Basic Fluid Mechanics Assignment

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Basics of Fluid Mechanics Genick Bar???Meir, Ph. D. 2729 West Jarvis Ave Chicago, IL 60645-1335 email:barmeir at gmail. com Copyright ?? 2010, 2009, 2008, 2007, and 2006 by Genick Bar-Meir See the ? le copying. fdl or copyright. tex for copying conditions. Version (0. 2. 4 March 2, 2010) ‘We are like dwarfs sitting on the shoulders of giants” from The Metalogicon by John in 1159 CONTENTS Nomenclature GNU Free Documentation License . . . . . . . . . . . . . . . . 1. APPLICABILITY AND DEFINITIONS . . . . . . . . . 2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . 3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . 4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . 5. COMBINING DOCUMENTS . . . . . . . . . . . . . . 6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . 7. AGGREGATION WITH INDEPENDENT WORKS . . . 8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . 9. TERMINATION . . . . . . . . . . . . . . . . . . . . . 10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . ADDENDUM: How to use this License for your documents How to contribute to this book . . . . . . . . . . . . . . . . . Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steven from artofproblemsolving. com . . . . . . . . . . . Dan H. Olson . . . . . . . . . . . . . . . . . . . . . . . Richard Hackbarth . . . . . . . . . . . . . . . . . . . . . . John Herbolenes . . . . . . . . . . . . . . . . . . . . . . . Eliezer Bar-Meir . . . . . . . . . . . . . . . . . . . . . . Henry Schoumertate . . . . . . . . . . . . . . . . . . . . . Your name here . . . . . . . . . . . . . . . . . . . . . . . Typo corrections and other “minor” contributions . . . . . Version 0. 1. 8 August 6, 2008 . . . . . . . . . . . . . . . . . . . pages 189 size 2. 6M . . . . . . . . . . . . . . . . . . . . . Version 0. 1 April 22, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii xix xx xxi xxi xxii xxiv xxiv xxv xxv xxv xxv xxvi xxvii xxvii xxvii xxviii xxviii xxviii xxviii xxviii xxviii xxix xxxix xxxix xxxix iii iv CONTENTS pages 151 size 1. 3M . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv 1 Introduction 1. 1 What is Fluid Mechanics? . . . . . 1. 2 Brief History . . . . . . . . . . . . 1. 3 Kinds of Fluids . . . . . . . . . . . 1. 4 Shear Stress . . . . . . . . . . . . 1. 5 Viscosity . . . . . . . . . . . . . . 1. 5. 1 General . . . . . . . . . . 1. 5. 2 Non???Newtonian Fluids . . 1. 5. 3 Kinematic Viscosity . . . . 1. 5. 4 Estimation of The Viscosity 1. 5. 5 Bulk Modulus . . . . . . . 1. 6 Surface Tension . . . . . . . . . . 1. 6. 1 Wetting of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 5 6 9 9 10 11 12 20 22 24 33 33 41 41 41 42 43 43 44 46 49 50 51 51 52 55 55 55 57 57 58 62 65 69 71 2 Review of Thermodynamics 2. 1 Basic De? nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Review of Mechanics 3. 1 Center of Mass . . . . . . . . . . . 3. 1. 1 Center of the Mass . . . . . . 3. 1. 2 Center of Area . . . . . . . . . 3. 2 Moment of Inertia . . . . . . . . . . . 3. 2. 1 Moment of Inertia for Mass . . 3. 2. 2 Moment of Inertia for Area . . 3. 2. 3 Examples of Moment of Inertia 3. 2. 4 Product of Inertia . . . . . . . 3. 2. 5 Principal Axes of Inertia . . . . 3. 3 Newton’s Laws of Motion . . . . . . . 3. 4 Angular Momentum and Torque . . . 3. 4. 1 Tables of geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fluids Statics 4. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 2 The Hydrostatic Equation . . . . . . . . . . . . . . . . . . . 4. 3 Pressure and Density in a Gravitational Field . . . . . . . . . 4. 3. 1 Constant Density in Gravitational Field . . . . . . . . 4. 3. Pressure Measurement . . . . . . . . . . . . . . . . 4. 3. 3 Varying Density in a Gravity Field . . . . . . . . . . 4. 3. 4 The Pressure E? ects Because Temperature Variations 4. 3. 5 Gravity Variations E? ects on Pressure and Density . 4. 3. 6 Liquid Phase . . . . . . . . . . . . . . . . . . . . . . CONTENTS 4. 4 Fluid in a Accelerated System . . . . . . . . . . 4. 4. 1 Fluid in a Linearly Accelerated System . 4. 4. 2 Angular Acceleration Systems: Constant Fluid Forces on Surfaces . . . . . . . . . . . . . 4. 5. 1 Fluid Forces on Straight Surfaces . . . . 4. 5. 2 Force on Curved Surfaces . . . . . . . . Buoyancy and Stability . . . . . . . . . . . . 4. 6. 1 Stability . . . . . . . . . . . . . . . . . 4. 6. 2 Surface Tension . . . . . . . . . . . . . Rayleigh???Taylor Instability . . . . . . . . . . . . . . . . . . . