Modeling of a non-Newtonian thin film passing a thin porous medium

This theoretical study deals with asymptotic behavior of a coupling between a thin film of fluid and an adjacent thin porous medium. We assume that the size of the microstructure of the porous medium is given by a small parameter $0<\varepsilon\ll 1$, the thickness of the thin porous medium is defined by a parameter $0<h_\varepsilon\ll 1$, and the thickness of the thin film is defined by a small parameter $0<η_\varepsilon\ll 1$, where $h_\varepsilon$ and $η_\varepsilon$ are devoted to tend to zero when $\varepsilon\to 0$. In this paper, we consider the case of a non-Newtonian fluid governed by the incompressible Stokes equations with power law viscosity of flow index $r\in (1, +\infty)$, and we prove that there exists a critical regime, which depends on $r$, between $\varepsilon$, $η_\varepsilon$ and $h_\varepsilon$. More precisely, in this critical regime given by $h_\varepsilon\approx η_\varepsilon^{2r-1\over r-1}\varepsilon^{-{r\over r-1}}$, we prove that the effective flow when $\varepsilon\to 0$ is described by a 1D Darcy law coupled with a 1D Reynolds law.

💡 Research Summary

This paper investigates the asymptotic behavior of a coupled system consisting of a thin non‑Newtonian fluid film and an adjacent thin porous medium. The porous medium is characterized by a periodic microstructure of size ε ≪ 1, a thickness hε ≪ 1, and the film has thickness ηε ≪ 1, with both hε and ηε tending to zero as ε→0. The fluid obeys the incompressible Stokes equations with a power‑law viscosity μ(|D(u)|)=μ0|D(u)|^{r‑2}, where the flow index r∈(1,∞).

The authors first establish global a‑priori estimates for velocity and pressure using energy inequalities and Korn’s inequality, then derive separate estimates in the porous region and the film. A novel restriction operator Rε^r is introduced to extend the pressure from the perforated porous domain to a fixed‑height domain without microstructure, which is essential for proving pressure continuity across the interface.

A key methodological contribution is a modified unfolding (periodic two‑scale) technique adapted to the thin geometry. By rescaling the vertical coordinate with hε in the porous medium and with ηε in the film, the authors map both subdomains onto fixed‑height reference domains. This allows simultaneous homogenization of the periodic microstructure and dimension reduction.

Through these tools they identify a critical scaling relation that balances the three small parameters:

  hε ≈ ηε^{(2r‑1)/(r‑1)} ε^{‑r/(r‑1)}.

When this relation holds, the three‑dimensional problem collapses to a coupled one‑dimensional system. In the porous medium the effective law is a nonlinear Darcy equation

  −d/dx