Basics of Fluid Mechanics Genick Bar–Meir, Ph. 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.com 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.php Example: If you are using the latest version Bar-Meir, Genick, “Basics of Fluid Mechanics”, Last modified: Version 0.php If you are using older version Bar-Meir, Genick, “Basics of Fluid Mechanics”, Accessed: Version 0.com iii ‘We are like dwarfs sitting on the shoulders of giants” from The Metalogicon by John in 1159 www.com iv www.com CONTENTS Nomenclature xxiii GNU Free Documentation License. APPLICABILITY AND DEFINITIONS. COPYING IN QUANTITY.
COLLECTIONS OF DOCUMENTS. AGGREGATION WITH INDEPENDENT WORKS. FUTURE REVISIONS OF THIS LICENSE. xxxix ADDENDUM: How to use this License for your documents.
xl How to contribute to this book. xli Steven from artofproblemsolving. xlii Richard Hackbarth. xlii John Herbolenes.
xlii Eliezer Bar-Meir. xlii Henry Schoumertate. xlii Your name here. xlii Typo corrections and other ”minor” contributions.
liii pages 617 size 4.com vi CONTENTS pages 400 size 3. liv pages 189 size 2. liv pages 151 size 1. lxi Open Channel Flow.
lxi 1 Introduction to Fluid Mechanics 1 1.1 What is Fluid Mechanics? .3 Kinds of Fluids .2 Non–Newtonian Fluids .4 Estimation of The Viscosity .1 Wetting of Surfaces. 35 2 Review of Thermodynamics 45 2. 45 3 Review of Mechanics 53 3.1 Kinematics of of Point Body .2 Center of Mass .1 Actual Center of Mass .2 Aproximate Center of Area .3 Moment of Inertia .1 Moment of Inertia for Mass .2 Moment of Inertia for Area .3 Examples of Moment of Inertia .4 Product of Inertia .5 Principal Axes of Inertia .4 Newton’s Laws of Motion .5 Angular Momentum and Torque .1 Tables of geometries .com CONTENTS vii 4 Fluids Statics 69 4.2 The Hydrostatic Equation .3 Pressure and Density in a Gravitational Field .1 Constant Density in Gravitational Field .3 Varying Density in a Gravity Field .4 The Pressure Effects Due To Temperature Variations .5 Gravity Variations Effects on Pressure and Density .4 Fluid in a Accelerated System .1 Fluid in a Linearly Accelerated System .2 Angular Acceleration Systems: Constant Density .3 Fluid Statics in Geological System .5 Fluid Forces on Surfaces .1 Fluid Forces on Straight Surfaces .2 Forces on Curved Surfaces .6 Buoyancy and Stability .7 Rayleigh–Taylor Instability. 143 I Integral Analysis 145 5 Mass Conservation 147 5.1 Non Deformable Control Volume .2 Constant Density Fluids .4 Reynolds Transport Theorem .5 Examples For Mass Conservation .6 The Details Picture – Velocity Area Relationship .7 More Examples for Mass Conservation .1 Momentum Governing Equation .1 Introduction to Continuous .3 Momentum Governing Equation .4 Momentum Equation in Acceleration System .5 Momentum For Steady State and Uniform Flow .2 Momentum Equation Application .com viii CONTENTS 6.1 Momentum for Unsteady State and Uniform Flow .2 Momentum Application to Unsteady State .3 Conservation Moment Of Momentum .4 More Examples on Momentum Conservation .1 The First Law of Thermodynamics .2 Limitation of Integral Approach .3 Approximation of Energy Equation .1 Energy Equation in Steady State .2 Energy Equation in Frictionless Flow and Steady State .4 Energy Equation in Accelerated System .1 Energy in Linear Acceleration Coordinate .2 Linear Accelerated System .3 Energy Equation in Rotating Coordinate System .4 Simplified Energy Equation in Accelerated Coordinate .5 Energy Losses in Incompressible Flow .5 Examples of Integral Energy Conservation.
