# Mechanics of Liquids and Gases

## Description

This 6th augmented edition of the classic text incorporates significant advances towards the solution of the problems of the boundary layer theory, dynamics of viscous fluids and theory of turbulence. The result is to bring its contents closer to that of a textbook and, on the other hand, to update it by adding new present-day problems.
As distinct from previous editions where only vector and tensor calculus formulas were given, the present edition provides a short background discussion on this area of mathematics. Newly written are three sections dealing with some general methods for numerical integration of differential equations and their application to Navier-Stokes viscous fluid dynamics equations.
Extensive revision of the chapter on turbulence culminated in the appearance of a new chapter dealing specially with the techniques for calculating turbulent boundary layers. The book is intended for students, postgraduate students, engineers, and research workers specializing in the field of fluid mechanics.

971 pages,
© 1995

## Table of Contents:

Preface to the 6th Edition

Preface to the English Edition

1 The Field of a Physical Quantity and Conditions for the Physical Objectivity of Quantities Specified Analytically: Basic Operations of the Field

1. Scalar and Vector Fields: Conditions for the Physical Objectivity of Scalar and Vector Quantities Specified Analytically

2. A Second-Rank Tensor Conditions for the Physical Objectivity of Its Analytical Specification — A Dyad and a Rotation Tensor

3. Basic Operations of Tensor Algebra

4. Expansion of a Tensor into Symmetric and Antisymmetric Parts. Invariants: Expansion of a Tensor into Spherical and Deviatoric Parts

5. Principal Axes and Principal Values of the Symmetric Tensor

6. Derivative with Respect to the Prescribed Direction. Space Derivatives in Scalar, Vector and Tensor Fields

7. A Measure of the Vector Field Nonuninformity — A Differential Dyad and Its Components: Deformation and Rotation of the Vector Field

8. Basic Integral Formulas of the Field: The Gauss-Ostrogradskiy and Stokes, Theorems

2 The Kinematics of a Continuous Medium

1. Specification of the Position and Motion of a Continuous Medium. Streamlines and Trajectories. Steam Tubes and Jets

2. Velocity Distribution in an Elementaiy Volume of the Medium. Helmholtz's First Theorem

3. Distorting Motion of a Liquid, Rate of Strain Tensor and the Kinematic Sense of Its Components. The Principal Axes of the Rate of Strain Tensor

4. A Vortex, a Vortex Line and a Vortex Tube. Helmholtz's Second Theorem

5. The Stokes, Theorem on the Relationship between the Strength of the Vortex Tube and Velocity Circulation Around the Encircling Contour

6. Acceleration of the Medium Particle. Local and Convective Components of Acceleration. Total Acceleration

7. Kelvin Kinematic Theorem

3 Distribution of Mass and Force in a Continuous Medium. Cauchy Equalities. Theorem on the Reciprocity of Tangential Stresses

