Mathematical derivation of a Reynolds equation for magneto-micropolar fluid flows through a thin domain

In this paper, we study the asymptotic behavior of the stationary 3D magneto-micropolar fluid flow through a thin domain, whose thickness is given by a parameter \(0<\varepsilon \ll 1\) $0<\epsilon \ll 1$ . Assuming that the magnetic Reynolds number is written in terms of the thickness \(\varepsilon \) $\epsilon$ , we prove that there exists a critical magnetic Reynolds number, namely \(Re_m^c=\varepsilon ^{-2}\) $R{e}_m^c={\epsilon }^{-2}$ , such that for every magnetic Reynolds number \(Re_m\) $R{e}_m$ with order smaller or equal than \(Re_m^c\) $R{e}_m^c$ , the magneto-micropolar fluid flow in the thin domain can be modeled asymptotically when \(\varepsilon \) $\epsilon$ tends to zero by a 2D Reynolds-like model, whose expression is also given.