Abstract
There are limits of validity of classical constitutive laws such as Fourier and Navier-Stokes equations. Phenomena beyond those limits have been experimentally found many decades ago. However, it is still not clear what theory would be appropriate to model different non-classical phenomena under different conditions considering either the low-temperature or composite material structure. In this paper, a modeling problem of rarefied gases is addressed. The discussion covers the mass density dependence of material parameters, the scaling properties of different theories and aspects of how to model an experiment. In the following, two frameworks and their properties are presented. One of them is the kinetic theory based Rational Extended Thermodynamics; the other one is the non-equilibrium thermodynamics with internal variables and current multipliers. In order to compare these theories, an experiment on sound speed in rarefied gases at high frequencies, performed by Rhodes, is analyzed in detail. It is shown that the density dependence of material parameters could have a severe impact on modeling capabilities and influences the scaling properties.
Highlights
The classical material laws such as Fourier and Navier-Stokes are acceptable for tasks concerning homogeneous materials, dense gases, and far from low-temperatures (20 K)
It is easier to understand the origin of the present continuum model of non-equilibrium thermodynamics with internal variables (NET-IV) for rarefied gases together with its properties if a purely heat conduction problem is presented first, since the essential attributes are inherited
It could be confusing it hides that the mass density ρ is the real independent variable in the equations in which all new coefficients depend on ρ
Summary
The classical material laws such as Fourier and Navier-Stokes are acceptable for tasks concerning homogeneous materials, dense gases, and far from low-temperatures (20 K). There are situations where some generalizations must be applied. Such a case could occur on small (micro or nano) length scales, short time scales, near low-temperature or far from equilibrium. The most visible differences are analogous in both cases, for instance, the available degrees of freedom, the structure of the equations and the interpretation of the parameters. All these differences originate at the roots of these approaches
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