*2.1. Density Dependence of Material Parameters*

Despite the fact that the experimental data are recorded as a function of *ω*/*p* only, it is also emphasized in the related paper of Rhodes [50] that the frequency was constant all along the measurement in an isothermal environment. It means that the pressure is varied by changing the density of the gas only. That is, changing the state of the gas from normal (or dense) state to a rarefied one must reflect the role of density dependence of material parameters.

Indeed, this is an efficient way to demonstrate the validity region of the classical Navier-Stokes-Fourier equations. As the experiment shows, some effects become essential when the gas reaches its rarefied state. Density-dependent material coefficients could represent it: some of them make the related memory and nonlocal effects (couplings) negligible for dense states and worthwhile to account in rarefied states while the others change accordingly. This is a natural expectation from theoretical point of view.

Considering the RET approach of Arima et al. [43], one can notice the following:


All the other parameters are fixed and do not depend on the mass density in any way. It is important to emphasize that the essential material parameters used in their analysis are constant for very different pressures, like 10<sup>3</sup> or 10<sup>5</sup> Pa. These viscosities are derived on kinetic theory considerations (see, e.g., [56–58]) and are independent of the mass density. Since it is a theoretical value, in the following it is referred to be "physical" or theoretical quantity. It is needed to be underlined that only the "effective" or apparent properties are measurable which may differ from the theoretical ones. As Michalis et al. [59] draw the attention to the "effective" viscosity, the theoretical one must be corrected for rarefied gas flows as a function of the Knudsen number. These corrections are devoted to facilitating the phenomenological descriptions and can be compared to the above-mentioned measurements [59–63].

As the measured effective viscosity shows mass dependence, it requires corrections in both directions: from the rarefied state to the extremely dense states (up to the magnitude of 103 MPa). One example is the Enskog-type correction [63–65]. It is in good agreement with particular experiments that are devoted to measuring the density dependence of shear viscosity for dense states [64,66–68]. These measurements do not aim to investigate low-pressure behavior. Extrapolating the "dense data" to zero, they show the presence of non-zero viscosities at zero density (see Figure 2 for details) [56,60,66,69].

On the other hand, the measurements of Itterbeek et al. [70–72] demonstrate decreasing viscosity by decreasing the pressure. It can be only piecewise linear, its steepness changes drastically at very low pressures (1–10 Pa), and the viscosity tends to zero (Figure 3). However, it is extremely difficult to perform viscosity measurements at such a rarefied state. The outcome of the experiment depends on the size of the apparatus and could influence the results. Furthermore, evaluating such

viscosity measurement that corresponds to high Kn number, the classical Navier-Stokes theory may be insufficient and a non-local theory should be used. So to say, there could be some uncertainty regarding the values of the viscosities. From practical point of view, the density dependence seems to be natural and the measurements of Itterbeek et al. also strengthen this expectation: at zero density, there is no viscosity. From a theoretical point of view, this is contradictory to the kinetic theory in the sense of the non-zero value for viscosity in the zero-density limit. Beskok and Karniadakis [73], to overcome that contradiction, suggested a correction that is inversely proportional with the Knudsen number. The correction of the "physical" viscosity is improved by Roohi and Darbandi [74], too.

From a practical point of view again, these effective quantities are essential for modeling, for instance see [59,75–79]. In these papers, generalized constitutive laws are proposed in which apparent quantities play a central role and are used as being the coefficients in the constitutive equations, this is the case also in NET-IV. These referred problems (rheology, non-Newtonian fluids, biological materials) demonstrates that the apparent transport coefficients can depend on various quantities, especially in complex materials. Since the theoretical coefficients differ from the apparent one, at least in the sense of density dependence, it could influence the scaling properties of a model. It is discussed in the following.

**Figure 2.** Density dependence of viscosity for dense gases when the non-zero viscosity at zero density appears. The original measurements can be found in [66]. Here, the red boxes show the region of interest together with the extrapolation to zero density.


**Figure 3.** Pressure dependence of viscosity for rarefied gases at room temperature. The original data can be found in [72] which is only partially depicted here.

For the sake of complete comparison with the work of Arima et al., the same assumptions are used, i.e.,

• both viscosities and the thermal conductivity are constant:

*<sup>ν</sup>* <sup>=</sup> 8.82 <sup>×</sup> <sup>10</sup>−<sup>6</sup> Pas, *<sup>η</sup>* <sup>=</sup> <sup>326</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> Pas and *<sup>λ</sup>* <sup>=</sup> 0.182 W/(mK), respectively.

• all relaxation times and the coupling coefficients have 1/*ρ* dependence.

These follow from a scaling requirement which is discussed in the next section.
