates imply that in any ball of radius R 0 in W 1 2 (Ω ε , β ε ) for a sufficiently small ε the operator B ρ ε β −1 ε is a contraction. Indeed, Choosing ε 0 so that we conclude that for any ε ≤ ε 0 the operator B is a contraction in the ball of radius R 0 in W 1 2 (Ω ε , β ε ) and maps this ball into itself. Theorem 3.1. For 0 < ε ≤ ε 0 with ε 0 satisfying (54), the nonlinear operator equation (45) corresponding to the system of equations (43) has a unique solution ρ ε β −1 ε ∈ W 1 2 (Ω ε , β ε ) .The system of equations (43) has a unique solution ρ ε β −1 ε , u ε with ρ ε β −1 ε ∈ W 1 2 (Ω ε , β ε ) and u ε ∈ J # 1 (Ω ε ) satisfying the following estimates with constants C independent of ε: We notice that P 1 − P 2 is the pressure drop between inflow and outflow parts of the boundary S ε 1 and S ε 2 , and the expression λ κ θ 2 is similar to the classical formula (8) for the osmotic pressure in the vicinity of an impermeable membrane. 4. Homogenization by two-scale convergence for the concentration of solute particles. This and the next section are devoted to passing to the two-scale limit [3] [26] in the system of equations (43) and obtaining a homogenized limit problem. Uniformity of the obtained estimates lets extend solutions u ε , ρ ε β −1 ε to the whole domain Ω in such a way that estimates (55)-(57) hold for the extended functions with a constant C that does not depend on ε, see [1]. We also extend β ε by zero outside Ω ε for convenience. We keep here the same notations for the extended functions.
Let µ be a periodic Borel measure normalized on the periodicity cell Y : µ (Y ) = 1 and µ ε be the scaled measure defined by for each Borel set B. The measure µ ε converges weakly to the Lebesgue measure dx in the sense that R N ϕdµ ε → R N ϕdx for any ϕ ∈ C 0 (R N ). We consider a sequence of measures µ ε and a sequence of functions z ε ∈ L 2 (Ω, dµ ε ) and test functions Φ(x, y) = ϕ(x)ψ(y) with ϕ ∈ C ∞ 0 (Ω) and ψ ∈ C ∞ per (Y ), where C ∞ per (Y ) stands for the space of smooth periodic functions on Y . Definition 4.1. The sequence z ε such that z ε L2(Ω,dµε) ≤ const is weakly twoscale convergent to a periodic in y ∈ Y function z = z(x, y) ∈ L 2 (Ω × Y, dxdµ) = for each test function Φ(x, y).

Proposition 1.
If the sequence z ε is bounded in L 2 (Ω, dµ ε ), then there is a subsequence that converges weakly two-scale to some z = z(x, y) ∈ L 2 (Ω × Y, dxdµ) periodic in y ∈ Y .
Definition 4.2. The sequence z ε is strongly two-scale convergent to a periodic in for any two-scale weakly convergent v ε (x) 2s v(x, y) . Taking Proposition 2. The following properties of weak two-scale convergence are useful.
Theorem 4.3. [39]. Let µ be an ergodic measure, and assume that the following conditions hold: Then the two-scale limit z(x, y) is independent of y: z(x, y) = z(x).
where v ∈ L 2 (Ω, V pot ), and V pot is the closure of gradients of smooth periodic functions on Y in norm L 2 (Y, βdy). Poincare inequality implies that in our case any such function is a gradient of a periodic function from W 1 2 (Y, βdy).
Turning to our problem notice that the measure dµ ε = β ε 1 Ωε dx converges weakly to the measure 1 Ω βdx with β = Y F β(y)dy in the sense that R N β ε 1 Ωε ϕdx → R N 1 Ω βϕdx for any ϕ ∈ C 0 (R N ). Theorem 4.5. The diffusion component ρ ε β −1 ε of the solution u ε , ρ ε β −1 ε to system (43) converges strongly in L 2 (Ω, dµ ε ) to a solution Θ 0 of the boundary value problem: div with boundary conditions for Θ 0 (x) the same as in the original problem: with a positive definite matrix A eff defined by Here χ(y) is the periodic solution to the cell problem div (β(y) (∇ y χ + I)) = 0 (64) ∂ ∂y n (χ) = −n(y), y ∈ ∂Y S ; Proof. The L 2 (Ω, dµ ε ) uniform estimates for ρ ε β −1 ε and ∇ ρ ε β −1 ε and the ergodicity of the measure 1 Ωε β ε dx imply according to the properties of the two-scale convergence above that for a subsequence ε → 0 it holds where Θ 0 ∈ W 1 2 (Ω, βdx), Θ 1 (x, y) ∈ L 2 (Ω, W 1 2 (Y, β)). Choosing in (58) a test function ψβ −1 ε = ε ϕ 1 x ε ϕ 2 (x) with smooth ϕ 1 (y) periodic in y ∈ Y , and ϕ 2 (x) ∈ C ∞ 0 (Ω) , and passing to the two-scale limit we obtain the following equation: where Y F is the fluid part of the periodic cell Y and 1 Y F (y) is its characteristic function. Zero limit for the right hand side is an immediate consequence of the estimates (55)-(57) for solutions. This yields that for almost all x ∈ Ω. Therefore with χ(y) being a periodic solution to the cell problem (64). The cell problem is well posed since apriori estimates on W 1 2 (Y, dµ) are fulfilled. Choosing now an arbitrary test function ϕ ∈ C ∞ (Ω), ϕ = 0 in the vicinity of S ε 1 ∪ S ε 2 , in the weak form of the advection-diffusion equation Integration with respect to y yields which is the weak formulation of (61). The boundary conditions (62) are evidently inherited from the original system. Strong convergence of ρ ε β −1 ε to Θ 0 follows from the apriori estimates (55) and the compactness of the embedding from W 1 2 (Ω ε , β ε ) to L 2 (Ω ε , β ε ).
