## A First Course in Dimensional Analysis

by Santiago

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An introduction to dimensional analysis, a method of scientific analysis used to investigate and simplify complex physical phenomena, demonstrated through a series of engaging examples.

This book offers an introduction to dimensional analysis, a powerful method of scientific analysis used to investigate and simplify complex physical phenomena. The method enables bold approximations and the generation of testable hypotheses. The book explains these analyses through a series of entertaining applications; students will learn to analyze, for example, the limits of world-record weight lifters, the distance an electric submarine can travel, how an upside-down pendulum is similar to a running velociraptor, and the number of Olympic rowers required to double boat speed.

The book introduces the approach through easy-to-follow, step-by-step methods that show how to identify the essential variables describing a complex problem; explore the dimensions of the problem and recast it to reduce complexity; leverage physical insights and experimental observations to further reduce complexity; form testable scientific hypotheses; combine experiments and analysis to solve a problem; and collapse and present experimental measurements in a compact form. Each chapter ends with a summary and problems for students to solve. Taken together, the analyses and examples demonstrate the value of dimensional analysis and provide guidance on how to combine and enhance dimensional analysis with physical insights. The book can be used by undergraduate students in physics, engineering, chemistry, biology, sports science, and astronomy.

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Contents (pg. vii)
Preface (pg. xi)
1. A Brief History of Dimensional Analysis (pg. 1)
2. Preliminary Concepts (pg. 5)
2.1 Physical Quantities (pg. 5)
2.2 Dimensions of Terms (pg. 6)
2.3 Unit Conversion Factors (pg. 8)
2.4 Notation and Manipulation of Unknown Functions (pg. 9)
2.4.1 Notations for Unknown Functions (pg. 9)
2.4.2 Useful Concepts Involving Unknown Functions (pg. 11)
2.5 Principle of Dimensional Homogeneity (pg. 11)
2.5.1 Check for Errors in a Derivation (pg. 12)
2.5.2 Guessing a Function from the Dimensions of Its Variables: How Much of This Martini Have I Had? (pg. 13)
2.6 Approximations and Asymptotic Limits (pg. 14)
2.7 Summary (pg. 15)
Problems (pg. 16)
3. Some Basic Physical Principles (pg. 19)
3.1 Newton’s Second Law (pg. 19)
3.2 Conservation of Energy in Dynamics (pg. 20)
3.3 Basic Concepts of Fluid Flow (pg. 21)
3.4 Summary (pg. 24)
Problems (pg. 25)
4. Dimensional Analysis: Motivation and Introduction (pg. 27)
4.1 Geometric Similarity (pg. 28)
4.2 The Drawbacks of Brute Force Experimentation: We Need a Bigger Submarine (pg. 29)
4.3 Comments on the Submarine Example (pg. 31)
4.4 Drag Coefficient as a Tool to Reduce Complexity (pg. 32)
4.4.1 Effect of Streamlining (pg. 32)
4.4.2 Drag on a Smooth Sphere: A First Example of Data Collapse (pg. 33)
4.4.3 Drag and Terminal Velocity of a Skydiver (pg. 35)
4.5 Summary (pg. 38)
Problems (pg. 39)
5. Dimensional Analysis Techniques (pg. 41)
5.1 Rules of Thumb for Initial Hypothesized Function: What to Include or Exclude? (pg. 41)
5.1.1 Rule D1: Formulate in Terms of Known Algebraic Combinations (pg. 42)
5.1.2 Rule D2: Exclude a Variable Expressible as a Function of the Others (pg. 43)
5.1.3 Rule D3: Keep Dimensional Constants but Absorb Nondimensional Constants (pg. 43)
5.1.4 Rule D4: Exclude Any Variable That Involves a Unique Dimension (No Blood from a Rock) (pg. 44)
5.1.5 Rule D5: If Invoked, Consistently Leverage Geometric Similarity (pg. 44)
5.2 Ipsen’s Method: A Step-by-Step Process (pg. 45)
5.3 Submarine Example Revisited (pg. 47)
5.4 An Inelegant Application of Ipsen’s Method (pg. 49)
5.5 Time for a Stone to Drop: Experimental Closure and Collapse of Data (pg. 50)
5.5.1 Supplementing Dimensional Analysis with Experiments (pg. 52)
5.5.2 A Note on Guiding the Nondimensional Parameters (pg. 54)
5.6 From Stones and Earth to Planets and Stars (pg. 55)
