An experimental investigation of boundary layer over permeable interfaces in Hele-Shaw micromodels

Transport phenomena at the interface between a porous medium and an adjacent boundary layer play a significant role in various natural and industrial processes, such as soil evaporation,¹ fuel cells,² transpiration cooling,³ and aircraft wing noise reduction,⁴ among others. However, a fundamental understanding of exchange processes across scales, which is crucial for effective upscaling of hydrodynamic dispersion, remains a challenge. The key difficulty lies in developing accurate mathematical models, as the use of Navier–Stokes equations across the entire domain to resolve all fine-scale heterogeneities is computationally infeasible. Without advanced techniques to reconstruct the internal porous structure, modeling the porous domain becomes computationally prohibitive unless the porous material is assumed to have a structured matrix with a predefined geometry.^5–10 In addition, two-equation turbulence models can be inadequate when the length scale of the turbulent eddies is similar to the pore size.¹¹ Consequently, efforts have focused on modeling coupled free flow and porous media systems at the representative elementary volume (REV) scale using volume-averaged techniques.^12,13 Nevertheless, pore-scale effects that may influence interfacial fluxes are not fully captured at this REV scale.

Darcy-based extended models, such as the Forchheimer equation,¹⁶ are applied to capture inertial effects when the relationship between velocity and the pressure gradient becomes nonlinear.

Consequently, accurately quantifying momentum transport at the interface requires either well-defined boundary conditions between the two domains or an accurate modeling of the transition zone, where both are challenging due to the limited availability of experimental data. Experimental studies, particularly at the microscale, remain limited, and thus, detailed investigations into pore-scale interactions and interfacial effects are still insufficient (see Sec. II). A multiscale approach is however essential, as knowledge at the microscopic level is crucial for accurately predicting macroscopic behavior. Without sufficient experimental evidence across multiple scales, establishing precise boundary conditions remains challenging, hindering a comprehensive understanding of flow dynamics at the interface between free flow and porous media.

This study addresses this gap by providing a detailed analysis of momentum transfer at the interface between a free-flow and a porous medium, using high-resolution experimental data from Hele-Shaw micromodels. Micro-particle image velocimetry (⁠ $μ$ PIV), with a spatial resolution of 5  $μ$m per vector, is used to precisely characterize the external boundary layer near the interface. The slip velocity is measured as a function of porosity and Reynolds number, offering new insights into interfacial flow dynamics. Additionally, the Brinkman equation is analytically solved to predict the thickness of the external boundary layer. Furthermore, the velocity field measurements revealed that flow in this region is influenced by a combination of shear and vorticity, depending on the solid fraction of the porous medium. These findings improve the understanding of coupled free-flow and porous-media systems by connecting pore-scale interactions to macroscopic transport processes.

Table I presents a chronological summary of key experimental studies investigating coupled free flow and porous media systems. These studies focus on single-phase systems, either gaseous or liquid, and examine flow behavior under laminar or turbulent regimes in isothermal conditions. The primary goal of these works is to understand how porosity influences flow behavior, either through direct observations and integral measurements of the global system (e.g., flow rate, pressure drop, torque) or by inferring from local velocity profiles and flow field measurements using techniques such as laser Doppler anemometry, hot-wire anemometry, and particle image velocimetry.