We notice that the limit equation for the scaled concentration ρ ε β −1 ε is decoupled from the flow equation. But ρ ε β −1 ε plays a role in the Stokes part of the system and its limit Θ 0 enters a homogenized Darcy type equation for flow. 5. Homogenization by two-scale convergence for the velocity and pressure of the solvent. Now we consider the two-scale limit for the Stokes equations (33). We need to extend the velocity field and pressure to the whole domain Ω to consider two-scale limits of solutions in a fixed domain. Velocity u ε is extended in a trivial way by zero with apriori estimates preserved for the extended function: The extension of the pressure p ε is more tricky and needs sophisticated estimates uniform with respect to ε to carry out a limit when ε → 0.

5.1.
Extension of pressure. The homogenization of the Stokes equations relies on an extension of pressure and on uniform with respect to ε estimates for pressure: [32], [2]. The following technical lemma from [11] is used here in the construction.
Lemma 5.1. Let g ∈ L 2 (Ω ε ) and Ωε gdx = 0, and assume that the cell domain Y F can be represented as a finite union of domains with Lipschitz boundaries. Then and the following estimates are satisfied: with C > 0 independent of ε and g.
Lemma 5.2. For the pressure p ε normalized by Ωε p ε dx = 0 and satisfying the equations (5) the following estimate holds: with C > 0 independent of ε.
Proof. Using lemma 5.1 and recalling our normalization for pressure we construct Multiplying the Stokes equation repeated here from (10) by w ε , integrating the resulting relation by parts over Ω ε we obtain and pointing out that Ωε div(w ε )dx = 0, we get Combining estimates with estimates (55), (56), (72) for ρ ε , u ε , ∇w ε , w ε we receive the following estimate for pressure p ε : and using the estimate (71) we can extend pressure from Ω ε to Ω by as in [2] for Y ε i,S ⊂ C ε . If the porous structure crosses the lateral boundary Γ 0 of Ω one can [2] complete this definition by extending p ε by zero on Ω\C ε : P ε = p ε in (Ω\C ε ) ∩ Ω ε , P ε = 0 in (Ω\C ε )\Ω ε 5.2. Homogenization for velocity and pressure in the Stokes equations with osmotic forces. In this section we deal with the Stokes part of the system (58)-(59) and consider properties of the two-scale limits of the extended velocity u ε and pressure P ε . The estimates (56), (57), (71) for velocity and pressure imply that there are functions u 0 (x, y) ∈ L 2 (Ω; ), and p 0 (x, y) ∈ L 2 (Ω × Y ) periodic with respect to y ∈ Y , such that extensions µε −2 u ε , µε −1 ∇ u ε , P ε converge two-scale to these functions: It means that and any Φ ∈ . Integrating by parts in the second of equations (76) and passing to the two-scale limit leads in a standard way to and after integration by parts with respect to y over the periodicity cell Y to the relation ξ 0 (x, y) = ∇ y u 0 .
The two-scale limit p 0 (x, y) has a specific structure that is one of the main results of the present paper. We express it in the following lemma.
Lemma 5.3. The two scale limit p 0 (x, y) of P ε is the sum of a function p(x) that can be interpreted as hydrodynamic pressure, and a term expressing local osmotic pressure: Proof. Multiplying the Stokes equation (10) by a test function εψ(x, x ε ) where ψ(x, y) ∈ C ∞ Ω; [C ∞ (Y )] N and has finite support in Ω × Y f , and integrating the resulting relation by parts we get Passing to the two-scale limit in and using (66) and (68) implies a relation between the two scale limit p 0 (x, y) of pressure P ε and the two scale limit Θ 0 of the scaled concentration ρ ε β −1 ε : Taking into account that ψ(x, y) is arbitrary we yield the desired formula (78).
