Acknowledgments ............................................... xix
1. A Preview .................................................... 1
A. A Brief Historical Perspective of Transport Phenomena
in Chemical Engineering ................................... 1
В. The Nature of the Subject ................................. 2
С. A Brief Description of the Contents of This Book .......... 4
Notes and References ......................................... 1
2. Basic Principles ............................................ 13
A. The Continuum Approximation .............................. 13
1. Foundations ........................................... 14
2. Consequences .......................................... 15
В. Conservation of Mass - The Continuity Equation ........... 18
С. Newton's Laws of Mechanics ............................... 25
D. Conservation of Energy and the Entropy Inequality ........ 31
E. Constitutive Equations ................................... 36
F. Fluid Statics - The Stress Tensor for a Stationary
Fluid .................................................... 37
G. The Constitutive Equation for the Heat Flux Vector -
Fourier's Law ............................................ 42
H. Constitutive Equations for a Flowing Fluid - The
Newtonian Fluid .......................................... 45
I. The Equations of Motion for a Newtonian Fluid - The
Navier-Stokes Equation ................................... 49
J. Complex Fluids - Origins of Non-Newtonian Behavior ....... 52
К. Constitutive Equations for Non-Newtonian Fluids .......... 59
L. Boundary Conditions at Solid Walls and Fluid
Interfaces ............................................... 65
1. The Kinematic Condition ............................... 67
2. Thermal Boundary Conditions ........................... 68
3. The Dynamic Boundary Condition ........................ 69
M. Further Considerations of the Boundary Conditions
at the Interface Between Two Pure Fluids - The Stress
Conditions ............................................... 74
1. Generalization of the Kinematic Boundary Condition
for an Interface ...................................... 75
2. The Stress Conditions ................................. 76
3. The Normal-Stress Balance and Capillary Flows ......... 79
4. The Tangential-Stress Balance and Thermocapillary
Flows ................................................. 84
N. The Role of Surfactants in the Boundary Conditions at
a Fluid Interface ........................................ 89
Notes and Reference ......................................... 96
Problems .................................................... 99
3. Unidirectional and One-Dimensional Flow and Heat Transfer
Problems ................................................... 110
A. Simplification of the Navier-Stokes Equations for
Unidirectional Flows .................................... 113
В. Steady Unidirectional Flows - Nondimensionalization
and Characteristic Scales I ............................. 115
С. Circular Couette Flow - A One-Dimensional Analog to
Unidirectional Flows .................................... 125
D. Start-Up Flow in a Circular Tube - Solution by
Separation of Variables ................................. 135
E. The Rayleigh Problem - Solution by Similarity
Transformation .......................................... 142
F. Start-Up of Simple Shear Flow ........................... 148
G. Solidification at a Planar Interface .................... 152
H. Heat Transfer in Unidirectional Flows ................... 157
1. Steady-State Heat Transfer in Fully Developed
Flow through a Heated (or Cooled) Section of
a Circular Tube ...................................... 158
2. Taylor Diffusion in a Circular Tube .................. 166
I Pulsatile Flow in a Circular Tube .................. 175
Notes ...................................................... 183
Problems ................................................... 185
4. An Introduction to Asymptotic Approximations ............... 204
A. Pulsatile Flow in a Circular Tube Revisited -
Asymptotic Solutions for High and Low Frequencies ....... 205
1. Asymptotic Solution for Rw << I ...................... 206
2. Asymptotic Solution for Rw << I ...................... 209
В. Asymptotic Expansions - General Considerations .......... 216
С. The Effect of Viscous Dissipation on a Simple Shear
Flow .................................................... 219
D. The Motion of a Fluid Through a Slightly Curved Tube -
The Dean Problem ........................................ 224
E. Flow in a Wavy-Wall Channel - "Domain Perturbation
Method" ................................................. 232
1. Flow Parallel to the Corrugation Grooves ............. 233
2. Flow Perpendicular to the Corrugation Grooves ........ 237
F. Diffusion in a Sphere with Fast Reaction - "Singular
Perturbation Theory" .................................... 242
G. Bubble Dynamics in a Quiescent Fluid .................... 250
1. The Rayleigh-Plesset Equation ........................ 251
2. Equilibrium Solutions and Their Stability ............ 255
3. Bubble Oscillations Due to Periodic Pressure