The table clearly shows that historically, research has primarily focused on highly porous materials, typically with porosity exceeding 80%, and particularly on macro-porous materials, characterized by pore sizes greater than 1 mm. These experimental studies were conducted under controlled conditions to isolate the effects of porosity, pore size, and flow regime, providing critical insights into the global flow resistance. For instance, as the porosity of the interface increases, studies have demonstrated that there is also an increase in slip velocity.^{29,34,37,40,45} On the other hand, the limited experiments available on microporous materials, where characteristic pore sizes are less than 0.5 mm, have mainly utilized industrial porous media, such as fiber networks^23,45 and sandpaper,⁴² typically employing integral or global flow measurements. However, this focus on global flow characteristics has limited the ability to experimentally capture the microscopic details of hydrodynamic dispersion at the interface. Furthermore, it has been shown that even small displacements of the porous interface can introduce significant errors in experimental results, with errors reaching up to 120% in the non-dimensionalized slip velocity and 50% in the interfacial slip coefficient.⁴² These uncertainties highlight the challenges of experimentally characterizing flow at the interface of microporous materials, where resolution is crucial for understanding the complex microscopic flow dynamics.⁴⁹ More recent studies⁴⁶ used micro-PIV to examine flow characteristics at pore scales of 240  $μm$⁠, offering a high-resolution perspective on flow dynamics in porous materials with 75% porosity. However, the free flow Reynolds number remained well within the Stokes regime, limiting the range of observable flow behaviors. Finally, to the best of our knowledge, only one study has experimentally investigated boundary layer behavior outside a porous material.⁵⁰ These results showed that the boundary layer thickness is reduced compared to the Blasius solution, while exhibiting similar spatial growth. This reduction is attributed to the material's porosity, where higher porosity values result in a thinner external boundary layer. Apparently, there is an imperative need for microscale experimental data to reveal the interfacial fluid dynamics and enable the derivation of accurate boundary conditions.

This study advances the current research by focusing on microporous materials with pore sizes as small as 50  $μm$⁠, an unexplored scale in previous investigations. Through $μ$ PIV, precise measurements of local velocity fields and interfacial dynamics in Hele-Shaw cells, are obtained. These high-quality experiments provide valuable insights into momentum transport at the microscale, addressing the gap between theoretical models and experimental data. The results are expected to significantly improve the prediction of external boundary layer behavior in coupled systems involving porous materials with varying permeability, contributing to a deeper understanding of flow dynamics at the microscopic scale.

The micromodel used in this study featured a four-port configuration with two inlets and two outlets, connecting separately to the free channel and the porous material, as shown in Fig. 1. A primary splitter in the initial section maintained the separation of the two domains, allowing the free flow to fully develop before interacting with the porous region. A secondary splitter at the outlet further isolated the porous medium from the free flow. The inlet and outlet ports of the porous material were sealed after initial saturation, ensuring independent fluid delivery to both domains while preserving the free flow conditions.

As shown in Fig. 1, the free channel height, h, was fixed at 2 mm for all models, while the porous material height, H, was 4  $±$ 0.4 mm, with minor variations attributed to the porosity range (⁠ $ϕ∈[50%,85%]$⁠). The porous material consisted of cubic pillars, each with a characteristic length, $lp$⁠, of 100  $μ$m. To accommodate different porosities, $ϕ$⁠, the total number of pillars was adjusted within the area of interest, L. Specifically, 272  $×$ 59 pillars were used for $ϕ=50%$⁠, 206  $×$ 23 pillars for $ϕ=75%$⁠, and 162  $×$ 18 pillars for $ϕ=85%$⁠. This adjustment allowed for consistent overall dimensions while varying the inter-pillar distance, $lt$⁠, to 50, 100, and 150  $μm$ for porosities of 50%, 75%, and 85%, respectively. The total axial length of the area of interest, L, remained constant at 40.6 mm across all configurations, ensuring a standardized region for flow analysis despite the varying porosity levels. This design ensured that the porosity could be systematically varied without changing the overall model dimensions, facilitating comparative studies across different porosity conditions.

The micromodels were fabricated using photolithography and soft lithography techniques. Initially, the pillars of the porous medium were created photolithographically in bas-relief to form a master. This master was then used as a mold to replicate the microfluidic models using Polydimethylsiloxane (PDMS). A photomask, with a pre-designed geometric pattern, was laser-printed on glass based on the model's computer-aided design. This pattern was transferred to a photoresist layer spin-coated onto a clean silicon wafer, with a uniform thickness of 100  $±$ 5  $μm$⁠. The sample was soft-baked, UV-exposed through a mask, and developed. Liquid PDMS, mixed with a curing agent at a 1:10 ratio, was poured onto the master, degassed, and soft-baked at 65°C. After curing, the PDMS was peeled off, and holes were punched to connect the microfluidic model to liquid reservoirs. The structured PDMS slab was bonded to a flat PDMS layer and a clean glass slide using oxygen plasma, ensuring the micromodel's rigidity and keeping the optical measurement plane-parallel to the glass. Two PDMS gaskets were added to each port using plasma corona treatment to enhance sealing, prevent leaks, and enable operation at high inlet pressures and flow rates.

