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

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) regime. Moreover, we obtain different nonlinear Reynolds-type equations in each case.Junta de AndalucíaMinisterio de Economía y Competitivida

Homogenization of parabolic problems with dynamical boundary conditions of reactive-diffusive type in perforated media

This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes ∇uε · ν + ε ∂uε ∂t = ε δ∆Γuε − ε g(uε), where ∆Γ denotes the Laplace-Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 0 plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results (see [3]) established in the case of a dynamical boundary condition of pure-reactive type, i.e., with δ = 0. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes