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Mathematics. Mechanics. Informatics

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Kovalev V. A., Radayev Y. N., Revinsky R. A. Generalized Cross-Coupled Type-III Thermoelastic Waves Propagating via a Waveguide under Sidewall Heat Interchange. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2011, vol. 11, iss. 1, pp. 59-70. DOI: 10.18500/1816-9791-2011-11-1-59-70

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Generalized Cross-Coupled Type-III Thermoelastic Waves Propagating via a Waveguide under Sidewall Heat Interchange

Kovalev Vladimir Aleksandrovich, Moscow City Government University of Management Moscow, Russia
Radayev Yuri Nickolaevich, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences
Revinsky R. A., Saratov State University

The paper is devoted to a study of cross-coupled type-III generalized thermoelastic waves propagation via a long cylindrical waveguide. The sidewall of the waveguide is assumed free from tractions and permeable to heat. The analysis is carried out in the framework of coupled generalized theory of GNIII- thermoelasticity consistent with the basic thermodynamic principles. The theory combines the both possible mechanisms of heat transfer: thermodiffusion and wave. Type-III generalized thermoelasticity includes classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity (GNII) as limiting cases. The GNII-theory can be formulated as a field theory and differential field equations are of hyperbolic analytical type. Closed solution of the coupled GNIII-thermoelasticity equations satisfying the required boundary conditions on the surface of waveguide including convective heat interchanging condition has been obtained. The paper provides numerical analysis of frequency equation. A scheme of frequency equation roots localization is described and wavenumbers of the coupled thermoelastic waves of the first azimuthal order are computed.

  1. Maxwell, J.C. On the Dynamical Theory of Gases / J.C. Maxwell // Phil. Trans. Royal Soc. London. 1867. Vol. 157. P. 49–88.
  2. Biot, M.A. Thermoelasticity and irreversible thermodynamics / M.A. Biot // J. Appl. Phys. 1956. Vol. l27. P. 240–253.
  3. McNelly, T.F. Heat pulses in NaF: Onset of second sound / T.F. McNelly [et al.] // Phys. Reviews. 1970. Vol. 24(3). P. 100–102.
  4. Jackson, H.E. Second sound in NaF / H.E. Jackson, C.T. Walker, T.F. McNelly // Phys. Reviews. 1970. Vol. 25(1). P. 26–28.
  5. Rogers, S.J. Transport of heat and approach to second sound in some isotopically pure Alkali-Halide crystals / S.J. Rogers // Phys. Reviews. 1971. Vol. 3(4). P. 1440– 1457.
  6. Pohl, D.W. Observation of second sound in NaF by means of light scattering / D.W. Pohl, V. Irniger // Phys. Review Letters. 1976. Vol. 36(9). P. 480–483.
  7. Hardy, R.J. Velocity of second sound in NaF / R.J. Hardy, S.S. Jaswal // Phys. Review. 1971. B. 3(12). P. 4385—4387.
  8. Narayanamurti, V. Observation of second sound in Bismuth / V. Narayanamurti, R.C. Dynes // Phys. Reviews. 1972. Vol. 28. P. 1461–1464.
  9. Lord, H. A generalized dynamical theory of thermoelasticity / H. Lord, Y. Shulman // J. Mech. Phys. Solid. 1967. Vol. 15. P. 299–309.
  10. Green, A.E. Thermoelasticity / A.E. Green, K.A. Lindsay // J. Elasticity. 1972. Vol. 2. P. 1–7.
  11. Green, A.E. On undamped heat waves in an elastic solid / A.E. Green, P.M. Naghdi // J. Thermal Stresses. 1992. Vol. 15. P. 253—264.
  12. Green, A.E. Thermoelasticity without energy dissipation / A.E. Green, P.M. Naghdi // J. Elasticity. 1993. Vol. 31. P. 189–208.
  13. Puri, P. On the propagation of plane waves in type-III thermoelastic media / P. Puri, P.M. Jordan // Proceedings of the Royal Society of London. 2004. A 460. P. 3203—3221.
  14. Ковалев, В.А. Волновые числа плоских GNIII- термоупругих волн и неравенства, обеспечивающие их нормальность / В.А. Ковалев, Ю.Н. Радаев // Изв. Сарат. ун-та. Нов. сер. 2010. Т. 10. Сер. Математика. Механика. Информатика, вып. 3. С. 46–53.
  15. Dhaliwal, R.S. Thermoelastic waves in an infinite solid caused by a line heat source / S.R. Dhaliwal, R.S. Majumdar, W. Jun // Intern. J. Math. & Math. Sci. 1997. Vol. 20, No 2. P. 323–334. Механика 69 Изв. Сарат. ун-та. Нов. сер. 2011. Т. 11. Сер. Математика. Механика. Информатика, вып. 1
  16. Ковалев, В.А. Волновые задачи теории поля и термомеханика / В.А. Ковалев, Ю.Н. Радаев. Саратов: Изд-во Сарат. ун-та, 2010. 328 с.
  17. Ковалев, В.А. Распространение связанных GNIII- термоупругих волн в длинном цилиндрическом волноводе / В.А. Ковалев, Ю.Н. Радаев // Вестн. ЧГПУ им. И.Я. Яковлева. Сер. Механика предельного состояния. 2010. No 2(8). С. 207–255.
  18. Ковалев, В.А. Прохождение теплового GNIII- волнового сигнала с высокой окружной гармоникой через цилиндрический волновод / В.А. Ковалев, Ю.Н. Радаев, А.Е. Романов // Актуальные проблемы прикладной математики, информатики и механики: сб. трудов междунар. конф., посвящ. 80-летию проф. Д.Д. Ивлева. Воронеж: Изд. центр Воронеж. гос. ун-та, 2010. С. 173–180.
  19. Ковалев, В.А. Элементы теории поля: вариационные симметрии и геометрические инварианты / В.А. Ковалев, Ю.Н. Радаев. М.: Физматлит, 2009. 156 с.
  20. Ковалев, В.А. Волновые задачи теории поля и термомеханика / В.А. Ковалев, Ю.Н. Радаев // Математическая физика и ее приложения: материалы междунар. конф. / под ред. чл.-кор. РАН И.В. Воловича и проф. Ю.Н. Радаева. Самара: Книга, 2010. С. 165–166.