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기타 참고사항

0 Summary of Fluid Mechanics 5
1 Vector and Tensor Analysis 9
1.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 Definition of Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Vector Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Vector Basis and Components . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 Dot (Scalar or Inner) Product of Two Vectors . . . . . . . . . . . . . . . . 11
1.1.5 Cross (Vector) Product of Two Vectors . . . . . . . . . . . . . . . . . . . 12

1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Definition of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Dyadic (Tensor or Outer) Product of Two Vectors . . . . . . . . . . . . . 14
1.2.3 Dot and Double Dot Products of Two Tensors . . . . . . . . . . . . . . . 14
1.2.4 Operations with Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . 16
1.2.5 Properties of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Vector Differential Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 Gradient of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 Gradient of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.3 Divergence of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.4 Divergence of a Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.5 Curl of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.6 Laplacian of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Vector Differential Operators in Other Coordinates . . . . . . . . . . . . . . . . . 21
1.4.1 Rectangular Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . 21
1.4.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Basic Concepts of Fluid Mechanics 27
2.1 Continuum Fluids and Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 Fluids and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Motion of a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Lagrangian and Eulerian Representations . . . . . . . . . . . . . . . . . . 28
2.2.2 Time Derivatives: Partial, Total, Material (Substantial) . . . . . . . . . . 30
2.2.3 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Steady and Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.5 Path Lines and Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Fundamental Physics Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Conservation of Mass . . . . . . . . . . . . .

3 Conservation of Mass 37
3.1 The Integral Form of Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 The Integral Mass Balance for a Steady-State Flow . . . . . . . . . . . . . 40
3.1.2 The Integral Mass Balance for an Incompressible Fluid . . . . . . . . . . . 40
3.2 The Differential Form of Mass Balance (Equation of Continuity) . . . . . . . . . 42
3.2.1 The Differential Mass Balance from the Integral Form . . . . . . . . . . . 42
3.2.2 The Differential Mass Balance from the Mass Balance over a Differential
Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Conservation of Linear Momentum and Angular Momentum 47
4.1 Linear Momentum and Angular Momentum (moment of momentum) . . . . . . . 48
4.2 The Integral Form of Linear Momentum Balance . . . . . . . . . . . . . . . . . . 49
4.3 Force, Stress and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.4 Types of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 The Differential Form of Linear Momentum Balance (Equation of Motion) . . . . 59
4.4.1 The Differential Linear Momentum Balance from the Integral Form . . . 60
4.4.2 The Differential Linear Momentum Balance from the Linear Momentum
Balance over a Differential Element . . . . . . . . . . . . . . . . . . . . . . 62

4.5 The Integral Form of Angular Momentum Balance . . . . . . . . . . . . . . . . . 65
4.6 The Differential Form of Angular Momentum Balance (Equation of Angular Momentum)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.7 Differential Mechanical Energy Balance (Equation of Mechanical Energy) . . . . 66
4.7.1 The Differential Mechanical Energy Balance for Incompressible Inviscid
Fluids - Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.7.2 Bernoulli Equation with Fluid Friction . . . . . . . . . . . . . . . . . . . . 69
4.7.3 Bernoulli Equation with a Pump . . . . . . . . . . . . . . . . . . . . . . . 70

5 Various Flows 73
5.1 Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1 Pressure Variation in a Static Fluid . . . . . . . . . . . . . . . . . . . . . 73
5.1.2 Uniform Rectilinear Acceleration . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Laminar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 No Slip Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.2 Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.3 Fully Developed Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.4 Flow in a Circular Pipe by Pressure Difference . . . . . . . . . . . . . . . 78

5.2.5 Flow down an Inclined-Plane Surface by Gravity . . . . . . . . . . . . . . 81
5.2.6 Flow between Two Plates by a Moving Plate (Couette Flow) . . . . . . . 84
5.2.7 Flow between Two Vertical Plates by a Moving Plate . . . . . . . . . . . 86

5.2.8 Flow in a Fluid by a Rotating Shaft . . . . . . . . . . . . . . . . . . . . . 86
5.3 Creeping Flow (Re ≪ 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Time-dependent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5 Inviscid and Irrotational Flows (ch. 10) . . . . . . . . . . . . . . . . . . . . . . . 94
5.5.1 Two-dimensional Irrotational Flow of an Incompressible Newtonian Fluid 94
5.6 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.6.1 Friction for Flow in Pipes and Channels . . . . . . . . . . . . . . . . . . . 98
5.7 Viscous Flows Past Solid Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7.1 Drag (Force) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.2 Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.7.3 Flow in a Packed Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.7.4 Flow in a Fluidized Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Dimensional Analysis 111
6.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3 The Buckingham Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.1 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A Incompressible Fluid 117
B Differential Mass and Momentum Balance for Incompressible Fluids 119
C Differential Mass and Momentum Balance for Incompressible Newtonian Fluids
121
D Irrotational Flow ∇× v = 0 125
E Derivation of Some of Equations 127
F Stream Function for Incompressible Fluids 129