Non-Isothermal Darcy-Brinkman Thin Film Flow

• Non-Isothermal Darcy-Brinkman thin-film flow is a coupled fluid and thermal transport system in porous media characterized by small aspect ratios and the interplay of Darcy drag with Brinkman viscous corrections.
• The modeling framework employs asymptotic reduction, non-dimensionalization, and periodic unfolding to derive effective macroscopic equations that capture the influence of surface roughness and multiscale interactions.
• This approach provides practical insights for applications in engineered porous systems, lubrication, and microfluidics by linking effective permeability and heat transfer with regulated roughness regimes.

Non-isothermal Darcy-Brinkman thin-film flow refers to the coupled fluid and thermal transport in saturated porous media of small aspect ratio, wherein momentum transfer is governed by both Darcy drag and Brinkman viscous correction, and the temperature field evolves under the influence of conduction, external boundary conditions, and internal viscous dissipation. The essential modeling framework addresses the multiscale interactions between geometric confinement, surface roughness, and nontrivial thermal sources, requiring rigorous asymptotic reduction, homogenization, and analysis of the resulting limit models.

1. Mathematical Formulation and Governing Equations

The non-isothermal Darcy-Brinkman thin-film system is described for a velocity field $u^\varepsilon(x)\in\mathbb{R}^3$, pressure $p^\varepsilon(x)\in\mathbb{R}$, and temperature $T^\varepsilon(x)\in\mathbb{R}$ in a thin oscillatory domain

$$\Omega^\varepsilon = \{ (x',x_3)\in\omega\times\mathbb{R}: 0 < x_3 < \varepsilon h(x'/\varepsilon^\ell) \},$$

where $0 < \varepsilon \ll 1$ is the non-dimensional thickness, $\varepsilon^\ell$ ($0<\ell\le1$) is the roughness period, and $h$ is a $Z'$-periodic boundary profile. The system's dimensional form reads, for $x\in\Omega^\varepsilon$, \begin{align*} &-\mu\,\Delta u^\varepsilon + \frac{\mu}{K^\varepsilon} u^\varepsilon + \nabla p^\varepsilon = f^\varepsilon,\ &\nabla\cdot u^\varepsilon = 0,\ &\rho\,c_p\,(u^\varepsilon\cdot\nabla T^\varepsilon) - k\,\Delta T^\varepsilon = \Phi^\varepsilon,\ &\Phi^\varepsilon = \mu\sum_{i,j=1}^3 (\partial_i u_j^\varepsilon + \partial_j u_i^\varepsilon)^2, \end{align*} with material parameters: $\mu$ (dynamic viscosity), $K^\varepsilon$ (permeability), $k$ (thermal conductivity), $\rho c_p$ (volumetric heat capacity), and a prescribed body force $f^\varepsilon$. Boundary conditions are no-slip for velocity and Dirichlet or flux-type for temperature on the lower and upper surfaces $x_3 = 0,\,\varepsilon h(\cdot)$ (Anguiano et al., 21 Dec 2025).

2. Non-dimensionalization and Thin-Film Asymptotics

Scaling the domain thickness with $\varepsilon$ and introducing the vertical coordinate $x_3 = \varepsilon z_3$ yields the stretched domain $$\widetilde\Omega^\varepsilon = \{ (x',z_3) : 0<z_3<h(x'/\varepsilon^\ell) \}$$. The flow is characterized by the following non-dimensional groups:

• $\mathrm{Da} = K/L^2$ (Darcy number)
• $\mathrm{Br} = \mu U^2/(k \Delta T)$ (Brinkman number)
• $\mathrm{Bi} = h_c L/k$ (Biot number)
• $\mathrm{Pe} = \rho c_p U L / k$ (Péclet number) (Pažanin et al., 6 Aug 2025)

Distinct regimes arise based on the roughness exponent $\ell$:

• Subcritical: $0<\ell<1$, roughness varies more slowly than film thickness.
• Critical: $\ell=1$, roughness period scales with thickness.
• Supercritical: $\ell>1$, roughness oscillations are so fine that the domain is macroscopically smooth.

The asymptotic analysis proceeds by expanding the unknowns in $\varepsilon$ and seeking leading-order reduced models (Anguiano et al., 21 Dec 2025, Pažanin et al., 6 Aug 2025).

3. Homogenization, Unfolding, and Limit Model Structure

Through periodic unfolding and appropriate rescalings, the asymptotics reveal effective equations in the limit $\varepsilon\to0$. The procedure involves:

• Dilation of the vertical coordinate to obtain a reference cell,
• Extension of unknowns by zero outside the physical domain,
• Definition of unfolded functions at the roughness scale $\varepsilon^\ell$.

Uniform a priori estimates in Sobolev spaces permit passage to limits for subsequences: $$\widehat u_\varepsilon \rightharpoonup \widehat u(x',z),\quad \widehat T_\varepsilon \rightharpoonup \widehat T(x',z),\quad \varepsilon^2 p^0_\varepsilon \rightharpoonup \widetilde p(x')$$ (Anguiano et al., 21 Dec 2025).

