# Mathematical Theory of Compressible Fluid Flow

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Mathematical Theory of Compressible Fluid Flow covers the conceptual and mathematical aspects of theory of compressible fluid flow. This five-chapter book specifically tackles the role of thermodynamics in the mechanics of compressible fluids. This text begins with a discussion on the general theory of characteristics of compressible fluid with its application. This topic is followed by a presentation of equations delineating the role of thermodynamics in compressible fluid mechanics. The discussion then shifts to the theory of shocks as asymptotic phenomena, which is set within the context of rational mechanics. The remaining two chapters is a thorough description of the hodograph method. These chapters provide a comparison of the modern integration theories. The features, characteristics, and application of transonic flow are also explored. This book is an ideal advanced textbook for both graduate students and research workers.

## Table of Contents

Preface

Chapter I Introduction

Article 1. The Three Basic Equations

1. Newton's Principle

2. Newton's equation for an inviscid fluid

3. Equation of continuity

4. Specifying equation

5. Adiabatic flow

Article 2. Energy Equation. Bernoulli Equation

1. Some transformations

2. The energy equation for an element of an inviscid perfect gas

3. Nonperfect (inviscid) gas

4. Energy equation for an elastic fluid

5. Bernoulli equation

6. Two integral theorems

7. Energy equation for a finite mass

Article 3. Influence of Viscosity. Heat Conduction

1. Viscous stresses and hydraulic pressure

2. Newton's equation for a viscous fluid

3. Work done by viscous forces. Dissipation

4. The energy equation for a viscous fluid

5. Heat conduction

6. General form of specifying equation

Article 4. Sound Velocity. Wave Equation

1. The problem

2. One-dimensional case. D'Alembert's solution

3. The wave equation in three dimensions

4. Poisson's solution

5. Discussion

6. Two-dimensional case

Article 5. Subsonic and Supersonic Motion. Mach Number, Mach Lines

1. Small perturbation of a state of uniform motion

2. Terminology

3. Propagation of the perturbation according to direction

4. Steady motion in two dimensions. Mach lines

5. Significance of the Mach lines

Chapter II General Theorems

Article 6. Vortex Theory of Helmholtz and Kelvin

1. Circulation

2. Mean rotation

3. Kelvin's theorem

4. Helmholtz' vortex theorems

5. Mean rotation and the Bernoulli function

6. Helmholtz' derivation of the vortex theorems

Article 7. Irrotational Motion

1. Potential

2. Equation for the potential

3. Steady radial flow

4. Nonsteady parallel flow

5. Steady plane motion

6. Transition between subsonic and supersonic flow. Limit line

7. Other particular cases of the general potential equation

Article 8. Steady Flow Relations

1. General relations among q, p, p, and Τ

2. Hodograph representation

3. Case of polytropic (p,p)-relation

4. Adiabatic (irrotational) airflow

Article 9. Theory of Characteristics

1. Introduction

2. General theory

3. Compatibility relations

4. First examples

5. Further examples

6. General case of fluid motion

Article 10. The Characteristics in the Case of Two Independent Variables.