Figure 2(a) shows the experimental setup, which consists of an inverted Nikon Eclipse Ti2 microscope paired with a Kinetix Scientific CMOS (sCMOS) camera, the PDMS micromodel, a pumping system connected to a hydraulic circuit, and a high-intensity LED (Spectra light engine—Lumencor^®). The entire setup was placed on an optical table to absorb external vibrations to ensure experimental stability. During the initial saturation phase, all four micromodel ports were connected to the hydraulic circuit. CO₂ was infused for 10 min to remove air particles, minimizing the hydrophobic effects of PDMS and preventing bubble formation. Following this, de-ionized water was injected through the inlet ports of both domains. Once saturation was complete, the porous material ports were sealed, while the free channel ports remained connected to the pumping system and discharge tank. A bubble trap with a microporous Polytetrafluoroethylene (PTFE) membrane was installed at the inlet to prevent air infiltration and allow system venting. Finally, the system was saturated with fluorescent particles to facilitate velocity measurements using micro-PIV techniques.

Velocity fields in both the free flow and porous medium were measured using micro-PIV techniques.^51,52 Deionized water, seeded with Fluoro-Max tracer particles (Thermo Fisher Scientific), was used as the working fluid. These particles, with an average diameter of 0.86  $μ$m and a density of approximately 1.05 g/cm³, closely followed the fluid streamlines without introducing gravitational effects. The particles were carefully selected to match the camera sensor's characteristics, achieving a resolution of 0.65 μm/pixel with a Plan Fluor 10× magnification lens (NA = 0.30, WD = 16 mm). This ensured that the particles were captured within 2–3 pixels on average, minimizing measurement uncertainties.⁵³ 

To achieve micrometer-scale spatial resolution, the particles were chosen for their ability to reliably track the flow without causing disruptions or clogging the device. When excited by green light at 542 nm, the particles emitted light at 612 nm, which was recorded through a MXR00709-Cy3-4040C optical filter. All tests were conducted under consistent lighting conditions to minimize uncertainties caused by background noise, with only the camera's acquisition speed being varied. Image acquisition speeds were adjusted according to the flow velocity in both the free channel and porous medium, ranging from 333 to 2300 fps. Each experiment recorded 4000 images, with data collected at an 8-bit digital resolution.

Velocity vectors were determined by analyzing the statistical displacements of particles carried by the flow. In this study, the displacement of 8–10 particles per smallest interrogation window (16 × 16 px²) was measured over a time interval defined by the camera's frame rate. A cross correlation function was computed for each interrogation window, with the peak value indicating the average particle displacement. The analysis was performed using PIVlab in MATLAB^54,55 which employed a fast Fourier Transform algorithm with window deformation to minimize experimental uncertainties. A multi-pass technique was applied, starting with a 32 px² interrogation window and reducing it to 16 px² in two steps with a 50% overlap. Subpixel-scale displacement vectors were calculated using a 2 × 3 points Gaussian fit. To accurately capture the flow within the porous material's interstitial space, at least 10 velocity vectors per 50  $μ$m were measured, resulting in a resolution of approximately 5.25  $μ$m/vector, as depicted in the vector field in Fig. 2(b).

