Derivation of an intermediate viscous Serre-Green-Naghdi equation

TL;DR

This paper discusses the ongoing derivation of a viscous Serre-Green-Naghdi model by combining inviscid Euler equations for the upper flow and Navier-Stokes equations for the boundary layer, within a specific flow regime.

Contribution

It presents a new approach to derive a viscous Serre-Green-Naghdi system considering a boundary layer modeled by a Prandtl-like equation.

Findings

Flow domain split into inviscid and viscous regions

Boundary layer reduces to a Prandtl-like equation

Further approximations needed for a practical model

Abstract

In this note 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 Prantdl-like equation. Further approximations seem to be needed to obtain a tractable model.