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . 72 . 72 . 74 . 76 . 76 . 86 . 92 . 100 . 110 . 110 4. 5 4. 6 4. 7 I Integral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 119 119 120 121 123 123 130 132 138 141 143 143 143 144 145 145 146 150 153 153 160 162 164 167 167 179 180 181 181 5 Mass Conservation 5. 1 Introduction . . . . . . . . . . . . . . . . . . . . 5. 2 Control Volume . . . . . . . . . . . . . . . . . . 5. 3 Continuity Equation . . . . . . . . . . . . . . . 5. 3. 1 Non Deformable Control Volume . . . . . 5. 3. 2 Constant Density Fluids . . . . . . . . . . 5. 4 Reynolds Transport Theorem . . . . . . . . . . . 5. 5 Examples For Mass Conservation . . . . . . . . . 5. 6 The Details Picture ??? Velocity Area Relationship 5. 7 More Examples for Mass Conservation . . . . . . 6 Momentum Conservation 6. 1 Momentum Governing Equation . . . . . . . . . . . . . 6. 1. 1 Introduction to Continuous . . . . . . . . . . . . 6. 1. 2 External Forces . . . . . . . . . . . . . . . . . . 6. 1. 3 The Momentum Governing Equation . . . . . . . 6. 1. 4 Momentum Equation in Acceleration System . . 6. 1. Momentum For Steady State and Uniform Flow . 6. 2 Momentum Equation Application . . . . . . . . . . . . . 6. 2. 1 Momentum for Unsteady State and Uniform Flow 6. 2. 2 Momentum Application to Unsteady State . . . . 6. 3 Conservation Moment Of Momentum . . . . . . . . . . 6. 4 More Examples on Momentum Conservation . . . . . . . 6. 4. 1 Qualitative Questions . . . . . . . . . . . . . . . 7 Energy Conservation 7. 1 The First Law of Thermodynamics . . . . . 7. 2 Limitation of Integral Approach . . . . . . . 7. 3 Approximation of Energy Equation . . . . . 7. 3. 1 Energy Equation in Steady State . . 7. 3. 2 Energy Equation in Frictionless Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Steady State vi 7. 4 Energy 7. 4. 1 7. 4. 2 7. 4. 3 7. 4. 4 CONTENTS Equation in Accelerated System . . . . . . . . . . . . . Energy in Linear Acceleration Coordinate . . . . . . . . Linear Accelerated System . . . . . . . . . . . . . . . . Energy Equation in Rotating Coordinate System . . . . . Energy Equation in Accelerated Coordinate with Uniform . . . . . . . . . . . . Flow . . . . 183 183 184 184 186 II Di? erential Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 189 189 189 190 191 192 193 197 198 201 202 204 205 206 208 209 211 212 219 221 221 221 221 222 223 225 225 226 228 228 230 8 Multi???Phase Flow 8. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8. 2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. 3 What to Expect From This Chapter . . . . . . . . . . . . . 8. 4 Kind of Multi-Phase Flow . . . . . . . . . . . . . . . . . . 8. 5 Classi? cation of Liquid-Liquid Flow Regimes . . . . . . . . . 8. 5. 1 Co???Current Flow . . . . . . . . . . . . . . . . . . . 8. 6 Multi???Phase Flow Variables De? nitions . . . . . . . . . . . . 8. 6. 1 Multi???Phase Averaged Variables De? nitions . . . . . 8. 7 Homogeneous Models . . . . . . . . . . . . . . . . . . . . . 8. 7. 1 Pressure Loss Components . . . . . . . . . . . . . . 8. 7. 2 Lockhart Martinelli Model . . . . . . . . . . . . . . 8. 8 Solid???Liquid Flow . . . . . . . . . . . . . . . . . . . . . . . 8. 8. 1 Solid Particles with Heavier Density ? S > ? L . . . . 8. 8. 2 Solid With Lighter Density ? S < ? nd With Gravity 8. 9 Counter???Current Flow . . . . . . . . . . . . . . . . . . . . . 8. 9. 1 Horizontal Counter???Current Flow . . . . . . . . . . . 8. 9. 2 Flooding and Reversal Flow . . . . . . . . . . . . . . 8. 10 Multi???Phase Conclusion . . . . . . . . . . . . . . . . . . . . A Mathematics For Fluid Mechanics A. 1 Ordinary Di? erential Equations . . . . . . . A. 1. 1 First Order Di? erential Equations . . A. 1. 2 Variable Separation . . . . . . . . . A. 1. 3 The Integral Factor . . . . . . . . . A. 1. 4 Non???Linear Equation . . . . . . . . A. 1. 5 Second Order Di? erential equations A. 1. 6 Non-Homogeneous Equation . . . . A. 1. Non-Linear Second Order Equation . A. 2 Vectors . . . . . . . . . . . . . . . . . . . . A. 2. 1 Vector Algebra . . . . . . . . . . . . A. 3 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index 233 Subjects Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 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. 2 1. 13 1. 14 1. 15 1. 16 1. 17 1. 18 3. 1 3. 2 3. 3 3. 4 3. 5 3. 6 3. 7 3. 