Very Thin Porous Medium (VTPM)

• VTPM is defined as a porous layer whose thickness is negligible relative to its in-plane dimensions, leading to effective lower-dimensional models.
• The analysis employs vertical rescaling, periodic unfolding, and compactness methods to derive varied effective laws such as Darcy, Carreau, and Bingham models.
• VTPM models find applications in engineering and biology, including tumour invasion membranes, microfluidic devices, and hemodynamic flow diverters.

Searching arXiv for recent and foundational papers on very thin porous media, thin porous layers, and effective interface models. Very Thin Porous Medium (VTPM) denotes, in the cited literature, a porous layer whose thickness is small relative to its in-plane dimensions and whose effective behavior is determined by a combined thin-domain and homogenization limit. Depending on the geometric scaling, a VTPM may appear as a three-dimensional perforated layer of thickness $\varepsilon$, as a thin porous medium with thickness $h_\varepsilon$ and pore scale $\varepsilon \ll h_\varepsilon$, or, in a singular limit, as a zero-thickness interface or screen carrying transmission conditions instead of a volumetric porous law (Anguiano et al., 5 Aug 2025, Anguiano et al., 6 Aug 2025, Ciavolella et al., 2021, Abdehkakha et al., 2021).

1. Geometric archetypes and scale regimes

A standard geometric realization of a VTPM is a domain of the form

$$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$

where $\omega_\varepsilon\subset\mathbb{R}^2$ is obtained by periodically removing solid cylinders from a bounded planar set $\omega$. In the Carreau formulation of VTPM, the cylinders are periodic at scale $\varepsilon^\ell$ with $0<\ell<1$, so that $\varepsilon\ll \varepsilon^\ell$; this regime is explicitly identified as “very thin porous medium (VTPM)” and contrasted with homogeneously thin porous media (HTPM), where $\varepsilon^\ell\ll \varepsilon$, and proportionally thin porous media (PTPM), where $h_\varepsilon$0 (Anguiano et al., 5 Aug 2025). Closely related micropolar and Bingham analyses use the ratio $h_\varepsilon$1, with the VTPM regime corresponding to $h_\varepsilon$2 (Suárez-Grau, 2020, Anguiano et al., 2019).

A second archetype is a thin porous medium whose thickness $h_\varepsilon$3 is small at the macroscopic scale but still much larger than the pore size, with

$h_\varepsilon$4

In this setting there are many pore layers across the thickness, and the thin-domain limit produces a lower-dimensional Darcy law while preserving three-dimensional cell problems in the periodic pore geometry (Anguiano et al., 6 Aug 2025). A coupled thin-film/thin-porous-medium configuration extends this geometry by adjoining a thin free-fluid film of thickness scale $h_\varepsilon$5 across an interface $h_\varepsilon$6, again under the assumptions $h_\varepsilon$7 and $h_\varepsilon$8 (Anguiano et al., 16 Dec 2025).

A third archetype replaces the thin porous layer by a collapsed interface. In the tumour-invasion model, a membrane of thickness $h_\varepsilon$9 separating two bulk domains shrinks to the plane $\varepsilon \ll h_\varepsilon$0, and the asymptotic problem is posed directly on the two bulk domains plus a transmission law on the limiting interface (Ciavolella et al., 2021). In hemodynamics, a flow diverter is modeled not as a volumetric porous block but as a zero-thickness internal surface, or “screen,” carrying a pressure-jump law whose coefficients vary along the device (Abdehkakha et al., 2021). Taken together, these configurations show that VTPM is not tied to a single geometry: the shared feature is the singular role of a porous structure whose thickness is negligible at the macroscopic scale.

2. Governing equations and constitutive descriptions

The governing equations associated with VTPM span several constitutive classes. They include porous-medium transport equations for cell populations, Newtonian Stokes flow with slip, micropolar Stokes systems, generalized Newtonian Carreau laws, viscoplastic Bingham laws, and thin-screen momentum jumps.