The total region of interest (ROI) of the model, schematized in Fig. 1, spans tens of millimeters in both axial and vertical directions. This area is significantly larger than the field of view of the camera. In order to achieve a spatial resolution of 5.25  $μ$m/vector, the velocity field was measured within smaller, discrete regions of the model (sub-ROIs), with the size of these sub-ROIs depending on the acquisition frame rate. As the Reynolds number or porosity increased, the measurement domain was divided into smaller sub-ROIs. This segmentation allowed the optical system to effectively capture particle dynamics while maintaining high-resolution imaging and precise velocity calculations across the entire region of interest, which was characterized by a wide range of length and time scales. The sub-ROIs were subsequently combined together using a pixel-scale alignment and patching process in both the axial and vertical directions. Therefore, the entire ROI was reconstructed, as illustrated in Fig. 3.

Statistical convergence in PIV studies is achieved when velocity measurements stabilize and no longer vary with additional data. In this study, convergence was analyzed across 4000 images per experiment, calculating the temporal mean of the flow field at intervals of 100, 500, 1000, 2000, 3000, and 4000 frames. Since the flow is stationary and laminar, the mean fields are considered converged when further data does not alter the results. The analysis primarily focused on the axial flow field, as the vertical flow at the interface is negligible, as shown later in this work.

Figure 5 shows the surface contours of the vertical velocity component in the vicinity of the interface (⁠ $y/lp$ = 0) and the middle of the micromodel (⁠ $x/lp$ = 0) for three porosities: 50%, 75%, and 85%. The Reynolds number is 0.94, and the vertical velocity is normalized by the Darcy velocity (⁠ $u¯FF$⁠) of the corresponding Hele-Shaw cell in the free flow region.

At a porosity of 85%, as depicted in Fig. 5(a), the vertical velocity shows pronounced fluctuations near the interface. This suggests that in regions with high porosity, the flow tends to channel through the spaces between the solid pillars, creating a non-uniform and non-parallel flow pattern at the interface. The larger voids allow for greater fluid redistribution, resulting in a U-shaped velocity profile, in alignment with findings from previous studies.⁴⁶ In contrast, as the solid fraction increases (i.e., porosity decreases), the flow becomes more constrained, reducing the fluctuations at the interface. This is particularly evident in Fig. 5(c), where at a porosity of 50%, the narrow gaps between the solid pillars, which are on the order of 50  $μm$ introduce greater resistance to flow, resulting in a more parallel flow pattern that closely approximates a no-slip boundary condition.

The vertical velocity contours across the different porosities clearly illustrate that as the solid fraction increases, the flow near the interface becomes more uniform and parallel. This indicates that as porosity decreases, the influence of the vertical velocity component diminishes, with the axial velocity component becoming more dominant in governing the overall flow direction.

Figure 6 illustrates the surface contours of normalized axial velocities relative to the free-stream velocity, accompanied by the fluid streamlines. The bluish-white regions just above the interface highlight the formation of a boundary layer, with its uniformity affected by the porosity of the porous medium. The bluish-white regions just above the interface reveal the development of a boundary layer, whose uniformity is influenced by the porosity of the porous medium. At a porosity (⁠ $ϕ$⁠) of 85%, as depicted in Fig. 6(a), the streamlines curve as they pass through the void spaces between the pillars, resulting in a U-shaped velocity profile and a corresponding reduction in axial velocity. However, as the porosity decreases, such as at 50% in Fig. 6(c), this curvature reduces, resulting in a more parallel flow along the interface. This effect is particularly evident near the permeable wall at 50% porosity, where the streamlines align closely with the free-stream velocity and the x-axis.

As the solid fraction of the porous medium increases, the boundary layer increasingly resembles that of a no-slip wall boundary condition. Since the no-slip condition is only met at the contact points with the solid pillars within the porous structure, it follows that, at the Representative Elementary Volume (REV) scale, the slip velocity increases with porosity. Consequently, as porosity rises, the void spaces at the interface enable greater fluid channeling.

The same phenomenon observed at the interface also occurs within the solid matrix. Between the pillars, the fluid moves slowly, while between successive rows of pillars, alternating high and low velocity zones emerge due to the Venturi effect. This effect is more pronounced at higher porosities, where the local velocity fluctuates between 50% and 80% of the free-stream velocity. However, at a porosity of 50%, these fluctuations nearly disappear. Furthermore, at lower porosities, the flow through the porous material is significantly reduced, resulting in a Darcy velocity that is approximately 30% of the free-stream velocity.