8 Diagram to explain part of relationships of ? uid mechanics branches. Density as a function of the size of sample. . . . . . . . . . . . . . Schematics to describe the shear stress in ? uid mechanics. . . . . . The deformation of ? uid due to shear stress. . . . . . . . . . . . . . The di? erence of power ? uids. . . . . . . . . . . . . . . . . . . . . Nitrogen and Argon viscosity. . . . . . . . . . . . . . . . . . . . . The shear stress as a function of the shear rate. . . . . . . . . . . . Air viscosity as a function of the temperature. . . . . . . . . . . . Water viscosity as a function temperature. . . . . . . . . . . . . . . Liquid metals viscosity as a function of the temperature. . . . . . . Reduced viscosity as function of the reduced temperature. . . . . . Reduced viscosity as function of the reduced temperature. . . . . . Surface Tension control volume analysis. . . . . . . . . . . . . . . . Forces in Contact angle. . . . . . . . . . . . . . . . . . . . . . . . . Description of wetting and non???wetting ? uids. . . . . . . . . . . . . Description of liquid surface. . . . . . . . . . . . . . . . . . . . . . The raising height as a function of the radii. . . . . . . . . . . . . The raising height as a function of the radius. . . . . . . . . . . . . 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 the element for calculation moment of inertia. . . . Description of rectangular in x???y plane. . . . . . . . . . . . . . . A square element for the calculations of inertia. . . . . . . . . . . The ratio of the moment of inertia 2D to 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 7 9 10 10 11 12 15 17 18 21 24 24 26 29 29 42 42 44 45 45 46 47 47 vii viii LIST OF FIGURES 3. 9 Description of parabola – moment of inertia and center of area. . . . . . 3. 10 Product of inertia for triangle . . . . . . . . . . . . . . . . . . . . . . . 4. 1 4. 2 4. 3 4. 4 4. 5 4. 6 4. 7 4. 8 4. 9 4. 10 4. 11 4. 12 4. 13 4. 14 4. 15 4. 16 4. 17 4. 18 4. 19 4. 20 4. 21 4. 22 4. 23 4. 24 4. 25 4. 26 4. 27 4. 28 4. 29 4. 30 4. 31 4. 32 4. 33 4. 34 4. 35 4. 36 4. 37 4. 38 4. 39 4. 40 4. 41 Description of a ? uid element in accelerated system. . . . . . . . . Pressure lines a static ? uid with a constant density. . . . . . . . . . . A schematic to explain the measure of the atmospheric pressure. . . . Schematic of gas measurement utilizing the “U” tube. . . . . . . . . Schematic of sensitive measurement device. . . . . . . . . . . . . . . Hydrostatic pressure when there is compressibility in the liquid phase. Two adjoin layers for stability analysis. . . . . . . . . . . . . . . . . . The varying gravity e? ects on density and pressure. . . . . . . . . . . The e? ective gravity is for accelerated cart. . . . . . . . . . . . . . . A cart slide on inclined plane. . . . . . . . . . . . . . . . . . . . . . Forces diagram of cart sliding on inclined plane. . . . . . . . . . . . . Schematic to explain the angular angle. . . . . . . . . . . . . . . . . Schematic angular angle to explain example 4. 5 . . . . . . . . . . . . Rectangular area under pressure. . . . . . . . . . . . . . . . . . . . . Schematic of submerged area. . . . . . . . . . . . . . . . . . . . . . The general forces acting on submerged area. . . . . . . . . . . . . . The general forces acting on non symmetrical straight area. . . . . . . The general forces acting on non symmetrical straight area. . . . . . . The e? ects of multi layers density on static forces. . . . . . . . . . . The forces on curved area. . . . . . . . . . . . . . . . . . . . . . . . Schematic of Net Force on ? oating body. . . . . . . . . . . . . . . . Dam is a part of a circular shape. . . . . . . . . . . . . . . . . . . . . Area above the dam arc subtract triangle. . . . . . . . . . . . . . . . Area above the dam arc calculation for the center. . . . . . . . . . . Moment on arc element around Point “O. ” . . . . . . . . . . . . . . Polynomial shape dam description. . . . . . . . . . . . . . . . . . . . The di? erence between the slop and the direction angle. . . . . . . . Schematic of Immersed Cylinder. . . . . . . . . . . . . . . . . . . . The ? oating forces on Immersed Cylinder. . . . . . . . . . . . . . . . Schematic of a thin wall ? oating body. . . . . . . . . . . . . . . . . . Schematic of ? oating bodies. . . . . . . . . . . . . . . . . . . . . . . Schematic of ? oating cubic. . . . . . . . . . . . . . . . . . . . . . . . Stability analysis of ? oating body. . . . . . . . . . . . . . . . . . . . Cubic body dimensions for stability analysis. . . . . . . . . . . . . . . Stability of cubic body in? nity long. . . . . . . . . . . . . . . . . . . The maximum height reverse as a function of density ratio. . . . . . .

