Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

Abstract

We consider a non-Newtonian fluid flow in a thin domain with thickness ηε and an oscillating top boundary of period ε. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index p, with 9/5 < p < +∞. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness (ηε ≈ ε), Reynolds roughness (ηε << ε) and high-frequency roughness (ηε >> ε) regime. Moreover, we obtain different nonlinear Reynolds-type equations in each case.