1. The Density of Mass Distribution in a Continuous Medium. Mass Conservation Law. Continuity Equation

2. Distribution of Forces in a Continuous Medium. Volumetric and Surface Forces. Cauchy Equalities. Stress Tensor

3. Theorem on the Reciprocity of Tangential Stresses

4 General Theorems of the Dynamics of a Continuous Medium

1. The Theorem of Momentum. Equation of Dynamics in Terms of “Stresses”

2. Theorem of Moments and Resulting Theorems on the Reciprocity of Tangential Stresses

3. Theorem of the Change in Kinetic Energy and General Law of Energy Conservation

4. Transport of a Physical Quantity by the Medium Flow through a Surface - The Euler Theorem

5. Statics of a Flowing Medium. Euler Equations for the Medium Equilibrium

6. Equilibrium State of an Incompressible Fluid. Archimedes' Law

7. Equilibrium State of a Uniformly Rotating Incompressible Fluid. Centrifuging Solid Particles

8. Barotropic Equilibrium of a Gas

5 The Dynamics of an Ideal Medium: Basic Equations and Theorems

1. The Equations of Euler, Gromeka-Lamb, and Helmholtz-Fridman. Theorem of Helmholtz

3. The Power of Internal Forces. The Equation of Energy Balance

4. Velocity of Propagation of Small Disturbances in an Ideal Gas: Speed of Sound

5. The Numbers M and λ: Isentropic Formulas

6 One-Dimensional Flow of an Ideal Gas

1. One-Dimensional Steady Motion of a Gas through a Tube of Variable Cross-Section

2. The Discharge of a Gas through a Nozzle

3. An Example of the Nonadiabatic Motion of Gas

4. The Nonisentropic Motion of a Gas through a Tube in the Presence of Resistance

5. Planar Disturbance Waves and a Shock Waves

6. Variation of the Velocity and Thermodynamic Parameters of the Gas in Passing across a Normal Shock Wave

7. Velocities of Propagation of a Shock Wave and the Accompanying Flow behind It

8. Elementary Theory of a Supersonic Diffuser

9. Measurement of Sub- and Supersonic Velocities by the Pneumatic Methods

10. Unsteady One-Dimensional Ideal Gas Flow: Propagation of Finite-Intensity Disturbances

11. Rarefaction Waves behind a Moving Piston. Centered Waves. Self-Similar and General Problems

12. An Elementary Theory of the Shock Tube

7 Irrotational Motions of an Ideal Medium. Plane Irrotational Motion of an Ideal Incompressible Fluid

1. The Theorems of Kelvin and Lagrange: Conditions for the Existence of Irrotational Flows

2. Velocity Potential and Its Definition from the Given Velocity Field

3. Lagrange-Cauchy Integral: Some General Properties of Irrotational Motion

4. A Plane Irrotational Motion of an Incompressible Fluid: Application of the Functions of a Complex Variable

5. Complex Potentials of Simple Flows

6. Solution of the Problem of Flow Past Bodies by the Method of Conformal Mappings: Zhukovskiy-Chaplygin Hypothesis - The Circulation Formula

7. Examples of the Use of Conformal Mappings: Flow Around an Ellipse and a Plate

8. The Zhukovskiy-Chaplygin Wing Profiles

9. Resultant Vector and Resultant Moment of Forces of Flow Pressure on the Closed Contour. Chaplygin Formulas. Zhukovskiy Theorem. Coefficients of the Lifting Force and of the Moment of the Plate

10. The Theory of a Thin Profile (Arc)

11. Zhukovskiy Theorem of the Lifting Force of the Profile in a Cascade

12. Application of the Method of Conformal Mappings to the Theory of Jet Rows

8 Plane Irrotational Motion of an Ideal Gas

1. Basic Equations of Motion and their Linearization

2. Subsonic Flow Past a Thin Profile: Prandtl-Glauert Rule

3. Supersonic Flow Past a Thin Profile: Ackeret's Formulas

4. The Similarity Laws of Plane Sub- and Supersonic Flows Past a Thin Profile: Case of Transonic Flow

5. A Converging Supersonic Flow. An Oblique Shock Wave

6. An Expanding Supersonic Flow. Gas Motion in the Sector of Rarefaction

7. The Case of Great Mach Numbers. The Law of Similarity of Hypersonic Flows

8. Equations of Gas Dynamics in the Hodograph Plane of Velocity

9. Effect of Compressibility on Velocity and Pressure Distributions in a Plane Subsonic Flow

10. Subcritical Flow Past a Wing Profile

11. A Plane Supersonic Flow. General Properties of Characteristics. The Graphic Method for Calculating Supersonic Flows

9 Three-Dimensional Irrotational Motion of Liquids and Gases

1. Velocity Potentials of the Simplest Three-Dimensional Flows

2. Velocity Field Induced by the Given System of Vortices in an Infinite Fluid. Biot-Savart Formula

3. Potential of the Velocity Field Induced by a Closed Vortex Filament

4. Stream Function in Three-Dimensional Motions

5. Flow Past a Sphere. D'Alembert's Paradox

6. Equation of the Longitudinal Axisymmetric Motion. Flow through Channels

7. Axisymmetric Longitudinal Flow Past Bodies of Revolution

8. Transverse Flow about the Bodies of Revolution

9. Application of the Method of Singularities for Calculating Longitudinal and Lateral Flows about the Bodies of Revolution. Slender Bodies

