Basics Of Fluid Mechanics
Basics of Fluid MechanicsGenick Bar–Meir, Ph. D.1107 16th Ave S. E.Minneapolis, MN 55414-2411email:barmeir@gmail.comCopyright 2008, 2007, and 2006 by Genick Bar-MeirSee the file copying.fdl or copyright.tex for copying conditions.Version (0.1.6August 11, 2008)
‘We are like dwarfs sitting on the shoulders of giants”from The Metalogicon by John in 1159
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iiCONTENTSOpen Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix1 Introduction1.1 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 Viscosity1.5.5 Bulk Modulus . . . . . . . .1.6 Surface Tension . . . . . . . . . .1.6.1 Wetting of Surfaces . . . .11356991011121922242 Review of Thermodynamics2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33333 Review of Mechanics3.1 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 Inertia3.2.4 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 . . . . .414141424343444648505051524 Fluids Statics4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 The Hydrostatic Equation . . . . . . . . . . . . . . . . . . . .4.3 Pressure and Density in a Gravitational Field . . . . . . . . .4.3.1 Constant Density in Gravitational Field . . . . . . . . .4.3.2 Pressure Measurement . . . . . . . . . . . . . . . . .4.3.3 Varying Density in a Gravity Field . . . . . . . . . . . .4.3.4 The Pressure Effects Because Temperature Variations4.3.5 Gravity Variations Effects on Pressure and Density . .4.3.6 Liquid Phase . . . . . . . . . . . . . . . . . . . . . . .4.4 Fluid in a Accelerated System . . . . . . . . . . . . . . . . . .4.4.1 Fluid in a Linearly Accelerated System . . . . . . . . .555555575759616569717272.
CONTENTS4.4.2 Angular Acceleration Systems: Constant Density4.5 Fluid Forces on Surfaces . . . . . . . . . . . . . . . . .4.5.1 Fluid Forces on Straight Surfaces . . . . . . . . .4.5.2 Force on Curved Surfaces . . . . . . . . . . . . .4.6 Buoyancy and Stability . . . . . . . . . . . . . . . . . . .4.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . .4.6.2 Surface Tension . . . . . . . . . . . . . . . . . .4.7 Rayleigh–Taylor Instability . . . . . . . . . . . . . . . . .iii. 74. 75. 75. 85. 92. 98. 107. 1085 Multi–Phase Flow5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 What to Expect From This Chapter . . . . . . . . . . . . .5.4 Kind of Multi-Phase Flow . . . . . . . . . . . . . . . . . .5.5 Classification of Liquid-Liquid Flow Regimes . . . . . . .5.5.1 Co–Current Flow . . . . . . . . . . . . . . . . . . .5.6 Multi–Phase Flow Variables Definitions . . . . . . . . . . .5.6.1 Multi–Phase Averaged Variables Definitions . . . .5.7 Homogeneous Models . . . . . . . . . . . . . . . . . . . .5.7.1 Pressure Loss Components . . . . . . . . . . . . .5.7.2 Lockhart Martinelli Model . . . . . . . . . . . . . .5.8 Solid–Liquid Flow . . . . . . . . . . . . . . . . . . . . . . .5.8.1 Solid Particles with Heavier Density ρS ρL . . .5.8.2 Solid With Lighter Density ρS ρ and With Gravity5.9 Counter–Current Flow . . . . . . . . . . . . . . . . . . . .5.9.1 Horizontal Counter–Current Flow . . . . . . . . . .5.9.2 Flooding and Reversal Flow . . . . . . . . . . . . .5.10 Multi–Phase Conclusion . . . . . . . . . . . . . . . . . . 36136143Index145Subjects Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Authors Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
ivCONTENTS
LIST OF 1.141.151.161.171.183.13.23.33.43.53.6Diagram to explain part of relationships of 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 as progression of time. .the difference of power fluids . . . . . . . . . . . . . . . . . . . . . .Nitrogen (left) and Argon (right) viscosity as a function of the temperature and pressure after Lemmon and Jacobsen. . . . . . . . . .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 fluids. . . . . . . . . . . . . .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 for calculation of momentinertia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v. . . . . .of. .2667910111212131718212424262929424244454546