Oscillations -Resonance and "Multiple-Time-Scale
Analysis" ............................................ 260
4. Stability to Nonspherical Disturbances ............... 269
Notes ...................................................... 282
Problems ................................................... 284
5. The Thin-Gap Approximation - Lubrication Problems .......... 294
A. The Eccentric Cylinder Problem .......................... 295
1. The Narrow-Gap Limit - Governing Equations and
Solutions ............................................ 297
2. Lubrication Forces ................................... 303
В. Derivation of the Basic Equations of Lubrication
Theory .................................................. 306
С. Applications of Lubrication Theory ...................... 315
1. The Slider-Block Problem ............................. 315
2. The Motion of a Sphere Toward a Solid, Plane
Boundary ............................................. 320
D. The Air Hockey Table .................................... 325
1. The Lubrication Limit, Re << I ....................... 328
2. The Uniform Blowing Limit, P*R >> I .................. 332
a Re << I ............................................ 334
b Re >> I ............................................ 336
с Lift on the Disk ................................... 345
Notes ...................................................... 346
Problems ................................................... 347
6. The Thin-Gap Approximation - Films with a Free Surface ..... 355
A. Derivation of the Governing Equations ................... 355
1. The Basic Equations and Boundary Conditions .......... 355
2. Simplification of the Interface Boundary
Conditions for a Thin Film ........................... 359
3. Derivation of the Dynamical Equation for the Shape
Function, h(x,s,t) ................................... 360
В. Self-Similar Solutions of Nonlinear Diffusion
Equations ............................................... 362
С. Films with a Free Surface - Spreading Films on
a Horizontal Surface .................................... 367
1. Gravitational Spreading .............................. 367
2. Capillary Spreading .................................. 371
D. The Dynamics of a Thin Film in the Presence of van
der Waals Forces ........................................ 376
1. Linear Stability ..................................... 378
2. Similarity Solutions for Film Rupture ................ 381
E. Shallow-Cavity Flows .................................... 385
1. The Horizontal, Enclosed Shallow Cavity .............. 386
2. The Horizontal Shallow Cavity with a Free Surface .... 391
a Solution by means of the classical thin-film
analysis ........................................... 392
b Solution by means of the method of domain
perturbations ...................................... 396
с The end regions .................................... 401
3. Thermocapillary Flow in a Thin Cavity ................ 404
a Thin-film solution procedure ....................... 410
b Solution by domain perturbation for S = I .......... 413
Notes ...................................................... 418
Problems ................................................... 418
7. Creeping Flow - Two-Dimensional and Axisymmetric
Problems ................................................... 429
A. Nondimensionalization and the Creeping-Flow Equations ... 430
В. Some General Consequences of Linearity and the
Creeping-Flow Equations ................................. 434
1. The Drag on Bodies That Are Mirror Images in
the Direction of Motion .............................. 434
2. The Lift on a Sphere That is Rotating in a Simple
Shear Flow ........................................... 436
3. Lateral Migration of a Sphere in Poiseuille Flow ..... 438
4. Resistance Matrices for the Force and Torque on
a Body in Creeping Flow .............................. 439
С. Representation of Two-Dimensional and Axisymmetric
Flows in Terms of the Streamfunction .................... 444
D. Two-Dimensional Creeping Flows: Solutions by Means of
Eigenfunction Expansions (Separation of Variables) ...... 449
1. General Eigenfunction Expansions in Cartesian and
Cylindrical Coordinates .............................. 449
2. Application to Two-Dimensional Flow near Corners ..... 451
E. Axisymmetric Creeping Flows: Solution by Means of
Eigenfunction Expansions in Spherical Coordinates
(Separation of Variables) ............................... 458
1. General Eigenfunction Expansion ...................... 459
2. Application to Uniform Streaming Flow past an
Arbitrary Axisymmetric Body .......................... 464
F. Uniform Streaming Flow past a Solid Sphere - Stokes'
Law ..................................................... 466
G. A Rigid Sphere in Axisymmetric, Extensional Flow ........ 470
1. The Flow Field ....................................... 470
2. Dilute Suspension Rheology - The Einstein Viscosity
Formula .............................................. 473
H. Translation of a Drop Through a Quiescent Fluid at
Low Re .................................................. 477