5.7 Ignoring the Rule That We Exclude Determined Parameters: Including Both G and g (pg. 58)
5.8 Period of a Pendulum for Any Angle: Experimental Closure and Collapse of Data (pg. 59)
5.9 Summary (pg. 63)
Problems (pg. 64)
6. Combining Dimensional Analysis with Physical Intuition and Experimental Observations (pg. 67)
6.1 Rules of Thumb for Manipulating Functions of Nondimensional Variables (pg. 67)
6.1.1 Rule ND1: Reorganize Expressions of Nondimensional Variables (pg. 68)
6.1.2 Rule ND2: Isolate, Then Evaluate, Known Dependence (pg. 69)
6.1.3 Rule ND3: Isolate a Variable with an Unknown but Weak Dependence (pg. 71)
6.1.4 Rule ND4: Replace a Nested Function with Its Independent Nondimensional Parameters (pg. 72)
6.1.5 Rule ND5: Consider Absorbing an Approximately Constant Nondimensional Variable into Function (pg. 73)
6.1.6 Rule ND6: Be Careful Eliminating Variables—Even If They Are Small (pg. 73)
6.2 Spine Patterns of Liquid Drop Impacts and Blood Drop Patterns in Forensics (pg. 74)
6.3 Atomic Explosions: An Example Confirmation of a Power Law (pg. 78)
6.3.1 History of Atomic Explosion Analysis by G. I. Taylor (pg. 78)
6.3.2 Atomic Explosion Analysis Using Dimensional Analysis (pg. 80)
6.4 World-class Weightlifters: Lifter-to-Lifter Comparisons and Lift Data Collapse (pg. 82)
6.5 Running (from) Dinosaurs: An Analysis That Does Not Assume Geometric Similarity (pg. 84)
6.6 Summary (pg. 90)
Problems (pg. 91)
7. Two Examples Combining Biomechanics and Fluid Mechanics (pg. 93)
7.1 Olympic Rowers and a Word of Caution on the Experimental Validation of Scaling Laws (pg. 93)
7.2 A Derivation for the Great Flight Diagram: Approximate Data Collapse (pg. 99)
7.2.1 The Great Flight Diagram (pg. 103)
7.3 Justification for Excluding Flier Power from the Formulation (pg. 105)
7.4 Including Viscosity in the Flight Analysis (pg. 106)
7.5 Summary (pg. 108)
Problems (pg. 109)
8. Buckingham Pi Theorem: An Alternate Method (pg. 111)
8.1 Buckingham Pi Theorem (pg. 111)
8.2 Pressure Drop in a Pipe Explored Using the Buckingham Pi Theorem (pg. 112)
8.3 Use of a Large Number of Scaled Pressure Drop Measurements to Close the Problem (pg. 116)
8.4 Buckingham Pi Theorem versus Ipsen’s Method (pg. 119)
8.5 Summary (pg. 119)
Problems (pg. 120)
9. Leveraging of Model Data to Build and Understand Prototypes (pg. 121)
9.1 Geometric, Kinematic, and Dynamic Similarity (pg. 121)
9.2 Experimental Design and Interpretation: To Match All Variables, Match All but One (pg. 122)
9.3 Examples of Model and Prototype Studies (pg. 123)
9.3.1 Submarine Prototype Scaling (pg. 124)
9.3.2 Boat Drag and How Scale Model Studies Are Not Always Possible (pg. 125)
9.4 Summary (pg. 128)
Problems (pg. 129)
10. Small Changes in Geometry Can Have Significant Effects: The Effect of Roughness on Drag (pg. 131)
10.1 Roughness and the Drag of Spheres (pg. 131)
10.2 Pressure Drop in Smooth versus Rough Pipes (pg. 135)
10.3 Estimating Drag from Different Shapes (pg. 135)
10.4 Example of Incorrect Dimensional Analysis: Neglecting a Parameter (pg. 136)
10.5 Summary (pg. 138)
Problems (pg. 139)
11. The Riabouchinsky–Rayleigh Paradox and the Rule of Relevance (pg. 141)
11.1 The Riabouchinsky–Rayleigh Paradox (pg. 141)
11.2 Resolution of the Apparent Paradox (pg. 143)
11.3 The Rule of Relevance (pg. 144)
11.4 Summary of Rules of Thumb for Combining Physical Insight with Dimensional Analysis (pg. 145)
12. Common Dimensionless Groups (pg. 147)
12.1 Mach Number (pg. 147)
12.2 Euler Number (pg. 149)
12.3 Table of Dimensionless Groups (pg. 149)
12.4 Summary (pg. 150)
Problems (pg. 150)
13. Scaling Using Approximate Equations (pg. 151)
13.1 Summary (pg. 153)
Problems (pg. 154)
Closing Note (pg. 155)
Appendix A: Properties of Common Fluids (pg. 157)
References (pg. 159)
Index (pg. 163)

#### Juan G. Santiago Santiago

Juan G. Santiago is Professor of Mechanical Engineering at Stanford University and Director of the Stanford Microfluidics Laboratory. He is cofounder of several companies in the microfluidics area and is a Fellow of the American Physical Society, the American Society of Mechanical Engineering, and the American Institute for Medical and Biological Engineering.

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