Basics of Fluid Mechanics Genick Bar–Meir, Ph. D. 7449 North Washtenaw Ave Chicago, IL 60645 email:genick at potto.org Copyright 2013, 2011, 2010, 2009, 2008, 2007, and 2006 by Genick Bar-Meir See the file copying.fdl or copyright.tex for copying conditions. Version (0.3.4.0 July 25, 2013) How to cite this book: Bar-Meir, Genick, “Basics of Fluid Mechanics”, {last modified or Accessed}: insert the date and version you are using, www.potto.org/downloads.php Example: If you are using the latest version Bar-Meir, Genick, “Basics of Fluid Mechanics”, Last modified: Version 0.3.4.0 March 17, 2013, www.potto.org/downloads.php If you are using older version Bar-Meir, Genick, “Basics of Fluid Mechanics”, Accessed: Version 0.3.0.0 Nov 17, 2010, www.potto.org/downloads.php iii ‘We are like dwarfs sitting on the shoulders of giants” from The Metalogicon by John in 1159 iv CONTENTS Nomenclature xxiii GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . . xxxiii 1. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . xxxiv 2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . xxxv 3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . xxxv 4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi 5. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . xxxviii 6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . xxxviii 7. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . . xxxix 8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix 9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix 10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . xxxix ADDENDUM: How to use this License for your documents . . . . . . . xl How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . xli Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli Steven from artofproblemsolving.com . . . . . . . . . . . . . . . . . . xli Dan H. Olson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Richard Hackbarth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii John Herbolenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Eliezer Bar-Meir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Henry Schoumertate . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii Typo corrections and other ”minor” contributions . . . . . . . . . . . . xliii Version 0.3.2.0 March 18, 2013 . . . . . . . . . . . . . . . . . . . . . . . . . liii pages 617 size 4.8M . . . . . . . . . . . . . . . . . . . . . . . . . . . . liii Version 0.3.0.5 March 1, 2011 . . . . . . . . . . . . . . . . . . . . . . . . . liii v vi CONTENTS pages 400 size 3.5M . . . . . . . . . . . . . . . . . . . . . . . . . . . . liii Version 0.1.8 August 6, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . liv pages 189 size 2.6M . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv Version 0.1 April 22, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv pages 151 size 1.3M . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxi Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxi 1 Intro duction to Fluid Mechanics 1 1.1 What is Fluid Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Kinds of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 ViscosityViscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 Non–Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . 10 1.5.3 Kinematic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.4 Estimation of The Viscosity . . . . . . . . . . . . . . . . . . . . 12 1.6 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.1 Fluid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6.2 Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.1 Wetting of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Review of Thermodynamics 45 2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Review of Mechanics 53 3.1 Kinematics of of Point Body . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 Actual Center of Mass . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Aproximate Center of Area . . . . . . . . . . . . . . . . . . . . 56 3.3 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1 Moment of Inertia for Mass . . . . . . . . . . . . . . . . . . . . 56 3.3.2 Moment of Inertia for Area . . . . . . . . . . . . . . . . . . . . 57 3.3.3 Examples of Moment of Inertia . . . . . . . . . . . . . . . . . . 59 3.3.4 Product of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.5 Principal Axes of Inertia . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Angular Momentum and Torque . . . . . . . . . . . . . . . . . . . . . 65 3.5.1 Tables of geometries . . . . . . . . . . . . . . . . . . . . . . . 66 CONTENTS vii 4 Fluids Statics 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 The Hydrostatic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Pressure and Density in a Gravitational Field . . . . . . . . . . . . . . . 