viLIST OF FIGURES3.7 A square element for the calculations of inertia of two-dimensionalto three–dimensional deviations. . . . . . . . . . . . . . . . . . . . . 473.8 The ratio of the moment of inertia of two-dimensional to three–dimensional. 473.9 Description of parabola for calculation of moment of inertia and center of area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.10 Product of inertia for triangle. . . . . . . . . . . . . . . . . . . . . . . 74.284.294.304.314.324.334.344.35Description of a fluid element in accelerated system under body forces. 55Pressure lines a static fluid with a constant density. . . . . . . . . . . 58A schematic to explain the measure of the atmospheric pressure. . . 58Schematic of gas measurement utilizing the “U” tube. . . . . . . . . 59Schematic of sensitive measurement device. . . . . . . . . . . . . . 60Hydrostatic pressure when there is compressibility in the liquid phase. 64Two adjoin layers for stability analysis. . . . . . . . . . . . . . . . . . 67The varying gravity effects on density and pressure. . . . . . . . . . 69The effective gravity is for accelerated cart. . . . . . . . . . . . . . . 73A cart slide on inclined plane . . . . . . . . . . . . . . . . . . . . . . 73Forces diagram of cart sliding on inclined plane . . . . . . . . . . . . 74Schematic to explain the angular angle. . . . . . . . . . . . . . . . . 74Rectangular area under pressure. . . . . . . . . . . . . . . . . . . . 75Schematic of submerged area to explain the center forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77The general forces acting on submerged area. . . . . . . . . . . . . 78The general forces acting on non symmetrical straight area. . . . . . 79The general forces acting on non symmetrical straight area. . . . . . 80The effects of multi layers density on static forces. . . . . . . . . . . 83The forces on curved area. . . . . . . . . . . . . . . . . . . . . . . . 85Schematic of Net Force on floating body. . . . . . . . . . . . . . . . . 86Dam is a part of a circular shape. . . . . . . . . . . . . . . . . . . . . 87Area above the dam arc subtract triangle. . . . . . . . . . . . . . . . 87Area above the dam arc calculation for the center. . . . . . . . . . . 88Moment on arc element around Point “O.” . . . . . . . . . . . . . . . 89Polynomial shape dam description for the moment around point “O”and force calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . 90The difference between the slop and the direction angle. . . . . . . . 90Schematic of Immersed Cylinder. . . . . . . . . . . . . . . . . . . . . 92The floating forces on Immersed Cylinder. . . . . . . . . . . . . . . . 93Schematic of a thin wall floating body. . . . . . . . . . . . . . . . . . 94Schematic of floating bodies. . . . . . . . . . . . . . . . . . . . . . . 98Schematic of floating cubic. . . . . . . . . . . . . . . . . . . . . . . . 98Stability analysis of floating body. . . . . . . . . . . . . . . . . . . . . 99Cubic body dimensions for stability analysis. . . . . . . . . . . . . . . 100Stability of cubic body infinity long. . . . . . . . . . . . . . . . . . . . 101The maximum height reverse as a function of density ratio. . . . . . 102