On the right-hand side of Fig. 6, the streamwise-averaged velocity profile across the surface contours is depicted for all Reynolds numbers. These plots provide several key insights. First, irrespective of the case, the averaged axial velocity shows that the velocity at the interface is never zero, confirming that a no-slip condition is not achieved, even for a porosity of 50%. Additionally, it is evident that flow channeling is heavily influenced by the material's porosity, which significantly impacts the flow within the porous medium. For instance, at a porosity of 85%, a large portion of the flow is channeled through the porous material, leading to a relatively low maximum velocity in the free-flow region. At a Reynolds number of 0.94, the maximum velocity in the free-flow region reaches only 4 mm/s for 85% porosity, while for 50% porosity, this velocity increases to 6 mm/s, a relative rise of 50%. This illustrates that as the solid fraction increases, less fluid is diverted into the porous medium. Moreover, adjusting the material's permeability by altering the porosity can result in an exponential reduction or increase in the discharged flow. Another notable feature is seen in the normalized streamwise-averaged velocity distributions (right-hand side plots), where, in the Reynolds number range of 0.16 to 0.94, the velocity distributions converge into a single curve across all porosities. This indicates that inertia effects are negligible within the system, and the examined Reynolds number range has no significant influence on the flow behavior.

Figure 7 illustrates the external boundary layer distribution for all porosities and Reynolds numbers examined in this study. The boundary layer initiates from the slip velocity at the interface (⁠ $y=0$⁠) and gradually transitions to the free-stream velocity in the main channel. The figure also shows that the boundary layer profiles for different Reynolds numbers do not perfectly collapse at $y=0$ for a given porosity, likely due to measurement errors. Nonetheless, it is clear that as the porosity of the porous material—and consequently its permeability—increases, the average slip velocity at the interface rises. For instance, the inset in Fig. 7 illustrates this trend, showing average slip velocities at the interface of 27%, 30%, and 36% for porosities of 50%, 75%, and 85%, respectively. This also suggests that the boundary condition proposed by Beavers and Joseph²³ is independent of the free-flow momentum, confirming that the Reynolds number does not influence the determination of the slip coefficient.

A key focus of this study is the characterization of the boundary layer thickness, beginning with its definition and followed by an analysis of its structure in tensorial terms. Understanding how the slip velocity at the interface varies with porosity is crucial, especially in relation to the ratio between the permeability of the porous material and that of the Hele-Shaw cell. To address this, an analytical solution to the Brinkman equation¹⁹ is provided, describing the external boundary layer and determining how its thickness varies based on the coupled free flow and porous medium system.

A scalar permeability $κ$ was adopted in place of the tensor $K$ to simplify the analysis. The system is confined by solid walls along the z-direction, rendering the longitudinal permeability component negligible due to the absence of flow. As the flow is predominantly aligned with the axial x-direction, the vertical permeability exerts no significant influence. The viscosity remains that of the fluid, since no additional modeling of effective permeability is necessary in the free-flow region.

The analytical solution for the external boundary layer thickness is shown in blue in Fig. 8, while in red, the ratio between the boundary layer thickness and the system's characteristic length, typically represented by the square root of the material's permeability, is plotted. From the graph, it is clear that the boundary layer thickness is consistently on the same order of magnitude as the characteristic length. Specifically, the ratio between these two quantities does not surpass the constant value of 4.6, which represents the limiting case of a no-slip condition, where the wall is entirely solid. In more practical scenarios, where the slip velocity is between 20% and 60% of the free-flow velocity, the boundary layer thickness is approximately three to four times the square root of the material's permeability.

