A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary [Fluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissement sur une frontière lègèrement rugueuse]

摘要

We consider a viscous fluid of small height ε on a periodic rough bottom Γ_Ε of period rΕand amplitude δ_ε, δ_ε?r_ε?ε, where we impose the slip boundary condition. When ε tends to zero we obtain a Reynolds system depending on the limit λ of (δ_εε)/(r_εr_ε). If λ=+∞, the fluid behaves as if we would impose the adherence condition on Γ_Ε 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 λε(0,+∞) it behaves as if Γ_Εwas flat but with a higher friction coefficient.