Model class	Representative relation or structure	Representative papers
Porous-medium transport	$\varepsilon \ll h_\varepsilon$1, $\varepsilon \ll h_\varepsilon$2	(Ciavolella et al., 2021)
Newtonian Stokes with slip	Stokes system in a thin porous domain with non-homogeneous slip depending on $\varepsilon \ll h_\varepsilon$3	(Anguiano et al., 2017)
Micropolar Stokes	Velocity $\varepsilon \ll h_\varepsilon$4, pressure $\varepsilon \ll h_\varepsilon$5, microrotation $\varepsilon \ll h_\varepsilon$6	(Suárez-Grau, 2020, Anguiano et al., 6 Aug 2025)
Carreau fluid	$\varepsilon \ll h_\varepsilon$7	(Anguiano et al., 2023, Anguiano et al., 5 Aug 2025)
Bingham fluid	$\varepsilon \ll h_\varepsilon$8	(Anguiano et al., 2019)
Thin screen law	$\varepsilon \ll h_\varepsilon$9	(Abdehkakha et al., 2021)

In the porous-medium membrane problem, the unknown is a cell density $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$0 with pressure $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$1, $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$2, and Darcy velocity $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$3. Mass conservation yields

$$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$4

equivalently

$$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$5

so the diffusion operator is of porous-medium type and degenerates where $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$6 (Ciavolella et al., 2021). In the thin-screen hemodynamic model, by contrast, the governing object is not a volumetric permeability field but an interfacial pressure drop

$$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$7

with local coefficients determined from porosity $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$8 and hydraulic diameter $$\Omega_\varepsilon=\omega_\varepsilon\times(0,\varepsilon),$$9 (Abdehkakha et al., 2021).

The Newtonian slip problem studies the Stokes system in a thin porous medium perforated by periodically distributed solid cylinders, with a Robin-type non-homogeneous slip law on the obstacle boundaries involving a parameter $\omega_\varepsilon\subset\mathbb{R}^2$0. The body force is horizontal, the surface force is periodic, and the limit law depends sensitively on the slip scaling $\omega_\varepsilon\subset\mathbb{R}^2$1 (Anguiano et al., 2017). The micropolar formulations augment velocity and pressure with a microrotation field $\omega_\varepsilon\subset\mathbb{R}^2$2, coupled through curl terms and controlled by a coupling number $\omega_\varepsilon\subset\mathbb{R}^2$3 and a characteristic micropolar length $\omega_\varepsilon\subset\mathbb{R}^2$4 (Suárez-Grau, 2020, Anguiano et al., 6 Aug 2025).

For generalized Newtonian fluids, the Carreau viscosity produces several distinct VTPM limits. In one thin porous setting the asymptotics depend on both the flow index $\omega_\varepsilon\subset\mathbb{R}^2$5 and a viscosity scaling $\omega_\varepsilon\subset\mathbb{R}^2$6, yielding linear Darcy laws, nonlinear Carreau-type Darcy laws, or nonlinear power-law Darcy laws according to the regime (Anguiano et al., 2023). In a VTPM with pore scale $\omega_\varepsilon\subset\mathbb{R}^2$7, the same rheology leads to reduced Hele–Shaw-type cell problems and explicit filtration laws for pseudoplastic and dilatant fluids (Anguiano et al., 5 Aug 2025). The Bingham case replaces shear-dependent viscosity by a yield-stress inequality, and the effective laws remain nonlinear in all three geometric regimes of $\omega_\varepsilon\subset\mathbb{R}^2$8 (Anguiano et al., 2019).

3. Asymptotic machinery: rescaling, unfolding, extension, and compactness

A common analytic step is vertical rescaling. For layers of thickness $\omega_\varepsilon\subset\mathbb{R}^2$9, one sets $\omega$0; for layers of thickness $\omega$1, one sets $\omega$2. This transforms the thin domain into a fixed-height set and introduces anisotropic differential operators such as

$\omega$3

or

$\omega$4

so that vertical derivatives explicitly carry the thinness scale (Anguiano et al., 2017, Anguiano et al., 2023, Anguiano et al., 6 Aug 2025).

Periodic unfolding is then used to separate slow and fast variables. In the Newtonian slip problem the unfolding acts only in the horizontal variables and converts bulk and boundary integrals in the thin perforated domain into integrals over $\omega$5 and $\omega$6 (Anguiano et al., 2017). In the micropolar thin porous medium with $\omega$7, unfolding maps the rescaled porous set into $\omega$8 and yields uniform control of unfolded velocity, microrotation, and pressure-corrector fields (Anguiano et al., 6 Aug 2025). In the VTPM Carreau problem, the unfolding is adapted to the mixed scales $\omega$9 and $\varepsilon^\ell$0, converting functions on $\varepsilon^\ell$1 into functions on $\varepsilon^\ell$2 and allowing the limit equations to be written on a fixed product domain (Anguiano et al., 5 Aug 2025). In the thin-film/VTPM coupling, an unfolding tailored to the porous layer is combined with separate rescaling in the film region (Anguiano et al., 16 Dec 2025).