LIST OF 5.65.75.85.95.105.115.125.135.145.155.16The effects of liquid movement on the GM . . . . . . . . . . . . . . .Measurement of GM of floating body. . . . . . . . . . . . . . . . . . .Calculations of GM for abrupt shape body. . . . . . . . . . . . . . .A heavy needle is floating on a liquid. . . . . . . . . . . . . . . . . .Description of depression to explain the Rayleigh–Taylor instability. .Description of depression to explain the instability. . . . . . . . . . .The cross section of the interface. The purple color represents themaximum heavy liquid raising area. The yellow color represents themaximum lighter liquid that are “going down.” . . . . . . . . . . . . .vii103105106107108110111lines a static fluid with a constant density. . . . . . . . . . . . . . . . 115Stratified flow in horizontal tubes when the liquids flow is very slow. . 117Kind of Stratified flow in horizontal tubes. . . . . . . . . . . . . . . . 118Plug flow in horizontal tubes when the liquids flow is faster slow. . . 118Modified Mandhane map for flow regime in horizontal tubes. . . . . . 119lines a static fluid with a constant density. . . . . . . . . . . . . . . . 120A dimensional vertical flow map under very low gravity against thegravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121The terminal velocity that left the solid particles. . . . . . . . . . . . . 131The flow patterns in solid-liquid flow. . . . . . . . . . . . . . . . . . . 132Counter–current flow in a can (the left figure) has only one hole thuspulse flow and a flow with two holes (right picture). . . . . . . . . . . 134Counter–flow in vertical tubes map. . . . . . . . . . . . . . . . . . . . 134Pictures of Counter-current flow in liquid–gas and solid–gas configurations. The container is made of two compartments. The uppercompartment is filled with the heavy phase (liquid, water solution,or small wood particles) by rotating the container. Even thoughthe solid–gas ratio is smaller, it can be noticed that the solid–gasis faster than the liquid–gas flow. . . . . . . . . . . . . . . . . . . . . 135Flood in vertical pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . 135A flow map to explain the horizontal counter–current flow. . . . . . . 136A diagram to explain the flood in a two dimension geometry. . . . . . 137General forces diagram to calculated the in a two dimension geometry.142
viiiLIST OF FIGURES
LIST OF TABLES11Books Under Potto Project . . . . . . . . . . . . . . . . . . . . . . . . xxixcontinue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx1.11.21.31.41.51.61.71.7Sutherland’s equation coefficients . . . . . . . . . . .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 . . . . . . . . . . . . . . . . . . . . . . . . . .13141415202531322.1 Properties of Various Ideal Gases [300K] . . . . . . . . . . . . . . .383.1 Moments of Inertia for various plane surfaces about their centergravity (full shapes) . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Moment of inertia for various plane surfaces about their centergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix.of. .of. .5354
xLIST OF TABLES
NOMENCLATURER̄Universal gas constant, see equation (2.26), page 38 Units length., see equation (2.1), page 33µviscosity at input temperature T, see equation (1.17), page 12µ0reference viscosity at reference temperature, Ti0 , see equation (1.17), page 12ΞMartinelli parameter, see equation (5.43), page 129AThe area of surface, see equation (4.117), page 85aThe acceleration of object or system, see equation (4.0), page 55BfBody force, see equation (2.9), page 35CpSpecific pressure heat, see equation (2.23), page 37CvSpecific volume heat, see equation (2.22), page 37EUInternal energy, see equation (2.3), page 34EuInternal Energy per unit mass, see equation (2.6), page 34EiSystem energy at state i, see equation (2.2), page 34GThe gravitation constant, see equation (4.62), page 70gGgeneral Body force, see equation (4.0), page 55HEnthalpy, see equation (2.18), page 36hSpecific enthalpy, see equation (2.18), page 36xi