218 II Differential Analysis 225 8 Differential Analysis 227 8.1 Mass Conservation Examples .2 Simplified Continuity Equation .3 Conservation of General Quantity .1 Generalization of Mathematical Approach for Derivations .2 Examples of Several Quantities .5 Derivations of the Momentum Equation .6 Boundary Conditions and Driving Forces .1 Boundary Conditions Categories .7 Examples for Differential Equation (Navier-Stokes) .2 Theory Behind Dimensional Analysis .3 Dimensional Parameters Application for Experimental Study .4 The Pendulum Class Problem .com CONTENTS ix 9.1 Construction of the Dimensionless Parameters .2 Basic Units Blocks .3 Implementation of Construction of Dimensionless Parameters .4 Similarity and Similitude .4 Summary of Dimensionless Numbers .1 The Significance of these Dimensionless Numbers .2 Relationship Between Dimensionless Numbers .3 Examples for Dimensional Analysis .6 Appendix summary of Dimensionless Form of Navier–Stokes Equations .1 Inviscid Momentum Equations .2 Potential Flow Function .1 Streamline and Stream function .2 Compressible Flow Stream Function .3 The Connection Between the Stream Function and the Potential Function338 10.3 Potential Flow Functions Inventory .1 Flow Around a Circular Cylinder .1 Complex Potential and Complex Velocity .5 Unsteady State Bernoulli in Accelerated Coordinates. 373 11 Compressible Flow One Dimensional 377 11.1 What is Compressible Flow? .2 Why Compressible Flow is Important? .3 Speed of Sound .2 Speed of Sound in Ideal and Perfect Gases .3 Speed of Sound in Almost Incompressible Liquid .4 Speed of Sound in Solids .5 The Dimensional Effect of the Speed of Sound .1 Stagnation State for Ideal Gas Model .2 Isentropic Converging-Diverging Flow in Cross Section .3 The Properties in the Adiabatic Nozzle .4 Isentropic Flow Examples .5 Mass Flow Rate (Number) .7 The Impulse Function .1 Solution of the Governing Equations .3 Operating Equations and Analysis .4 The Moving Shocks .5 Shock or Wave Drag Result from a Moving Shock .6 Tables of Normal Shocks, k = 1.1 The Control Volume Analysis/Governing equations .3 The Entrance Limitation of Supersonic Branch .5 Figures and Tables .6 Isothermal Flow Examples .2 Non–Dimensionalization of the Equations .3 The Mechanics and Why the Flow is Choked? .4 The Working Equations .5 Examples of Fanno Flow .7 The Pressure Ratio, P2 / P1 , effects .8 Practical Examples for Subsonic Flow .9 Subsonic Fanno Flow for Given 4 D fL and Pressure Ratio .10 Subsonic Fanno Flow for a Given M1 and Pressure Ratio .11 More Examples of Fanno Flow .8 The Table for Fanno Flow .2 Rayleigh Flow Tables and Figures .3 Examples For Rayleigh Flow. 478 12 Compressible Flow 2–Dimensional 485 12.1 Preface to Oblique Shock .1 Solution of Mach Angle .2 When No Oblique Shock Exist or the case of D > 0 .3 Application of Oblique Shock .3 Prandtl-Meyer Function .3 Alternative Approach to Governing Equations .4 Comparison And Limitations between the Two Approaches .4 The Maximum Turning Angle .com CONTENTS xi 12.5 The Working Equations for the Prandtl-Meyer Function .7 Flat Body with an Angle of Attack .8 Examples For Prandtl–Meyer Function .9 Combination of the Oblique Shock and Isentropic Expansion. 530 13 Multi–Phase Flow 535 13.3 What to Expect From This Chapter .4 Kind of Multi-Phase Flow .5 Classification of Liquid-Liquid Flow Regimes .1 Co–Current Flow .6 Multi–Phase Flow Variables Definitions .1 Multi–Phase Averaged Variables Definitions .1 Pressure Loss Components .2 Lockhart Martinelli Model .8 Solid–Liquid Flow .1 Solid Particles with Heavier Density ρS > ρL .2 Solid With Lighter Density ρS < ρ and With Gravity .9 Counter–Current Flow .1 Horizontal Counter–Current Flow .2 Flooding and Reversal Flow .10Multi–Phase Conclusion.
