Microhydrodynamics and Complex Fluids

254 Pages·2012·16.009 MB·English

by  Barthès-Biesel, Dominique

B
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Mechanical Engineering r
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h
è
s
-B
ie
s
e
l Microhydrodynamics
Microhydrodynamics M
i and Complex Fluids
c
r
and Complex Fluids o
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y
d
r
o
A self-contained textbook, Microhydrodynamics and Complex
d
Fluids deals with the main phenomena that occur in slow,
y
inertialess viscous flows often encountered in various industrial,
n
biophysical and natural processes. It examines a wide range of
a
situations, from flows in thin films, porous media and narrow
m
channels to flows around suspended particles. Each situation is
illustrated with examples that can be solved analytically so that i
c
the main physical phenomena are clear. It also discusses a range
s
of numerical modelling techniques.

a
Two chapters deal with the flow of complex fluids, presented n
first with the formal analysis developed for the mechanics of d
suspensions and then with the phenomenological tools of non-

Newtonian fluid mechanics. All concepts are presented simply, C
with no need for complex mathematical tools. End-of-chapter o
exercises and exam problems help you test yourself. m
p
l
e
Dominique Barthès-Biesel has taught this subject for over 15
x
years and is well known for her contributions to low Reynolds

number hydrodynamics. Building on the basics of continuum F
mechanics, this book is ideal for graduate students specializing l
u
in chemical or mechanical engineering, material science,
i
bioengineering, and physics of condensed matter. d
s
K13875
ISBN 978-1-4398-8196-5
90000
9 781439 881965
K13875_COVER_final.indd 1 5/1/12 2:11 PM
Microhydrodynamics
and Complex Fluids
Dominique Barthès-Biesel
K13875_FM.indd 1 4/24/12 11:42 AM
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
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Version Date: 20120518
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Contents
List of Figures ix
List of Tables xv
Foreword xvii
About the Author xix
Symbol Description xxi
1 Fundamental Principles 1
1.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . 4
1.5 Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . 6
2 General Properties of Stokes Flows 7
2.1 Stationary Stokes Equations . . . . . . . . . . . . . . . . . . 8
2.1.1 Pressure–Velocity Relation . . . . . . . . . . . . . . . 8
2.1.2 Pressure–Vorticity Relation . . . . . . . . . . . . . . . 9
2.1.3 Boundary Conditions on a Solid Surface . . . . . . . . 9
2.2 Simple Stokes Flow Problem . . . . . . . . . . . . . . . . . . 10
2.3 Linearity and Reversibility . . . . . . . . . . . . . . . . . . . 11
2.3.1 Linearity Theorem . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Reversibility . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . 12
2.4.2 Demonstration . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Minimum Energy Dissipation . . . . . . . . . . . . . . . . . . 14
2.5.1 Minimum Energy Theorem . . . . . . . . . . . . . . . 14
2.5.2 Demonstration . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Reciprocal Theorem . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.1 Reciprocal Theorem . . . . . . . . . . . . . . . . . . . 16
iii
iv
2.6.2 Demonstration . . . . . . . . . . . . . . . . . . . . . . 16
2.6.3 Particular Cases of the Reciprocal Theorem . . . . . . 17
2.7 Solution in Terms of Harmonic Functions . . . . . . . . . . . 17
2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.1 Symmetry of Stokes Flow . . . . . . . . . . . . . . . . 19
2.8.2 EnergyDissipationDuetotheMotionofaSolidParticle
in a Quiescent Fluid . . . . . . . . . . . . . . . . . . . 19
2.8.3 Drag on a Particle in a Reservoir . . . . . . . . . . . . 20
3 Two-Dimensional Stokes Flows 21
3.1 Stream Function . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . 21
3.1.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . 23
3.2 Two-Dimensional Stokes Momentum Equation . . . . . . . . 23
3.3 Wedge with a Moving Boundary . . . . . . . . . . . . . . . . 24
3.3.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . 28
3.3.3 Force on the Scraper . . . . . . . . . . . . . . . . . . . 28
