Derivation of a viscous Serre—Green—Naghdi equation: an impasse?

Abstract

In this article we present the current status of the derivation of a viscous Serre-Green-Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier-Stokes equations. We consider a particular regime linking the Reynolds number and the shallowness parameter. The computations presented in this note are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations seem to be needed to obtain a tractable model.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2021derivationviscous,
  author = {Dutykh, D. and Le Meur, H.},
  title = {Derivation of a viscous Serre—Green—Naghdi equation: an impasse?},
  journal = {Fluids},
  year = {2021},
  volume = {6},
  number = {4},
  pages = {135},
  doi = {10.3390/fluids6040135},
  abstract = {In this article we present the current status of the derivation of a viscous Serre-Green-Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier-Stokes equations. We consider a particular regime linking the Reynolds number and the shallowness parameter. The computations presented in this note are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations seem to be needed to obtain a tractable model.},
  keywords = {Prandtl theory, boundary layer, Prandtl equation, shallowness parameter, viscous Serre-Green-Naghdi equation, Serre-Green-Naghdi equations, Reynolds number, Navier-Stokes equations, asymptotic analysis, Visco-potential flows, shallow water equations, boundary layers, fully nonlinear regime}
}