Stability of two triangles put tougher. . . . . . . . . . . . . . . . . . The e? ects of liquid movement on the GM . . . . . . . . . . . . . . . Measurement of GM of ? oating body. . . . . . . . . . . . . . . . . . Calculations of GM for abrupt shape body. . . . . . . . . . . . . . . A heavy needle is ? oating on a liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 50 55 58 58 59 60 64 67 69 73 73 74 74 75 76 78 78 80 81 84 86 86 88 88 89 90 91 91 93 93 94 100 100 100 102 103 103 104 106 107 108 110 LIST OF FIGURES 4. 42 4. 43 4. 44 4. 45 5. 1 5. 5. 3 5. 4 5. 5 5. 6 5. 7 5. 8 5. 9 5. 10 5. 11 5. 12 6. 1 6. 2 6. 3 6. 4 6. 5 6. 6 6. 7 6. 8 6. 9 6. 10 6. 11 7. 1 7. 2 7. 3 7. 4 8. 1 8. 2 8. 3 8. 4 8. 5 8. 6 8. 7 8. 8 8. 9 8. 10 Description of depression to explain the Rayleigh???Taylor instability. Description of depression to explain the instability. . . . . . . . . . The cross section of the interface for max liquid. . . . . . . . . . Three liquids layers under rotation . . . . . . . . . . . . . . . . . Control volume and system in motion . . . . . . . . . . Piston control volume . . . . . . . . . . . . . . . . . . . Schematics of velocities at the interface . . . . . . . . Schematics of ? ow 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 averaged velocity Control volume and system in motion . . . . . . . . . . Circular cross section for Finding Ux . . . . . . . . . . . Velocity for a circular shape . . . . . . . . . . . . . . . . Boat for example 5. 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 111 112 114 115 119 120 121 122 125 128 133 138 139 140 141 142 144 147 148 150 151 153 155 156 161 162 165 168 170 171 180 191 193 194 194 195 196 197 207 208 209 The explain for the direction relative to surface . . . . . . . . . Schematics of area impinged by a jet . . . . . . . . . . . . . . . Nozzle schematic for forces calculations . . . . . . . . . . . . Propeller schematic to explain the change of momentum . . . . Toy Sled pushed by the liquid jet . . . . . . . . . . . . . . . . . A rocket a with moving control volume . . . . . . . . . . . . . . Schematic of a tank seating on wheels . . . . . . . . . . . . . . A new control volume to ? nd the velocity in discharge tank . . . The impeller of the centrifugal pump and the velocities diagram Nozzle schematics water rocket . . . . . . . . . . . . . . . . . . Flow out of un symmetrical tank . . . . . . . . . . . . . . . . . The work on the control volume . . . Discharge from a Large Container . Kinetic Energy and Averaged Velocity The work on the control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Di? erent ? elds of multi phase ? ow. . . . . . . . . . . . . . . . . . . Strati? ed ? ow in horizontal tubes when the liquids ? ow is very slow. Kind of Strati? ed ? ow in horizontal tubes. . . . . . . . . . . . . . . Plug ? ow in horizontal tubes with the liquids ? ow is faster. . . . . . Modi? ed Mandhane map for ? ow regime in horizontal tubes. . . . . Gas and liquid in Flow in verstical tube against the gravity. . . . . A dimensional vertical ? ow map low gravity against gravity. . . . . . The terminal velocity that left the solid particles. . . . . . . . . . . The ? ow patterns in solid-liquid ? ow. . . . . . . . . . . . . . . . . . Counter???? ow in vertical tubes map. . . . . . . . . . . . . . . . . . x 8. 11 8. 12 8. 13 8. 14 8. 15 8. 16 LIST OF FIGURES Counter???current ? ow in a can. . . . . . . . . . . . . . . . . . . . . . Image of counter-current ? ow in liquid???gas/solid???gas con? gurations. . Flood in vertical pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . A ? ow map to explain the horizontal counter???current ? ow. . . . . . A diagram to explain the ? ood in a two dimension geometry. . . . . . General forces diagram to calculated the in a two dimension geometry. . . . . . . 210 210 211 212 212 218 A. 1 The right hand rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 LIST OF TABLES 1 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 7 2. 1 3. 1 3. 2 Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . xxxvi Sutherland’s equation coe? cients . . . . . . . . . . . . Viscosity of selected gases . . . . . . . . . . . . . . . . Viscosity of selected liquids . . . . . . . . . . . . . . . Properties at the critical stage . . . . . . . . . . . . Bulk modulus for selected materials . . . . . . . . . . The contact angle for air/water with selected materials. The surface tension for selected materials. . . . . . . . continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 14 15 20 25 31 32 38 53 54 Properties of Various Ideal Gases [300K] . . . . . . . . . . . . . . . . . Moments of Inertia full shape. . . . . . . . . . . . . . . . . . . . . . . Moment of inertia for various plane surfaces . . . . . . . . . . . . . . xi xii LIST OF TABLES NOMENCLATURE ? R ? Universal gas constant, see equation (2. 26), page 37 The shear stress Tenser, see equation (6. 7), page 144 Units length. , see equation (2. 1), page 33 M ?? ??0 F ext U ? A a Bf c. v. Cp Cv EU Eu Angular Momentum, see equation (6. 38), page 160 viscosity at input temperature T, see equation (1. 17), page 12 reference viscosity at reference temperature, Ti0 , see equation (1. 17), page 12 External forces by non???? uids means, see equation (6. 11), page 145 The velocity taken with the direction, see equation (6. 1), page 143 Martinelli parameter, see equation (8. 3), page 205 The area of surface, see equation (4. 117), page 86 The acceleration of object or system, see equation (4. 0), page 55 Body force, see equation (2. 9), page 35 subscribe for control volume, see equation (5. 0), page 120 Speci? c pressure heat, see equation (2. 23), page 37 Speci? c volume heat, see equation (2. 22), page 37 Internal energy, see equation (2. 3), page 34 Internal Energy per unit mass, see equation (2. 6), page 34 xiii xiv Ei G gG H h k kT L System energy at state i, see equation (2. 2), page 34 The gravitation constant, see equation (4. 62), page 70 general Body force, see equation (4. ), page 55 Enthalpy, see equation (2. 18), page 36 Speci? c enthalpy, see equation (2. 18), page 36 the ratio of the speci? c heats, see equation (2. 24), page 37 Fluid thermal conductivity, see equation (7. 3), page 168 Angular momentum, see equation (3. 38), page 51 LIST OF TABLES Patmos Atmospheric Pressure, see equation (4. 85), page 78 q Q12 R S Suth T? Ti0 Tin U w W12 z says Energy per unit mass, see equation (2. 6), page 34 The energy transfered to the system between state 1 and state 2, see equation (2. 2), page 34 Speci? c gas constant, see equation (2. 27), page 38 Entropy of the system, see equation (2. 3), page 36 Suth is Sutherland’s constant and it is presented in the Table 1. 1, see equation (1. 17), page 12 Torque, see equation (3. 40), page 52 reference temperature in degrees Kelvin, see equation (1. 17), page 12 input temperature in degrees Kelvin, see equation (1. 17), page 12 velocity , see equation (2. 4), page 34 Work per unit mass, see equation (2. 6), page 34 The work done by the system between state 1 and state 2, see equation (2. 2), page 34 the coordinate in z direction, see equation (4. 14), page 57 Subscribe says, see equation (5. 0), page 120 The Book Change Log Version 0. 2. 4 March 01, 2010 (2. 9 M 280 pages) The energy conservation chapter was released. ? Some additions to mass conservation chapter on averaged velocity. ? Some additions to momentum conservation chapter. ? Additions to the mathematical appendix on vector algebra. ? Additions to the mathematical appendix on variables separation in second order ode equations. ? Add the macro picText to insert ? gure in lower right corner thanks to Steven from www. artofproblemsolving. com. ? Add the macro to improve emphases equation thanks to Steven from www. artofproblemsolving. com. ? Add example about the the third component of the velocity. ? English corrections, Thanks to Eliezer Bar-Meir