When the coefficients degenerate inside the thin layer, compactness may fail unless the solutions are extended. In the membrane problem, the mobility in the membrane satisfies $\varepsilon^\ell$3, so the natural $\varepsilon^\ell$4-bound controls only $\varepsilon^\ell$5, not $\varepsilon^\ell$6. The analysis therefore constructs an explicit extension operator $\varepsilon^\ell$7 by reflecting functions from the bulk domains into the thin layer; this yields strong convergence of $\varepsilon^\ell$8 and $\varepsilon^\ell$9 and weak convergence of their gradients on $0<\ell<1$0 (Ciavolella et al., 2021). In the Carreau VTPM, pressure decomposition $0<\ell<1$1 and extension operators provide compactness for the macro-pressure while showing that the micro-pressure vanishes in the limit (Anguiano et al., 5 Aug 2025). The power-law thin-film/VTPM coupling uses a restriction operator preserving divergence-free fields in the porous layer and a De Rham-type pressure reconstruction (Anguiano et al., 16 Dec 2025).

For nonlinear constitutive laws, limit passage relies on monotonicity. The Carreau analyses explicitly use Minty’s method to identify the nonlinear limit operator in the critical scaling $0<\ell<1$2 and in the dilatant power-law regime $0<\ell<1$3 (Anguiano et al., 2023, Anguiano et al., 5 Aug 2025). The Bingham analysis passes to the limit in convex plastic dissipation functionals and obtains variational inequalities of Bingham type in the homogenized problems (Anguiano et al., 2019). A plausible implication is that VTPM analysis is less a single theorem than a recurring technical pattern: anisotropic rescaling identifies the dominant gradients, unfolding resolves the pore geometry, and compactness or monotonicity arguments decide which nonlinearities survive.

4. Effective laws and lower-dimensional closures

The effective models produced by VTPM asymptotics are uniformly lower-dimensional, but they are not uniform in form. Linear Darcy laws, generalized Darcy laws with microrotation, nonlinear power-law and Carreau filtrations, Bingham-type constitutive maps, Kedem–Katchalsky transmission conditions, and quadratic screen jumps all arise in the cited literature.

For generalized Newtonian Carreau flow in a VTPM with $0<\ell<1$4, the pseudoplastic problem with $0<\ell<1$5 and the dilatant problem with $0<\ell<1$6 reduce to a linear Darcy law

$0<\ell<1$7

where $0<\ell<1$8 is a permeability tensor defined by 2D Hele–Shaw cell problems on $0<\ell<1$9, and $\varepsilon\ll \varepsilon^\ell$0 is $\varepsilon\ll \varepsilon^\ell$1 or $\varepsilon\ll \varepsilon^\ell$2 depending on the scaling regime (Anguiano et al., 5 Aug 2025). In the proportionally thin Carreau medium, the corresponding linear regimes yield

$\varepsilon\ll \varepsilon^\ell$3

with $\varepsilon\ll \varepsilon^\ell$4 computed from 3D cell Stokes problems in $\varepsilon\ll \varepsilon^\ell$5 (Anguiano et al., 2023). These are Darcy laws in the strict sense: the mean in-plane flux is linearly related to the in-plane pressure gradient and forcing.

Micropolar VTPM limits retain coupled translational and rotational response. In the thin porous medium with $\varepsilon\ll \varepsilon^\ell$6, the averaged fields satisfy

$\varepsilon\ll \varepsilon^\ell$7

together with

$\varepsilon\ll \varepsilon^\ell$8

in $\varepsilon\ll \varepsilon^\ell$9 (Anguiano et al., 6 Aug 2025). In the VTPM regime $\varepsilon^\ell\ll \varepsilon$0, the earlier micropolar model yields

$\varepsilon^\ell\ll \varepsilon$1

with a generalized Darcy equation for $\varepsilon^\ell\ll \varepsilon$2 and with $\varepsilon^\ell\ll \varepsilon$3 (Suárez-Grau, 2020).

Nonlinear rheologies survive in critical scalings. In the VTPM Carreau model, the case $\varepsilon^\ell\ll \varepsilon$4 produces a nonlinear Darcy law of Carreau type, with $\varepsilon^\ell\ll \varepsilon$5 given by an explicit nonlinear function of $\varepsilon^\ell\ll \varepsilon$6 through a 2D nonlinear cell problem (Anguiano et al., 5 Aug 2025). In the proportionally thin Carreau model, the same critical scaling yields

$\varepsilon^\ell\ll \varepsilon$7

where $\varepsilon^\ell\ll \varepsilon$8 is defined through 3D Carreau cell problems in $\varepsilon^\ell\ll \varepsilon$9 (Anguiano et al., 2023). For dilatant fluids with $h_\varepsilon$00, the VTPM limit becomes a nonlinear power-law Darcy law governed by a monotone coercive map $h_\varepsilon$01 (Anguiano et al., 5 Aug 2025).

