Numerical investigation of a homogenized model with effective interface conditions for Stokes flow through porous membranes

1 Introduction

• Phenomenological interface models can require scenario-specific parameter estimates, while direct microscopic simulations are computationally expensive.The paper numerically validates the effective model against microscopic simulations for a specific scenario.Source: §1 · pp. 2, 4
• The micro-macro model couples bulk Stokes flow to cell problems, with effective interface coefficients determined by the cell solutions.This one-way coupling ties the transmission conditions to microscopic geometry, but produces a nonstandard transmission problem.Source: §1 · p. 2
• The new Taylor–Hood-type discretization permits discontinuous pressure and tangential velocity across the interface, with stability and optimal-order error bounds.The model ensures continuity only of normal velocity, motivating a discretization that allows the other components to jump.Source: §1 · p. 3
• As cylinder height approaches its maximum, the effective coefficients grow and interface velocity tends to zero; asymmetric geometry deflects the effective velocity.The numerical results also indicate continuous dependence of interface velocity on microscopic geometry, though this has not been proved analytically.Source: §1 · p. 4
• The numerical study considers symmetric cylinder-based and asymmetric cells, several outer boundary conditions, and sensitivity to cylinder height and radius.It also compares microscopic and effective solutions for a specific computational scenario.Source: §1 · p. 4

2.1 Notations and function spaces

• The cell and bulk function spaces use periodicity in the two directions parallel to the membrane, denoted Y-periodic and Σ-periodic, respectively.The paper uses the index # for periodic functions in the first two coordinates.Source: §2.1 · p. 5

2 Statement of the mathematical model

• The transmission model couples bulk Stokes problems across the interface using conditions determined by solutions to cell problems on the fluid part of the periodic cell.The cell comprises a solid skeleton and a fluid pore region; the model assumes the solid does not touch the cell's top or bottom.Source: §2 · p. 5, §2.2 · p. 6

2.2 The transmission problem

• For a nonempty solid phase, the weak transmission problem has a unique solution under the stated boundary-data assumptions.The proof uses a saddle-point formulation and inf-sup stability.Source: §2.2 · p. 9
• Pure Neumann boundary conditions require a compatibility condition on the data for well-posedness.The stated proposition otherwise covers the prescribed boundary conditions.Source: §2.2 · p. 11

3.1 Mesh generation and numerical environment

• Because splitting failed to preserve periodicity on the cylindrical cross, the authors regenerated meshes by target size and paired degrees of freedom on opposite periodic boundaries.Mesh construction and refinement use gmsh; periodic constraints are then imposed in the finite-element computation.Source: §3.1 · pp. 12–13

3.2 Numerical convergence study

• The selected coefficients converge globally at order α ≈ 2 on successively refined meshes for a cuboid-built cross-shaped domain.The cuboid geometry permits refine-by-splitting, unlike the cylindrical cross, for which that approach could not preserve periodicity on very fine meshes.Source: §3.2 · pp. 13–14, Figure 4 · p. 14

3.3 Computation of cell solutions. Sensitivity study with respect to geometric parameters

• Changing h and r visibly changes q+1, while q3 is relatively unchanged as h increases but develops sharper gradients as r increases.Figures 7 and 8 show these cell solutions for varying h at fixed r and varying r at fixed h, respectively.Source: Figure 7 · p. 16, Figure 8 · p. 16, §3.3 · p. 17
• For the asymmetric cell with an inclined segment, q3 visibly deflects at the inclined plane, while q±i remain qualitatively similar to solutions for the symmetric obstacle.The computations cover a symmetric cylindrical-cross geometry and an asymmetric geometry containing an inclined segment.Source: §3 · p. 12, §3.3 · p. 17

3.4 Computation of effective coefficients and sensitivity study

• For the symmetric geometry, K+ = K− and K± are diagonal with K±11 = K±22; M± are diagonal with M±11 = M±22.The text also gives M− = (M+)T.Source: §3.4 · pp. 17–18
• M±11 = M±22 are almost vanishing, consistent with energy localization near the boundaries where nonzero Dirichlet conditions are imposed.The authors report similar coefficient behavior for the asymmetric geometry, where energy localization also occurs.Source: §3.4 · pp. 18–19
• K±33 remain nearly constant up to h = 1.85 and appear to diverge as h approaches 2; increasing r increases all coefficients.Source: §3.4 · pp. 18–19
• As a centered spherical obstacle shrinks to zero radius, M+11 converges to −1/4, its value for the cell without an obstacle.The comparison isolates the effect of the obstacle on this coefficient.Source: §3.4 · p. 19

