Incompressible Flow

Incompressible Flow
Contents
Preface
About the Companion Website
1 Continuum Mechanics
	1.1 Continuum Assumption
	1.2 Fundamental Concepts, Definitions, and Laws
	1.3 Space and Time: Control Regions
	1.4 Density, Velocity, and Internal Energy
	1.5 Interface between Phases
	1.6 Conclusions
	Problems
2 Thermodynamics
	2.1 Systems, Properties, and Processes
	2.2 Independent Variables
	2.3 Temperature and Entropy
	2.4 Fundamental Equations of Thermodynamics
	2.5 Euler’s Equation for Homogenous Functions
	2.6 Gibbs–Duhem Equation
	2.7 Intensive Forms of Basic Equations
	2.8 Dimensions of Temperature and Entropy
	2.9 Working Equations
	2.10 Ideal Gas
	2.11 Incompressible Substance
	2.12 Compressible Liquids
	2.13 Conclusions
	Problems
3 Vector Calculus and Index Notation
	3.1 Index Notation Rules and Coordinate Rotation
	3.2 Definition of Vectors and Tensors
	3.3 Special Symbols and Isotropic Tensors
	3.4 Direction Cosines and the Laws of Cosines
	3.5 Algebra with Vectors
	3.6 Symmetric and Antisymmetric Tensors
	3.7 Algebra with Tensors
	3.8 Vector Cross-Product
	3.9 Alternative Definitions of Vectors and Tensors
	3.10 Principal Axes and Values
	3.11 Derivative Operations on Vector Fields
	3.12 Integral Formulas of Gauss and Stokes
	3.13 Leibnitz’s Theorem
	3.14 Conclusions
	Problems
4 Kinematics of Local Fluid Motion
	4.1 Lagrangian Viewpoint
	4.2 Eulerian Viewpoint
	4.3 Substantial Derivative
	4.4 Decomposition of Motion
	4.5 Elementary Motions in a Linear Shear Flow
	4.6 Proof of Vorticity Characteristics
	4.7 Rate-of-Strain Characteristics
	4.8 Rate of Expansion
	4.9 Streamline Coordinates
	4.10 Conclusions
	Problems
5 Basic Laws
	5.1 Continuity Equation
	5.2 Momentum Equation
	5.3 Surface Forces
	5.4 Stress Tensor Derivation
	5.5 Interpretation of the Stress Tensor Components
	5.6 Pressure and Viscous Stress Tensor
	5.7 Differential Momentum Equation
	5.8 Moment of Momentum, Angular Momentum, and Symmetry of Tij
	5.9 Energy Equation
	5.10 Mechanical and Thermal Energy Equations
	5.11 Energy Equation with Temperature as the Dependent Variable
	5.12 Second Law of Thermodynamics
	5.13 Integral Form of the Continuity Equation
	5.14 Integral Form of the Momentum Equation
	5.15 Momemtum Equation for a Deformable Particle of Variable Mass
	5.16 Integral Form of the Energy Equation
	5.17 Integral Mechanical Energy Equation
	5.18 Jump Equations at Interfaces
	5.19 Conclusions
	Problems
6 Newtonian Fluids and the Navier–Stokes Equations
	6.1 Newton’s Viscosity Law
	6.2 Molecular Model of Viscous Effects
	6.3 Non-Newtonian Liquids
	6.4 Wall Boundary Conditions; The No-Slip Condition
	6.5 Fourier’s Heat Conduction Law
	6.6 Navier–Stokes Equations
	6.7 Conclusions
	Problems
7 Some Incompressible Flow Patterns
	7.1 Pressure-Driven Flow
	7.2 Mechanical Energy, Head Loss, Pipe Flow, and Bernoulli Equation
	7.3 Plane Couette Flow
	7.4 Pressure-Driven Flow in a Slot with a Moving Wall
	7.5 Double Falling Film on a Wall
	7.6 Outer Solution for Rotary Viscous Coupling
	7.7 The Rayleigh Problem
	7.8 Windmills
	7.9 Conclusions
	Problems
8 Dimensional Analysis
	8.1 Measurement, Dimensions, and Scale Change Ratios
	8.2 Physical Variables and Functions
	8.3 Pi Theorem and Its Applications
	8.4 Pump or Blower Analysis: Use of Extra Assumptions
	8.5 Number of Primary Dimensions
	8.6 Proof of Bridgman’s Equation