For Bingham fluids the limit law remains nonlinear in all three geometric regimes $h_\varepsilon$02, $h_\varepsilon$03, and $h_\varepsilon$04. The macroscopic filtration velocity takes the form

$h_\varepsilon$05

where $h_\varepsilon$06 is a nonlinear map determined by a Bingham cell problem that is 3D in the critical regime, 2D in the subcritical regime, and 1D in the supercritical regime (Anguiano et al., 2019).

The interface and screen formulations are lower-dimensional in a different sense. In the porous-medium membrane problem, the limit bulk equations are posed on the two sides of $h_\varepsilon$07, while the vanished membrane is encoded through

$h_\varepsilon$08

which is identified as a nonlinear generalized Kedem–Katchalsky condition (Ciavolella et al., 2021). In the flow-diverter screen model, the lower-dimensional closure is

$h_\varepsilon$09

with $h_\varepsilon$10 and $h_\varepsilon$11 (Abdehkakha et al., 2021). The literature therefore does not support a single universal “VTPM law”; it supports a family of asymptotically consistent closures whose form is dictated by geometry, rheology, and scale separation.

5. Role of scale ratios and critical parameters

The central structural variable in VTPM theory is the ratio between thickness and pore scale. One formulation uses $h_\varepsilon$12: in micropolar and Bingham models, $h_\varepsilon$13 yields a critical regime, $h_\varepsilon$14 a homogeneously thin regime, and $h_\varepsilon$15 the VTPM regime proper (Suárez-Grau, 2020, Anguiano et al., 2019). Another formulation uses $h_\varepsilon$16 for the pore scale in a layer of thickness $h_\varepsilon$17; then $h_\varepsilon$18 implies $h_\varepsilon$19 and defines VTPM, while $h_\varepsilon$20 and $h_\varepsilon$21 correspond to PTPM and HTPM respectively (Anguiano et al., 5 Aug 2025). A third formulation uses a thin porous thickness $h_\varepsilon$22 with $h_\varepsilon$23, so the medium is thin at the macro-scale yet contains many pore layers across its thickness (Anguiano et al., 6 Aug 2025, Anguiano et al., 16 Dec 2025).

These ratios determine the local problem dimension. In the micropolar thin porous medium, the VTPM regime leads to a micropolar Reynolds-type system with derivatives only in the vertical coordinate $h_\varepsilon$24 and 2D cross-sectional cell problems, whereas the PTPM regime requires fully 3D local Stokes–micropolar problems and the HTPM regime yields 2D micropolar local problems (Suárez-Grau, 2020). In the Bingham case, the same trichotomy leads to 3D, 2D, and 1D Bingham-type cell problems respectively (Anguiano et al., 2019). In the Carreau VTPM, the authors state that compared to proportionally thin media, the microproblems simplify to Hele–Shaw-type problems involving only pressure, not a full velocity–pressure Stokes system (Anguiano et al., 5 Aug 2025).

A second class of critical parameters concerns constitutive scaling. In the slip Stokes problem, the Robin coefficient scales like $h_\varepsilon$25, and three limit regimes are derived for $h_\varepsilon$26, $h_\varepsilon$27, and $h_\varepsilon$28. These regimes determine whether the limit Darcy law is driven only by pressure, by pressure plus body and surface forces, or by a tensorial permeability coming from 2D cell Stokes problems (Anguiano et al., 2017). In Carreau flow, the viscosity prefactor $h_\varepsilon$29 plays the analogous role: $h_\varepsilon$30 yields a linear Darcy law with viscosity $h_\varepsilon$31, $h_\varepsilon$32 preserves the full Carreau nonlinearity, and $h_\varepsilon$33 produces either a linear law with viscosity $h_\varepsilon$34 or a nonlinear power-law Darcy law depending on whether $h_\varepsilon$35 or $h_\varepsilon$36 (Anguiano et al., 2023, Anguiano et al., 5 Aug 2025).

The coupled thin-film/VTPM problem adds a third scale and a genuine critical balance. For a power-law fluid of flow index $h_\varepsilon$37, the critical regime is

$h_\varepsilon$38

under which the effective model becomes a coupled $h_\varepsilon$39D Darcy law in the porous layer and $h_\varepsilon$40D Reynolds law in the film (Anguiano et al., 16 Dec 2025). This suggests that “critical” in VTPM theory has a precise meaning: it designates scale balances under which multiple substructures remain visible in the macroscopic law instead of one asymptotic mechanism dominating the others.