71 4.3.1 Constant Density in Gravitational Field . . . . . . . . . . . . . . 71 4.3.2 Pressure Measurement . . . . . . . . . . . . . . . . . . . . . . 75 4.3.3 Varying Density in a Gravity Field . . . . . . . . . . . . . . . . 79 4.3.4 The Pressure Effects Due To Temperature Variations . . . . . . 86 4.3.5 Gravity Variations Effects on Pressure and Density . . . . . . . 90 4.3.6 Liquid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4 Fluid in a Accelerated System . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.1 Fluid in a Linearly Accelerated System . . . . . . . . . . . . . . 93 4.4.2 Angular Acceleration Systems: Constant Density . . . . . . . . 95 4.4.3 Fluid Statics in Geological System . . . . . . . . . . . . . . . . 97 4.5 Fluid Forces on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5.1 Fluid Forces on Straight Surfaces . . . . . . . . . . . . . . . . . 100 4.5.2 Forces on Curved Surfaces . . . . . . . . . . . . . . . . . . . . 109 4.6 Buoyancy and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.7 Rayleigh–Taylor Instability . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.8 Qualitative questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 I Integral Analysis 145 5 Mass Conservation 147 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.2 Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.3 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3.1 Non Deformable Control Volume . . . . . . . . . . . . . . . . . 151 5.3.2 Constant Density Fluids . . . . . . . . . . . . . . . . . . . . . . 151 5.4 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . 158 5.5 Examples For Mass Conservation . . . . . . . . . . . . . . . . . . . . . 160 5.6 The Details Picture – Velocity Area Relationship . . . . . . . . . . . . 166 5.7 More Examples for Mass Conservation . . . . . . . . . . . . . . . . . . 169 6 Momentum Conservation 173 6.1 Momentum Governing Equation . . . . . . . . . . . . . . . . . . . . . 173 6.1.1 Introduction to Continuous . . . . . . . . . . . . . . . . . . . . 173 6.1.2 External Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.1.3 Momentum Governing Equation . . . . . . . . . . . . . . . . . 175 6.1.4 Momentum Equation in Acceleration System . . . . . . . . . . 175 6.1.5 Momentum For Steady State and Uniform Flow . . . . . . . . . 176 6.2 Momentum Equation Application . . . . . . . . . . . . . . . . . . . . . 180 viii CONTENTS 6.2.1 Momentum for Unsteady State and Uniform Flow . . . . . . . . 183 6.2.2 Momentum Application to Unsteady State . . . . . . . . . . . . 183 6.3 Conservation Moment Of Momentum . . . . . . . . . . . . . . . . . . 190 6.4 More Examples on Momentum Conservation . . . . . . . . . . . . . . . 192 6.4.1 Qualitative Questions . . . . . . . . . . . . . . . . . . . . . . . 194 7 Energy Conservation 197 7.1 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . 197 7.2 Limitation of Integral Approach . . . . . . . . . . . . . . . . . . . . . . 209 7.3 Approximation of Energy Equation . . . . . . . . . . . . . . . . . . . . 211 7.3.1 Energy Equation in Steady State . . . . . . . . . . . . . . . . . 211 7.3.2 Energy Equation in Frictionless Flow and Steady State . . . . . 212 7.4 Energy Equation in Accelerated System . . . . . . . . . . . . . . . . . 213 7.4.1 Energy in Linear Acceleration Coordinate . . . . . . . . . . . . 213 7.4.2 Linear Accelerated System . . . . . . . . . . . . . . . . . . . . 214 7.4.3 Energy Equation in Rotating Coordinate System . . . . . . . . . 215 7.4.4 Simplified Energy Equation in Accelerated Coordinate . . . . . . 216 7.4.5 Energy Losses in Incompressible Flow . . . . . . . . . . . . . . 216 7.5 Examples of Integral Energy Conservation . . . . . . . . . . . . . . . . 218 II Differential Analysis 225 8 Differential Analysis 227 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.2 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.2.1 Mass Conservation Examples . . . . . . . . . . . . . . . . . . . 231 8.2.2 Simplified Continuity Equation . . . . . . . . . . . . . . . . . . 233 8.3 Conservation of General Quantity . . . . . . . . . . . . . . . . . . . . . 238 8.3.1 Generalization of Mathematical Approach for Derivations . . . . 238 8.3.2 Examples of Several Quantities . . . . . . . . . . . . . . . . . . 239 8.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.5 Derivations of the Momentum Equation . . . . . . . . . . . . . . . . . 244 8.6 Boundary Conditions and Driving Forces . . . . . . . . . . . . . . . . . 255 8.6.1 Boundary Conditions Categories . . . . . . . . . . . . . . . . . 255 8.7 Examples for Differential Equation (Navier-Stokes) . . . . . . . . . . . 259 8.7.1 Interfacial Instability . . . . . . . . . . . . . . . . . . . . . . . . 269 9 Dimensional Analysis 273 9.