Remark 1. We point out that a similar formula was derived in [8] in one dimensional case for an infinitely long cylindric channel.
The same argument in situation without osmotic forces leads to the conclusion that the two scale limit p 0 (x, y) = p(x) is independent of y.
We proceed with clarifying properties of the two-scale limit u 0 (x, y) of velocity. The incompressibility conditions for u 0 (x, y) and u(x) = Y u 0 (x, y)dy and boundary conditions for u 0 (x, y) and u(x) are formulated in the following lemma.
Proof. Integrating by parts the equation div ( u ε ) = 0 with a test function λ(x) that is zero on ∂Ω, passing to the two-scale limit and integrating by parts again we obtain and conclude that div x (u) = div x Y u 0 (x, y)dy = 0. Integrating by parts the equation div ( u ε ) = 0 with a test function λ(x) that is zero only on the inflow and outflow part S 1 ∪ S 2 of the boundary ∂Ω, taking into account the boundary condition u ε = 0 on Γ 0 , passing to the two-scale limit, and integrating by parts again Integrating by parts the equation div ( u ε ) = 0 with the test function ελ(x, x/ε) that is zero on ∂Ω, passing to the two-scale limit and integrating by parts again Y Ω div y (u 0 (x, y)) λ(x, y)dxdy we conclude that div y (u 0 (x, y)) = 0.
Proof. We follow the way of reasoning from [4]. Choose a test function ψ(x, y) ∈ We suppose also that ψ(x, y) satisfies incompressibility conditions div y ψ(x, y) = 0, div x Y ψ(x, y)dy = 0. Multiplication of the Stokes equation in form (10) by the test function ψ(x, x ε ), taking into account the incompressibility condition for ψ in y, and integration by parts yields We can replace the integration domain in the last equation with Ω and p ε with P ε since the test function ψ(x, x ε ) is zero outside Ω ε . Passing to the two-scale limit in the first term in (84) gives the expression − Ω×Y λ κ Θ 0 (x)β(y)div x (ψ(x, y)) dxdy because the first term in the two-scale limit p 0 (x, y) = p(x)− λ κ Θ 0 (x)β(y) of P ε does not depend on y and ψ satisfies div x Y ψ(x, y)dy = 0. Passing to the two-scale limit in other terms in (84) and using (66) and (68) gives The boundary term disappears because of the boundedness of ψ(x, x ε ) and its support. The last term in the right hand side of (84) disappears by the estimates for ∇u ε (x). Finally after integration by parts, taking into account boundary conditions for p(x) and Θ 0 (x) and cancelling two integrals with λ κ Θ 0 (x)β(y), the variational form of the homogenized equation with osmotic forces reads By density the last equation holds for ψ(x, y) in the Hilbert space V of functions periodic in y ∈ Y , defined by One can check that the Lax-Milgram lemma holds for the problem (86) and that it has therefore a unique solution u 0 (x, y) ∈ V. Let L 2,per (Y ) be the space of periodic on Y , square integrable functions with standard scalar product. By a variant of the Weyl decomposition, see [4] we conclude that the orthogonal complement V ⊥ of V with respect to the scalar product in L 2 Ω; [L 2,per (Y )] N coincides with vector fields of the form ∇ x q(x) + ∇ y q 1 (x, y) with q(x) ∈ W 1 2 (Ω) and q 1 (x, y) ∈ L 2 [Ω; L 2,per (Y F )] having zero mean values over Ω and Y F correspondingly. Using this statement and integrating by parts in (86) we get the strong form (83) of the two-scale homogenized limit for our problem. We must show that the pressure like expression p(x) − λ κ Θ 0 (x)β(y) arising from the incompressibility constraint div x Y u 0 (x, y)dy = 0 is the same as the two-scale limit p 0 (x, y) of the pressure P ε . We multiply the Stokes equation (10) by a test function ψ(x, y) that is divergence free only in y: div y ψ(x, y) = 0, integrate the resulting expression by parts as in (85) and identify two-scale limits: Ω×Y p(x)div x ψ(x, y)dxdy = Ω×Y ∇ y u 0 (x, y) · ∇ y ψ(x, y)dxdy Since the system (83) has a unique solution (u 0 (x, y), p(x)), the entire sequence ( u ε , P ε ) converges to u 0 (x, y), p(x) − λ κ Θ 0 (x)β ε (y) . We are in the position to separate variables in the two-scale homogenized system (83) and reduce it to a periodic cell problem of y variable on Y and a homogenized problem of x variable only in the domain Ω.
