Applied Dimensional Analysis and Modeling (eBook)

Front Cover 1
Applied Dimensional Analysis and Modeling 4
Copyright Page 5
Contents 8
List of Titled Examples and Problems 14
Foreword to the First Edition 20
Foreword to the Second Edition 22
Acknowledgments 24
Preface to the First Edition 26
Preface to the Second Edition 28
Organization, Notation, and Conventions 30
Chapter 1. Mathematical Preliminaries 34
1.1. Matrices and Determinants 34
1.2. Operations with Matrices 39
1.3. The Rank of a Matrix 47
1.4. Systems of Linear Equations 52
1.5. List of Selected Publications Dealing with Linear Algebra and Matrices 59
Chapter 2. Formats and Classification 60
2.1. Formats for Physical Relations 60
2.2. Classification of Physical Quantities 65
Chapter 3. Dimensional Systems 70
3.1. General Statements 70
3.2. Classification 75
3.3. The SI 76
3.4. Other Than SI Dimensional Systems 99
3.5. A Note on the Classification of Dimensional Systems 101
Chapter 4. Transformation of Dimensions 102
4.1. Numerical Equivalences 102
4.2. Technique 106
4.3. Examples 107
4.4. Problems 124
Chapter 5. Arithmetic of Dimensions 128
Chapter 6. Dimensional Homogeneity 132
6.1. Equations 132
6.2. Graphs 143
6.3. Problems 159
Chapter 7. Structure of Physical Relations 166
7.1. Monomial Power Form 166
7.2. The Dimensional Matrix 167
7.3. Generating Products of Variables of Desired Dimension 168
7.4. Number of Independent Sets of Products of Given Dimension (I) 172
7.5. Completeness of the Set of Products of Variables 176
7.6. Special Case: Matrix A is Singular 177
7.7. Number of Independent Sets of Products of Given Dimension (II) Buckingham’s Theorem
7.8. Selectable and Nonselectable Dimensions in a Product of Variables 184
7.9. Minimum Number of Independent Products of Variables of Given Dimension 185
7.10. Constancy of the Sole Dimensionless Product 186
7.11. Number of Dimensions Equals or Exceeds the Number of Variables 190
7.12. Problems 193
Chapter 8. Systematic Determination of Complete Set of Products of Variables 196
8.1. Dimensional Set Derivation of Products of Variables of a Given Dimension
8.2. Checking the Results 204
8.3. The Fundamental Formula 209
Chapter 9. Transformations 214
9.1. Theorems Related to Some Specific Transformations 214
9.2. Transformation Between Systems of Different Matrics 235
9.3. Transformation Between Dimensional Sets 244
9.4. Independence of Dimensionless Products of the Dimensional System Used 258
Chapter 10. Number of Sets of Dimensionless Products of Variables 262
10.1. Distinct and Equivalent Sets 262
10.2. Changes in a Dimensional Set Not Affecting the Dimensionless Variables 264
10.3. Prohibited Changes in a Dimensional Set 271
10.4. Number of Distinct Sets 277
10.5. Exceptions 283
10.6. Problems 292
Chapter 11. Relevancy of Variables 296
11.1. Dimensional Irrelevancy 296
11.2. Physical Irrelevancy 307
11.3. Problems 345
Chapter 12. Economy of Graphical Presentation 350
12.1. Number of Curves and Charts 350
12.2. Problems 362
Chapter 13. Forms of Dimensionless Relations 366
13.1. General Classification 366
13.2. Monomial is Mandatory 368
13.3. Monomial is Impossible—Proven 371
13.4. Monomial is Impossible— Not Proven 381
13.5. Reconstructions 386
13.6. Problems 406
Chapter 14. Sequence of Variables in the Dimensional Set 414
14.1. Dimensionless Physical Variable is Present 414
14.2. Physical Variables of Identical Dimensions are Present 418
14.3. Independent and Dependent Variables 422
14.4. Problems 432
Chapter 15. Alternate Dimensions 434
Chapter 16. Methods of Reducing the Number of Dimensionless Variables 446
16.1. Reduction of the Number of Physical Variables 447
16.2. Fusion of Dimensionless Variables 460
16.3. Increasing the Number of Dimensions 466
16.4. Problems 493
Chapter 17. Dimensional Modeling 496
17.1. Introductory Remarks 496
17.2. Homology 500
17.3. Specific Similarities 501
17.4. Dimensional Similarity 505
17.5. Scale Effects 544
17.6. Problems 556
Chapter 18. Fifty-two Additional Applications 560
References: 692
Numerical Order 692
Alphabetical Order of Authors’ Surnames 700
Appendices: 708
Appendix 1: Recommended Names and Symbols for Some Physical Quantities 710
Appendix 2: Some More-Important Physical Constants 714
Appendix 3: Some More-Important Named Dimensionless Variables 716
Appendix 4: Notes Attached to Figures 726
Appendix 5: Acronyms 754
Appendix 6: Solutions of Problems 756
Appendix 7: Proofs of Selected Theorems and Equations 830
Appendix 8: Blank Modeling Data Table 836
Indixes 838
Subject Index 838
Surname Index 850