Kinetic and thermodynamic approach to precisely solve the unsteady Rayleigh flow problem of a rarefied homogeneous charged gas under external force influence

Taha Zakaraia Abdel Wahid; Zaki Mrzog Alaofi

doi:10.1515/jnet-2024-0022

Article

Kinetic and thermodynamic approach to precisely solve the unsteady Rayleigh flow problem of a rarefied homogeneous charged gas under external force influence

Taha Zakaraia Abdel Wahid and Zaki Mrzog Alaofi

Published/Copyright: June 13, 2024

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 49 Issue 4

Abstract

An extension and further development of our previous article [J. Non-equilibrium Thermodyne. 37 (2012), 119–141] is presented. We study the irreversible non-equilibrium thermodynamics (INT) properties of the exact solution to the dilute homogeneously charged gas problem with unsteady Rayleigh flow. In contrast to previous research, the charged gas flows under the influence of an external force, the flat plate oscillates, and the displacement current term is considered, leading to significant advancements in understanding natural plasma dynamics. We are solving the Boltzmann kinetic equation (BKE) Krook model supplemented by Maxwell’s equations. We used a travelling wave and moments method with an electron velocity distribution function (EVDF). To the best of our knowledge, as three new scientific achievements, we introduced a new mathematical model for calculating the thermodynamic forces, kinetic coefficients, and fluxes variables, Equations (28–40) and (50–54). Second, we determined, with reasonable accuracy, the thermodynamic equilibrium time of electrons, t _equ = 26.7955, under an external force. We clarify the difference between equilibrium EVDF and perturbed EVDF and take advantage of BKE to account for non-equilibrium thermodynamic principles. For diamagnetic and paramagnetic plasmas, the extended Gibbs equation predicts ratios between various contributions to the internal energy change (IEC) is presented. A standard laboratory argon plasma model is used to apply the results.

Keywords: irreversible non-equilibrium thermodynamic; external magnetic field; Boltzmann kinetic equation; entrop generation; gaseous plasma; internal energy change

PACS Codes: 05.70.Ln; 52.25.Kn; 52.25.Dg; 51. 30. +i; 05.70.-a; 52.35.Sb; 88.05.De

Corresponding author: Taha Zakaraia Abdel Wahid, Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Kom 32511, Egypt, E-mail: Taha.Zakaria@science.menofia.edu.eg

Acknowledgments

Professor Taha Z. Abdel Wahid would like to thank Professors A. M. Abourabia and M. Abdel-Latif Ramadan for their inspirational words, helpful guidance, and assistance. The authors thank Professor Karl Heinz Hoffmann, Editor in Chief, and all reviewers for their valuable comments that enhanced the paper. The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through the Large Research Project under grant number RGP2/222/45.

Research ethics: Not applicable.
Author contributions: The authors contributed equally to the conception, design, implementation, analysis, and interpretation of the research presented in this paper. Each author played a significant role in drafting and revising the manuscript, ensuring its intellectual integrity and accuracy.
Competing interests: The authors state no competing interests.
Research funding: The authors have no research funding.
Data availability: The raw data can be obtained on request from the corresponding author.

Nomenclature

B z E ⃗: external magnetic field
B z I ⃗: induced magnetic field
dU _S: IEC for entropy
dU _P: IEA for polarization
dU _dia: IEA for magnetic field
E ⃗: electric vector
g _e: non-equilibrium EVDF
g ₀: EVDF
g ₁: non-equilibrium EVDF for ξ _y < 0
g ₂: non-equilibrium EVDF for ξ _y > 0
J: current density
J _y ^(S): entropy flux
J ₁, J ₂, J ₃: thermodynamic fluxes
K _B: Boltzmann constant
L _ij: nine kinetic coefficients
M: Mach number
m: specific magnetization
P: polarization
R: gas constant
S: entropy
T: temperature
U: IEC
U ₀: plate velocity
V _x: mean velocity
V: gas volume
V _T: thermal velocity
X ₁: flow velocity-related thermodynamic force.
X ₂: magnetic field-related thermodynamic force.
X ₃: electric field-related thermodynamic force.
ξ ₀: speed of light
ξ: particles velocity
d: particle diameter
e: electron charge
F ⃗: Lorantz’s force
m _e: electron mass
n _e: concentration
p: pressure
t: time variable
y: displacement
Z: ionization

Superscripts
′: dimensionless variable

Subscripts
1: due to g ₁
2: connected to g ₂
e: connected to electrons
i: connected to ions
x: connected to x-coordinate

Greek letters
τ: relaxation time
τ _xz: shear stress
μ: viscosity coefficient
λ: mean free path
α ₀: dimensionless parameter
β: constant
ν: collision frequency
ε: mass ratio
λ: mean free path
λ _D: Debye radius

Abbreviations
BKE: Boltzmann kinetic equation
BGK: Bhatnagar, Gross, and Krook
EVDF: Electron velocity distribution function
IEC: Internal energy change
INT: Irreversible non-equilibrium thermodynamics
LMM: Lees moment method
ME: Maxwell’s equations
Kn: Knudsen number
MEMS: Micro-electro-mechanical-system.

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Received: 2024-04-01

Accepted: 2024-05-24

Published Online: 2024-06-13

Published in Print: 2024-10-28

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https://doi.org/10.1515/jnet-2024-0022

Keywords for this article

irreversible non-equilibrium thermodynamic; external magnetic field; Boltzmann kinetic equation; entrop generation; gaseous plasma; internal energy change