Version 0. 2. 3 Jan 01, 2010 (2. 8 M 241 pages) ? The momentum conservation chapter was released. ? Corrections to Static Chapter. xv xvi LIST OF TABLES ? Add the macro ekes to equations in examples thanks to Steven from www. artofproblemsolving. ? English corrections, Thanks to Eliezer Bar-Meir Version 0. 1. 9 Dec 01, 2009 (2. 6 M 219 pages) ? The mass conservation chapter was released. ? Add Reynold’s Transform explanation. ? Add example on angular rotation to statics chapter. ? Add the open question concept. Two open questions were released. ? English corrections, Thanks to Eliezer Bar-Meir Version 0. 1. 8. 5 Nov 01, 2009 (2. M 203 pages) ? First true draft for the mass conservation. ? Improve the dwar? ng macro to allow ? exibility with sub title. ? Add the ? rst draft of the temperature-velocity diagram to the Therm’s chapter. Version 0. 1. 8. 1 Sep 17, 2009 (2. 5 M 197 pages) ? Continue ? xing the long titles issues. ? Add some examples to static chapter. ? Add an example to mechanics chapter. Version 0. 1. 8a July 5, 2009 (2. 6 M 183 pages) ? Fixing some long titles issues. ? Correcting the gas properties tables (thanks to Heru and Micheal) ? Move the gas tables to common area to all the books. LIST OF TABLES xvii Version 0. 1. 8 Aug 6, 2008 (2. M 189 pages) ? Add the chapter on introduction to muli???phase ? ow ? Again additional improvement to the index (thanks to Irene). ? Add the Rayleigh???Taylor instability. ? Improve the doChap scrip to break up the book to chapters. Version 0. 1. 6 Jun 30, 2008 (1. 3 M 151 pages) ? Fix the English in the introduction chapter, (thanks to Tousher). ? Improve the Index (thanks to Irene). ? Remove the multiphase chapter (it is not for public consumption yet). Version 0. 1. 5a Jun 11, 2008 (1. 4 M 155 pages) ? Add the constant table list for the introduction chapter. ? Fix minor issues (English) in the introduction chapter.