xiiLIST OF TABLESkthe ratio of the specific heats, see equation (2.24), page 37LAngular momentum, see equation (3.38), page 51Patmos Atmospheric Pressure, see equation (4.85), page 77qEnergy per unit mass, see equation (2.6), page 34Q12The energy transfered to the system between state 1 and state 2, see equation (2.2), page 34RSpecific gas constant, see equation (2.27), page 38SEntropy of the system, see equation (2.13), page 36SuthSuth is Sutherland’s constant and it is presented in the Table 1.1, see equation (1.17), page 12TτTorque, see equation (3.40), page 51Ti0reference temperature in degrees Kelvin, see equation (1.17), page 12Tininput temperature in degrees Kelvin, see equation (1.17), page 12Uvelocity , see equation (2.4), page 34wWork per unit mass, see equation (2.6), page 34W12The work done by the system between state 1 and state 2, see equation (2.2), page 34zthe coordinate in z direction, see equation (4.14), page 58
The Book Change LogVersion 0.1.8Aug 6, 2008 (2.6 M 189 pages) Add the chapter on introduction to muli–phase flow 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 0.1.6Jun 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 0.1.5aJun 11, 2008 (1.4 M 155 pages) Add the constant table list for the introduction chapter. Fix minor issues (English) in the introduction chapter.xiii
xivLIST OF TABLESVersion 0.1.5Jun 5, 2008 (1.4 M 149 pages) Add the introduction, viscosity and other properties of fluid. Fix very minor issues (English) in the static chapter.Version 0.1.1May 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, 2008Version 0.1aApril 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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LIST OF FIGURES 1.1 Diagram to explain part of relationships of fluid mechanics branch
Fluid Mechanics Fluid Engineers basic tools Experimental testing Computational Fluid Theoretical estimates Dynamics Fluid Mechanics, SG2214 Fluid Mechanics Definition of fluid F solid F fluid A fluid deforms continuously under the action of a s
Fluid Mechanics 63 Chapter 6 Fluid Mechanics _ 6.0 Introduction Fluid mechanics is a branch of applied mechanics concerned with the static and dynamics of fluid - both liquids and gases. . Solution The relative density of fluid is defined as the rate of its density to the density of water. Thus, the relative density of oil is 850/1000 0.85.
Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and Bernoulli's Equation 7. General Energy Equation 8. Reynolds Number, Laminar Flow, Turbulent Flow and Energy Losses Due to Friction
Fundamentals of Fluid Mechanics. 1 F. UNDAMENTALS OF . F. LUID . M. ECHANICS . 1.1 A. SSUMPTIONS . 1. Fluid is a continuum 2. Fluid is inviscid 3. Fluid is adiabatic 4. Fluid is a perfect gas 5. Fluid is a constant-density fluid 6. Discontinuities (shocks, waves, vortex sheets) are treated as separate and serve as boundaries for continuous .
L M A B CVT Revision: December 2006 2007 Sentra CVT FLUID PFP:KLE50 Checking CVT Fluid UCS005XN FLUID LEVEL CHECK Fluid level should be checked with the fluid warmed up to 50 to 80 C (122 to 176 F). 1. Check for fluid leakage. 2. With the engine warmed up, drive the vehicle to warm up the CVT fluid. When ambient temperature is 20 C (68 F .
Chapter 06 Fluid Mechanics _ 6.0 Introduction Fluid mechanics is a branch of applied mechanics concerned with the static and dynamics of fluid - both liquids and gases. The analysis of the behavior of fluids is based on the fundamental laws of mechanics, which relate continuity of
Motion of a Fluid ElementMotion of a Fluid Element 1. 1. Fluid Fluid Translation: The element moves from one point to another. 3. 3. Fluid Fluid Rotation: The element rotates about any or all of the x,y,z axes. Fluid Deformation: 4. 4. Angular Deformation:The element's angles between the sides Angular Deformation:The element's angles between the sides
Jul 09, 2015 · Tiny-Fogger/Tiny F07, Tiny-Compact/Tiny C07 Tiny-Fluid 42 Tiny FX Tiny-Fluid 42 Tiny S Tiny-Fluid 43 Unique 2.1 Unique-Fluid 43 Viper NT Quick-Fog Fluid 44 Viper NT Regular-Fog Fluid 45 Viper NT Slow-Fog Fluid 46 Martin K-1 Froggy’s Fog K-razy Haze Fluid 47 Magnum 2000 Froggy’s Fog Backwood Bay Fluid 48
