Qualitative and quantitative homogenization of some non-Newtonian flows in perforated domains: case of `small holes'
We consider the homogenization of three dimensional viscous incompressible non-Newtonian flows satisfying certain general $r$-structure in perforated domains. We focus on the case of `small holes' by assuming the holes under consideration are of size $\varepsilon^α$ with $α>3$, where $\varepsilon$ is the perforation parameter used to measure the mutual distance between the holes. We show the limit equations remain unchanged in the homogenization limit under the constraint $ \frac{6(α- 1)}{4α-5}< r<3-\frac{3}α$, which seems optimal in the sense of Sobolev capacity of holes as explained in Remark 1.3. Quantitative convergence rates are further derived for both the velocity field and the pressure. To the best of our knowledge, both the qualitative and quantitative homogenization results are firstly given for non-Newtonian flows in the `small holes' case.