# Acceleration Waves in Rational Extended Thermodynamics of Rarefied Monatomic Gases

^{2}, University of Bologna, via Saragozza 8, 40123 Bologna, Italy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rational Extended Thermodynamics Models for Monatomic Gases

## 3. Acceleration Waves

- The normal velocity $V=-{\phi}_{t}/|\nabla \phi |$ of the wave front coincides with a characteristic speed of system (10) evaluated in the unperturbed field: $V=\lambda \left({\mathbf{u}}_{0}\right)$.
- The jump vector $\mathbf{A}$ is proportional to the right eigenvector $\mathbf{r}$ (of matrix $\mathbf{B}$) corresponding to $\lambda $, evaluated in ${\mathbf{u}}_{0}$, so that $\mathbf{A}=A\mathbf{r}\left({\mathbf{u}}_{0}\right)$.
- The scalar amplitude A satisfies the Bernoulli equation, if $d/dt$ denotes the time derivative along the characteristic line (in our case, $dx/dx=\lambda \left({\mathbf{u}}_{\mathbf{0}}\right)$ and $a\left(t\right)$ and $b\left(t\right)$ are suitable function of the time t):

## 4. Acceleration Waves in RET Theories

#### 4.1. The ET${}_{20}$ Model

#### 4.2. The ET${}_{35}$ Model

#### 4.3. The ET${}_{56}$ Model

#### 4.4. The ET${}_{84}$ Model

#### 4.5. The ET${}_{120}$ Model

## 5. Results, Conclusions, and Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. The ET 56 Case

#### Appendix A.2. The ET 84 Case

#### Appendix A.3. The ET 120 Case

## References

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**Figure 1.**The coefficient b of the fastest acceleration wave, multiplied by the relaxation time $\tau $ is plotted as a function of n. The values were calculated in the previous sections and reported in the previous tables.

**Table 1.**The values of a, b and ${G}_{cr}$ of the acceleration waves propagating with different characteristic velocities.

${\tilde{\mathit{\lambda}}}_{\mathit{i}}{|}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{G}}_{\mathbf{cr}}$ |
---|---|---|---|

${\tilde{\lambda}}_{1,2}^{\left(20\right)}{|}_{E}$ | $\mp \frac{2.52727}{{c}_{0}}$ | $\frac{0.280584}{\tau}$ | $\pm \frac{0.111023{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{3,4}^{\left(20\right)}{|}_{E}$ | $\mp \frac{0.722222}{{c}_{0}}$ | $\frac{0.458333}{\tau}$ | $\pm \frac{0.634615{c}_{0}}{\tau}$ |

$\pm {\tilde{\lambda}}_{max}^{\left(20\right)}={\tilde{\lambda}}_{5,6}^{\left(20\right)}{|}_{E}$ | $\mp \frac{1.3862}{{c}_{0}}$ | $\frac{0.552749}{\tau}$ | $\pm \frac{0.398751{c}_{0}}{\tau}$ |

**Table 2.**The table collects the values of a, b and ${G}_{cr}$ corresponding to the non-exceptional acceleration waves of ET${}_{35}$.

${\tilde{\mathit{\lambda}}}_{\mathit{i}}{|}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{G}}_{\mathbf{cr}}$ |
---|---|---|---|

${\tilde{\lambda}}_{4,5}^{\left(35\right)}{|}_{E}$ | $\mp \frac{1.68132}{{c}_{0}}$ | $\frac{0.343836}{\tau}$ | $\mp \frac{0.204503{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{6,7}^{\left(35\right)}{|}_{E}$ | $\mp \frac{0.529238}{{c}_{0}}$ | $\frac{0.401235}{\tau}$ | $\mp \frac{0.758137{c}_{0}}{\tau}$ |

$\pm {\tilde{\lambda}}_{max}^{\left(35\right)}={\tilde{\lambda}}_{8,9}^{\left(35\right)}{|}_{E}$ | $\mp \frac{1.53569}{{c}_{0}}$ | $\frac{0.800609}{\tau}$ | $\mp \frac{0.521334{c}_{0}}{\tau}$ |

**Table 3.**Values of the coefficients ${a}_{i}$ and ${b}_{i}$, together with the critical value of $G\left(0\right)$ in the case of one-dimensional ET${}_{56}$ theory.

${\tilde{\mathit{\lambda}}}_{\mathit{i}}{|}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{G}}_{\mathbf{cr}}$ |
---|---|---|---|

${\tilde{\lambda}}_{1,2}^{\left(56\right)}{|}_{E}$ | $\mp \frac{0.854423}{{c}_{0}}$ | $\frac{0.55994}{\tau}$ | $\pm \frac{0.655343{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{3,4}^{\left(56\right)}{|}_{E}$ | $\mp \frac{0.33333}{{c}_{0}}$ | $\frac{0.813333}{\tau}$ | $\pm \frac{2.44{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{5,6}^{\left(56\right)}{|}_{E}$ | $\mp \frac{0.528062}{{c}_{0}}$ | $\frac{0.383322}{\tau}$ | $\pm \frac{0.725904{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{7,8}^{\left(56\right)}{|}_{E}$ | $\mp \frac{2.9081}{{c}_{0}}$ | $\frac{0.409518}{\tau}$ | $\pm \frac{0.14082{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{9,10}^{\left(56\right)}{|}_{E}$ | $\mp \frac{1.51194}{{c}_{0}}$ | $\frac{0.497643}{\tau}$ | $\pm \frac{0.329141{c}_{0}}{\tau}$ |

