A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
Juan Casado‐Díaz, Manuel Luna‐Laynez, Francisco J. Suárez‐Grau · Comptes Rendus Mathématique · 2010
We consider a viscous fluid of small height ε on a periodic rough bottom Γ ε of period r ε and 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 ε ε ) / ( r ε r 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 Γ ε was plane. Finally, for λ ∈ ( 0 , + ∞ ) it behaves as if Γ ε was flat but with a higher friction coefficient.