Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Continuum Mechanics and Thermodynamics
Published in: Continuum Mechanics and Thermodynamics 5/2023

20-06-2023 | Original Article

Equivalence group and exact solutions of the system of nonhomogeneous Boltzmann equations

Authors: Yurii N. Grigoryev, Sergey V. Meleshko

Published in: Continuum Mechanics and Thermodynamics | Issue 5/2023

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The article is devoted to the construction of exact solutions of a system of two Boltzmann kinetic inhomogeneous equations. The source functions in the equations simulate the integrals of double and triple inelastic collisions. An extension of the Lie group \(L_4\) admitted by the system of homogeneous equations is carried out. In the present paper, the Lie group \(L_4\) is considered as an equivalence group for inhomogeneous equations. Conditions are found under which transformations from the extended group vanish the sources in the transformed equations. A class of sources linear in the distribution functions is obtained for which the generalized Bobylev–Krook–Wu solutions hold in explicit form. Physical interpretations are also presented.
previous article On the thermal stresses in chiral porous elastic beams

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Springer, New York (1994)MATH
2.
Grigoriev, Y.N., Meleshko, S.V.: The complete lie group and invariant solutions of the Boltzmann system of equations for a multicomponent mixture of gases. Sibirsk. Math. J. 38(3), 510–525 (1997)MathSciNet
3.
Bobylev, A.V., Gamba, I.M.: Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails. J. Stat. Phys. 124, 497–516 (2006)MathSciNetCrossRefMATHADS
4.
Torrilhon, M.: Special issues on moment methods in kinetic gas theory. Contin. Mech. Thermodyn. 21, 341–343 (2009)MathSciNetCrossRef
5.
6.
Nagnibeda, E., Kustova, E.: Kinetic Theory of Transport and Relaxation Processes in Flows of Nonequilibrium Reacting Gases. Springer, Berlin (2009)MATH
7.
Boffi, V.C., Spiga, G.: Nonlinear diffusion of test particles in the presence of an external conservative force. J. Phys. Fluids 25, 1987–1992 (1982)MathSciNetCrossRefMATHADS
8.
Suriyawichitseranee, A., Grigoriev, Y.N., Meleshko, S.V.: Group analysis of the Fourier transform of the spatially homogeneous and isotropic Boltzmann equation with a source term. Commun. Nonlinear Sci. Numer. Simul. 20, 719–730 (2015)MathSciNetCrossRefMATHADS
9.
Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V.: Symmetries of Integro-differential Equation with Applications in Mechanics and Plasma Physics. Lecture Notes in Physics, vol. 806. Springer, Berlin (2010)MATH
10.
Zhou, L., Meleshko, S.V.: Invariant and partially invariant solutions for a linear thermoviscoelasticity. Contin. Mech. Thermodyn. 29, 207–224 (2017)MathSciNetCrossRefMATHADS
11.
Long, F.-S., Karnbanjong, A., Suriyawichitseranee, A., Grigoriev, Y.N., Meleshko, S.V.: Application of a Lie group admitted by a homogeneous equation for group classification of a corresponding inhomogeneous equation. Commun. Nonlinear Sci. Numer. Simul. 48, 350–360 (2017)MathSciNetCrossRefMATHADS
12.
13.
Santos, A., Brey, J.J.: Comments on ‘a generalized BKW solution of the nonlinear Boltzmann equation with removal’. Phys. Fluids 29(5), 1750 (1985)CrossRefMATHADS
14.
Grigoriev, Y.N., Meleshko, S.V., Suriyawichitseranee, A.: Exact solutions of the Boltzmann equation with a source. J. Appl. Mech. Tech. Phys. 59(2), 189–196 (2018)MathSciNetCrossRefMATHADS
15.
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, Cambridge (1978)MATH
16.
Grigoriev, Y.N., Meleshko, S.V.: Group analysis of the integrodifferential Boltzmann equation. Dokl. AS USSR 297(2), 323–327 (1987)ADS
Metadata
Title
Equivalence group and exact solutions of the system of nonhomogeneous Boltzmann equations
Authors
Yurii N. Grigoryev
Sergey V. Meleshko
Publication date
20-06-2023
Publisher
Springer Berlin Heidelberg
Published in
Continuum Mechanics and Thermodynamics / Issue 5/2023
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI
https://doi.org/10.1007/s00161-023-01238-4

Other articles of this Issue 5/2023

Continuum Mechanics and Thermodynamics 5/2023 Go to the issue

Original Article

Extended hydrodynamical models for plasmas

Original Article

Particle-based approach to the Eulerian distortion field and its dynamics

Original Article

Damping behavior of a particulate composite with interface slip and diffusion under a deviatoric far-field load

Original Article

Weakly nonlocal Rayleigh waves with impedance boundary conditions

Review Article

The effects of chemical and mechanical interactions on the thermodynamic pressure for mineral solid solutions

Original Article

Propagation of axisymmetric electroelastic waves in a homogeneous and a continuously inhomogeneous piezoceramic solid cylinder under the mechanical excitation

Premium Partners

    Image Credits