From quasi-incompressible to semi-compressible fluids

Tomáš Roubíček

doi:10.3934/dcdss.2020414

Article Contents

2021, Volume 14, Issue 11: 4069-4092. Doi: 10.3934/dcdss.2020414 open access

This issue Previous Article A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion Next Article Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions

From quasi-incompressible to semi-compressible fluids

Tomáš Roubíček^1,2,

1.
Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic
2.
Institute of Thermomechanics, Czech Acad. Sci., Dolejškova 5, CZ-18200 Praha 8, Czech Republic

Received: November 30, 2019

Revised: March 31, 2020

Early access: July 2020

Published: November 2021

This research has been partially supported from the grants 19-04956S of Czech Science Foundation, and the FWF/CSF project I 4052 N32 with BMBWF through the OeAD-WTZ project CZ04/2019, and also from the institutional support RVO: 61388998 (ČR)

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Abstract

A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wave-length (and possibly also pressure) dependent velocity.

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Mathematics Subject Classification: 35K55, 76A10, 76N10, 76R50.

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Figure 1. Dependence of the velocity of sinusoidal waves on the angular wavenumber k (left) and on the wave length λ = 2π/k (right); an illustration of the normal dispersion due to (30) and (28) for K=9 and ϱ = 1, and D = 3. Waves with ultra short lengths (or with ultra high wave numbers) have zero velocity, i.e. cannot propagate

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Figure 2. Dependence of the velocity of sinusoidal waves on the angular wavenumber k (left) and on the wave length λ = 2π/k (right); an illustration of the anormalous dispersion due to (38) and (37) for K=1, ϱ = 1, and D > 0 very small

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