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 9.1.2 Theory Behind Dimensional Analysis . . . . . . . . . . . . . . . 275 9.1.3 Dimensional Parameters Application for Experimental Study . . 277 9.1.4 The Pendulum Class Problem . . . . . . . . . . . . . . . . . . . 278 9.2 Buckingham–π–Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 280 CONTENTS ix 9.2.1 Construction of the Dimensionless Parameters . . . . . . . . . . 281 9.2.2 Basic Units Blocks . . . . . . . . . . . . . . . . . . . . . . . . 282 9.2.3 Implementation of Construction of Dimensionless Parameters . . 285 9.2.4 Similarity and Similitude . . . . . . . . . . . . . . . . . . . . . 294 9.3 Nusselt’s Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.4 Summary of Dimensionless Numbers . . . . . . . . . . . . . . . . . . . 308 9.4.1 The Significance of these Dimensionless Numbers . . . . . . . . 312 9.4.2 Relationship Between Dimensionless Numbers . . . . . . . . . . 315 9.4.3 Examples for Dimensional Analysis . . . . . . . . . . . . . . . . 316 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.6 Appendix summary of Dimensionless Form of Navier–Stokes Equations . 319 10 Potential Flow 325 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 10.1.1 Inviscid Momentum Equations . . . . . . . . . . . . . . . . . . 326 10.2 Potential Flow Function . . . . . . . . . . . . . . . . . . . . . . . . . . 332 10.2.1 Streamline and Stream function . . . . . . . . . . . . . . . . . 333 10.2.2 Compressible Flow Stream Function . . . . . . . . . . . . . . . 336 10.2.3 The Connection Between the Stream Function and the Potential Function338 10.3 Potential Flow Functions Inventory . . . . . . . . . . . . . . . . . . . . 342 10.3.1 Flow Around a Circular Cylinder . . . . . . . . . . . . . . . . . 357 10.4 Conforming Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 10.4.1 Complex Potential and Complex Velocity . . . . . . . . . . . . 369 10.5 Unsteady State Bernoulli in Accelerated Coordinates . . . . . . . . . . 373 10.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 11 Compressible Flow One Dimensional 377 11.1 What is Compressible Flow? . . . . . . . . . . . . . . . . . . . . . . . 377 11.2 Why Compressible Flow is Important? . . . . . . . . . . . . . . . . . . 377 11.3 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 11.3.2 Speed of Sound in Ideal and Perfect Gases . . . . . . . . . . . . 380 11.3.3 Speed of Sound in Almost Incompressible Liquid . . . . . . . . . 381 11.3.4 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . 382 11.3.5 The Dimensional Effect of the Speed of Sound . . . . . . . . . 382 11.4 Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 11.4.1 Stagnation State for Ideal Gas Model . . . . . . . . . . . . . . 384 11.4.2 Isentropic Converging-Diverging Flow in Cross Section . . . . . 386 11.4.3 The Properties in the Adiabatic Nozzle . . . . . . . . . . . . . . 387 11.4.4 Isentropic Flow Examples . . . . . . . . . . . . . . . . . . . . . 391 11.4.5 Mass Flow Rate (Number) . . . . . . . . . . . . . . . . . . . . 394 11.4.6 Isentropic Tables . . . . . . . . . . . . . . . . . . . . . . . . . 401 11.4.7 The Impulse Function . . . . . . . . . . . . . . . . . . . . . . . 403 11.5 Normal Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 11.5.1 Solution of the Governing Equations . . . . . . . . . . . . . . . 408 x CONTENTS 11.5.2 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . . . . . . 411 11.5.3 Operating Equations and Analysis . . . . . . . . . . . . . . . . 413 11.5.4 The Moving Shocks . . . . . . . . . . . . . . . . . . . . . . . . 414 11.5.5 Shock or Wave Drag Result from a Moving Shock . . . . . . . . 416 11.5.6 Tables of Normal Shocks, k = 1.4 Ideal Gas . . . . . . . . . . . 418 11.6 Isothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 11.6.1 The Control Volume Analysis/Governing equations . . . . . . . 421 11.6.2 Dimensionless Representation . . . . . . . . . . . . . . . . . . 422 11.6.3 The Entrance Limitation of Supersonic Branch . . . . . . . . . 426 11.6.4 Supersonic Branch . . . . . . . . . . . . . . . . . . . . . . . . 428 11.6.5 Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . 429 11.6.6 Isothermal Flow Examples . . . . . . . . . . . . . . . . . . . . . 430 11.7 Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 11.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 11.7.2 Non–Dimensionalization of the Equations . . . . . . . . . . . . 438 11.7.3 The Mechanics and Why the Flow is Choked? . . . . . . . . . . 441 11.7.4 The Working Equations . . . . . . . . . . . . . . . . . . . . . . 442 11.7.5 Examples of Fanno Flow . . . . . . . . . . . . . . . . . . . . . 445 11.7.6 Working Conditions . . . . . . . . . . . . . . . . . . . . . . . . 451 11.7.7 The Pressure Ratio, P 2 / P 1 , effects . . . . . . . . . . . . . . . 