10. Elementary Theory of the Wing of Finite Span

11. General Case of Solid Body Motion in an Infinite Ideal Incompressible Fluid - Kirchhoff s Problem

12. Axisymmetric Sub-and Supersonic Flow Past a Thin Body of Revolution

13. Similarity Laws for the Flows Past Thin Bodies of Revolution and Thin Wings of Finite Span

14. Longitudinal Supersonic Flow Past a Circular Cone. Conical Shock Wave

15. Supersonic Flow Past a Thin Body of Revolution at Great Mach Numbers

10 The Dynamics of an Incompressible Viscous Fluid (Linear Problems)

1. Newtonian Viscous Fluid and Its Rheological Equation

2. Rheological Laws of Non-Newtonian Viscous Incompressible Fluids

3. The Navier-Stokes Equations of the Dynamics of a Newtonian Incompressible Medium

4. Similarity of Flows of a Viscous Incompressible Fluid

5. Foundations of the Dimensionality Theory: The II Theorem

6. Examples of the Solution of the Navier-Stokes Equations. Simplest Linear Problems

7. Developed Motion of a Viscoplastic Fluid through a Cylindrical Tube of Circular Profile

8. Developed Flow of an Electrically Conducting Viscous Fluid through Tubes in the Presence of Transverse Magnetic Fields

9. Pulsating Laminar Motion of a Viscous Fluid through a Cylindrical Tube of Circular Profile

10. Dissipation of Mechanical Energy in the Flow of a Viscous Fluid: Variational Principle of Helmholtz

11. Diffusion of Vorticity

12. Diffusion of Heat and Mass in Incompressible Viscous Fluid Flows

11 Integration of the Navier-Stokes Equations: Linearized and Self-Similar Solutions

1. Flow Past a Sphere at Small Reynolds Numbers; The Stokes Equation and Its Generalizations

2. Spatial Viscous Fluid Flow between Close Parallel Planes. Hele-Shaw “Spectra”. The Darcy Law

3. Viscous Fluid Flow between Rotating Coaxial Cylinders. The Problem of Slider Motion Along the Plane Coated with Viscous Fluid

4. Hydrodynamic Theory of Bearing Lubrication. Plane Problem

5. Spatial Problem of the Hydrodynamic Theory of Lubrication of Bearings. Ball Bearings and Suspensions

6. Application of the Dimensionality Theory to Determination of the Structure of the Solution of the Navier-Stokes Equations. Self-Similar Solutions

12 Laminar Boundary Layer in Incompressible Fluid

1. Interaction between Convection and Diffusion in an Incompressible Viscous Fluid Flow. Laminar Boundary Layer

2. Derivation of the Prandtl Equation for Viscous Fluid Flow in a Laminar Boundary Layer. Separation Phenomenon

3. Different Forms of the Prandtl Equation. Mises and Crocco Equations

4. Simplest Self-Similar Solutions of Laminar Boundary Layer Equations. Boundary Layer on a Plate in a Longitudinal Flow

5. Two-dimensional “Free” Boundary Layers: Far Laminar Wake, “Submerged” Jet Issuing from the Point Source

6. Problem on a Two-dimensional "Near-Wall" Jet

7. General Case of Exact Asymptotic Solutions to the Equations of a Stationary Two-Dimensional Near-Wall Boundary Layer

8. Approximate Methods for Calculating Laminar Boundary Layers. von Karman Integral Relation. Pohlhausen's and Kocbin-Loitsyanskiy's Methods

9. Generalization of Affine Similarity to the Case of Nonself-Similar Flows in Boundary Layers. Method of Generalized Similarity

10. Examples of the Application of the Method of Generalized Similarity

11. Spatial Near-Wall Boundary Layers. Free Spatial Jets

12. Plane Nonstationary Boundary Layer

13. Temperature and Concentration Boundary Layers in Incompressible Fluid

13 Turbulent Flows of Incompressible Viscous Fluid

1. Instability of Laminar Flows and Occurrence of Turbulence

2. Transient Phenomena in a Boundary Layer. Resistance Crisis of Blunt Bodies

3. The Reynolds Equations of the Averaged Turbulent Motion

4. Some Data on the Inner Structure of Turbulent Hows

5. Transfer of Momentum, Heat, and Admixtures in Turbulent Flows. Hypotheses of Boussinesq, Fourier, and Fick

6. The Prandtl "Mixing Length" Theory

7. “Free” Turbulence. Submerged Jets. Far Wake

8. Two-Layer Scheme of “Near-Wall” Turbulence. Logarithmic Velocity Profile

9. Logarithmic and Power-Law Formulas for the Velocity Profiles and the Resistance of Smooth Tubes

10. Empirical Method for Calculating a Turbulent Boundary Layer on a Plate with Smooth and Rough Surfaces in a Longitudinal Flow

