R. von Mises - Mathematical theory of compressible fluid flow

APPLIED MATHEMATICSAND MECHANICSAn International Scries of MonographsEDITORSF. N. FRENKIELUniversityof MinnesotaMinneapolis,MinnesotaG. TEMPLEMathematicalOxfordOxford,InstituteUniversityEnglandVolume 1. K. OSWATITSCH: Gas Dynamics, English version by G.Kuerti(1956)Volume 2.

G. BIRKHOFF and Ε. H . ZARANTONELLO: Jet, Wakes, and Cavities(1957)Volume 3. R . VON MISES: Mathematical Theory of Compressible Fluid Flow,Revised and completed by Hilda Geiringer and G. S. S. Ludford(1958)Volume 4. F. L. A L T : Electronic Digital Computers—Their Use in Scienceand Engineering (1958)Volume 5A. W. D. HAYES and R .

F. PROBSTEIN: Hypersonic Flow Theory,second edition, Volume I, Inviscid Flows (1966)Volume 6. L. M . BREKHOVSKIKH: Waves in Layered Media, Translated fromthe Russian by D. Lieberman (1960)Volume 7. S. FRED SINGER (ed.): Torques and Attitude Sensing in EarthSatellites (1964)Volume 8. M I L T O N V A N D Y K E : Perturbation Methods in Fluid Mechanics(1964)Volume 9. ANGELO M I E L E (ed.): Theory of Optimum Aerodynamic Shapes(1965)MATHEMATICAL THEORYOF COMPRESSIBLE FLUID FLOWRICHARD VO N MISESlate Gordon McKay Professor of Aerodynamicsand Applied Mathematics, Harvard UniversityCompleted byHILDA G E I R I N G E RG .

S. S. L U D F O R D1958ACADEMIC PRESS INC ·PUBLISHERS·NEW YORKE D I T I N G SUPPORTED B Y THE B U R E A U OF O R D N A N C EU . S. N A V Y , U N D E R C O N T R A C T N O R D 7386.C O P Y R I G H T ©, 1958BYACADEMIC PRESS INC.IllFIFTHAVENUEN E W Y O R K 3, Ν .Y.A L L RIGHTS RESERVEDNO PART OF T H I S BOOK M A Y BE REPRODUCED I N A N Y FORM,B Y PHOTOSTAT, MICROFILM, OR A N Y OTHER M E A N S , W I T H O U TW R I T T E N PERMISSION FROM T H E P U B L I S H E R S .R E P R O D U C T I O N I N W H O L E OR I N PART I S PERMITTED FOR A N YPURPOSE OF T H E U N I T E D STATES G O V E R N M E N T .L I B R A R Y OF C O N G R E S S C A T A L O G C A R DNUMBER:57-14531First Printing, 1958Second Printing, 1966P R I N T E D I N T H E U N I T E D STATES OF AMERICAPREFACEWhen Richard von Mises died suddenly in July, 1953, he left the firstthree chapters (Arts.

1-15) of what was intended to be a comprehensivework on compressible flow. B y themselves these did not form a completebook, and it was decided to augment them with the theory of steady planeflow, which according to von Mises' plan was the next and last topic ofthe first part.This last work of Richard von Mises embodies his ideas on a centralbranch of fluid mechanics. Characteristically, he devotes special care tofundamentals, both conceptual and mathematical. T h e novel concept ofa specifying equation clarifies the role of thermodynamics in the mechanicsof compressible fluids. T h e general theory of characteristics is treated in anunusually complete and simple manner, with detailed applications.

T h etheory of shocks as asymptotic phenomena is set within the context ofrational mechanics.Chapters I V and V (Arts. 16-25) were written with the author's papersand lecture notes as guide. A thorough presentation of the hodographmethod includes a discussion and comparison of the modern integrationtheories. Shock theory once more receives special attention. T h e text endswith a study of transonic flow, the last subject to engage von Mises' in­terest.In revising the existing three chapters great restraint was exercised, soas not to impair the author's distinctive presentation; a few sections wereadded (in Arts.

7, 9, 15). M o r e than forty pages of Notes and Addenda,partly bibliographical and historical, and partly in the nature of appendices,follow the text. This is in line with von Mises' practice of keeping text freefrom distraction, while at the same time providing a fuller background.T h e text is, however, completely independent of the Notes.E v e r y facet of the work was studied jointly by us, in an attempt to con­tinue in the author's spirit. Final responsibility for the text and the Notesto Arts. 16-21, 25 and the Notes to Chapters I , I I lies, however, with HildaGeiringer (Mrs. R.

v. Mises) and for the text and Notes to Arts. 22-24and the Notes to Chapter I I I with G. S. S. Ludford.T h e present book contains no extensive discussion of the approximationtheories, which have proved to be so fruitful. I t was the author's intentionto discuss these in the second part of his work, along with various otherνviPREFACEtopics. T h e book has been written as an advanced text-book in the hopethat both graduate students and research workers will find it useful.W e are greatly indebted to many people for help given in various phasesof our task. Helen K . Nickerson, who was the much appreciated assistantof von Mises in the writing of the first three chapters, helped in theirlater revision and read Chapters I V , V with constructive criticism. T h ewhole manuscript was read by G.

