Effective Flow of Incompressible Micropolar Fluid Through a System of Thin Pipes

In this paper, we consider the incompressible micropolar fluid flowing through a multiple pipe system via asymptotic analysis. Introducing the ratio between pipes thickness and its length as a small parameter , we propose an approach leading to a macroscopic model describing the effective flow. In the interior of each pipe (far from the junction), we deduce that the fluid behavior is different depending on the magnitude of viscosity coefficients with respect to . In particular, we prove the solvability of the critical case characterized by the strong coupling between velocity and microrotation. In the vicinity of junction, an interior layer is observed so we correct our asymptotic approximation by solving an appropriate micropolar Leray's problem. The error estimates are also derived providing the rigorous mathematical justification of the constructed approximation. We believe that the obtained result could be instrumental for understanding the microstructure effects on the fluid flow in pipe networks.