I. Marangoni Effects on the Motion of Bubbles and Drops .... 486
J. Surfactant Effects on the Buoyancy-Driven Motion
of a Drop ............................................... 490
1. Governing Equations and Boundary Conditions for a
Translating Drop with Surfactant Adsorbed at the
Interface ............................................ 493
2. The Spherical-Cap Limit .............................. 497
3. The Limit of Fast Adsorption Kinetics ................ 503
Notes ...................................................... 510
Problems ................................................... 512
8. Creeping Flow - Three-Dimensional Problems ................. 524
A. Solutions by Means of Superposition of Vector Harmonic
Functions ............................................... 525
1. Preliminary Concepts ................................. 525
a Vector "equality" - pseudo-vectors ................. 525
b Representation theorem for solution of the
creeping-flow equations ............................ 526
с Vector harmonic functions .......................... 527
2. The Rotating Sphere in a Quiescent Fluid ............. 528
3. Uniform Flow past a Sphere ........................... 529
В. A Sphere in a General Linear Flow ....................... 530
С. Deformation of a Drop in a General Linear Flow .......... 537
D. Fundamental Solutions of the Creeping-Flow Equations .... 545
1. The "Stokeslet": A Fundamental Solution for the
Creeping-Flow Equations .............................. 545
2. An Integral Representation for Solutions of
the Creeping-Flow Equations due to Ladyzhenskaya ..... 547
E. Solutions for Solid Bodies by Means of Internal
Distributions of Singularities .......................... 550
1. Fundamental Solutions for a Force Dipole and Other
Higher-Order Singularities ........................... 551
2. Translation of a Sphere in a Quiescent Fluid
(Stokes' Solution) ................................... 554
3. Sphere in Linear Flows: Axisymmetric Extensional
Flow and Simple Shear ................................ 555
4. Uniform Flow past a Prolate Spheroid ................. 557
5. Approximate Solutions of the Creeping-Flow
Equations by Means of Slender-Body Theory ............ 560
F. The Boundary Integral Method ............................ 564
1. A Rigid Body in an Unbounded Domain .................. 565
2. Problems Involving a Fluid Interface ................. 565
3. Problems in a Bounded Domain ......................... 568
G. Further Topics in Creeping-Flow Theory .................. 570
1. The Reciprocal Theorem ............................... 571
2. Faxen's Law for a Body in an Unbounded Fluid ......... 571
3. Inertial and Non-Newtonian Corrections to the Force
on a Body ............................................ 573
4. Hydrodynamic Interactions Between Widely Separated
Particles - The Method of Reflections ................ 576
Notes ...................................................... 580
Problems ................................................... 582
9. Convection Effects in Low-Reynolds-Number Flows ............ 593
A. Forced Convection Heat Transfer - Introduction .......... 593
1. General Considerations ............................... 594
2. Scaling and the Dimensionless Parameters for
Convective Heat Transfer ............................. 596
3. The Analogy with Single-Solute Mass Transfer ......... 598
В. Heat Transfer by Conduction (Pe → 0) ................... 600
С. Heat Transfer from a Solid Sphere in a Uniform
Streaming Flow at Small, but Nonzero, Peclet Numbers .... 602
1. Introduction - Whitehead's Paradox ................... 602
2. Expansion in the Inner Region ........................ 605
3. Expansion in the Outer Region ........................ 606
4. A Second Approximation in the Inner Region ........... 611
5. Higher-Order Approximations .......................... 613
6. Specified Heat Flux. 615 D. Uniform Flow past a
Solid Sphere at Small, but Nonzero, Reynolds
Number ............................................... 616
E. Heat Transfer from a Body of Arbitrary Shape in a
Uniform Streaming Flow at Small, but Nonzero, Peclet
Numbers ................................................. 627
F. Heat Transfer from a Sphere in Simple Shear Flow at
Low Peclet Numbers ...................................... 633
G. Strong Convection Effects in Heat and Mass Transfer
at Low Reynolds Number - An Introduction ................ 643
H. Heat Transfer from a Solid Sphere in Uniform Flow for
Re << I and Pe >> I ..................................... 645
1. Governing Equations and Rescaling in the Thermal
Boundary-Layer Region ................................ 648
2. Solution of the Thermal Boundary-Layer Equation ...... 652
I. Thermal Boundary-Layer Theory for Solid Bodies of
Nonspherical Shape in Uniform Streaming Flow ............ 656
1. Two-Dimensional Bodies ............................... 659
2. Axisymmetric Bodies .................................. 661
3. Problems with Closed Streamlines (or Stream
Surfaces) ............................................ 662
J. Boundary-Layer Analysis of Heat Transfer from a Solid