Version 0. 1. 5 Jun 5, 2008 (1. 4 M 149 pages) ? Add the introduction, viscosity and other properties of ? uid. ? Fix very minor issues (English) in the static chapter. Version 0. 1. 1 May 8, 2008 (1. 1 M 111 pages) ? Major English corrections for the three chapters. ? Add the product of inertia to mechanics chapter. ? Minor corrections for all three chapters. Version 0. 1a April 23, 2008 xviii LIST OF TABLES Version 0. 1a April 23, 2008 ? The Thermodynamics chapter was released. ? The mechanics chapter was released. ? The static chapter was released (the most extensive and detailed chapter).

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If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with… Texts. ” line with this: with the Invariant Sections being LIST THEIR TITLES, with the FrontCover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

CONTRIBUTOR LIST How to contribute to this book As a copylefted work, this book is open to revisions and expansions by any interested parties. The only “catch” is that credit must be given where credit is due. This is a copyrighted work: it is not in the public domain! If you wish to cite portions of this book in a work of your own, you must follow the same guidelines as for any other GDL copyrighted work. Credits All entries have been arranged in alphabetical order of surname (hopefully. Major contributions are listed by individual name with some detail on the nature of the contribution(s), date, contact info, etc.

Minor contributions (typo corrections, etc. ) are listed by name only for reasons of brevity. Please understand that when I classify a contribution as “minor,” it is in no way inferior to the e? ort or value of a “major” contribution, just smaller in the sense of less text changed. Any and all contributions are gratefully accepted. I am indebted to all those who have given freely of their own knowledge, time, and resources to make this a better book! ? Date(s) of contribution(s): 1999 to present ? Nature of contribution: Original author. ? Contact at: barmeir at gmail. com Steven from artofproblemsolving. om ? Date(s) of contribution(s): June 2005, Dec, 2009 xxvii xxviii LIST OF TABLES ? Nature of contribution: LaTeX formatting, help on building the useful equation and important equation macros. ? Nature of contribution: In 2009 creating the exEq macro to have di? erent counter for example. Dan H. Olson ? Date(s) of contribution(s): April 2008 ? Nature of contribution: Some discussions about chapter on mechanics and correction of English. Richard Hackbarth ? Date(s) of contribution(s): April 2008 ? Nature of contribution: Some discussions about chapter on mechanics and correction of English.

John Herbolenes ? Date(s) of contribution(s): August 2009 ? Nature of contribution: Provide some example for the static chapter. Eliezer Bar-Meir ? Date(s) of contribution(s): Nov 2009, Dec 2009 ? Nature of contribution: Correct many English mistakes Mass. ? Nature of contribution: Correct many English mistakes Momentum. Henry Schoumertate ? Date(s) of contribution(s): Nov 2009 ? Nature of contribution: Discussion on the mathematics of Reynolds Transforms. Your name here ? Date(s) of contribution(s): Month and year of contribution ?

Nature of contribution: Insert text here, describing how you contributed to the book. ? Contact at: my email@provider. net CREDITS xxix Typo corrections and other “minor” contributions ? R. Gupta, January 2008, help with the original img macro and other ( LaTeX issues). ? Tousher Yang April 2008, review of statics and thermo chapters. xxx LIST OF TABLES About This Author Genick Bar-Meir holds a Ph. D. in Mechanical Engineering from University of Minnesota and a Master in Fluid Mechanics from Tel Aviv University. Dr. Bar-Meir was the last student of the late Dr. R. G. E. Eckert.

Much of his time has been spend doing research in the ? eld of heat and mass transfer (related to renewal energy issues) and this includes ? uid mechanics related to manufacturing processes and design. Currently, he spends time writing books (there are already three very popular books) and softwares for the POTTO project (see Potto Prologue). The author enjoys to encourage his students to understand the material beyond the basic requirements of exams. In his early part of his professional life, Bar-Meir was mainly interested in elegant models whether they have or not a practical applicability.

