Asymptotic Analysis - Volume 98, issue 3

Authors: Duchêne, Vincent

Article Type: Research Article

Abstract: We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial …data, accounting for rapidly propagating “acoustic” waves (here, the so-called barotropic mode) which interact only weakly with the “incompressible” component (here, baroclinic). Show more

Keywords: internal waves, multilayer shallow water, small density contrast, singular limit, mode decomposition, rigid-lid approximation, Boussinesq approximation

DOI: 10.3233/ASY-161366

Citation: Asymptotic Analysis, vol. 98, no. 3, pp. 189-235, 2016

Authors: Ersoy, M.

Article Type: Research Article

Abstract: We present the full derivation of a one dimensional “Saint–Venant” like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ -pressurized model where γ is the adiabatic constant. It is obtained through the three dimensional barotropic Navier–Stokes equations under “thin layer” assumptions as a first order approximation. Prescribing suitable boundary conditions, one can introduce a general friction law and then explicitly show its geometrical (w.r.t. the hydraulic radius) and hydrodynamical (w.r.t. the Oser number) dependencies in the reduced model. In particular, for linear pressure law (γ = 1 ), we …justify the one dimensional pressurized model (called P -model) introduced by the author in the context of unsteady mixed flows in closed water pipes. For non linear pressure law (γ ≠ 1 ), the γ -pressurized model describes the evolution of a compressible (almost) gravityless flow. Show more

Keywords: pressurized flow, compressible Navier–Stokes, barotropic laws, thin layer approximation, hydrostatic approximation, friction law, Oser number

DOI: 10.3233/ASY-161367

Citation: Asymptotic Analysis, vol. 98, no. 3, pp. 237-255, 2016

Authors: Nadia, Bengouga | Fadila, Bentalha

Article Type: Research Article

Abstract: The aim of this paper is to provide corrector results associated to the homogenization of a diffusion process which takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small spheres distributed in a ε -periodic network. The suspension has a total mass of unity order and a vanishing volume. The results obtained here complete the earlier study by Bentalha et al. [Revue Roumaine de Math. Pures Appl. 52 (2) (2007), 129–149] on the asymptotic behavior of this problem.

Keywords: diffusion, periodic homogenization, corrector, non-local effects, fine-scale substructure

DOI: 10.3233/ASY-161368

Citation: Asymptotic Analysis, vol. 98, no. 3, pp. 257-284, 2016