Modeling Carreau fluid flows through a very thin porous medium

This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect the plates and are distributed periodically in perpendicular directions. We denote the order of magnitude of the thickness of the domain by $\epsilon$ and define the period and order of magnitude of the cylinders' diameter by $\epsilon$^l, where 0 < l < 1 is fixed. In other words, we consider the regime $\epsilon$ $\ll$ $\epsilon$^l. We assume that the viscosity of the non-Newtonian fluid follows Carreau's law and is scaled by a factor of $\epsilon$^$\gamma$, where $\gamma$ is a real number. Using asymptotic techniques with respect to the thickness of the domain, we perform a new, complete study of the asymptotic behaviour of the fluid as $\epsilon$ tends to zero. Our mathematical analysis is based on deriving sharp a priori estimates through pressure decomposition, and on compactness results for the rescaled velocity and pressure, obtained using the unfolding method. Depending on $\gamma$ and the flow index r, we rigorously derive different linear and nonlinear reduced limit systems. These systems allow us to obtain explicit expressions for the filtration velocity and simpler Darcy's laws for limit pressure.