# Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration

^{1}

^{2}

^{2}), Department of Mathematics, University of Bologna, Bologna 40123-I, Italy

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rational Extended Thermodynamics: Present Status

- Phenomenological RET: The closure is obtained by using the universal principles of continuum thermomechanics—(I) the Galilean invariance and the objectivity principle; (II) the entropy principle; and (III) the causality and thermodynamic stability (i.e., convexity of the entropy)—to select admissible constitutive equations (see [8,18] for monatomic gases and [9,19] for polyatomic ones);
- Molecular Extended Thermodynamics (molecular ET): The fields are the moments of a distribution function and the closure is done by using the maximum entropy principle (MEP) . This principle has its root in statistical mechanics. It is developed by Jaynes [20,21] in the context of the theory of information basing on the Shannon entropy. Nowadays the importance of MEP is recognized fully due to the numerous applications in many fields [22], for example, in the field of computer graphics. MEP states that the probability distribution that represents the current state of knowledge in the best way is the one with the largest entropy. Concerning the applicability of MEP in nonequilibrium thermodynamics, this was originally raised by the observation made by Kogan [23] that Grad’s distribution [24] function maximizes the entropy. The MEP was proposed in RET for the first time by Dreyer [25]. The MEP procedure was then generalized by Müller and Ruggeri to the case of any number of moments in the first edition of 1993 of their book [8] proving that the closed system is symmetric hyperbolic and coining the name of molecular ET for this kind of closure process.

#### 2.1. RET of Rarefied Monatomic Gases

#### 2.2. RET of Rarefied Polyatomic Gases

#### 2.3. RET of Dense Gases

## 3. RET Theory with Molecular Internal Variables

#### 3.1. Molecular Internal Variables

#### 3.2. System of Balance Equations with Triple Hierarchy

#### 3.3. Entropy Law

#### 3.4. Equilibrium Distribution Function

#### 3.5. Molecular ET Theory with 7 Independent Fields (ET${}_{7}$)

#### 3.6. Molecular ET Theory with 15 Independent Fields (ET${}_{15}$)

#### 3.6.1. Galilean Invariance and Intrinsic Variables

#### 3.6.2. Nonequilibrium Distribution Function Derived from MEP

#### 3.6.3. Physical Meaning of the Parameters ${\theta}^{K}$, ${\theta}^{R}$, and ${\theta}^{V}$

#### 3.6.4. Closure of the Differential System

#### 3.6.5. Entropy Density, Flux and Production

**Remark**

**1.**

**Remark**

**2.**

#### 3.7. Production Terms in the Generalized BGK-Model

#### 3.7.1. Generalized BGK-Model

#### 3.7.2. Production Terms of ET${}_{15}$

#### 3.8. Maxwellian Iteration and Phenomenological Coefficients: Shear and Bulk Viscosities, and Heat Conductivity

## 4. Summary and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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$\left(\mathfrak{b}\mathfrak{c}\right)$-Process | $(\mathfrak{a},\mathfrak{b},\mathfrak{c})$ | Relaxation Time | Collision Term |
---|---|---|---|

$\left(KR\right)$-process | $(V,K,R)$ | ${\tau}_{KR}$ | ${Q}^{KR}\left(f\right)$ |

$\left(KV\right)$-process | $(R,K,V)$ | ${\tau}_{KV}$ | ${Q}^{KV}\left(f\right)$ |

$\left(RV\right)$-process | $(K,R,V)$ | ${\tau}_{RV}$ | ${Q}^{RV}\left(f\right)$ |

$\left(\mathfrak{b}\mathfrak{c}\right)$ | $\mathit{\delta}$ | ${\Delta}$ |
---|---|---|

$\left(KR\right)$ | $-\frac{{c}_{v}^{R}}{{c}_{v}^{K+R}}{\Delta}^{V}-{\Delta}^{R}$ | ${\Delta}^{V}$ |

$\left(KV\right)$ | $-\frac{{c}_{v}^{V}}{{c}_{v}^{K+V}}{\Delta}^{R}-{\Delta}^{V}$ | ${\Delta}^{R}$ |

$\left(RV\right)$ | $\frac{{c}_{v}^{V}{\Delta}^{R}-{c}_{v}^{R}{\Delta}^{V}}{{c}_{v}^{R+V}}$ | $-{\Delta}^{R}-{\Delta}^{V}$ |

$\left(\mathfrak{b}\mathfrak{c}\right)$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | $\mathbf{\Pi}$ |
---|---|---|---|

$\left(KR\right)$ or $\left(KV\right)$ | $\frac{{c}_{v}^{\mathfrak{c}}}{{c}_{v}^{\mathfrak{b}+\mathfrak{c}}}$ | 0 | $\frac{2}{3}\rho \left(\right)open="("\; close=")">\delta -\frac{{c}_{v}^{\mathfrak{b}}}{{c}_{v}^{\mathfrak{b}+\mathfrak{c}}}\Delta$ |

$\left(RV\right)$ | 0 | ${c}_{v}$ | $\frac{2}{3}\rho \Delta$ |

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Arima, T.; Ruggeri, T.; Sugiyama, M.
Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration. *Entropy* **2018**, *20*, 301.
https://doi.org/10.3390/e20040301

**AMA Style**

Arima T, Ruggeri T, Sugiyama M.
Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration. *Entropy*. 2018; 20(4):301.
https://doi.org/10.3390/e20040301

**Chicago/Turabian Style**

Arima, Takashi, Tommaso Ruggeri, and Masaru Sugiyama.
2018. "Extended Thermodynamics of Rarefied Polyatomic Gases: 15-Field Theory Incorporating Relaxation Processes of Molecular Rotation and Vibration" *Entropy* 20, no. 4: 301.
https://doi.org/10.3390/e20040301