4.1 Discretization and numerical analysis of the effective transmission model

• The subdomain-wise Taylor–Hood scheme permits tangential velocity and pressure jumps while enforcing normal-velocity continuity, and is stable with mesh-independent inf-sup constants under the stated refinement assumption.The discrete problem has a unique weak solution and an error estimate; the convergence estimate assumes a sufficiently smooth solution.Source: §4.1 · pp. 20–22

4.2 Implementation aspects

• The implementation uses DUNE modules to construct and assemble the subdomain-restricted, piecewise Taylor–Hood space and enforce normal-velocity continuity; the application advanced support for restricted bases and interface variational forms.Source: §4.2 · pp. 22–23

4.3 Numerical convergence study

• After a brief pre-asymptotic phase, first-order Taylor–Hood elements attain cubic L2 velocity, quadratic L2 pressure, and quadratic subdomain H1 velocity error convergence.The study uses effective coefficients from the cylindrical cross cell geometry; Table 1 presents the convergence study.Source: §4.3 · p. 24, Table 1 · p. 24

4.4 Simulation of the effective transmission model

• Asymmetry in the cell obstacle deflects flow, pressure jumps occur in several scenarios, and the interface can reduce velocity when boundary conditions do not enforce through-flow.The pressure jump is especially pronounced when Dirichlet conditions enforce normal volume flux; the velocity reduction is seen in Scenarios C and D.Source: Figure 12 · p. 26, §4.4 · p. 28
• For Scenario A, the symmetric cylindrical-cross cell yields constant velocity and piecewise constant pressure, matching the explicit solution; the asymmetric cell produces a stronger pressure drop and flow deflection aligned with its inclined segment.In Scenario B, the parabolic inflow is essentially maintained across geometries, with only slight profile widening or narrowing near the interface.Source: §4.4 · p. 27
• In Scenario C, parallel inflow establishes flux across the interface, especially for the obstacle-free cell, while a local pressure drop arises where the prescribed inflow profile vanishes.The pressure variation is attributed to the abrupt boundary-velocity-gradient change at the inflow edge, which induces large local stresses.Source: §4.4 · p. 27
• Scenario D is not well-posed for the obstacle-free cell; for the inclined-plane cell, the approximately constant flow direction is close to the plane’s orientation, with remaining deviation attributed to other geometry and finite plane extent.For the symmetric cell, the numerical solution agrees with an explicit solution.Source: §4.4 · pp. 27–28

4.5 Parameter study with respect to the parameter h

• The parameter study varies the cylindrical obstacle height from 1.85 to 1.99 in increments of 0.01 and evaluates velocity norms and pressure along the x3-axis for four scenarios.Source: §4.5 · p. 28
• As h approaches 2 in Scenario D, velocity decreases while the pressure jump remains constant; in Scenario A, prescribed flow keeps velocity constant while the pressure jump increases, especially near h = 2.The homogenization assumptions on the periodicity-cell geometry are violated as h approaches 2, but the parameter study numerically shows continuous velocity dependence in that limit.Source: §4.5 · pp. 28, 30
• In Scenario D, increasing h from 1.85 toward 1.99 makes the velocity at the origin decrease monotonically toward zero, following an almost straight log-log relation with K33 of slope −1.The computed values lie almost exactly on the reference curve f(K33), and the relation is validated by the explicit solution; the velocity dependence is slightly sigmoidal.Source: §4.5 · pp. 28, 30, Figure 14 · p. 30

4.6 Numerical study of the approximation accuracy by comparison with the microscopic model

• In the microscopic model, pressure appears continuous between the porous layer and lower bulk but discontinuous at the upper interface; interchanging cap boundary conditions reverses this pattern, producing jumps at both interfaces.The microscopic velocity is locally increased in the membrane pores.Source: §4.6 · pp. 31–32
• The microscopic and effective solutions match qualitatively, especially for small ε; the pressure-discontinuity location in the microscopic solution approaches the effective solution’s location at x3 = 0.As ε decreases, the negative microscopic velocity peak remains almost constant in magnitude while narrowing in width.Source: §4.6 · p. 32

• The paper provides numerical methods for analyzing and implementing effective transmission models whose coefficients are computed from microscopic cell solutions.Source: §5 · p. 32
• Figure 16 compares effective and microscopic solutions for decreasing ε using velocity and pressure profiles along the x3-axis.Source: Figure 16 · p. 33
• Simulations provide insights into cell-solution localization and symmetry, effective-coefficient structure, and microstructure's impact on macroscopic flow.Source: §5 · p. 33
• The microscale analysis assumes a rigid solid; extending the methods to elastic fluid–structure interaction and its generalized cell problems remains future work.Source: §5 · p. 33