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.

Contenu

Preface

Chapter I Introduction

Article 1. The Three Basic Equations

Newton's Principle
Newton's equation for an inviscid fluid
Equation of continuity
Specifying equation
Adiabatic flow
Article 2. Energy Equation. Bernoulli Equation
Some transformations
The energy equation for an element of an inviscid perfect gas
Nonperfect (inviscid) gas
Energy equation for an elastic fluid
Bernoulli equation
Two integral theorems
Energy equation for a finite mass
Article 3. Influence of Viscosity. Heat Conduction
Viscous stresses and hydraulic pressure
Newton's equation for a viscous fluid
Work done by viscous forces. Dissipation
The energy equation for a viscous fluid
Heat conduction
General form of specifying equation
Article 4. Sound Velocity. Wave Equation
The problem
One-dimensional case. D'Alembert's solution
The wave equation in three dimensions
Poisson's solution
Discussion
Two-dimensional case
Article 5. Subsonic and Supersonic Motion. Mach Number, Mach Lines
Small perturbation of a state of uniform motion
Terminology
Propagation of the perturbation according to direction
Steady motion in two dimensions. Mach lines
Significance of the Mach lines

Chapter II General Theorems

Article 6. Vortex Theory of Helmholtz and Kelvin

Circulation
Mean rotation
Kelvin's theorem
Helmholtz' vortex theorems
Mean rotation and the Bernoulli function
Helmholtz' derivation of the vortex theorems
Article 7. Irrotational Motion
Potential
Equation for the potential
Steady radial flow
Nonsteady parallel flow
Steady plane motion
Transition between subsonic and supersonic flow. Limit line
Other particular cases of the general potential equation
Article 8. Steady Flow Relations
General relations among q, p, p, and
Hodograph representation
Case of polytropic (p,p)-relation
Adiabatic (irrotational) airflow
Article 9. Theory of Characteristics
Introduction
General theory
Compatibility relations
First examples
Further examples
General case of fluid motion
Article 10. The Characteristics in the Case of Two Independent Variables.
Characteristic directions
Compatibility relations
Two important theorems
The linear case
Riemann's solution
Interchange of variables
Geometrical interpretation

Chapter III One-Dimensional FLOW

Article 11. Steady Flow with Viscosity and Heat Conduction

General equations for parallel nonsteady flow
Equations for steady motion
Steady flow without heat conduction
The complete problem
Numerical data
Conclusions
Article 12. Nonsteady Flow of an Ideal Fluid
General equations
Potential and particle function
Interchange of variables. Speedgraph
General integral in the adiabatic case
Application of the speedgraph. Initial-value problem
Analytic solution: values given on two characteristics
Analytic solution: given and at t = 0
Article 13. Simple Waves. Examples
Simple waves: definition and basic relations
Centered waves
Other examples of simple waves
Combination of simple waves
Article 14. Theory of Shock Phenomena
Nonexistence of solutions. Effect of viscosity
The shock conditions for a perfect gas
Some properties of shocks
The algebra of the shock conditions
Representation of a shock in the speedgraph plane
Example of a shock phenomenon. The Riemann problem
Article 15. Further Shock Problems
Behavior of a shock at the end of a tube or a wall (shock reflection )
Discontinuous solutions of the equations for an ideal fluid
Example of a contact discontinuity: collison of two shocks
Numerical method of integration
Some remarks on the application of the preceding method
The inviscid flow behind a curved shock line
A second approach
Nonisentropic simple waves. Linearization

Chapter IV PLane Steady Potential Flow

Article 16. Basic Relations

Direct approach
Equations for the potential and stream functions
Subsonic and supersonic flow. Characteristics
Basic boundary-value problems
Hodograph
Characteristics in the hodograph plane
The nets of characteristics in the physical and hodograph planes
Article 17. Further Discussion of the Hodograph Method
Differential equations for the Legendre transforms
Other linear differential equations
Transition from the hodograph to the physical plane
Radial flow, vortex flow, and spiral flow obtained as exact solutions in the hodograph
The Chaplygin-Karman-Tsien approximation
Continuation
Article 18. Simple Waves
Definition and basic properties
Numerical data. Streamlines and cross Mach lines
Examples of simple waves
More elaborate examples involving simple waves
Article 19. Limit Lines and Branch Lines
Singularities of the hodograph transformation
Some basic formulas. Subsonic cases
Limit lines £1 and £2
Special points of the limit line
Limit singularities for = 1
Branch lines
Final remarks
Article 20. Chaplygin's Hodograph Method
Separation of variables
Relation to incompressible flow solutions
A flow with imbedded supersonic region
Further comments and generalizations
Compressible doublet
Subsonic jet