	8.7 Proof of the Pi Theorem
	8.8 Dynamic Similarity and Scaling Laws
	8.9 Similarity with Geometric Distortion
	8.10 Nondimensional Formulation of Physical Problems
	8.11 Conclusions
	Problems
9 Elements of Compressible Flow
	9.1 Compressible Couette Flow: Adiabatic Wall
	9.2 Flow with Power Law Transport Properties
	9.3 Inviscid Compressible Waves: Speed of Sound
	9.4 Conclusions
	Problems
10 Incompressible Flow
	10.1 Characterization
	10.2 Incompressible Flow as Low-Mach-Number Flow with Adiabatic Walls
	10.3 Nondimensional Problem Statement
	10.4 Characteristics of Incompressible Flow
	10.5 Splitting the Pressure into Kinetic and Hydrostatic Parts
	10.6 Mathematical Aspects of the Limit Process M2 → 0
	10.7 Invariance of Incompressible Flow Equations under Unsteady Motion
	10.8 Low-Mach-Number Flows with Constant- Temperature Walls
	10.9 Energy Equation Paradox
	10.10 Conclusions
	Problems
11 Some Solutions of the Navier–Stokes Equations
	11.1 Pressure-Driven Flow in Tubes of Various Cross Sections: Elliptical Tube
	11.2 Flow in a Rectangular Tube
	11.3 Asymptotic Suction Flow
	11.4 Stokes’s Oscillating Plate
	11.5 Wall under an Oscillating Free Stream
	11.6 Transient for a Stokes Oscillating Plate
	11.7 Flow in a Slot with a Steady and Oscillating Pressure Gradient
	11.8 Decay of an Ideal Line Vortex (Oseen Vortex)
	11.9 Plane Stagnation Point Flow (Hiemenz Flow)
	11.10 Burgers Vortex
	11.11 Composite Solution for the Rotary Viscous Coupling
	11.12 Von Kármán Viscous Pump
	11.13 Conclusions
	Problems
12 Streamfunctions and the Velocity Potential
	12.1 Streamlines
	12.2 Streamfunction for Plane Flows
	12.3 Flow in a Slot with Porus Walls
	12.4 Streamlines and Streamsurfaces for a Three-Dimensional Flow
	12.5 Vector Potential and the E2 Operator
	12.6 Stokes’s Streamfunction for Axisymmetric Flow
	12.7 Velocity Potential and the Unsteady Bernoulli Equation
	12.8 Flow Caused by a Sphere with Variable Radius
	12.9 Conclusions
	Problems
13 Vorticity Dynamics
	13.1 Vorticity
	13.2 Kinematic Results Concerning Vorticity
	13.3 Vorticity Equation
	13.4 Vorticity Diffusion
	13.5 Vorticity Intensification by Straining Vortex Lines
	13.6 Production of Vorticity at Walls
	13.7 Typical Vorticity Distributions
	13.8 Development of Vorticity Distributions
	13.9 Helmholtz’s Laws for Inviscid Flow
	13.10 Kelvin’s Theorem
	13.11 Vortex Definitions
	13.12 Inviscid Motion of Point Vortices
	13.13 Circular Line Vortex
	13.14 Fraenkel–Norbury Vortex Rings
	13.15 Hill’s Spherical Vortex
	13.16 Breaking and Reconnection of Vortex Lines
	13.17 Vortex Breakdown
	13.18 Conclusions
	Problems
14 Flows at Moderate Reynolds Numbers
	14.1 Some Unusual Flow Patterns
	14.2 Entrance Flows
	14.3 Entrance Flow into a Cascade of Plates: Computer Solution by the Streamfunction–Vorticity Method
	14.4 Entrance Flow into a Cascade of Plates: Pressure Solution
	14.5 Entrance Flow into a Cascade of Plates: Results
	14.6 Flow around a Circular Cylinder
	14.7 Jeffrey–Hamel Flow in a Wedge
	14.8 Limiting Case for Re → 0; Stokes Flow
	14.9 Limiting Case for R → − ∞
	14.10 Conclusions
	Problems
15 Asymptotic Analysis Methods
	15.1 Oscillation of a Gas Bubble in a Liquid
	15.2 Order Symbols, Gauge Functions, and Asymptotic Expansions
	15.3 Inviscid Flow Over a Wavy Wall
	15.4 Nonuniform Expansions: Friedrich’s Problem
