Modeling Carreau Fluid Flows Through a Very Thin Porous Medium

ABSTRACT 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 and define the period and order of magnitude of the cylinders' diameter by , where is fixed. In other words, we consider the regime . We assume that the viscosity of the non‐Newtonian fluid follows Carreau's law and is scaled by a factor of , where is a real number. Using asymptotic techniques with respect to the thickness of the domain, we perform a new, complete study of the asymptotic behavior of the fluid as tends to zero. From a mathematical perspective, the main novelty of our approach is the use of sharp estimates on the pressure, based on a recent decomposition result for functions defined over a thin layer. Depending on and the flow index , 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.