By analyzing Eq. (12), it becomes evident that the velocity characteristic ratio, $uslip/uFF$⁠, plays a crucial role in understanding the flow dynamics. This ratio, due to physical constraints, must lie between 0 and 1. Geometrically, the permeability of the porous material can never match or exceed that of the Hele-Shaw cell, as the porous medium introduces additional resistance through the arrangement of pillars, whether ordered or disordered. The presence of the pillars inherently increases the flow resistance, thereby reducing the permeability of the porous material compared to that of the Hele-Shaw cell. Consequently, the ratio $uslip/uFF$ is bounded within this range, excluding the extremes. By maintaining a constant slip velocity and increasing the free-flow velocity, the solution inevitably approaches that of a system with a no-slip condition, i.e., $δFF≈4.6·kFF$⁠. A similar outcome occurs when the free-flow velocity is constrained, but the porosity of the porous material is reduced. This reduction in porosity decreases both the permeability and the slip velocity at the interface, leading to the same limiting behavior.

Figure 8 also illustrates the behavior of the boundary layer, measured using micro-PIV. The experimental results show good agreement with the proposed analytical solution, emphasizing that the maximum boundary layer thickness occurs at low porosity values (⁠ $ϕ=50%$⁠). This corresponds to low slip velocity and high free-flow velocity, attributed to the low permeability of the porous material and the high channeling of flow through the free-flow region. Within a maximum error margin of 10%, the experimental and analytical results show a reasonable match. However, to fully resolve the problem analytically, it is necessary to predict the behavior of the characteristic ratio $uslip/uFF$ as a function of the coupled system's domain properties, particularly the characteristic permeability ratio $kFF/kPM$⁠. Establishing such a relationship requires defining a suitable boundary condition along the interface, allowing for the analytical or numerical resolution of the Brinkman equation. Additionally, it is essential to develop a consistent volume averaging method that facilitates the transformation of numerical or experimental data, as shown in Fig. 6, from a microscopic, two-dimensional description to a representative elementary volume (REV) and one-dimensional format. This step is crucial for comparing results unambiguously with the analytical solutions, which are inherently based on the REV scale.

Figures 9(a)–9(c) show the surface contours of normalized Q values for porosities of 85%, 75%, and 50%, respectively. These contours illustrate the distribution of vorticity and strain-rate across the interface flow field, highlighting how the flow characteristics evolve with varying porosity levels.

In the free-flow region (e.g., $y/lp>1$⁠), regardless of the porosity, the normalized Q values approach zero, indicating that the flow streamlines remain parallel, with no vortex structures or fluid deformation forming in the system. However, near the interface, distinct structures emerge, particularly at the corners of the pillars. In these regions, where the flow interacts with the pillars, local flow rotation dominates, potentially leading to flow separation. Despite this, no flow separation was observed during measurements, likely due to the strongly laminar nature of the flow. Between the leading and trailing edges of the pillars, the flow remains predominantly parallel, with neither deformation nor rotation dominating. However, when the flow passes through the interstitial spaces between the pillars, $Q/|Qmax|$ becomes negative, indicating that deformation dominates over rotation. Inside the porous material, similar periodic structures appear, although with reduced intensity, especially as the solid fraction increases. As porosity decreases, the deformation component becomes more prominent, balancing with the rotational component. This transition results in flow behavior resembling Couette or Poiseuille flow, where Q values tend to average around zero, reflecting an equilibrium between deformation and rotation.

Figures 9(d)–9(f) show the average distribution of the $Q/|Qmax|$ parameter above the interface, providing a quantitative comparison of the flow characteristics for the examined porosities. These plots highlight how the balance between local flow rotation and deformation varies with porosity, offering a clearer view of how porosity influences the overall structure of the external boundary layer. From the interface toward the free-flow region, $Q/|Qmax|$ asymptotically approaches zero outside the boundary layer thickness, e.g., $y/lp>1$⁠. However, within the boundary layer, the strain-rate tensor dominates, resulting in negative $Q/|Qmax|$ values. As the flow approaches the interface, $Q/|Qmax|$ becomes positive. Notably, in the fluid layers very close to the interface, $Q/|Qmax|$ turns negative again a few micrometers from the interface, where shear stresses dominate and deformation prevails. These trends are consistently observed across all experiments performed in this study.

