Asymptotic behavior of the Navier-Stokes system in a thin domain with Navier condition on a slightly rough boundary

We study the asymptotic behavior of the solutions of the Navier–Stokes system in a thin domain Ωε of thickness ε satisfying the Navier boundary condition on a periodic rough set Γε ⊂ ∂Ωε of period rε and amplitude δε, with δε rε ε. We prove that the limit behavior as ε goes to zero depends on the limit λ of δεε 1 2 /r 3 2 ε . Namely, if λ = +∞, the roughness is so strong that the fluid behaves as if we had imposed the adherence condition on Γε. If λ = 0, the roughness is too weak and the fluid behaves as if Γε were a plane. Finally, if λ ∈ (0, +∞), the roughness is strong enough to make a new friction term appear in the limit.Ministerio de Economía y Competitividad (España) MTM2011- 24457Junta de Andalucía FQM30

On the Navier boundary condition for viscous fluids in rough domains

In this paper we review some recent results concerning the study of the asymptotic behavior of viscous fluids in rough domains assuming Navier boundary conditions on the rough boundary. Our main interest is to study the relation between both the adherence and the Navier boundary conditions in the case of a boundary with weak rugosities. We show that the roughness acts on the fluid as a friction term. In particular, if the roughness is sufficiently strong, Navier condition implies adherence condition. This generalizes previous results of other authors.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Fluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissement sur une frontière lègèrement rugueuse

We consider a viscous fluid of small height ε on a periodic rough bottom Γε of period rε and amplitude δε, δε rε ε, where we impose the slip boundary condition. When ε tends to zero we obtain a Reynolds system depending on the limit λ of (δε √ε )/(rε √rε ). If λ = +∞, the fluid behaves as if we would impose the adherence condition on Γε. This justifies why this is the usual boundary condition for viscous fluids. If λ = 0 the fluid behaves as if Γε was plane. Finally, for λ ∈ (0,+∞) it behaves as if Γε was flat but with a higher friction coefficient.On considère un fluide visqueux de faible épaisseur ε sur un fond rugueux Γε, périodique de période rε et amplitude δε, δε rε ε, où on impose la condition de glissement. Quand ε converge vers zéro on obtient un système de type Reynolds qui dépend de la limite λ de (δε √ε )/(rε √rε ). Si λ = +∞, le fluide se comporte comme si on aurait imposé la condition d’adhérence sur Γε. Ceci justifie la condition usuelle pour un fluide visqueux. Si λ = 0 le fluide se comporte comme si Γε était plate. Enfin, pour λ ∈ (0,+∞), tout se passe comme si Γε était plate, mais avec un coefficient de frottement plus élevé

Radioguided adrenal surgery access in complex situations: Technical notes

The laparoscopic adrenalectomy is considered as the procedure of choice for the treatment of adrenal hyperplasia and tumor lesions. However, some special situations may limit the use of this method due to the difficulty to locate the gland and perform the lesion excision. We analyze 2 patients of a left adrenal tumor, explaining how they have overcome the difficulties in both situations. The first case was a patient with a history of intra-abdominal surgery and the other patient suffered from severe obesity. We performed with the use of the gamma probe, and the 2 cases, was of great help to access and glandular localization. The help of gamma probe test was achieved in the surgical bed, that removal was complete. The use of the portable gamma probe facilitated the access to the left adrenal gland as well as conducting the glandular excision without delay, despite the difficulties due to the intra abdominal surgery caused by the previous surgery, and in the case of severe obesity