$\pm {\tilde{\lambda}}_{max}^{\left(56\right)}={\tilde{\lambda}}_{11,12}^{\left(56\right)}{|}_{E}$ | $\mp \frac{1.60945}{{c}_{0}}$ | $\frac{0.926172}{\tau}$ | $\pm \frac{0.575458{c}_{0}}{\tau}$ |

**Table 4.**The values of a, b, and ${G}_{cr}$ corresponding to the non-exceptional acceleration waves of ET${}_{84}$.

${\tilde{\mathit{\lambda}}}_{\mathit{i}}{|}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{G}}_{\mathbf{cr}}$ |
---|---|---|---|

${\tilde{\lambda}}_{5,6}^{\left(84\right)}{|}_{E}$ | $\mp \frac{0.642969}{{c}_{0}}$ | $\frac{0.519028}{\tau}$ | $\pm \frac{0.807236{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{7,8}^{\left(84\right)}{|}_{E}$ | $\mp \frac{0.3849}{{c}_{0}}$ | $\frac{0.797101}{\tau}$ | $\pm \frac{2.07093{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{9,10}^{\left(84\right)}{|}_{E}$ | $\mp \frac{0.497686}{{c}_{0}}$ | $\frac{0.38631}{\tau}$ | $\pm \frac{0.776212{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{11,12}^{\left(84\right)}{|}_{E}$ | $\mp \frac{1.82855}{{c}_{0}}$ | $\frac{0.435448}{\tau}$ | $\pm \frac{0.238139{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{13,14}^{\left(84\right)}{|}_{E}$ | $\mp \frac{1.49312}{{c}_{0}}$ | $\frac{0.688412}{\tau}$ | $\pm \frac{0.461057{c}_{0}}{\tau}$ |

$\pm {\tilde{\lambda}}_{max}^{\left(84\right)}={\tilde{\lambda}}_{15,16}^{\left(84\right)}{|}_{E}$ | $\mp \frac{1.69454}{{c}_{0}}$ | $\frac{0.97614}{\tau}$ | $\pm \frac{0.576051{c}_{0}}{\tau}$ |

${\tilde{\mathit{\lambda}}}_{\mathit{i}}{|}_{\mathit{E}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{G}}_{\mathbf{cr}}$ |
---|---|---|---|

${\tilde{\lambda}}_{1,2}^{\left(120\right)}{|}_{E}$ | $\mp \frac{0.813203}{{c}_{0}}$ | $\frac{0.78892}{\tau}$ | $\pm \frac{0.970139{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{3,4}^{\left(120\right)}{|}_{E}$ | $\mp \frac{2.83333}{{c}_{0}}$ | $\frac{0.869608}{\tau}$ | $\pm \frac{0.30692{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{5,6}^{\left(120\right)}{|}_{E}$ | $\mp \frac{0.398372}{{c}_{0}}$ | $\frac{0.742313}{\tau}$ | $\pm \frac{1.86337{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{7,8}^{\left(120\right)}{|}_{E}$ | $\mp \frac{1.11852}{{c}_{0}}$ | $\frac{0.614625}{\tau}$ | $\pm \frac{0.549497{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{9,10}^{\left(120\right)}{|}_{E}$ | $\mp \frac{0.593506}{{c}_{0}}$ | $\frac{0.491156}{\tau}$ | $\pm \frac{0.827551{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{11,12}^{\left(120\right)}{|}_{E}$ | $\mp \frac{0.447707}{{c}_{0}}$ | $\frac{0.361391}{\tau}$ | $\pm \frac{0.807203{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{13,14}^{\left(120\right)}{|}_{E}$ | $\mp \frac{3.27138}{{c}_{0}}$ | $\frac{0.487302}{\tau}$ | $\pm \frac{0.148959{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{15,16}^{\left(120\right)}{|}_{E}$ | $\mp \frac{1.56393}{{c}_{0}}$ | $\frac{0.513738}{\tau}$ | $\pm \frac{0.328492{c}_{0}}{\tau}$ |

${\tilde{\lambda}}_{17,18}^{\left(120\right)}{|}_{E}$ | $\mp \frac{1.52887}{{c}_{0}}$ | $\frac{0.839253}{\tau}$ | $\pm \frac{0.548936{c}_{0}}{\tau}$ |

$\pm {\tilde{\lambda}}_{max}^{\left(120\right)}={\tilde{\lambda}}_{19,20}^{\left(120\right)}{|}_{E}$ | $\mp \frac{1.78365}{{c}_{0}}$ | $\frac{0.993041}{\tau}$ | $\pm \frac{0.556746{c}_{0}}{\tau}$ |

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Brini, F.; Seccia, L.
Acceleration Waves in Rational Extended Thermodynamics of Rarefied Monatomic Gases. *Fluids* **2020**, *5*, 139.
https://doi.org/10.3390/fluids5030139

**AMA Style**

Brini F, Seccia L.
Acceleration Waves in Rational Extended Thermodynamics of Rarefied Monatomic Gases. *Fluids*. 2020; 5(3):139.
https://doi.org/10.3390/fluids5030139

**Chicago/Turabian Style**

Brini, Francesca, and Leonardo Seccia.
2020. "Acceleration Waves in Rational Extended Thermodynamics of Rarefied Monatomic Gases" *Fluids* 5, no. 3: 139.
https://doi.org/10.3390/fluids5030139