1. Characteristic directions

2. Compatibility relations

3. Two important theorems

4. The linear case

5. Riemann's solution

6. Interchange of variables

7. Geometrical interpretation

Chapter III One-Dimensional FLOW

Article 11. Steady Flow with Viscosity and Heat Conduction

1. General equations for parallel nonsteady flow

2. Equations for steady motion

3. Steady flow without heat conduction

4. The complete problem

5. Numerical data

6. Conclusions

Article 12. Nonsteady Flow of an Ideal Fluid

1. General equations

2. Potential and particle function

3. Interchange of variables. Speedgraph

4. General integral in the adiabatic case

5. Application of the speedgraph. Initial-value problem

6. Analytic solution: values given on two characteristics

7. Analytic solution: given υ and ν at t = 0

Article 13. Simple Waves. Examples

1. Simple waves: definition and basic relations

2. Centered waves

3. Other examples of simple waves

4. Combination of simple waves

Article 14. Theory of Shock Phenomena

1. Nonexistence of solutions. Effect of viscosity

2. The shock conditions for a perfect gas

3. Some properties of shocks

4. The algebra of the shock conditions

5. Representation of a shock in the speedgraph plane

6. Example of a shock phenomenon. The Riemann problem

Article 15. Further Shock Problems

1. Behavior of a shock at the end of a tube or a wall (shock reflection )

2. Discontinuous solutions of the equations for an ideal fluid

3. Example of a contact discontinuity: collison of two shocks

4. Numerical method of integration

5. Some remarks on the application of the preceding method

6. The inviscid flow behind a curved shock line

7. A second approach

8. Nonisentropic simple waves. Linearization

Chapter IV PLane Steady Potential Flow

Article 16. Basic Relations

1. Direct approach

2. Equations for the potential and stream functions

3. Subsonic and supersonic flow. Characteristics

4. Basic boundary-value problems

5. Hodograph

6. Characteristics in the hodograph plane

7. The nets of characteristics in the physical and hodograph planes

Article 17. Further Discussion of the Hodograph Method

1. Differential equations for the Legendre transforms

2. Other linear differential equations

3. Transition from the hodograph to the physical plane

4. Radial flow, vortex flow, and spiral flow obtained as exact solutions in the hodograph

5. The Chaplygin-Karman-Tsien approximation

6. Continuation

Article 18. Simple Waves

1. Definition and basic properties

2. Numerical data. Streamlines and cross Mach lines

3. Examples of simple waves

4. More elaborate examples involving simple waves

Article 19. Limit Lines and Branch Lines

1. Singularities of the hodograph transformation

2. Some basic formulas. Subsonic cases

3. Limit lines £1 and £2

4. Special points of the limit line

5. Limit singularities for Μ = 1

6. Branch lines

7. Final remarks

Article 20. Chaplygin's Hodograph Method

1. Separation of variables

2. Relation to incompressible flow solutions

3. A flow with imbedded supersonic region

4. Further comments and generalizations

5. Compressible doublet

6. Subsonic jet

Chapter V Integration Theory and Shocks

Article 21. Development of Chaplygin's Method

1. The problem

2. Replacement of Chaplygin's factor [ψn(τ1)]-1

3. Flow around a circular cylinder

4. General solution for the subsonic region

5. Bergman's integration method

6. Convergence

7. Integral transformation

8. Relation of the two methods

Article 22. Shock Theory

1. Nonexistence of solutions

2. The oblique shock conditions for a perfect gas

3. Analysis of the shock conditions

4. Representation of a shock in the hodograph plane

5. Shock diagram and pressure hills

6. The deflection of a streamline by a shock

7. Strong and weak shocks

Article 23. Examples Involving Shocks

1. Comparison of deflections caused by shocks and simple waves

2. Supersonic flow along a partially inclined wall

3. Supersonic flow past a straight line profile: contact discontinuity

4. Behavior of a shock at a wall (oblique shock reflection)

5. Properties of the reflection

6. Intersection of two shocks

Article 24. Nonisentropic Flow

1. Strictly adiabatic flow of an inviscid fluid

2. Equation for the stream function

3. Substitution principle. Modified stream function

4. A second approach

5. The sufficiency of the shock conditions

6. Asymptotic solutions of the equations of viscous flow

Article 25. Transonic Flow

1. On some additional boundary-value problems

2. Problem of existence of flow past a profile

3. Apparent conflict between mathematical evidence and experiment

4. Limit-line conjecture

5. The local approach

6. Conjectures on existence and uniqueness in the large

Notes and Addenda

Chapter 1

Chapter II

Chapter III

Chapter IV

Chapter V

Selected Reference Books

Author Index

Subject Index

## Product details

- No. of pages: 528
- Language: English
- Copyright: © Academic Press 1958
- Published: January 1, 1958
- Imprint: Academic Press
- eBook ISBN: 9780323146999

## About the Author

### Richard Von Mises

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