nach oben

Erschienen in:

2016 | OriginalPaper | Buchkapitel

2. Fundamental Equations

verfasst von : Roger J. Hosking, Robert L. Dewar

Erschienen in: Fundamental Fluid Mechanics and Magnetohydrodynamics

Verlag: Springer Singapore

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Conservation equations are the foundation for Fluid Mechanics and MHD, but others are needed to close the mathematical models. Although fluid pressure was at first assumed to be isotropic, when viscous stress was considered early in the eighteenth century it was evident that the assumption of incompressibility or the inclusion of a simple equation of state was no longer sufficient. The classical macroscopic equations (for mass, momentum and energy) follow from underlying microscopic theory, which also provides the relevant pressure tensor to incorporate viscosity. Except near magnetic null points, the pressure tensor for a plasma in a magnetic field is found to differ significantly from the classical shear viscosity form for a neutral fluid. There is also a brief introduction to the additional equation of magnetic induction required in MHD. The bibliography includes some references that provide further background to our presentation in this chapter, a worthy source on thermodynamics, and two books by Lamb and Prager particularly recommended for supplementary reading (more books on Fluid Mechanics are listed for Chap. 3).

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Jetzt informieren

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!

Jetzt informieren

Vorheriges Kapitel Vectors and Tensors

Nächstes Kapitel Basic Fluid Dynamics

This outcome may be recognised by considering the motion of an infinitesimal cylinder of fluid, slanted in the direction of $\mathbf{v}$, crossing the surface S. The cylindrical volume $\Delta \tau = dS\ {\hat{{\mathbf {n}}}} \varvec{\,\cdot \,}\mathbf{{v}} \, \Delta t$ carries the mass $\Delta m = \rho \Delta \tau = \rho \mathbf{v} \varvec{\,\cdot \,}d \mathbf{S} \Delta t$, with (2.2) obtained from the Divergence Theorem (1.60) in this case.

In solid mechanics the dyadic ${\varvec{\mathsf T}}$ in the equation for the displacement field corresponding to (2.5) is usually called the stress tensor—but in Fluid Mechanics and MHD we prefer to emphasise the pressure tensor ${\varvec{\mathsf p}}$, which is the pressure component of ${\varvec{\mathsf T}}$.

The operator $\{\}$ introduced here and often invoked later is defined by

$$\{ {\varvec{\mathsf F}} \} \equiv {\frac{1}{2}}({\varvec{\mathsf F}}+{\varvec{\mathsf F}}^{T}\;) - {\frac{1}{3}} {\varvec{\mathsf F}} \varvec{:}{\varvec{\mathsf I}}\; {\varvec{\mathsf I}}$$

for any dyadic ${\varvec{\mathsf F}}$, where ${\varvec{\mathsf F}}^{T}$ denotes its transpose. Since $\mathrm{Tr} \; {\varvec{\mathsf F}}\equiv {\varvec{\mathsf F}} \varvec{:}{\varvec{\mathsf I}}$ and $\mathrm{Tr} \; {\varvec{\mathsf I}}= 3$, the resulting symmetric dyadic is also traceless.

P. Debye and E. Hückel, Physikalische Zeitschrift 24, 183 and 305 (1923).

In the case of a plasma consisting of electrons and only one ion species, the mean velocity is effectively the ion velocity ($\mathbf{v} \simeq \mathbf{v}_i$ corresponding to $m_e \!\ll \!m_i$ and $\mathbf{u}_{i}\simeq 0$) so the defined fields such as ${\varvec{\mathsf P}}_{i}$ and $\mathbf{Q}_{i}$ for example are essentially the same as ${\varvec{\mathsf {p}}}_{i}$ and $\mathbf{q}_{i}$, respectively. However, this is not so for the electrons, since the alternative reference velocities for the electron component are quite distinct corresponding to inter-species (electron) diffusion—i.e. $\mathbf{u}_{e}\ne 0$.

The corresponding extension of the hierarchy of equations (where the moments are defined using the species mean flow velocity $\mathbf{v}_s$ as the reference velocity) was originally obtained by J.J. Ramos (Physics of Plasmas 14, 052506, 2007).

H. Grad (Communications in Pure and Applied Mathematics 2, 231– 407, 1949—cf. also Handbuch der Physik, S. Flügge (Editor), Volume 12, Chapter X, Springer, Berlin, 1958).