Now, this author’s views had changed and the virtue of the practical part of any model becomes the essential part of his ideas, books and software. He developed models for Mass Transfer in high concentration that became a building blocks for many other models. These models are based on analytical solution to a family of equations1 . As the change in the view occurred, Bar-Meir developed models that explained several manufacturing processes such the rapid evacuation of gas from containers, the critical piston velocity in a partially ? led chamber (related to hydraulic jump), application of supply and demand to rapid change power system and etc. All the models have practical applicability. These models have been extended by several research groups (needless to say with large research grants). For example, the Spanish Comision Interministerial provides grants TAP97-0489 and PB98-0007, and the CICYT and the European Commission provides 1FD97-2333 grants for minor aspects of that models. Moreover, the author’s models were used in numerical works, in GM, British industry, Spain, and Canada. In the area of compressible ? w, it was commonly believed and taught that there is only weak and strong shock and it is continue by Prandtl???Meyer function. Bar??? 1 Where the mathematicians were able only to prove that the solution exists. xxxi xxxii LIST OF TABLES Meir discovered the analytical solution for oblique shock and showed that there is a quiet bu? er between the oblique shock and Prandtl???Meyer. He also build analytical solution to several moving shock cases. He described and categorized the ? lling and evacuating of chamber by compressible ? uid in which he also found analytical solutions to cases where the working ? id was ideal gas. The common explanation to Prandtl???Meyer function shows that ? ow can turn in a sharp corner. Engineers have constructed design that based on this conclusion. Bar-Meir demonstrated that common Prandtl???Meyer explanation violates the conservation of mass and therefor the turn must be around a ? nite radius. The author’s explanations on missing diameter and other issues in fanno ? ow and “”naughty professor’s question”” are used in the industry. In his book “Basics of Fluid Mechanics”, Bar-Meir demonstrated that ? uids must have wavy surface when the materials ? ow together.

All the previous models for the ? ooding phenomenon did not have a physical explanation to the dryness. He built a model to explain the ? ooding problem (two phase ? ow) based on the physics. He also constructed and explained many new categories for two ? ow regimes. The author lives with his wife and three children. A past project of his was building a four stories house, practically from scratch. While he writes his programs and does other computer chores, he often feels clueless about computers and programing. While he is known to look like he knows about many things, the author just know to learn quickly.

The author spent years working on the sea (ships) as a engine sea o? cer but now the author prefers to remain on solid ground. Prologue For The POTTO Project This books series was born out of frustrations in two respects. The ? rst issue is the enormous price of college textbooks. It is unacceptable that the price of the college books will be over $150 per book (over 10 hours of work for an average student in The United States). The second issue that prompted the writing of this book is the fact that we as the public have to deal with a corrupted judicial system.

As individuals we have to obey the law, particularly the copyright law with the “in? nite2 ” time with the copyright holders. However, when applied to “small” individuals who are not able to hire a large legal ? rm, judges simply manufacture facts to make the little guy lose and pay for the defense of his work. On one hand, the corrupted court system defends the “big” guys and on the other hand, punishes the small “entrepreneur” who tries to defend his or her work. It has become very clear to the author and founder of the POTTO Project that this situation must be stopped. Hence, the creation of the POTTO Project.

As R. Kook, one of this author’s sages, said instead of whining about arrogance and incorrectness, one should increase wisdom. This project is to increase wisdom and humility. The Potto Project has far greater goals than simply correcting an abusive Judicial system or simply exposing abusive judges. It is apparent that writing textbooks especially for college students as a cooperation, like an open source, is a new idea3 . Writing a book in the technical ? eld is not the same as writing a novel. The writing of a technical book is really a collection of information and practice.

There is always someone who can add to the book. The study of technical material isn’t only done by having to memorize the material, but also by coming to understand and be able to solve 2 After the last decision of the Supreme Court in the case of Eldred v. Ashcro? (see http://cyber. law. harvard. edu/openlaw/eldredvashcroft for more information) copyrights practically remain inde? nitely with the holder (not the creator). 3 In some sense one can view the encyclopedia Wikipedia as an open content project (see http://en. wikipedia. org/wiki/Main Page).

The wikipedia is an excellent collection of articles which are written by various individuals. xxxiii xxxiv LIST OF TABLES related problems. The author has not found any technique that is more useful for this purpose than practicing the solving of problems and exercises. One can be successful when one solves as many problems as possible. To reach this possibility the collective book idea was created/adapted. While one can be as creative as possible, there are always others who can see new aspects of or add to the material. The collective material is much richer than any single person can create by himself.

The following example explains this point: The army ant is a kind of carnivorous ant that lives and hunts in the tropics, hunting animals that are even up to a hundred kilograms in weight. The secret of the ants’ power lies in their collective intelligence. While a single ant is not intelligent enough to attack and hunt large prey, the collective power of their networking creates an extremely powerful intelligence to carry out this attack4 . When an insect which is blind can be so powerful by networking, so can we in creating textbooks by this powerful tool. Why would someone volunteer to be an author or organizer of such a book?

