Small Darcy number limit of the Navier–Stokes–Darcy system

We study the small Darcy number behavior of the Navier–Stokes–Darcy system with the conservation of mass, Beavers–Joseph–Saffman–Jones condition, and the Lions balance of the normal-force interface boundary conditions imposed on the interface separating the Navier–Stokes flow and Darcy flow. We show that the asymptotic behavior of the coupled system, at small Darcy number, can be captured by two semi-decoupled Darcy number independent sequences: a sequence of (linearized) Navier–Stokes equations, and a sequence of Darcy equations with appropriate initial and boundary data. Approximate solutions to any order of the small parameter (Darcy number) can be constructed via the two sequences. The local in time validity of the asymptotic expansion up to second order is presented. And the global in time convergence is derived under the assumption that the Reynolds number is below a threshold value.