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Table of Contents

ALGEBRA OF VECTORS AND TENSORS ~ 1
VECTOR MULTIPLICATION ~ 1
Definition of Dyadic Product ~ 2
DECOMPOSITION INTO SCALAR COMPONENTS ~ 3
SCALAR FIELDS ~ 3
GRADIENT OF A SCALAR ~ 4
Geometric Meaning of the Gradient ~ 6
Applications of Gradient ~ 7
CURVILINEAR COORDINATES ~ 7
Cylindrical Coordinates ~ 7
Spherical Coordinates ~ 8
DIFFERENTIATION OF VECTORS W.R.T. SCALARS ~ 9
VECTOR FIELDS ~ 11
Fluid Velocity as a Vector Field ~ 11

CALCULUS OF VECTOR FIELDS ~ 14
GRADIENT OF A SCALAR (EXPLICIT) ~ 14
DIVERGENCE, CURL, AND GRADIENT ~ 16
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 ~ 19

VECTOR FIELD THEORY ~ 22
DIVERGENCE THEOREM ~ 23
Corollaries of the Divergence Theorem ~ 24
The Continuity Equation ~ 24
Reynolds Transport Theorem ~ 26
STOKES THEOREM ~ 27
Velocity Circulation: Physical Meaning ~ 28
DERIVABLE FROM A SCALAR POTENTIAL ~ 29
THEOREM III ~ 31
TRANSPOSE OF A TENSOR, IDENTITY TENSOR ~ 31

INTRODUCTION TO CONTINUUM MECHANICS* ~ 34
CONTINUUM HYPOTHESIS ~ 34

FLOW OF IDEAL FLUIDS ~ 37
EULER'S EQUATION ~ 38
KELVIN'S THEOREM ~ 41
IRROTATIONAL FLOW OF AN INCOMPRESSIBLE FLUID ~ 42
Potential Flow Around a Sphere ~ 45
d'Alembert's Paradox ~ 50

STREAM FUNCTION ~ 53
TWO-D FLOWS ~ 54
AXISYMMETRIC FLOW (CYLINDRICAL) ~ 55
AXISYMMETRIC FLOW (SPHERICAL) ~ 56
ORTHOGONALITY OF y=CONST AND f=CONST ~ 57
STREAMLINES, PATHLINES AND STREAKLINES ~ 57
PHYSICAL MEANING OF STREAMFUNCTION ~ 58
INCOMPRESSIBLE FLUIDS ~ 60

VISCOUS FLUIDS ~ 62
TENSORIAL NATURE OF SURFACE FORCES ~ 62
GENERALIZATION OF EULER'S EQUATION ~ 66
MOMENTUM FLUX ~ 68
RESPONSE OF ELASTIC SOLIDS TO UNIAXIAL STRESS ~ 70
RESPONSE OF ELASTIC SOLIDS TO PURE SHEAR ~ 72
GENERALIZED HOOKE'S LAW ~ 73
RESPONSE OF A VISCOUS FLUID TO PURE SHEAR ~ 75
GENERALIZED NEWTON'S LAW OF VISCOSITY ~ 76
NAVIER-STOKES EQUATION ~ 77
BOUNDARY CONDITIONS ~ 78

EXACT SOLUTIONS OF N-S EQUATIONS ~ 80
PROBLEMS WITH ZERO INERTIA ~ 80
Flow in Long Straight Conduit of Uniform Cross Section ~ 81
Flow of Thin Film Down Inclined Plane ~ 84
PROBLEMS WITH NON-ZERO INERTIA ~ 89
Rotating Disk* ~ 89
CREEPING FLOW APPROXIMATION ~ 91
CONE-AND-PLATE VISCOMETER ~ 91

CREEPING FLOW AROUND A SPHERE (Re®0) ~ 96
Scaling ~ 97
Velocity Profile ~ 99
Displacement of Distant Streamlines ~ 101
Pressure Profile ~ 103
CORRECTING FOR INERTIAL TERMS ~ 106
FLOW AROUND CYLINDER AS RE®0 ~ 109

BOUNDARY-LAYER APPROXIMATION ~ 110
FLOW AROUND CYLINDER AS Re® ¥ ~ 110
MATHEMATICAL NATURE OF BOUNDARY LAYERS ~ 111
MATCHED-ASYMPTOTIC EXPANSIONS ~ 115
MAE’S APPLIED TO 2-D FLOW AROUND CYLINDER ~ 120
Outer Expansion ~ 120
Inner Expansion ~ 120
Boundary Layer Thickness ~ 120
PRANDTL’S B.L. EQUATIONS FOR 2-D FLOWS ~ 120
ALTERNATE METHOD: PRANDTL’S SCALING THEORY ~ 120
SOLUTION FOR A FLAT PLATE ~ 120
Time Out: Flow Next to Suddenly Accelerated Plate ~ 120
Time In: Boundary Layer on Flat Plate ~ 120
Boundary-Layer Thickness ~ 120
Drag on Plate ~ 120
06-703 3 Fall, 2000
SOLUTION FOR A SYMMETRIC CYLINDER ~ 120
Boundary-Layer Separation ~ 120
Drag Coefficient and Behavior in the Wake of the Cylinder ~ 120

THE LUBRICATION APPROXIMATION ~ 157
TRANSLATION OF A CYLINDER ALONG A PLATE ~ 163
CAVITATION ~ 166
SQUEEZING FLOW ~ 167
REYNOLDS EQUATION ~ 171
TURBULENCE ~ 176
GENERAL NATURE OF TURBULENCE ~ 176

TURBULENT FLOW IN PIPES ~ 177
TIME-SMOOTHING ~ 179
TIME-SMOOTHING OF CONTINUITY EQUATION ~ 180
TIME-SMOOTHING OF THE NAVIER-STOKES EQUATION ~ 180
ANALYSIS OF TURBULENT FLOW IN PIPES ~ 182
PRANDTL’S MIXING LENGTH THEORY ~ 184
PRANDTL’S "UNIVERSAL" VELOCITY PROFILE ~ 187
PRANDTL’S UNIVERSAL LAW OF FRICTION ~ 189

ELECTROHYDRODYNAMICS ~ 120
ORIGIN OF CHARGE ~ 120
GOUY-CHAPMAN MODEL OF DOUBLE LAYER ~ 120
ELECTROSTATIC BODY FORCES ~ 120
ELECTROKINETIC PHENOMENA ~ 120
SMOLUCHOWSKI'S ANALYSIS (CA. 1918) ~ 120
ELECTRO-OSMOSIS IN CYLINDRICAL PORES ~ 120
ELECTROPHORESIS ~ 120
STREAMING POTENTIAL ~ 120

SURFACE TENSION ~ 120
MOLECULAR ORIGIN ~ 120
BOUNDARY CONDITIONS FOR FLUID FLOW ~ 120

INDEX ~ 211