Summary
The problem of the dynamical effects of heat absorption or rejection on the flow of an electrically charged fluid such as an electron or ion gas is considered within the framework of special relativity. The source or sink for the heat transferred to or from the fluid may be imagined to be a heat reservoir whose velocity is in general not the same as the fluid velocity. The analysis of a relativistic Carnot cycle that operates between two heat reservoirs having relative motion with respect to each other provides a simple way to study the dynamical effects of heat transfer and to incorporate the moving heat reservoir model into the formalism of fluid dynamics. All the dynamical effects that are functions of the 4-gradient of the specific entropy of the fluid can be expressed in terms of an antisymmetric tensor, the entropy-force tensor, whose role in the equation of fluid motion is analogous to that of the electromagnetic field tensor.
Riassunto
Nel quadro della relatività ristretta si studia il problema degli effetti dinamici dell’assorbimento o della emissione di calore sul flusso di un fluido elettricamente carico quale un gas di elettroni o di ioni. Si può immaginare che la sorgente o il ricevitore del calore trasferito al (o dal) fluido sia un serbatoio di calore la cui velocità in generale non è la stessa della velocità del fluido. L’analisi di un ciclo di Carnot relativistico che opera fra due serbatoi di calore che hanno un moto relativo uno rispetto all’altro fornisce un semplice modo di studiare gli effetti dinamici del trasferimento di calore e di incorporare il modello con serbatoio di calore mobile nel formalismo della dinamica dei fluidi. Si possono esprimere tutti gli effetti dinamici che sono funzioni del quadrigradiente dell’entropia specifica del fluido in base ad un tensore antisimmetrico, il tensore entropia-forza, il cui ruolo nell’equazione di moto del fluido è analogo a quello del tensore del campo elettromagnetico.
Реэюме
В рамках общей теории относительности рассматривается проб-лема динамических зффектов поглошения и отвода тепла от потока злектрически эаряженной жидкости, такой как злектронный или ионный гаэ. Исток или сток для тепла передаваемого к или от жидкости можно представить как источник тепла, скорость которого, вообще говоря, не является той же самой, как скорость жидкости. Аналиэ релятивистского цикла Карно, который происходит между двумя источни-ками тепла, которые движутся друг относительно друга, проиэводится обычным о браэом, чтобы иэучить динамические зффекты передачи тепла, и чтобы применить модель движушегося источника тепла в формалиэме динамики жидкости. Все дина-мические зффекты, которые являются функциями четырех-градиента от удельной знтропии жидкости, могут быть выражены череэ выражения от антисимметричного тенэора, тенэора знтропии и знергии, роль которого в уравнении движения жидкости аналогична роли тенэора злектромагнитного поля.
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References
Surveys of relativistic fluid dynamics can be found in most relativity textbooks. Of the available brief surveys, probably the most relevant to the present work is the one given in Chap. XV ofFluid Mechanics byL. D. Landau andE. M. Lifshitz (Reading, Mass., 1959). More detailed discussions are given byA. Lichnerowicz:Théories Relativistes de la Gravitation et de l’Electromagnetisme (Paris, 1955), Chap. IV–VI, and byF. Halbwachs:Théorie Relativiste des Fluides à Spin (Paris, 1960), Chap. IV, Sect. 2, and Appendix B. The relativistic equations of fluid dynamics were first derived independently byG. Herglotz:Ann. d. Phys., (4),36, 493 (1911);E. Lamla:Ann. d. Phys., (4),37 772 (1912). Lamla’s equations were less general in that they were limited to the case of isentropic flow, whereas this restriction was not imposed in Herglotz’s derivation. Subsequent research has been limited almost exclusively either to isentropic flow (e.g. I. M. Khalatnikov:Žurn. Ėksp. Teor. Fiz.,27, 529 (1954), a brief review of which is given inLandau andLifshitz cited above, andA. H. Taub:Phys. Rev.,103, 454 (1956)) or to the somewhat more general case of a barotropic fluid,i.e. one for which the pressure may be regarded as a function of the density alone (e.g. J. L. Synge:Proc. London Math. Soc., (2),43, 376 (1937), and the work ofLichnerowicz, which is summarized in his book cited above). The papers cited in footnotes (2,3) are not, however, restricted to either of these two special cases.
