Nonlinear Brinkman-Type Law in Thin Porous Media

• Non-linear Brinkman-type law is a homogenized model that captures vertical shear-thinning effects and microstructural drag in thin, perforated porous media.
• The model integrates non-Newtonian power-law rheology with two-scale asymptotic analysis to derive effective macroscopic flow equations.
• The scaling parameter λ governs the transition between Darcy, Brinkman, and Reynolds-type behaviors, providing actionable insights for microfluidic design.

A non-linear Brinkman-type law describes the macroscopic behavior of generalized Newtonian fluids, especially those with power-law viscosity, in complex domains such as thin porous media perforated with small obstacles. This law emerges as the homogenized limit of the Stokes momentum equations with non-linear (typically shear-thinning, $1$h_\varepsilon$ of the domain and the obstacle diameter $\varepsilon\delta_\varepsilon$. The critical balance between the characteristic size of the microstructure and the thinness of the domain is captured by the parameter $$\lambda = \lim_{\varepsilon\to0} \sigma_\varepsilon/h_\varepsilon$$, where $$\sigma_\varepsilon = \varepsilon/\delta_\varepsilon^{(2-r)/r}$$. When $\lambda \in (0,+\infty)$, both vertical and microscopic diffusive effects persist, leading to a homogenized lower-dimensional non-linear Brinkman-type law.

1. Mathematical Setting and Scaling Regimes

Consider the flow of a generalized Newtonian fluid in a thin domain $h_\varepsilon \to 0$ perforated by $\varepsilon$-periodically arranged vertical cylinders, each cylinder of diameter $\varepsilon\delta_\varepsilon \to 0$ as $\varepsilon \to 0$. The fluid is governed by the power-law Stokes system with viscosity $\eta_r(\mathbf{D}[u]) = \mu |\mathbf{D}[u]|^{r-2}$ for flow index $r$ satisfying $1 < r < 2$ (shear-thinning regime). A crucial parameter is

$$\sigma_\varepsilon = \varepsilon / \delta_\varepsilon^{(2-r)/r}.$$

The behavior as $\varepsilon \to 0$ falls into three regimes, depending on the limiting value

$$\lambda = \lim_{\varepsilon \to 0} \frac{\sigma_\varepsilon}{h_\varepsilon} \in [0, +\infty].$$

The value of $\lambda$ determines the form of the limiting model:

• $\lambda=0$: non-linear Darcy law,
• $\lambda \in (0,+\infty)$: non-linear Brinkman-type law,
• $\lambda=+\infty$: non-linear Reynolds (or thin Stokes) law.

This classification arises because the ratio $\sigma_\varepsilon/h_\varepsilon$ quantifies the competition between the obstacle-induced microstructural (microscopic) effects and the transverse diffusive effects associated with domain thinness.

2. Critical Scaling and the Limit Non-linear Brinkman-Type Law

For the critical case $\lambda \in (0, +\infty)$ (i.e., $\sigma_\varepsilon$ and $h_\varepsilon$ are of the same order), the vertical and horizontal diffusive effects are retained in the homogenized limit. After rescaling and taking the limit, the velocity field $$u' = (u_1, u_2): \Omega = \omega \times (0,1) \to \mathbb{R}^2$$ (with $u_3 \equiv 0$) satisfies

$$\begin{cases} -\mu \lambda^r 2^{-r/2} \,\partial_{y_3}\left(|\partial_{y_3}u'(x',y_3)|^{r-2}\partial_{y_3}u'(x',y_3)\right) + \mu\,\mathcal{G}(u'(x',y_3)) + \nabla_{x'}p(x') = f'(x') &\text{in }\Omega\ u_3(x',y_3) \equiv 0,\ u'(x',0) = u'(x',1) = 0 &\text{for } x'\in\omega\ \operatorname{div}_{x'}\left(\int_0^1 u'(x',y_3)dy_3\right) = 0 &\text{in }\omega\ \left(\int_0^1 u'(x',y_3)dy_3\right)\cdot n' = 0 &\text{on }\partial\omega. \end{cases}$$

Here, $\mathcal{G}:\mathbb{R}^2 \to \mathbb{R}^2$ is a nonlinear drag operator defined below, and the constant $2^{-r/2}$ arises from the thin domain symmetric gradient.

This law demonstrates two dissipative mechanisms: a vertical diffusion term with non-linear dependence on the vertical derivative of $u'$, and a nonlinear drag term $\mathcal{G}(u')$ modeling the microscopic resistance due to the array of small cylinders.

3. Nonlinear Power-Law Viscosity and Stress

The extra-stress tensor for a generalized Newtonian (shear-thinning) fluid is

$$\bm{\tau}(u) = \mu|\mathbf{D}[u]|^{r-2}\mathbf{D}[u],$$

where $\mathbf{D}[u]$ is the symmetric gradient. The non-linearity comes from the shear-thinning index $1 < r < 2$, which means that as the local strain rate increases, the effective viscosity decreases. Upon homogenization, this non-linearity is preserved:

• The vertical diffusion acting on $u'$ in the thin region involves the operator $S(\xi) = |\xi|^{r-2}\xi$.
• The effective 1D vertical dissipative term thus reads $$-\mu\lambda^r 2^{-r/2}\partial_{y_3}[|\partial_{y_3}u'|^{r-2}\partial_{y_3}u']$$.

