Non-linear Continuum Theories In Mechanics And Physics And Their Applications

R. S. Rivlin ( E d.) Non-linear Continuum Theories in Mechanics and Physics and their Applications Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, September 3-11, 1969 C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected] ISBN 978-3-642-11089-4 e-ISBN: 978-3-642-11090-0 DOI:10.1007/978-3-642-11090-0 Springer Heidelberg Dordrecht London New York ©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1970 With kind permission of C.I.M.E. Printed on acid-free paper Springer.com CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.) 2" Ciclo - Bressanone - dal 3 all'll Settembre 1969 x NON LINEAR CONTINUUM THEORIES IN MECHANICS AND PHYSICS AND THEIR APPLICATIONS n Coordinatore: Prof. R. S. RIVLIN P. A. BLYTHE: Non-linear far-field theories in relaxing gas flows Page 1 J. MAIXNER: Thermodynamics of deformable materials n 29 A. C. PIPKIN: Non-linear phenomena in continua n 51 R. S. RIVLIN: An introduction to non-linear continuum mechanics n 151 G. F. SMITH: The generation of integrity bases n 311 E. VARLEY: Testo non pervenuto n 353 CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. ) P. A. BLYTHE NON-LINEAR FAR-FIELD THEORIES IN RELAXING GAS FLOWS t e n u t o a B r e s s a n o n e d a l 3 a l l ' l l ' s e t t e m b r e 1969 NON-LINEAR FAR-FIELD THEORIES IN RELAXING GAS FLO\VS by P.A. Blythe (Lehigh - University ) Summary In the introduction the small amplitude non-linear far-field theory for one-dimensional isentropic wave propagation is briefly reviewed. The extension to non-equilibrium situations is then discussed for both high frequency and low frequency disturbances and the limitations of these classical theories are examined. It is shown that a suitable small-energy approach can be used both to remove these limitations and to provide a simplified description over the whole frequency range. P . A . Blythe - 4 - 1. Introduction: isentropic far-field theory The purpose of this lecture is to present a unified non-linear far-field theory* for relaxing or reacting gas flows. Attention will be restricted to small amplitude one-dimensional progressing waves and, for simplicity, only rate processes which involve a single internal mode or reaction will be considered. The corresponding far-field signalling problem in an inviscid gas which is in thermodynamic equilibrium has been well understood for some time. (Whitham 1950, Lighthill It is bseful first to briefly review this problem 1955). before discussing the non-equilibrium situations which are of interest here. In general the mass-conservation, momentum and energy equations take the form (adiabatic flow) where p is the density, p is the pressure, u is the particle speed, e is the internal energy and a t is the convective operator * i.e., the theory must be capable of providing a valid result for '