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

On the non‐stationary non‐Newtonian flow through a thin porous medium

“…P r o o f . Taking into account the estimate of the velocity (27) for every case, we argue as the proof of Lemma 5.1 in [26] and Lemma 5.2 in [16]. Now, we give a convergence result for the pressureP ε .…”

“…It would remain to prove the Y -periodicity ofû in y . This can be obtain by proceeding as in Lemma 5.4 in [26].…”

Boundary Layer for a Non-Newtonian Flow Over a Rough Surface

“…• For S n,1 , we use the inequality (26). It gives |S n,1 | ≤ C|Du bl,n |1l |Du bl,n |<M , which provides a uniform bound in L 2 ∩ L ∞ , and so in particular in L p ′ and in L 2 .…”

“…• For S n,2 , we use first that |S n,2 | ≤ C|Du bl,n | p−1 1l |Du bl,n |≥M , so that it is uniformly bounded in L p ′ . We use then (26), so that |S n,2 | ≤ C|Du bl,n |1l |Du bl,n |≥M , which yields a uniform bound in L p , in particular…”

“…In this setting, one has A = 0, and the authors refer to [24] for well-posedness and qualitative behaviour. We also refer to the recent article [26] dedicated to power-law fluids in thin domains, with Navier condition and anisotropic roughness (with a wavelength that is larger than the amplitude). In this setting, no boundary layer analysis is needed, and the author succeeds to describe the limit asymptotics by the unfolding method.…”