Theorem 6.2. The extension ( u ε , P ε ) of the velocity and pressure (u ε , p ε ) satisfying the system (5)-(6) converges weakly in [L 2 (Ω)] N × L 2 (Ω) to the unique solution (u, p) of the homogenized problem where u(x) = Y F u 0 (x, y)dy, the values P 1 and P 2 are uniquely defined by the normalization Ω pdx = 0 and the pressure drop δP . B D and B osm are constant symmetric matrices with entries defined by where for 1 ≤ i ≤ N , w i (y) and W i (y) are unique periodic solutions to the cell Stokes problems Proof. The two-scale homogenized problem (83) is equivalent to (89) through the relation The incompressibility condition for u 0 (x, y) implies After integrating the last expression over Y and recalling the problem (61), (62) for Θ 0 we arrive at the following macroscopic system of equations for p(x) and Θ 0 : where the values P 1 and P 2 are uniquely defined by the normalization Ω pdx = 0 and the pressure drop δP . An expression for u(x) follows: Remark. We notice that the limit macroscopic system (95)  Appendix. Poincare and Friedrichs inequalities in weighted Sobolev spaces. Potentials V ε (x) acting in the problems of interest are natural to interpret as functions of the distance d Γ (x) from the solid boundary Γ ε : V ε (x) = V ε (d Γ (x)).
The weight that appears in our problems is β ε (x) = exp {−V ε (d Γ (x))} depends on the point x through d Γ (x). If the potential V ε (x) goes to infinity when x approaches the solid boundary Γ ε , that can naturally happen in applications, the weight β ε (x) degenerates at Γ ε . We provide below some specific results about conditions implying Friedrichs inequality (20) and the Poincare inequality (21) as well as embedding of W 1 2 (Ω ε , β ε ) into L 6 (Ω ε , [β ε ] 6 ) in weighted Sobolev spaces and give nontrivial examples of potentials V ε (d Γ (x)) such that these conditions are satisfied for the weight β ε (x) = exp − κ λ V ε (x) .
General Hardy inequalities in one dimension. The most flexible and practical results for embedding, and Poincare and Fridrichs inequalities in weighted Sobolev spaces with weights degenerate only on the boundary follow from one dimensional Hardy inequalities on a finite interval and estimates on a thin stripe along the boundary. For Lipschitz domains and weights depending on the distance from the boundary corresponding estimates are similar to ones for the interval because the distance to the graph of a Lipschitz function along the corresponding coordinate direction and the usual distance d(x) are equivalent for small distances.
The following results on one-dimensional Hardy inequalities from [27] are useful both for estimates and for embedding results in weighted Sobolev spaces with particular choice of weights degenerating on the boundary of Lipschitz domains. Global results (20), (21) can be reduced in this case to local one-dimensional estimates by considering a thin stripe along the boundary similarly as in [22].
In domains with periodic perforated structure these results can be gained by combining the Friedrichs and Poincare inequalities in reference domains that do not depend on the small parameter, and by scaling arguments.
Let W (a, b) be the set of measurable positive functions finite almost everywhere on (a, b).   Example. We check that conditions of Theorem 6.4 are fulfilled for weights w(x) = exp (−β/x n ) and v(x) = exp (−α/x n ) with α ≤ β and p = q = 2 on the interval (0, b) for functions equal to zero on the right endpoint. It implies as in [22] that Poincare and Friedrichs inequalities are valid in the Lipschitz domain Ω ε for weights w(x) = exp (−β/ (d Γ (x)) n ) and v(x) = exp (−α/ (d Γ (x)) n ) corresponding to potentials with singularity V (d) ∼ 1/d n , n > 0, at the boundary.
It is sufficient to verify compactness of the operator H R (101) acting from L 2 (0, b; v) to L 2 (0, b; w). Corresponding value of the function F R (x) is: We estimate integral F 1,n (x) = x 0 exp (−β/s n ) ds by introducing variable y = β/s n , s = (β/y) 1/n . We estimate integral F 2,n (x) = b x exp (α/s n ) ds by introducing variable y = α/s n , s = (α/y) 1/n , d dy s = d dy (α/y) 1/n = − (α) It is easy to observe that we have the following relation for x → 0+ Therefore lim We also observe that F R (x) is bounded, and lim x→b− F R (x) = 0.
It implies that the operator H R is compact from L 2 (0, b; exp (−β/x n )) to itself. Checking boundedness of H R acting from the space L p (0, b; exp (−α/x n )) to the space L q (0, b; exp (−β/x n )) we observe that The main term to estimate is e − β x n q e α x n p . It is bounded in the case α/p ≤ β/q. Similarly as above lim x→0+ F R (x) = 0 and lim x→b− F R (x) = 0 in this case. It means that H R will be both bounded and compact from L p (0, b; exp (−α/x n )) to L q (0, b; exp (−β/x n )) if α/p ≤ β/q.