11. Empirical Method for Calculating Turbulent Boundary Layer with Arbitrary Velocity Distribution on the Outer Edge

12. Semiempirical Method for Calculating Turbulent Boundary Layer on a Plate in a Longitudinal How

13. Integral Methods for Calculation of Turbulent Boundary Layers

14. Drag of a Body Moving in Fluid. Profile Drag, Friction Drag, Pressure Drag

15. Approximate Formulae for the Profile Drag

14 Some Modem Methods for Calculation of a Turbulent Boundary Layer in an Incompressible Fluid

1. “Moment” Methods of the Turbulent Boundary Layer Theory

2. Method of the Moments of the First Order. Two-Layer Procedure

3. “Near-Wall” Subregion of a Turbulent Boundary Layer. Interaction between Molecular and Turbulent Viscosities

4. Heat and Mass Transfer in a “Near-Wall” Subregion

5. “Outer” Subregion of a Turbulent Boundary Layer. Clauser Hypothesis

6. Results of Numerical Calculations of Turbulent Boundary Layers by the Method of First Order Moments

7. Review of the Application of Second Order Models. Models “k − ε”, “u'v' − k − ε”

8. Relaxation Phenomena in Turbulent Boundary Layers. Hinze Relaxation Equation

15 Dynamics of a Viscous Gas

1. Basic Equations of Viscous Gas Flows

2. Conditions for the Similarity of Viscous Gas Rows

3. The Laminar Boundary Layer in High-Speed Gas Flows

4. The Laminar Boundary Layer on a Plate in a Parallel High Speed Gas Flow

5. Laminar Boundary Layer on a Cone Parallel to a Supersonic Flow

6. The Laminar Boundary Layer in High-Speed Row with External Pressure Gradients

7. Modification of the Laminar Boundary Layer Equations to Equations for an Incompressible Fluid

8. The Method of Generalized Similarity in the Theory of a Laminar Boundary Layer in a High Speed Gas Flow

9. Laminar Boundary Layer in a Supersonic Flow of Reacting Gases

10. Interaction between a Laminar Boundary Layer and an External Nonviscous Supersonic Flow

11. Turbulent Boundary Layer on a Plate in a Gas Flow Parallel to It

12. Turbulent Boundary Layer in a Gas Flow in the Presence of the Longitudinal Pressure Gradient

16 Some Numerical Methods for Solving Equations of Hydrogas Dynamics

1. Introduction: Basic Concepts of Theory of Difference Schemes

1.1. Concerning the Construction of Difference Grids

1.2. Difference Approximation

1.3. Convergence, Approximation Stability

1.4. Amplitude and Phase Errors.

1.5. Differential (Approximation of the Difference Scheme) Scheme Viscosity

1.8. Explicit and Implicit Schemes

1.9. Schemes Based on the Time-Dependent Technique

2. Methods for Solving the Euler Equations

2.1. Some Specific Features of Numerical Simulation of Inviscid Gas flows

2.3. Method of Large Particles

2.4. The MonotOnized McCormack Scheme

3. Methods for Solving the Boundary Layer Equations

3.1. Brailovskaya and Chudov Scheme

3.2. Compact Integro-Interpolation Scheme

3.3. Difference Schemes with Exponential Operator

4. Methods for Solving the Navier-Stokes Equations of Incompressible Fluid

4.1. An Implicit Scheme of Alternate Directions

4.2. Explicit Scheme of Second Order of Accuracy in Terms of Time

4.3. Problems of the Determination of the Pressure Field on the Example

4.4. Implicit Time-Dependent Scheme

4.5. Calculation of Viscous Incompressible Fluid Flow in a Rectangular Cavity with the Upper Wall Moving in its Plane with the Velocity V0 (Figure 317)

5. Methods for Solving the Navier-Stokes Equations for a Gas

5.1. Implicit Factorized Scheme

5.2. Scheme of Constant Direction