565 A Mathematics For Fluid Mechanics 567 A.2 Differential Operators of Vectors .3 Differentiation of the Vector Operations .2 Ordinary Differential Equations (ODE) .1 First Order Differential Equations .2 Variables Separation or Segregation .3 Non–Linear Equations .4 Second Order Differential Equations .5 Non–Linear Second Order Equations .6 Third Order Differential Equation .7 Forth and Higher Order ODE .8 A general Form of the Homogeneous Equation .3 Partial Differential Equations .1 First-order equations .com xii CONTENTS Index 597 Subjects Index .com LIST OF FIGURES 1.1 Diagram to explain fluid mechanics branches .2 Density as a function of the size of sample .3 Schematics to describe the shear stress in fluid mechanics .4 The deformation of fluid due to shear stress .5 The difference of power fluids .6 Nitrogen and Argon viscosity.7 The shear stress as a function of the shear rate .8 Air viscosity as a function of the temperature .9 Water viscosity as a function temperature.10 Liquid metals viscosity as a function of the temperature .11 Reduced viscosity as function of the reduced temperature .12 Reduced viscosity as function of the reduced temperature .13 Concentrating cylinders with the rotating inner cylinder .14 Rotating disc in a steady state .15 Water density as a function of temperature .16 Two liquid layers under pressure .17 Surface tension control volume analysis .18 Surface tension erroneous explanation .19 Glass tube inserted into mercury .20 Capillary rise between two plates .21 Forces in Contact angle .22 Description of wetting and non–wetting fluids .23 Description of the liquid surface .24 The raising height as a function of the radii .25 The raising height as a function of the radius .1 Description of the extinguish nozzle .com xiv LIST OF FIGURES 3.2 Description of how the center of mass is calculated .3 Thin body center of mass/area schematic.4 The schematic that explains the summation of moment of inertia.5 The schematic to explain the summation of moment of inertia.6 Cylinder with an element for calculation moment of inertia .7 Description of rectangular in x–y plane.8 A square element for the calculations of inertia.9 The ratio of the moment of inertia 2D to 3D.10 Moment of inertia for rectangular .11 Description of parabola - moment of inertia and center of area .12 Triangle for example 3.13 Product of inertia for triangle .1 Description of a fluid element in accelerated system.2 Pressure lines in a static constant density fluid .3 A schematic to explain the atmospheric pressure measurement .4 The effective gravity is for accelerated cart .5 Tank and the effects different liquids .6 Schematic of gas measurement utilizing the “U” tube .7 Schematic of sensitive measurement device .10 Hydrostatic pressure under a compressible liquid phase .11 Two adjoin layers for stability analysis .12 The varying gravity effects on density and pressure .13 The effective gravity is for accelerated cart .14 A cart slide on inclined plane .15 Forces diagram of cart sliding on inclined plane .16 Schematic to explain the angular angle .17 Schematic angular angle to explain example 4.18 Earth layers not to scale .19 Illustration of the effects of the different radii .20 Rectangular area under pressure .21 Schematic of submerged area .22 The general forces acting on submerged area .23 The general forces acting on non symmetrical straight area .24 The general forces acting on a non symmetrical straight area .25 The effects of multi layers density on static forces .26 The forces on curved area .27 Schematic of Net Force on floating body .28 Circular shape Dam .29 Area above the dam arc subtract triangle .30 Area above the dam arc calculation for the center .31 Moment on arc element around Point “O” .com LIST OF FIGURES xv 4.32 Polynomial shape dam description .33 The difference between the slop and the direction angle .34 Schematic of Immersed Cylinder .35 The floating forces on Immersed Cylinder .36 Schematic of a thin wall floating body .37 Schematic of floating bodies .38 Schematic of floating cubic .39 Stability analysis of floating body .40 Cubic body dimensions for stability analysis .41 Stability of cubic body infinity long .42 The maximum height reverse as a function of density ratio .43 Stability of two triangles put tougher .44 The effects of liquid movement on the GM .45 Measurement of GM of floating body .46 Calculations of GM for abrupt shape body .47 A heavy needle is floating on a liquid.48 Description of depression to explain the Rayleigh–Taylor instability .49 Description of depression to explain the instability .50 The cross section of the interface for max liquid.51 Three liquids layers under rotation .1 Control volume and system in motion .2 Piston control volume .3 Schematics of velocities at the interface .4 Schematics of flow in a pipe with varying density .5 Filling of the bucket and choices of the control volumes .6 Height of the liquid for example 5.7 Boundary Layer control mass .