456 11.7.8 Practical Examples for Subsonic Flow . . . . . . . . . . . . . . 463 11.7.9 Subsonic Fanno Flow for Given 4 f L D and Pressure Ratio . . . . 463 11.7.10 Subsonic Fanno Flow for a Given M 1 and Pressure Ratio . . . . 466 11.7.11 More Examples of Fanno Flow . . . . . . . . . . . . . . . . . . 468 11.8 The Table for Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . 469 11.9 Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 11.10Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 11.10.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 472 11.10.2 Rayleigh Flow Tables and Figures . . . . . . . . . . . . . . . . . 475 11.10.3 Examples For Rayleigh Flow . . . . . . . . . . . . . . . . . . . 478 12 Compressible Flow 2–Dimensional 485 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 12.1.1 Preface to Oblique Shock . . . . . . . . . . . . . . . . . . . . . 485 12.2 Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 12.2.1 Solution of Mach Angle . . . . . . . . . . . . . . . . . . . . . . 489 12.2.2 When No Oblique Shock Exist or the case of D > 0 . . . . . . 492 12.2.3 Application of Oblique Shock . . . . . . . . . . . . . . . . . . . 508 12.3 Prandtl-Meyer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 520 12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 12.3.2 Geometrical Explanation . . . . . . . . . . . . . . . . . . . . . 521 12.3.3 Alternative Approach to Governing Equations . . . . . . . . . . 522 12.3.4 Comparison And Limitations between the Two Approaches . . . 525 12.4 The Maximum Turning Angle . . . . . . . . . . . . . . . . . . . . . . . 526 [...]... xiv LIST OF FIGURES 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 Description of how the center of mass is calculated Thin body center of mass/area schematic The schematic that explains the summation of moment of inertia The schematic to explain the summation of moment of inertia Cylinder with an element for calculation moment of inertia Description of rectangular... rectangular in x–y plane A square element for the calculations of inertia The ratio of the moment of inertia 2D to 3D Moment of inertia for rectangular Description of parabola - moment of inertia and center of area Triangle for example 3.7 Product of inertia for triangle ... Schematic of floating cubic Stability analysis of floating body Cubic body dimensions for stability analysis Stability of cubic body infinity long The maximum height reverse as a function of density ratio Stability of two triangles put tougher The effects of liquid movement on the GM Measurement of GM of floating... 416 417 421 427 xviii LIST OF FIGURES 11.19Control volume of the gas flow in a constant cross section for Fanno 11.20Various parameters in fanno flow 11.21Schematic of Example 11.18 11.22The schematic of Example (11.19) f 11.23The effects of increase of 4 DL on the Fanno line f 11.24The effects of the increase of 4 DL on the Fanno Line... 603 LIST OF FIGURES 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 Diagram to explain fluid mechanics branches Density as a function of the size of sample Schematics to describe the shear stress in fluid mechanics The deformation of fluid due to shear stress The difference of power fluids ... discussion change of bulk modulus of mixture ˆ Addition of several examples ˆ Improve English in several chapters Version 0.3.0.1 Nov 12, 2010 (3.3 M 358 pages) ˆ Build the chapter log file for latex (macro) process Steven from www.artofproblemsolving.com LIST OF TABLES xxix ˆ Add discussion change of density on buck modulus calculations as example as integral equation ˆ Minimal discussion of converting... sound wave and gas moves relative to the pulse 11.3 Moving object at three relative velocities (a) Object travels at 0.005 of the speed of sound (b) Object travels at 0.05 of the speed of sound (c) Object travels at 0.15 of the speed of sound 11.4 Flow through a converging diverging nozzle 11.5 Perfect gas flows through a tube 11.7... updated version of the Document 4 MODIFICATIONS You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it In addition,... 5.8 5.9 5.10 5.11 5.12 Control volume and system in motion Piston control volume Schematics of velocities at the interface Schematics of flow in a pipe with varying density Filling of the bucket and choices of the control volumes Height of the liquid for example 5.4 Boundary Layer control mass Control volume usage to calculate local... a function of the 4f L ˙ D 11.26M1 as a function M2 for various 4f L D 11.27 M1 as a function M2 f 11.28 The pressure distribution as a function of 4 DL f 11.29Pressure as a function of long 4 DL 11.30 The effects of pressure variations on Mach number profile f f 11.31 Pressure ratios as a function of 4 DL when