Sphere in Generalized Shear Flows at Low Reynolds
Number .................................................. 663
К. Heat (or Mass) Transfer Across a Fluid Interface for
Large Peclet Numbers .................................... 666
1. General Principles ................................... 666
2. Mass Transfer from a Rising Bubble or Drop in
a Quiescent Fluid .................................... 668
L. Heat Transfer at High Peclet Number Across Regions of
Closed-Streamline Flow .................................. 671
1. General Principles ................................... 671
2. Heat Transfer from a Rotating Cylinder in Simple
Shear Flow ........................................... 672
Notes ...................................................... 680
Problems ................................................... 681
10.Laminar Boundary-Layer Theory .............................. 697
A. Potential-Flow Theory ................................... 698
В. The Boundary-Layer Equations ............................ 704
С. Streaming Flow past a Horizontal Flat Plate -
The Blasius Solution .................................... 713
D. Streaming Flow past a Semi-Infinite Wedge -
The Falkner-Skan Solutions .............................. 719
E. Streaming Flow past Cylindrical Bodies - Boundary-
Layer Separation ........................................ 725
F. Streaming Flow past Axisymmetric Bodies -
A Generalizaiton of the Blasius Series .................. 733
G. The Boundary-Layer on a Spherical Bubble ................ 739
Notes ...................................................... 754
Problems ................................................... 756
11.Heat and Mass Transfer at Large Reynolds Number ............ 767
A. Governing Equations (Re 3 > I, Pe 3 > I, with
Arbitrary Pr or Sc numbers) ............................. 769
В. Exact (Similarity) Solutions for Pr (or Sc) ≈ 0(I) ...... 771
С. The Asymptotic Limit, Pr (or Sc) >> I ................... 773
D. The Asymptotic Limit, Pr (or Sc) << I ................... 780
E. Use of the Asymptotic Results at Intermediate
Pe (or Sc) .............................................. 787
F. Approximate Results for Surface Temperature with
Specified Heat Flux or Mixed Boundary Conditions ........ 788
G. Laminar Boundary-Layer Mass Transfer for Finite
Interfacial Velocities .................................. 793
Notes ...................................................... 797
Problems ................................................... 797
12.Hydrodynamic Stability ..................................... 800
A. Capillary Instability of a Liquid Thread ................ 801
1. The Inviscid Limit ................................... 804
2. Viscous Effects on Capillary Instability ............. 808
3. Final Remarks ........................................ 811
В. Rayleigh-Taylor Instability (The Stability of a Pair
of Immiscible Fluids That Are Separated by
a Horizontal Interface) ................................. 812
1. The Inviscid Fluid Limit ............................. 816
2. The Effects of Viscosity on the Stability of
a Pair of Superposed Fluids .......................... 818
3. Discussion ........................................... 822
С. Saffman-Taylor Instability at a Liquid Interface ........ 823
1. Darcy's Law .......................................... 823
2. The Taylor-Saffman Instability Criteria .............. 826
D. Taylor-Couette Instability .............................. 829
1. A Sufficient Condition for Stability of an
Inviscid Fluid ....................................... 832
2. Viscous Effects ...................................... 835
E. Nonisothermal and Compositionally Nonuniform Systems .... 840
F. Natural Convection in a Horizontal Fluid Layer Heated
from Below - The Rayleigh-Benard Problem ................ 845
1. The Disturbance Equations and Boundary Conditions .... 845
2. Stability for Two Free Surfaces ...................... 851
3. The Principle of Exchange of Stabilities ............. 853
4. Stability for Two No-Slip, Rigid Boundaries .......... 855
G. Double-Diffusive Convection ............................. 858
H. Marangoni Instability ................................... 867
I. Instability of Two-Dimensional Unidirectional Shear
Flows ................................................... 872
1. Inviscid Fluids ...................................... 873
a The Rayleigh stability equation .................... 873
b The Inflection-point theorem ....................... 875
2. Viscous Fluids ....................................... 876
a The Orr-Sommerfeld equation ........................ 876
b A sufficient condition for stability ............... 877
Notes ...................................................... 878
Problems ................................................... 880
Appendix A: Governing Equations and Vector Operations in
Cartesian, Cylindrical, and Spherical
Coordinate Systems ................................ 891
Appendix B: Cartesian Component Notation ...................... 897
Index ......................................................... 899
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