The inversion points in these trends are not fixed but follow a consistent pattern: they shift downward and become more compressed near the interface as the porous solid fraction increases. Concurrently, the distribution flattens and approaches an asymptotic null value as porosity decreases. As previously discussed, the boundary layer thickness grows as the characteristic velocity ratio at the interface decreases, leading to a thicker boundary layer at lower porosities. The distribution of $Q/|Qmax|$ further supports these observations by providing additional insights. As porosity decreases, the boundary layer becomes more organized and uniform, with fewer spatial fluctuations in deformation and rotational components. In contrast, higher porosities show significant spatial variations, resulting in more pronounced curvature of the streamlines near the interface. The observed $Q/|Qmax|$ distribution remains consistent across all measured Reynolds numbers and porosity values. Flow conditions have minimal impact on the normalized Q distribution; instead, the structure of the material at the interface is the primary influencing factor. Porosity plays a crucial role in shaping both the distribution of Q and its effective values within the domain. In highly porous materials (⁠ $ϕ=85%$⁠), the negative Q values range between 25% and 50% of the positive values. As porosity decreases (⁠ $ϕ=50%$⁠), this trend reverses, with negative Q values becoming dominant, indicating that local flow rotation is less significant compared to deformation. This behavior reflects the vortex and deformation characteristics of the boundary layer in ordered, periodic, homogeneous, and structured porous materials. However, further studies are recommended to investigate this behavior in disordered and random porous materials, where the flow dynamics may differ significantly.

This study provided a comprehensive analysis of the boundary layer that develops over a permeable interface using highly-resolved micro-PIV. The findings are relevant to both systems coupling a Hele-Shaw cell with a porous material and those involving two porous materials, as the mathematical formulation remains the same.

The results demonstrated that flow behavior at the interface is strongly influenced by the properties of the porous material. Specifically, flow alignment at the microscale becomes less uniform as porosity increases, while decreasing permeability leads to a more parallel and uniform flow pattern within the external boundary layer. Additionally, the experiments revealed that the interaction between the free-flow and the porous medium flow resembles a parallel electrical circuit, where the flow preferentially follows the path of least resistance. The examined Reynolds number range (0.14–0.94) has minimal impact on the velocity distribution, as all velocity profiles converge into a common pattern governed primarily by the permeability and geometry of the porous material.

The analytical solution for the outer boundary layer showed excellent agreement with experimental data, confirming that the boundary layer thickness is typically 2 to 4 times the square root of the permeability of the Hele-Shaw cell. Moreover, the use of the Q-criterion provided an in-depth analysis of the rotational and deformation components of the external boundary layer, whose structure was found to consist of three distinct zones: an outer zone dominated by deformation (⁠ $Q<0$⁠), a central zone dominated by rotation (⁠ $Q>0$⁠), and a thin strip near the interface where deformation prevails. As the resistance of the porous material increases, this structure compresses toward the interface, and the Q distribution flattens to a null value with increasing solid fraction. This behavior is analogous to flows confined by solid walls, such as in Poiseuille or Couette flows.

Despite the many mathematical models in the literature, experimental validation remains challenging due to the complexity of porous structures and fluid-structure interactions. This study addresses these gaps by enabling refined experimentation on microporous systems, offering new insights into fluid interactions at the microscale. A key advancement is the model's ability to predict the external boundary layer thickness, which facilitates the identification of the transition zone in three-layer models (free flow, transition zone, porous media). Further investigation into the boundary layer's rotational and deformational components could enhance mixing performance, with implications for filtration, fuel cells, and evaporation processes.

The findings of this study offer valuable insights into the interplay between permeability, porosity, and flow characteristics in coupled free-flow and porous media systems, and highlight the significance of material properties in shaping the external boundary layer behavior. Future research is needed to extend this analysis to disordered or random porous materials.

The authors gratefully acknowledge the financial support provided by the Israel Science Foundation (ISF) under Grant Agreement No. 2084/21.

The authors have no conflicts to disclose.

Mario Del Mastro: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Alexandros Terzis: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.