This term embodies the principal effect of the power-law rheology at the macroscopic scale and distinguishes the non-linear Brinkman-type law from its linear counterpart.

4. Microstructure-Induced Nonlinear Drag and the Cell Problem

The drag function $\mathcal{G}$ arises from the microscopic geometry through a nonlinear cell (or "exterior") problem. For a fixed direction $\xi' \in \mathbb{R}^2$, the auxiliary problem is

$$\begin{cases} -\operatorname{div}_{z'}\left( |\mathbf{D}_{z'}[w^{(\xi')}|^{r-2} \mathbf{D}_{z'}[w^{(\xi')}] \right) + \nabla_{z'} \pi^{(\xi')} = 0 &\text{in } \mathbb{R}^2 \setminus Y_s',\ \operatorname{div}_{z'} w^{(\xi')} = 0 &\text{in } \mathbb{R}^2 \setminus Y_s',\ w^{(\xi')} = 0 &\text{on } \partial Y_s',\ w^{(\xi')}(z') \to \xi' &\text{as } |z'| \to \infty. \end{cases}$$

Here, $Y_s'$ is the solid obstacle cross-section. The nonlinear drag is then prescribed by

$$\mathcal{G}(\xi') \cdot \tau' = \int_{\mathbb{R}^2 \setminus Y_s'} \left| \mathbf{D}_{z'}[w^{(\xi')}] \right|^{r-2} \mathbf{D}_{z'}[w^{(\xi')}] : \mathbf{D}_{z'}[w^{(\tau')}] dz',\quad \forall \tau' \in \mathbb{R}^2.$$

This construction ensures that non-Newtonian, geometry-induced resistance is faithfully encapsulated in the macroscopic law.

5. Domain Geometry, Thinness, and the Homogenization Procedure

The model arises in a setting where the global domain is thin ($h_\varepsilon\to0$) and densely perforated with vanishingly small vertical cylinders ($\varepsilon\delta_\varepsilon \to 0$). The geometry defines the scaling parameter $\lambda$:

• If $\lambda=0$ ($\sigma_\varepsilon \ll h_\varepsilon$), the microscopic constraints are dominant, and one recovers a non-linear Darcy law.
• If $\lambda=+\infty$ ($\sigma_\varepsilon \gg h_\varepsilon$), the thinness is dominant, yielding a nonlinear Reynolds dominance and a reduced 2D nonlinear Stokes system.
• Precisely when $$\sigma_\varepsilon/h_\varepsilon \to \lambda \in (0, +\infty)$$, the critical balance gives rise to the non-linear Brinkman-type law, capturing both effects.

The emergence of the vertical nonlinear diffusion and the microstructural drag is a direct result of two-scale homogenization and unfolding methods applied in this geometric regime.

6. Physical and Modeling Implications

The non-linear Brinkman-type law derived in this setting is significant for several reasons:

• It accounts for both vertical (thin domain) shear-thinning dissipation and horizontal/microscale drag, relevant in filtration, microfluidics, and biological transport.
• The vertical operator inherits nonlinearity from the microscopic rheology; the horizontal drag is completely dictated by the microstructure and flow index.
• The analysis provides explicitly how the effective macroscopic law transitions between classical non-linear Darcy, Brinkman, and Reynolds-type behaviors as geometric parameters change.
• The structure is particularly well-suited for numerical simulation, as the lower-dimensional model effectively reduces computational costs while incorporating essential nonlinear mechanisms.

This framework can be succinctly represented in LaTeX as: $\begin{cases} -\mu \lambda^r 2^{-r/2}\partial_{y_3}\left(|\partial_{y_3}u'|^{r-2}\partial_{y_3}u'\right) + \mu\,\mathcal{G}(u') + \nabla_{x'}p = f',\[2mm] u_3\equiv 0,\ u'(x',0)=u'(x',1)=0,\ \operatorname{div}_{x'}\left(\int_0^1 u'(x',y_3)\,dy_3\right) = 0,\ \left(\int_0^1 u'(x',y_3)\,dy_3\right)\cdot n' = 0, \end{cases}$ where $\lambda$ quantifies the ratio between the geometric microstructure and the domain thickness.

7. Broader Context and Connections

The critical non-linear Brinkman-type law in thin perforated domains generalizes classical Brinkman and Darcy models to account for power-law (non-Newtonian) effects and intricate geometry. It illustrates how homogenization and matched asymptotic expansions yield reduced macroscopic models with explicit dependence on both rheological parameters and geometric configuration. Such nonlinear laws are crucial for accurately modeling transport in thin tissue scaffolds, membrane separation processes, or engineered microporous materials where non-Newtonian fluid effects and multiscale structural features must be taken into account (Anguiano et al., 6 Aug 2025).