	15.5 Matching Process: Van Dyke’s Rule
	15.6 Composite Expansions
	15.7 Characteristics of Overlap Regions and Common Parts
	15.8 Composite Expansions and Data Analysis
	15.9 Lagerstrom’s Problems
	15.10 Conclusions
	Problems
16 Characteristics of High-Reynolds-Number Flows
	16.1 Physical Motivation
	16.2 Inviscid Main Flows: Euler Equations
	16.3 Pressure Changes in Steady Flows: Bernoulli Equations
	16.4 Boundary Layers
	16.5 Conclusions
	Problems
17 Kinematic Decomposition of Flow Fields
	17.1 General Approach
	17.2 Helmholtz’s Decomposition; Biot–Savart Law
	17.3 Line Vortex and Vortex Sheet
	17.4 Complex Lamellar Decomposition
	17.5 Conclusions
	Problems
18 Ideal Flows in a Plane
	18.1 Problem Formulation For Plane Ideal Flows
	18.2 Simple Plane Flows
	18.3 Line Source and Line Vortex
	18.4 Flow Over a Nose or a Cliff
	18.5 Doublets
	18.6 Cylinder in a Stream
	18.7 Cylinder with Circulation in a Uniform Stream
	18.8 Lift and Drag on Two-Dimensional Shapes
	18.9 Magnus Effect
	18.10 Conformal Transformations
	18.11 Joukowski Transformation: Airfoil Geometry
	18.12 Kutta Condition
	18.13 Flow Over a Joukowski Airfoil: Airfoil Lift
	18.14 Numerical Method For Airfoils
	18.15 Actual Airfoils
	18.16 Schwarz–Christoffel Transformation
	18.17 Diffuser or Contraction Flow
	18.18 Gravity Waves in Liquids
	18.19 Conclusions
	Problems
19 Three-Dimensional Ideal Flows
	19.1 General Equations and Characteristics of Three-Dimensional Ideal Flows
	19.2 Swirling Flow Turned into an Annulus
	19.3 Flow Over a Weir
	19.4 Point Source
	19.5 Rankine Nose Shape
	19.6 Experiments on the Nose Drag of Slender Shapes
	19.7 Flow From a Doublet
	19.8 Flow Over a Sphere
	19.9 Work to Move a Body in a Still Fluid
	19.10 Wake Drag of Bodies
	19.11 Induced Drag: Drag Due to Lift
	19.12 Lifting Line Theory
	19.13 Winglets
	19.14 Added Mass of Accelerating Bodies
	19.15 Conclusions
	Problems
20 Boundary Layers
	20.1 Blasius Flow Over a Flat Plate
	20.2 Displacement Thickness
	20.3 Von Kármán Momentum Integral
	20.4 Von Kármán–Pohlhausen Approximate Method
	20.5 Falkner–Skan Similarity Solutions
	20.6 Arbitrary Two-Dimensinoal Layers: Crank–Nicolson Difference Method
	20.7 Vertical Velocity
	20.8 Joukowski Airfoil Boundary Layer
	20.9 Boundary Layer on a Bridge Piling
	20.10 Boundary Layers Beginning at Infinity
	20.11 Plane Boundary Layer Separation
	20.12 Axisymmteric Boundary Layers
	20.13 Jets
	20.14 Far Wake of Nonlifting Bodies
	20.15 Free Shear Layers
	20.16 Unsteady and Erupting Boundary Layers
	20.17 Entrance Flow into a Cascade, Parabolized Navier–Stokes Equations
	20.18 Three-Dimensional Boundary Layers
	20.19 Boundary Layer With a Constant Transverse Pressure Gradient
	20.20 Howarth’s Stagnation Points
	20.21 Three-Dimensional Separation Patterns
	20.22 Conclusions
	Problems
21 Flow at Low Reynolds Numbers
	21.1 General Relations for R→0: Stokes’s Equations
	21.2 Global Equations for Stokes Flow
	21.3 Streamfunction for Plane and Axisymmetric Flows
	21.4 Local Flows, Moffatt Vortices
	21.5 Plane Internal Flows
	21.6 Flows Between Rotating Cylinders
	21.7 Flows in Tubes, Nozzles, Orifices, and Cones
	21.8 Sphere in a Uniform Stream
	21.9 Composite Expansion for Flow Over a Sphere
	21.10 Stokes Flow Near a Circular Cylinder
	21.11 Axisymmetric Particles
	21.12 Oseen’s Equations
	21.13 Interference Effects
	21.14 Conclusions