C. Eckart:Phys. Rev.,58, 919 (1940). This paper concentrates on the thermodynamical aspects of relativistic fluid dynamics. Of all the fluid dynamical references cited, it and Khalatnikov’s paper are the most relevant to the present work. Both of these papers, like the present work, are within the framework of special, rather than general, relativity.
G. Pichon:Ann. Inst. Poincaré, A2, 21 (1965). This work, like that ofSynge, Lichnerowicz andTaub cited in footnote (1), is within the framework of general relativity. Its relevance to the present work lies in the fact that it presents alternatives to the treatment of viscosity and heat conduction in a charged fluid that differ from the approach taken in the present work.
See, for example,L. Landau andE. Lifshitz (cited in footnote (1)), Chap. XV.
A. H. Taub:Phys. Rev.,103, 454 (1956).Eckart (footnote (2)) derived his formulation of the stress-energy tensor by means of macroscopic thermodynamic arguments.Taub arrived at the same form for the tensor through a kinetic-theory argument.
H. Ott:Zeits. Phys.,175, 70 (1963).
H. Arzeliès:Nuovo Cimento,35, 792 (1965);41 B, 81 (1966).
A. Gamba:Nuovo Cimento,37, 1792 (1965);41 B, 79 (1966).
T. W. B. Kibble:Nuovo Cimento,41 B, 72, 83, 84 (1966).Kibble disagrees withArzeliès andGamba as to the appropriate definition of energy and work in the case of a nonisolated system, but all agree that heat and temperature should be regarded as 4-vectors, which is the only aspect of the Ott formulation that enters into the present work.
A. Børs:Proc. Phys. Soc.,86, 1141 (1965). The authors cited in footnotes (7–10) present thermodynamical arguments in favor of a 4-vector temperature, whereasBørs works within the framework of statistical mechanics.
M. Planck:Ann. d. Phys., (4),76, 1 (1908). (Planck first communicated this work at a meeting of the Preussische Akademie der Wissenschaften on 13 June 1907.) Further references are given byW. Pauli:Theory of Relativity (London, 1958), p. 134.
M. von Laue:Die Relativitätstheorie, I (Braunschweig, 1961), 7th ed. (first edition, 1911). See footnotes on p. 138 and p. 177, 178.
See, for example,W. Pauli (cited in footnote (12)), p. 134;R. Tolman:Relativity, Thermodynamics and Cosmology (Oxford, 1934), Chap. V;C. Møller:The Theory of Relativity (Oxford, 1952), Sect. 78.
See, for example,A. Sommerfeld:Mechanics (New York, 1952), p. 29, eq. (4).
See, for example,C. Møller (cited in footnote (14)), p. 106.
Cited in footnote (1). Khalatnikov’s equation amounts to a rediscovery of the equations ofHerglotz andLamla (cited in footnote (1)), except that the role of the specific enthalpy as the thermal potential is more explicit in Khalatnikov’s work than in that of eitherHerglotz orLamla. The relativistic equation of motion for an ideal chargedbarotropic fluid is given in Lichnerowicz’s book, which was cited in footnote (1) (Chap. VI, eq. (55.7)). This equation is equivalent to (5.14) above (for∂ j S jk=0) for the isentropic case (Θ jk=0) except that the thermal potential is not the specific enthalpy, but rather a generalized barotropic thermal potential called theindex-function, which was first introduced by Synge (cited in footnote (1)) and further developed byLicherowicz. (A similar function had already been introduced byL. P. Eisenhart:Trans. Am. Math. Soc.,26, 205 (1924).)
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Schmid, L.A. Heat transfer in relativistic charged-fluid flow. Nuovo Cimento B (1965-1970) 47, 1–28 (1967). https://doi.org/10.1007/BF02712304
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DOI: https://doi.org/10.1007/BF02712304