Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary

We consider a non-Newtonian flow in a thin domain of thickness εε. The flow is described by the 3D incompressible Navier–Stokes (Stokes) system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index pp.The bottom of the domain is irregular by the presence of slight roughness of amplitude εδεδ and period εβεβ, satisfying the relation 1<β<δ1<β<δ. Assuming pure slip or partial slip with a friction coefficient ε−γε−γ, with γ>0γ>0, on the rough boundary, we consider the limit when domain thickness tends to zero and we obtain different models depending on the magnitude δδ with respect to 2p−1pβ−p−1p, and the magnitude γγ with respect to 1p−1.