Kuerti, who suggested important im­provements. S. Goldstein at times gave us the benefit of his unique in­sight into the whole field of mechanics. M . Schiffer was always ready withdiscussion and advice on delicate questions of a more mathematical nature.T h e influence of C. TruesdelFs important contributions to the history ofmechanics is obvious in many Notes; he also readily provided more specificinformation.Very able assistance was rendered by W . Gibson and S.

Schot, whom wethank cordially for their dedicated interest and valuable help. Thanks arealso due to M . Murgai who prepared the subject index and to D . Rubenfeldwho made the final figures.W e are particularly grateful to F. N . Frenkiel as an understanding andpatient advisor. T h e work of Hilda Geiringer at the Division of Engineeringand Applied Physics, Harvard University, was generously supported bythe Office of N a v a l Research; that of G.

S. S. Ludford, was carried outunder the sympathetic sponsorship of the Institute for Fluid Dynamicsand Applied Mathematics, University of Maryland, and its director Μ . H .Martin.Finally, much more than a formal acknowledgment is due to GarrettBirkhofT, who enabled H . Geiringer to carry out her task under ideal work­ing conditions. I t is mainly due to his vision and understanding that thislast work of Richard von Mises has been preserved.HILDAG.Cambridge, MassachusettsFall 1957S.GEIRINGERS.LUDFORDThe leitmotif, the ever recurringindispensablein any reasoning, in any descriptionof a segment of reality:the languagemelody, is that two things areto submitto experiencethat is used, with unceasingwe shapeand to facelogicalcriticism.from an unpublished paper of R.

v. MisesCHAPTER IINTRODUCTIONArticle 1The Three Basic Equations1. Newton's PrincipleT h e theory of fluid flow (for an incompressible or compressible fluid,whether liquid or gas) is based on the Newtonian mechanics of a smallsolid body. T h e essential part of Newton's Principle can be formulated intothe following statements:(a) T o each small solid body can be assigned a positive number ra,invariant in time and called its mass; and(b) T h e body moves in such a way that at each moment the productof its acceleration vector by ra is equal to the sum of certain othervectors, called forces, which are determined by the circumstancesunder which the motion takes place (Newton's Second L a w ) .1For example, if a bullet moves through the atmosphere, one force is gravityrag,* directed vertically downward (g = 32.17 ft/sec at latitude 4 5 ° N ) ;another is the air resistance, or drag, opposite in direction to the velocityvector, with magnitude depending upon that of the velocity, etc.2B y means of a limiting process, this principle can be adapted to the caseof a continuum in which a velocity vector q and an acceleration vectordq/dt exist at each point.

Let Ρ be a point with coordinates (x, y, z ) , orposition vector r, and dV a volume element in the neighborhood of P ; tothis volume element will be assigned a mass pdV, where ρ is the density,or mass per unit volume. Density will be measured in slugs per cubic foot.For air under standard conditions (temperature 59°F, pressure 29.92 in. Hg,or 2116 lb/ft ), ρ = 0.002378 slug/ft , as compared with ρ = 1.94 slug/ftfor water. T h e forces acting upon this element are, the external force of233* Vectors will be identified by means of boldface type a, v, etc.; the same letter inlightface italic denotes the absolute value ox the vector: a = | a |; components willbe indicated by subscripts, as a , a .

T h e scalar and vector products of a and b willbe represented by a»b and a X b respectively. In the figures the bar notationfor vectors will be used namely a, g, etc.xy12I.gravity pgdVINTRODUCTIONand the internal forces resulting from interaction with ad­jacent volume elements. Thus, after dividing by dV, the relation(1)ρ= Pg + internal force per unit volumeatis a first expression of statement ( b ) .T o formulate part ( a ) of Newton's Principle, note that the mass t o beassigned to any finite portion of the continuum is given by fpdV and there­fore, since this mass is invariant with respect to time,(2)These two relations will be developed further in succeeding sections, butfirst the meaning of the differentiation symbol d/dt occurring in Eqs. (1)and (2) must be clarified.T h e density ρ and the velocity vector q are each considered as functionsof the four variables x, y, z, and t, so that partial derivatives with respectto time and with respect to the space coordinates may be taken, as wellas the directional derivative corresponding to any direction I, given by- = cos (/, x) ^- + cos (Z, y)+ cos (Ζ, ζ) |- ,θίdxdydzdwhere cos (Ϊ, x), cos (I, y), and cos (Z, z) are the direction cosines definingthe direction I.