Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

Low-Mach type approximation of the Navier–Stokes system with temperature and salinity for free surface flows

Pages: 151 – 172

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a7

Authors

Léa Boittin (INRIA Paris, France; and Laboratoire Jacques-Louis Lions, Sorbonne Université, Université Paris Diderot, Paris, France)

Marie-Odile Bristeau (INRIA Paris, France; and Laboratoire Jacques-Louis Lions, Sorbonne Université, Université Paris Diderot, Paris, France)

François Bouchut (Laboratoire d’Analyse et de Mathématiques Appliquées, Université Gustave Eiffel, Marne-la-Vallée, France)

Anne Mangeney (Institut de Physique du Globe de Paris, Université Paris Cité, Paris, France)

Jacques Sainte-Marie (INRIA Paris, France; and Laboratoire Jacques-Louis Lions, Sorbonne Université, Université Paris Diderot, Paris, France)

Fabien Souillé (Laboratoire National d’Hydraulique et Environnement, Électricité de France (EDF) R&D, Chatou, France)

Abstract

We are interested in free surface flows where density variations coming, for example, from temperature or salinity differences, play a significant role in the hydrodynamic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising from compressible fluid mechanics often requires a decoupling between these waves. Starting from the compressible Navier–Stokes system, we derive the so-called Navier–Stokes–Fourier system in an “incompressible” regime using the low-Mach scaling, hence filtering the acoustic waves, neglecting the density dependency on the fluid pressure but keeping its variations in terms of temperature and salinity. A slightly modified low-Mach asymptotics is proposed to obtain a model with thermo-mechanical compatibility. The case when the density depends only on the temperature is studied first. Then the variations of the fluid density with respect to temperature and salinity are considered, and it seems to be the first time that salinity dependency is considered in this low Mach limit. The obtained models conserve the mass of the fluid but not the volume and satisfy the second principle of thermodynamics.

Keywords

Navier–Stokes equations, compressible and incompressible fluids, free surface flows, variable density flows, low-Mach approximation

2010 Mathematics Subject Classification

35Q30, 35Q35, 76D05

Full Text (PDF format)

Received 9 January 2022

Received revised 7 April 2022

Accepted 9 April 2022

Published 27 December 2022