The limiting coupled system typically comprises:

• A Reynolds-type equation for macroscopic pressure,
• Darcy-Brinkman cell problems on the microstructure domain for effective permeability and dissipation kernels,
• A reduced energy equation incorporating viscous dissipation, with macroscopic temperature obtained by averaging over the cell.

This homogenized structure depends critically on the nature of the roughness regime.

4. Impact of Surface Roughness and Coupling Phenomena

The significance of the surface roughness emerges through its influence on the limit equations:

• For $\ell<1$, coefficients in the reduced equations are averaged only in $z_3$, leading to standard Reynolds-Brinkman-heat systems.
• For $\ell=1$, the so-called "critical regime," the cell problems are fully three-dimensional in $(z',z_3)$ and must be solved simultaneously, resulting in strong coupling between the Reynolds equation, permeability tensor, and macro-scale heat generation. Specifically, both the flow permeability tensor and the heat-generation term are functionals of the detailed roughness geometry through the same microscopic cell functions $(w^i,\widehat T)$.
• For $\ell>1$, the oscillations average out entirely, and the model reduces to the smooth-wall Darcy-Brinkman-heat limit (Anguiano et al., 21 Dec 2025).

This coupling, present only in the critical regime, means that the temperature cannot generally be decoupled from the pressure field; their interplay is governed by the nonlinear dependence of the cell solutions on the macro-scale gradients.

5. Reduced Model Quantities and Physical Interpretation

Key explicit structures and interpretations of the limiting model components are as follows:

• The macroscopic pressure satisfies a closed Reynolds- or Reynolds-Brinkman-type PDE, e.g.,

$$-\nabla_{x'} \cdot \bigg( \frac{h^3(x')}{12\,\mu} \nabla_{x'} p^0(x') \bigg) + F(x',T^0(x')) = 0,$$

with $F$ representing the effect of dissipation after cell averaging.

• The effective permeability tensor $A$ is constructed by solving the cell problems for local Brinkman flows driven by tangential unit vectors, entering the upscaled model as

$$-\nabla_{x'}\cdot\left( \frac{K}{\mu}A(f' - \nabla_{x'}p^0)\right) = 0.$$

• The limit temperature $T^0(x')$ results from integrating the solution of the cell-scale energy equation, including both the Darcy and Brinkman viscous heating contributions.

For the thin-film regime without oscillatory roughness, the limit reduces to a one-dimensional cell problem, while in the critical oscillatory regime, the full multidimensional cell problem is retained.

In the energy balance, physical interpretations include:

• The Brinkman correction ($$\propto -2\,(\mu_{\rm eff}/\mu)\,\mathrm{Da}\,\partial_z^2 u_0$$) represents viscous-shear enhancement to Darcy law,
• Classical Darcy drag ($\propto (1/\mathrm{Da})\,u_0$) persists,
• The energy dissipation comprises terms for both bulk friction (Darcy) and local shear (Brinkman),
• Non-homogeneous thermal boundary conditions at the substrate introduce additional temperature gradients (Pažanin et al., 6 Aug 2025).

6. Mathematical Convergence and Regime Classification

Rigorous justification of the above reductions employs compactness results and uniform bounds in Sobolev (and Bochner) spaces for the unknowns. As $\varepsilon\to0$, one has weak convergence in velocity and temperature and strong convergence in pressure, with regime-dependent limit models (Anguiano et al., 21 Dec 2025). The possible limiting cases are:

Roughness Regime	Cell Problem Dimensionality	Coupling Structure
$\ell<1$ (subcritical)	1D ($z_3$)	Weak (decoupled after averaging)
$\ell=1$ (critical)	3D ($z',z_3$)	Strong (fully coupled, nonlinear system)
$\ell>1$ (supercritical)	— (no effect)	Decoupled, smooth-wall limit

A plausible implication is that engineered manipulation of surface roughness at the scale of the film thickness enables macroscopic control of both effective permeability and heat transfer characteristics, whereas roughness at much larger or smaller scales is homogenized out and does not affect the bulk transport properties.

7. Applications and Practical Implications in Porous Media

The rigorous models for non-isothermal Darcy-Brinkman thin-film flow, incorporating roughness and viscous dissipation, are directly applicable in a range of systems:

• Lubrication in porous bearings or microfluidic devices with engineered surface texture,
• Enhanced heat exchange and flow control in membrane filtration technologies,
• Groundwater and oil reservoir simulations where both small-scale heterogeneity and thermal effects are non-negligible.

The methodology, particularly the unfolded homogenization, offers a systematic approach to derive effective boundary and bulk laws from microscale geometry and physics. The regime classification provides guidance for the design of materials or fluidic devices aiming to exploit or suppress roughness-induced effects (Anguiano et al., 21 Dec 2025, Pažanin et al., 6 Aug 2025).