6. Applications, modeling choices, and limitations

The motivating applications are diverse. The porous-medium membrane problem is explicitly motivated by biological applications on tumour invasion through thin membranes, with cell-density transport under Darcy’s law and an effective nonlinear generalized Kedem–Katchalsky transmission condition on the collapsed membrane (Ciavolella et al., 2021). The Newtonian slip problem lists manufacturing of fibre-reinforced polymer composites via liquid moulding processes, passive mixing in microfluidic devices, paper-making processes, block copolymer thin-film self-assembly on nanometer scales, and heat and mass transfer equipment using fibrous or porous layers (Anguiano et al., 2017). The micropolar thin porous formulations point to filters, coatings, gaskets, microfluidic devices, lubrication with structured fluids, and biological tissues (Anguiano et al., 6 Aug 2025). The coupled thin-film/VTPM model is motivated by flows in the oil industry, such as non-Newtonian drilling muds or fracturing fluids infiltrating porous rock around a wellbore (Anguiano et al., 16 Dec 2025).

The hemodynamic application is the most quantitatively developed. The thin inhomogeneous porous medium (iPM) model represents a cerebral aneurysm flow diverter as a zero-thickness internal surface whose resistance is assigned pore by pore from local device geometry. In three patient-specific aneurysms, iPM predictions of aneurysm-averaged velocity, inflow rate, shear rate, and turnover time differed by only $h_\varepsilon$41–$h_\varepsilon$42 from explicit-device CFD, summarized as $h_\varepsilon$43–$h_\varepsilon$44 accuracy, while “iPM CFD ran 500% faster” than explicit CFD; the reported mesh reduction is about $h_\varepsilon$45 and runtime reduction about $h_\varepsilon$46 (Abdehkakha et al., 2021). This does not imply that every VTPM model is a screen model, but it does show that interface-type representations can be both asymptotically appropriate and computationally decisive when the porous structure is extremely thin.

A recurrent modeling distinction concerns whether the porous structure should be resolved as a volume or collapsed to a surface. The cited works show both possibilities. Very thin perforated domains with no-slip, no-spin, slip, Carreau, micropolar, or Bingham constitutive laws are treated by volume homogenization and dimension reduction (Anguiano et al., 2017, Suárez-Grau, 2020, Anguiano et al., 2019, Anguiano et al., 2023, Anguiano et al., 5 Aug 2025, Anguiano et al., 6 Aug 2025). By contrast, the tumour membrane and the flow-diverter screen are replaced in the limit by interfacial laws on zero-thickness sets (Ciavolella et al., 2021, Abdehkakha et al., 2021). A common misconception is therefore that a VTPM must always be represented by a bulk Darcy–Forchheimer block. The literature instead supports a more conditional statement: when the thickness becomes asymptotically negligible, an interface law or screen law may be more faithful than a volumetric porous approximation.

The limitations are likewise systematic. The thin porous analyses assume periodic microstructure, smooth obstacles, rigid solid matrices, and steady or quasi-steady low-Reynolds-number settings in the Stokes regime (Anguiano et al., 2017, Suárez-Grau, 2020, Anguiano et al., 2023, Anguiano et al., 6 Aug 2025). The aneurysm iPM model assumes incompressible Newtonian blood, rigid no-slip walls, and steady-state flow, and it was validated only for a single type of flow diverter, the Pipeline Embolization Device, with extension to other devices or overlapping layers left open (Abdehkakha et al., 2021). The micropolar and Carreau studies fix specific constitutive scalings such as $h_\varepsilon$47 or $h_\varepsilon$48, and the resulting effective laws are rigorously valid within those regimes rather than universally (Suárez-Grau, 2020, Anguiano et al., 6 Aug 2025, Anguiano et al., 2023). The Bingham work similarly ties the persistence of nonlinear yield behavior to carefully chosen scalings of $h_\varepsilon$49 with $h_\varepsilon$50 or $h_\varepsilon$51 (Anguiano et al., 2019).

Across these applications and limitations, VTPM emerges not as a single equation but as a modeling framework for singularly thin porous structures. Its defining mathematical content is the replacement of a three-dimensional microscopic problem by an effective lower-dimensional description whose coefficients, constitutive maps, or transmission laws still encode pore geometry, rheology, and dominant scale ratios.