A Course in Fluid Mechanics
with Vector Field Theory
by

Dennis C. Prieve
Department of Chemical Engineering
Carnegie Mellon University
Pittsburgh, PA 15213

An electronic version of this book in Adobe PDF® format was made available to students of 06-703, Department of Chemical Engineering, Carnegie Mellon University, Fall, 1998.

Table of Contents

Algebra of Vectors and Tensors 1

Vector Multiplication 1

Definition of Dyadic Product 2

Decomposition into Scalar Components 3

Scalar Fields 4

Gradient of a Scalar 4

Geometric Meaning of the Gradient 6

Applications of Gradient 7

Curvilinear coordinates 7

Cylindrical Coordinates 7

Differentiation of Vectors w.r.t. Scalars 8

Vector Fields 10

Fluid Velocity as a Vector Field 11

Partial & Material Derivatives 12

Calculus of Vector Fields 14

Gradient of a Scalar (Explicit) 14

Divergence, Curl, and Gradient 15

Physical Interpretation of Divergence 16

Calculation of Ñ.v in R.C.C.S. 16

Evaluation of Ñ´v and Ñv in R.C.C.S. 18

Evaluation of Ñ.v, Ñ´v and Ñv in Curvilinear Coordinates 18

Physical Interpretation of Curl 19
Vector Field Theory 21

Divergence Theorem 22

Corollaries of the Divergence Theorem 22

The Continuity Equation 23

Reynolds Transport Theorem 24

Stokes Theorem 26

Velocity Circulation: Physical Meaning 26

Derivable from a Scalar Potential 28

Theorem III 29

Transpose of a Tensor, Identity Tensor 29

Divergence of a Tensor 31

Introduction to Continuum Mechanics* 32

Continuum Hypothesis 33

Classification of Forces 34

Hydrostatic Equilibrium 35

Flow of Ideal Fluids 36

Euler's Equation 36

Kelvin's Theorem 39

Irrotational Flow of an Incompressible Fluid 40

Potential Flow Around a Sphere 43

d'Alembert's Paradox 48
Stream Function 51

Two-D Flows 52

Axisymmetric Flow (Cylindrical) 53

Axisymmetric Flow (Spherical) 54

Orthogonality of y=const and f=const 55

Streamlines, Pathlines and Streaklines 55

Physical Meaning of Streamfunction 55

Incompressible Fluids 58

Viscous Fluids 60

Tensorial Nature of Surface Forces 60

Generalization of Euler's Equation 64

Momentum Flux 66

Response of Elastic Solids to Uniaxial Stress 68

Response of Elastic Solids to Pure Shear 70

Generalized Hooke's Law 71

Response of a Viscous Fluid to Pure Shear 73

Generalized Newton's Law of Viscosity 74

Navier-Stokes Equation 75

Boundary Conditions 76

Exact Solutions of N-S Equations 78

Problems with Zero Inertia 78

Flow in Long Straight Conduit of Uniform Cross Section 79

Flow of Thin Film Down Inclined Plane 82

Problems with Non-Zero Inertia 87

Rotating Disk* 87
Creeping Flow Approximation 89

Cone-and-Plate Viscometer 89

Creeping Flow Around a Sphere (Re®0) 94

Scaling 94

Velocity Profile 96

Displacement of Distant Streamlines 99

Pressure Profile 100

Correcting for Inertial Terms 103

Flow Around Cylinder as Re®0 106

Boundary-Layer Approximation 107

Flow Around Cylinder as Re® ¥ 107

Mathematical Nature of Boundary Layers 108

Matched-Asymptotic Expansions 112

MAE’s Applied to 2-D Flow Around Cylinder 117

Outer Expansion 118

Inner Expansion 119

Boundary Layer Thickness 122

Prandtl’s B.L. Equations for 2-D Flows 124

Alternate Method: Prandtl’s Scaling Theory 125

Solution for a Flat Plate 131

Time Out: Flow Next to Suddenly Accelerated Plate 133

Time In: Boundary Layer on Flat Plate 134

Boundary-Layer Thickness 135

Drag on Plate 138

Solution for a Symmetric Cylinder 139

Boundary-Layer Separation 142

Drag Coefficient and Behavior in the Wake of the Cylinder 143
The Lubrication Approximation 145

Translation of a Cylinder Along a Plate 152

Cavitation 154

Squeezing Flow 155

Reynolds Equation 159

Turbulence 164

General Nature of Turbulence 164

Turbulent Flow in Pipes 165

Time-Smoothing 167

Time-Smoothing of Continuity Equation 167

Time-Smoothing of the Navier-Stokes Equation 168

Analysis of Turbulent Flow in Pipes 169

Prandtl’s Mixing Length Theory 171

Prandtl’s "Universal" Velocity Profile 175

Prandtl’s Universal Law of Friction 178

Electrohydrodynamics 178

Origin of Charge 178

Gouy-Chapman Model of Double Layer 179

Electrostatic Body Forces 183

Electrokinetic Phenomena 184

Smoluchowski's Analysis (ca. 1918) 185

Electro-Osmosis in Cylindrical Pores 188

Electrophoresis 189

Streaming Potential 190

Surface Tension 192

Molecular Origin 192

Boundary Conditions for Fluid Flow 194

Index 199