The result (2.69) was originally obtained by B.S. Liley, and the successive terms (2.71)–(2.73) for a sufficiently large magnetic field (when $|\mathbf{a}| \gg 1$) were identified by R.J. Hosking and G.M. Marinoff (Plasma Physics 15, 327–341, 1971). The alternative forms (2.76)–(2.78) were presented by J.D. Callen, W.X. Qu, K.D. Siebert, B.A. Carreras, K.C. Shaing and D.A. Spong (Plasma Physics and Controlled Nuclear Fusion Research, Volume II, IAEA, Vienna, 1987), and essentially earlier by S.I. Braginskii (Reviews of Plasma Physics 1, 205–311, 1965) in the collisional limit—cf. the traceless component of his pressure tensor. In the absence of collisions, an anisotropic plasma pressure ${\varvec{\mathsf p}} = p_{\parallel }{\hat{\mathbf{{b}}}}{\hat{\mathbf{{b}}}} + p_{\perp }{\varvec{\mathsf I}}_{\perp }$ was first discussed by G.E. Chew, M.L. Goldberger and F.E. Low (Proceedings of the Royal Society of London A 236, 112–118, 1956).

The definition of all ion field quantities following Braginskii can therefore be considered to coincide with ours in an ion-electron plasma. On the other hand, if the electron temperature is so much higher than the ion temperature such that the electron viscosity contribution remains of interest, recall that the Braginskii pressure tensor for the electrons is ${\varvec{\mathsf {p}}}_e - m_e n_e \mathbf{u}_e \mathbf{u}_e$ in our notation—cf. Exercise (Q1) of Sect. 2.6.

Most authors have associated the generalised Ohm’s law with the electron momentum equation in some way. Another viewpoint that produces the terms of interest as in (2.96), but with an alternative identification of the conductivity coefficient $\sigma $, was proposed by H.S. Green (Physics of Fluids 2, 341–349, 1959).

R. Balescu, Transport Processes in Plasmas, vol. 1, Classical Transport Theory, vol. 2, Neoclassical Transport (North Holland, 1989). (Provides a unified treatment of plasma transport, with underlying concepts from kinetic theory and physics discussed, making the two volumes largely self-contained)

C. Borgnakke, R.E. Sonntag, Fundamentals of Thermodynamics, 8th edn. (Wiley, New York, 2013). (Discusses the principles of thermodynamics, with notable applications to diverse fields and problem solving)

S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edn. (Cambridge University Press, Cambridge, 1991). (Classic book providing a detailed development of the Maxwell-Boltzmann approach to kinetic theory and the resulting macroscopic equations, with ionised gases in electric and magnetic fields considered in the later chapters)

H. Goedbloed, S. Poedts, Principles of Magnetohydrodynamics (Cambridge University Press, Cambridge, 2004). (Graduate textbook emphasising the physical background and detailed development of the ideal MHD model, in the context of thermonuclear fusion research and plasma astrophysics)

R.D. Hazeltine, J.D. Meiss, Plasma Confinement (Dover, New York, 2003). (Graduate textbook that presents fundamental theory for the magnetic confinement of plasma at or near thermonuclear conditions)

Yu.L Klimontovich, Kinetic Theory of Non-Ideal Gases and Non-Ideal Plasmas (Pergamon Press, Oxford, 1982). (Specialist exposition by a respected researcher in kinetic theory, treating strongly coupled non-ideal gases in the first part and fully ionised non-ideal plasmas in the second, followed by a third part on quantum aspects)

H. Lamb, Hydrodynamics, 6th edn. (Cambridge University Press, Cambridge, 1993). (The remarkable classic text, noted for its excellent detailed presentation of the fundamentals of fluid mechanics and various solution methods)

W. Prager, An Introduction to Mechanics of Continua (Dover, New York, 2004). (Originally published in 1961, this now classic textbook systematically addresses the foundations of fluid mechanics, plasticity and elasticity)

I.P. Shkarovsky, T.W. Johnston, M.P. Bachynski, The Particle Kinetics of Plasmas (Addison-Wesley, Reading, 1966). (A thorough treatment of the kinetic theory of plasmas, with some interesting material and results not readily accessible elsewhere)

10.

L. Spitzer, Physics of Fully Ionized Gases, 2nd edn. (Dover, New York, 2006). (Second edition of a slim volume on plasmas and fusion research by one of the founders of the field, emphasising the macroscopic equations but with an appendix on kinetic theory)

Titel: Fundamental Equations
verfasst von: Roger J. Hosking
Robert L. Dewar
Verlag: Springer Singapore
Buch: Fundamental Fluid Mechanics and Magnetohydrodynamics
Print ISBN: 978-981-287-599-0

Electronic ISBN: 978-981-287-600-3

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-981-287-600-3_2

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht