Analysis of localized damping effects in channel flows with a periodic rough boundary

In this study, we justify a modified stationary Navier–Stokes system involving the damping effect for a channel flow with periodic roughness. We solve the modified Navier–Stokes system in a smooth extended channel after reducing the rough boundary to a parameter in a linear damping term. We find that the proposed damping term is valid when the size and amplitude of the imperfection tend to zero. Furthermore, the term yields an O(ε^3/2) quadratic approximation of real flow, i.e. O(ε^3/2) approximation in the L²-norm, for a small pressure difference. An effective mass flow is also determined up to an order of O(ε^3/2).