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.
The present paper deals with the modeling and numerical approximation of bed load transport under the action of water.
A new shallow water type model is derived from the stratified two-fluid Navier-Stokes equations.
Its novelty lies in the magnitude of a viscosity term that leads to a momentum equation of elliptic type.
The full model, sediment and water, verifies a dissipative energy balance for smooth solutions.
The numerical resolution of the sediment layer is not trivial since the viscosity introduces a non-local term in the model.
Adding a transport threshold makes the resolution even more challenging.
A schema based on a staggered discretization is proposed for the full model, sediment and water.
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