	Problems
22 Lubrication Approximation
	22.1 Basic Characteristics: Channel Flow
	22.2 Flow in a Channel With a Porous Wall
	22.3 Reynolds Equation For Bearing Theory
	22.4 Slipper Pad Bearing
	22.5 Squeeze-Film Lubrication: Viscous Adhesion
	22.6 Journal Bearing
	22.7 Hele-Shaw Flow
	22.8 Conclusions
	Problems
23 Surface Tension Effects
	23.1 Interface Concepts and Laws
	23.2 Statics: Plane Interfaces
	23.3 Statics: Cylindrical Interfaces
	23.4 Statics: Attached Bubbles and Drops
	23.5 Constant-Tension Flows: Bubble in an Infinite Stream
	23.6 Constant-Tension Flows: Capillary Waves
	23.7 Moving Contact Lines
	23.8 Constant-Tension Flows: Coating Flows
	23.9 Marangoni Flows
	23.10 Conclusions
	Problems
24 Introduction to Microflows
	24.1 Molecules
	24.2 Continuum Description
	24.3 Compressible Flow in Long Channels
	24.4 Simple Solutions with Slip
	24.5 Gases
	24.6 Couette Flow in Gases
	24.7 Poiseuille Flow in Gases
	24.8 Gas Flow Over a Sphere
	24.9 Liquid Flows in Tubes and Channels
	24.10 Liquid Flows Near Walls; Slip Boundaries
	24.11 Conclusions
25 Stability and Transition
	25.1 Linear Stability and Normal Modes as Perturbations
	25.2 Kelvin–Helmholtz Inviscid Shear Layer Instability
	25.3 Stability Problems for Nearly Parallel Viscous Flows
	25.4 Orr–Sommerfeld Equation
	25.5 Inviscid Stability of Near ParallelFlows
	25.6 Viscous Stability of Nearly Parallel Flows
	25.7 Experiments on Blasius Boundary Layers
	25.8 Transition, Secondary, Instability, and Bypass
	25.9 Spatially Developing Open Flows
	25.10 Transition in Free Shear Flows
	25.11 Poiseuille and Plane Couette Flows
	25.12 Inviscid Instability of Flows with Curved Streamlines
	25.13 Taylor Instability of Couette Flow
	25.14 Stability of Regions of Concentrated Vorticity
	25.15 Other Instabilities: Taylor, Curved, Pipe, Capillary Jets, and Görtler
	25.16 Conclusions
26 Turbulent Flows
	26.1 Types of Turbulent Flows
	26.2 Characteristics of Turbulent Flows
	26.3 Reynolds Decomposition
	26.4 Reynolds Stress
	26.5 Counterrotating Vortices (Mushroom Shapes) on the Braids between Vortices Correlation of Fluctuations
	26.6 Mean and Turbulent Kinetic Energy
	26.7 Energy Cascade: Kolmogorov Scales and Taylor Microscale
	26.8 Wall Turbulence: Channel Flow Analysis
	26.9 Channel and Pipe Flow Experiments
	26.10 Boundary Layers
	26.11 Wall Turbulence: Fluctuations
	26.12 Turbulent Structures
	26.13 Free Turbulence: Plane Shear Layers
	26.14 Free Turbulence: Turbulent Jet
	26.15 Bifurcating and Blooming Jets
	26.16 Conclusions
27 Gas Dynamics
	27.1 Perfect Gases
	27.2 Speed of Sound and Mach Waves
	27.3 Classification of Flow Fields
	27.4 Gas Dynamics and the Navier-Stokes Equations
	27.5 Steady Flow Characteristics; Linear Theory
	27.6 Prandtl-Glauert Rule; Subsonic Flow
	27.7 Calculators and Reference States
	27.8 Quasi-One-Dimensional Flow with Area Change
	27.9 Normal Shock Waves
	27.10 Convergent-Divergent Nozzle Flow
	27.11 Rocket Motor Analysis
	27.12 Performance of the F1 Rocket Motor
	27.13 Rayleigh Flow; Heating-Cooling in Flow
	27.14 Fanno Flow; Adiabatic Pipe Flow with Wall Friction
	27.15 Pipe Flow; Isothermal Flow with Wall Friction
	27.16 Oblique Shock Waves
	27.17 Prandtl-Meyer Expansion Waves
	27.18 Shock-Expansion Method
	27.19 Combustion Waves; Detonation and Deflagration
	27.20 Conclusions
	Problems
References
Index
EULA