This is the ? rst question the undersigned was asked. The answer varies from individual to individual. It is hoped that because of the open nature of these books, they will become the most popular books and the most read books in their respected ? eld. For example, the books on compressible ? ow and die casting became the most popular books in their respective area. In a way, the popularity of the books should be one of the incentives for potential contributors. The desire to be an author of a well???known book (at least in his/her profession) will convince some to put forth the e? rt. For some authors, the reason is the pure fun of writing and organizing educational material. Experience has shown that in explaining to others any given subject, one also begins to better understand the material. Thus, contributing to these books will help one to understand the material better. For others, the writing of or contributing to this kind of books will serve as a social function. The social function can have at least two components. One component is to come to know and socialize with many in the profession.

For others the social part is as simple as a desire to reduce the price of college textbooks, especially for family members or relatives and those students lacking funds. For some contributors/authors, in the course of their teaching they have found that the textbook they were using contains sections that can be improved or that are not as good as their own notes. In these cases, they now have an opportunity to put their notes to use for others. Whatever the reasons, the undersigned believes that personal intentions are appropriate and are the author’s/organizer’s private a? air.

If a contributor of a section in such a book can be easily identi? ed, then that contributor will be the copyright holder of that speci? c section (even within question/answer sections). The book’s contributor’s names could be written by their sections. It is not just for experts to contribute, but also students who happened to be doing their homework. The student’s contributions can be done by adding a question and perhaps the solution. Thus, this method is expected to accelerate the creation of these high quality books. These books are written in a similar manner to the open source software 4 see also in Franks, Nigel R. “Army Ants: A Collective Intelligence,” American Scientist, 77:139, 1989 (see for information http://www. ex. ac. uk/bugclub/raiders. html) CREDITS xxxv process. Someone has to write the skeleton and hopefully others will add “? esh and skin. ” In this process, chapters or sections can be added after the skeleton has been written. It is also hoped that others will contribute to the question and answer sections in the book. But more than that, other books contain data5 which can be typeset in A LTEX. These data (tables, graphs and etc. ) can be redone by anyone who has the time to do it.

Thus, the contributions to books can be done by many who are not experts. Additionally, contributions can be made from any part of the world by those who wish to translate the book. It is hoped that the books will be error-free. Nevertheless, some errors are possible and expected. Even if not complete, better discussions or better explanations are all welcome to these books. These books are intended to be “continuous” in the sense that there will be someone who will maintain and improve the books with time (the organizer(s)). These books should be considered more as a project than to ? the traditional de? nition of “plain” books. Thus, the traditional role of author will be replaced by an organizer who will be the one to compile the book. The organizer of the book in some instances will be the main author of the work, while in other cases only the gate keeper. This may merely be the person who decides what will go into the book and what will not (gate keeper). Unlike a regular book, these works will have a version number because they are alive and continuously evolving. In the last 5 years three textbooks have been constructed which are available for download.

These books contain innovative ideas which make some chapters the best in the world. For example, the chapters on Fanno ? ow and Oblique shock contain many original ideas such as the full analytical solution to the oblique shock, many algorithms for calculating Fanno ? ow parameters which are not found in any other book. In addition, Potto has auxiliary materials such as the gas dynamics tables (the largest compressible ? ow tables collection in the world), Gas Dynamics Calculator (Potto-GDC), etc. The combined number downloads of these books is over half a million (December 2009) or in a rate of 20,000 copies a month.

Potto books on compressible ? ow and ? uid mechanics are used as the main textbook or as a reference book in several universities around the world. The books are used in more than 165 di? erent countries around the world. Every month people from about 110 di? erent countries download these books. The book on compressible ? ow is also used by “young engineers and scientists” in NASA according to Dr. Farassat, NASA Langley Research Center. The undersigned of this document intends to be the organizer/author/coordinator of the projects in the following areas: 5 Data are not copyrighted. xxvi Table -1. Books under development in Potto project. LIST OF TABLES Project Name Compressible Flow Die Casting Dynamics Fluid Mechanics Heat Transfer Progress beta alpha NSY alpha NSY Remarks Version 0. 4. 8. 2 0. 0. 3 0. 0. 0 0. 1. 1 0. 0. 0 Based on Eckert Availability for Public Download Mechanics Open Channel Flow Statics Strength of Material Thermodynamics Two/Multi ? ow phases NSY NSY early alpha NSY early alpha NSY ?rst chapter 0. 0. 0 0. 0. 0 0. 0. 1 0. 0. 0 0. 0. 01 TelAviv’notes 0. 0. 0 NSY = Not Started Yet The meaning of the progress is as: The Alpha Stage is when some of the chapters are already in a rough draft; ? in Beta Stage is when all or al