### Mathematical Modeling in Science and Engineering. An Axiomatic Approach

- Language: English
- 264 Pages
- Published: March 2012

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- ID: 2183527
- February 2011
- 518 Pages
- John Wiley and Sons Ltd

Written by an experienced author with a strong background in applications of this field, this monograph provides a comprehensive and detailed account of the theory behind hydromechanics. He includes numerous appendices with mathematical tools, backed by extensive illustrations. The result is a must-have for all those needing to apply the methods in their research, be it in industry or academia.

Dedication V

Preface XIII

List of Symbols XVII

1 Introduction 1

1.1 Goals and Methods of Continuum Mechanics 1

1.2 The Main Hypotheses of ContinuumMechanics 3

2 Kinematics of the Deformed Continuum 5

2.1 Dynamics of the Continuum in the Lagrangian Perspective 5

2.2 Dynamics of the Continuum in the Eulerian Perspective 8

2.3 Scalar and Vector Fields and Their Characteristics 8

2.4 Theory of Strains 13

2.5 The Tensor of Strain Velocities 24

2.6 The Distribution of Velocities in an Infinitesimal Continuum Particle 25

2.7 Properties of Vector Fields. Theorems of Stokes and Gauss 30

3 Dynamic Equations of Continuum Mechanics 39

3.1 Equation of Continuity 39

3.2 Equations of Motion 43

3.3 Equation of Motion for the AngularMomentum 51

4 Closed Systems of Mechanical Equations for the Simplest Continuum Models 55

4.1 Ideal Fluid and Gas 55

4.2 Linear Elastic Body and Linear Viscous Fluid 58

4.3 Equations in Curvilinear Coordinates 63

4.3.1 Equation of Continuity 64

4.3.2 Equation of Motion 65

4.3.3 Gradient of a Scalar Function 66

4.3.4 Laplace Operator 66

4.3.5 Complete System of Equations of Motion for a Viscous, Incompressible Medium in the Absence of Heating 67

5 Foundations and Main Equations of Thermodynamics 69

5.1 Theorem of the Living Forces 69

5.2 Law of Conservation of Energy and First Law of Thermodynamics 72

5.3 Thermodynamic Equilibrium, Reversible and Irreversible Processes 76

5.4 Two Parameter Media and Ideal Gas 77

5.5 The Second Law of Thermodynamics and the Concept of Entropy 80

5.6 Thermodynamic Potentials of Two-Parameter Media 83

5.7 Examples of Ideal and Viscous Media, and Their Thermodynamic Properties, Heat Conduction 86

5.7.1 The Model of the Ideal, Incompressible Fluid 87

5.7.2 The Model of the Ideal, Compressible Gas 88

5.7.3 The Model of Viscous Fluid 90

5.8 First and Second Law of Thermodynamics for a Finite Continuum Volume 93

5.9 Generalized Thermodynamic Forces and Currents, Onsager’s Reciprocity Relations 94

6 Problems Posed in Continuum Mechanics 97

6.1 Initial Conditions and Boundary Conditions 97

6.2 Typical Simplifications for Some Problems 101

6.3 Conditions on the Discontinuity Surfaces 105

6.4 Discontinuity Surfaces in Ideal Compressible Media 111

6.5 Dimensions of Physical Quantities 118

6.6 Parameters that Determine the Class of the Phenomenon 120

6.7 Similarity and Modeling of Phenomena 127

7 Hydrostatics 131

7.1 Equilibrium Equations 131

7.2 Equilibrium in the Gravitational Field 132

7.3 Force and Moment that Act on a Body from the Surrounding Fluid 133

7.4 Equilibrium of a Fluid Relative to a Moving System of Coordinates 135

8 Stationary Continuum Movement of an Ideal Fluid 137

8.1 Bernoulli’s Integral 137

8.2 Examples of the Application of Bernoulli’s Integral 139

8.3 Dynamic and Hydrostatic Pressure 141

8.4 Flow of an Incompressible Fluid in a Tube of Varying Cross Section 142

8.5 The Phenomenon of Cavitation 143

8.6 Bernoulli’s Integral for Adiabatic Flows of an Ideal Gas 144

8.7 Bernoulli’s Integral for the Flow of a Compressible Gas 147

9 Application of the Integral Relations on Finite Volumes 151

9.1 Integral Relations 151

9.2 Interaction of Fluids and Gases with Bodies Immersed in the Flow 153

10 Potential Flows for Incompressible Fluids 159

10.1 The Cauchy–Lagrange Integral 160

10.2 Some Applications for the General Theory of Potential Flows 161

10.3 Potential Movements for an Incompressible Fluid 163

10.4 Movement of a Sphere in the Unlimited Volume of an Ideal, Incompressible Fluid 171

10.5 Kinematic Problem of the Movement of a Solid Body in the Unlimited Volume of an Incompressible Fluid 176

10.6 Energy, Movement Parameters and Moments of Movement Parameters for a Fluid during the Movement of a Solid Body in the Fluid 177

11 Stationary Potential Flows of an Incompressible Fluid in the Plane 181

11.1 Method of Complex Variables 181

11.2 Examples of Potential Flows in the Plane 183

11.3 Application of the Method of Conformal Mapping to the Solution of Potential Flows around a Body 192

11.4 Examples of the Application of the Method of Conformal Mapping 195

11.5 Main Moment and Main Vector of the Pressure Force Exerted on a Hydrofoil Profile 199

12 Movement of an Ideal Compressible Gas 203

12.1 Movement of an Ideal Gas Under Small Perturbations 203

12.2 Propagation of Waves with Finite Amplitude 207

12.3 Plane Vortex-Free Flow of an Ideal Compressible Gas 211

12.4 Subsonic Flow around a Thin Profile 215

12.5 Supersonic Flow around a Thin Profile 216

13 Dynamics of the Viscous Incompressible Fluid 219

13.1 Rheological Laws of the Viscous Incompressible Fluid 219

13.2 Equations of the Newtonian Viscous Fluid and Similarity Numbers 221

13.3 Integral Formulation for the Effect of Viscous Fluids on a Moving Body 223

13.4 Stationary Flow of a Viscous Incompressible Fluid in a Tube 226

13.5 Oscillating Laminar Flow of a Viscous Fluid through a Tube 231

13.6 Simplification of the Navier–Stokes Equations 233

14 Flow of a Viscous Incompressible Fluid for Small Reynolds Numbers 237

14.1 General Properties of Stokes Flows 237

14.2 Flow of a Viscous Fluid around a Sphere 240

14.3 Creeping Spatial Flow of a Viscous Incompressible Fluid 247

15 The Laminar Boundary Layer 251

15.1 Equation of Motion for the Fluid in the Boundary Layer 251

15.2 Asymptotic Boundary Layer on a Plate 255

15.3 Problem of the Injected Beam 257

16 Turbulent Flow of Fluid 263

16.1 General Information on Laminar and Turbulent Flows 263

16.2 Momentum Equation of a Viscous Incompressible Fluid 264

16.3 Equations of Heat Inflow, Heat Conduction and Diffusion 267

16.4 The Condition for the Beginning of Turbulence 269

16.5 Hydrodynamic Instability 270

16.6 The Reynolds Equations 272

16.7 The Equation of Turbulent Energy Balance 277

16.8 Isotropic Turbulence 281

16.9 The Local Structure of Fully Developed Turbulence 291

16.10 Models of Turbulent Flow 301

16.10.1 Semi-empirical Theories of Turbulence 302

16.10.2 The Use of Transport Equations 308

References 312

Appendix A Foundations of Vectorial and Tensorial Analysis 315

A.1 Vectors 316

A.2 Tensors 325

A.3 Curvilinear Systems of Coordinates and Physical Components 338

A.4 Calculation of Lengths, Surface Areas and Volumes 341

A.5 Differential Operators and Integral Theorems 344

Appendix B Some Differential Geometry 349

B.1 Curves on a Plane 349

B.2 Vectorial Definition of Curves 350

B.3 Curvature of a Curve in the Plane 353

B.4 Curves in Space 355

B.5 Curvature of Spatial Curves 358

B.6 Surfaces in Space 360

B.7 Fundamental Forms of the Surface 363

B.8 Curvature of a Curve on the Surface 367

B.9 Internal Geometry of a Surface 371

B.10 Surface Vectors 376

B.11 Geodetic Lines on a Surface 379

B.12 Vector Fields on the Surface 384

B.13 Hybrid Tensors 386

Appendix C Foundations of Probability Theory 389

C.1 Events and Set of Events 389

C.2 Probability 390

C.3 Common and Conditional Probability, Independent Events 391

C.4 Random Variables 392

C.5 Distribution of Probability Density and Mean Values 393

C.6 Generalized Functions 394

C.7 Methods of Averaging 396

C.8 Characteristic Function 398

C.9 Moments and Cumulants of Random Quantities 400

C.10 Correlation Functions 402

C.11 Poisson, Bernoulli and Gaussian Distributions 404

C.12 Stationary Random Functions and Homogeneous Random Fields 408

C.13 Isotropic Random Fields 410

C.14 Stochastic Processes, Markovian Processes and Chapman–Kolmogorov Integral Equation 412

C.15 Differential Equations of Chapman–Kolmogorov et al. 415

C.16 Stochastic Differential Equations and the Langevin Equation 427

Appendix D Basics of Complex Analysis 433

D.1 Complex Numbers 433

D.1.1 Operations with Complex Numbers 433

D.1.2 Geometrical Interpretation of Complex Numbers 434

D.2 Complex Variables 436

D.2.1 Geometrical Notions 436

D.2.2 Functions of a Complex Variable 437

D.2.3 Differentiation and Analyticity of Complex Functions 438

D.3 Elementary Functions 439

D.3.1 Functions 439

D.3.2 Joukowski Function 442

D.4 Integration of Complex Variable Functions 443

D.4.1 Integral of Complex Variable Functions 443

D.4.2 Some Theorems of Integral Calculus in Simply Connected Regions 444

D.4.3 Extension of Integral Calculus to Multiply Connected Regions 446

D.4.4 Cauchy Formula 448

D.5 Representation of a Function as a Series 450

D.5.1 Taylor Series 450

D.5.2 Laurent Series 450

D.6 Singular Points 452

D.6.1 Theorem about Residues 453

D.6.2 Infinitely Remote Point 456

D.7 Conformal Transformations 458

D.7.1 Notion of Conformal Transformation 458

D.7.2 Main Problem 461

D.7.3 Correspondence of Boundaries 462

D.7.4 Linear Fractional Function 462

D.7.5 Particular Cases 464

D.8 Application of the Theory of Complex Variables to Boundary-Value Problems 467

D.8.1 Harmonic Functions 467

D.8.2 Dirichlet Problem 468

D.9 Physical Representations and Formulation of Problems 470

D.9.1 Plane Field and Complex Potential 470

D.9.2 Examples of Plane Fields 474

References to Appendix 481

Index 483

Emmanuil G. Sinaiski Emmanreceived a doctorate in petroleum engineering from Gubkin-State University of Oil&Gas, Moscow, Russia, where he was later appointed a full professor. He has published numerous books and scientific articles. Professor Sinaiski's fields of interests are applied mathematics, fluid mechanics, physicochemical hydrodynamics, chemical and petroleum engineering.

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