3.3.4 Other Wedge Flows with Moving Boundary . . . . . . 29
3.4 Flow in Fixed Wedges . . . . . . . . . . . . . . . . . . . . . . 30
3.4.1 Large Wedge Angle (2α>146.3o) . . . . . . . . . . . 32
3.4.2 Small Wedge Angle (2α<146.3o) . . . . . . . . . . . 34
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.1 Closing a Wedge . . . . . . . . . . . . . . . . . . . . . 37
3.5.2 Flow in an Unbounded Cavity . . . . . . . . . . . . . 38
3.5.3 Flow near a Stagnation Point on a Plane Wall . . . . 38
4 Lubrification Flows 41
4.1 Two-Dimensional Lubrication Flows . . . . . . . . . . . . . . 42
4.1.1 Orders of Magnitude and Approximations . . . . . . . 42
4.1.2 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.3 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.4 Two-Dimensional Reynolds Equation. . . . . . . . . . 45
4.1.5 Example: Slider Bearing . . . . . . . . . . . . . . . . . 46
4.2 Three-Dimensional Lubrication Flows . . . . . . . . . . . . . 49
4.2.1 Three-Dimensional Reynolds Equation . . . . . . . . . 49
4.2.2 Example: Squeezing a Drop . . . . . . . . . . . . . . . 51
4.3 Flow between Fixed Solid Boundaries . . . . . . . . . . . . . 53
4.3.1 Hele–Shaw Flow . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Flow in Microfluidic Channels . . . . . . . . . . . . . . 55
4.4 Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Definition of a Porous Medium . . . . . . . . . . . . . 59
4.4.2 Flow in a Porous Medium . . . . . . . . . . . . . . . . 60
v
4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.1 Lubrication Flow in a Journal Bearing . . . . . . . . 63
4.5.2 Flow in an Ideal Porous Membrane . . . . . . . . . . . 64
4.5.3 Flow along a Porous Wall . . . . . . . . . . . . . . . . 66
4.5.4 Motion of a Cylinder between Two ParallelWalls . . . 67
5 Free Surface Films 71
5.1 Interface between Two Immiscible Fluids . . . . . . . . . . . 72
5.1.1 Geometry of a Free Surface . . . . . . . . . . . . . . . 72
5.1.2 Kinematic Conditions . . . . . . . . . . . . . . . . . . 73
5.1.3 Dynamic Conditions: Laplace’s Law . . . . . . . . . . 73
5.1.4 Air–Liquid Interface . . . . . . . . . . . . . . . . . . . 74
5.2 Gravity Spreading of a Fluid on a Horizontal Plane . . . . . 74
5.2.1 Spreading with No Surface Tension . . . . . . . . . . . 75
5.2.2 Effect of Surface Tension . . . . . . . . . . . . . . . . 78
5.3 Stability of a Film Flowing Down an Inclined Plane . . . . . 79
5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 Gravity Spreading of a Film with Continuous Flux . . 82
5.4.2 Two-Dimensional Thermocapillary Flow . . . . . . . 83
5.4.3 Motion of a Contact Lens on the Eye . . . . . . . . . 84
6 Motion of a Solid Particle in a Fluid 87
6.1 Motion of a Solid Particle in a Quiescent Fluid . . . . . . . . 88
6.1.1 Resistance and Mobility Tensors . . . . . . . . . . . . 89
6.1.2 Relations between the Resistance and Mobility Tensors 90
6.1.3 Translation without Rotation . . . . . . . . . . . . . . 92
6.1.4 Rotation without Translation . . . . . . . . . . . . . . 93
6.2 Isotropic Particles . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Flow around a Translating Sphere . . . . . . . . . . . . . . . 95
6.3.1 Stream Function . . . . . . . . . . . . . . . . . . . . . 96
6.3.2 Velocity and Vorticity . . . . . . . . . . . . . . . . . . 98
6.3.3 Force Exerted on the Sphere . . . . . . . . . . . . . . 98
6.3.4 Streamline Pattern . . . . . . . . . . . . . . . . . . . . 100
6.4 Flow around a Rotating Sphere . . . . . . . . . . . . . . . . 101
6.5 Slender Particles . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.1 Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.2 Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.6.1 Sedimentation of a Particle with a Symmetry Plane . 105
6.6.2 Swimming Motion of Spheres and Bacteria . . . . . . 105
6.6.3 Micro-OrganismPropulsion by Means of a Flagellum. 107
6.6.4 Solid Particle in a General Flow Field . . . . . . . . . 109
vi
7 Flow of Bubbles and Droplets 111
7.1 Freely Suspended Liquid Drop . . . . . . . . . . . . . . . . . 112
7.1.1 General Problem Statement . . . . . . . . . . . . . . . 112
7.1.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . 113
7.2 Translation of a Bubble in a Quiescent Fluid . . . . . . . . . 114
7.2.1 Problem Equations . . . . . . . . . . . . . . . . . . . . 114
7.2.2 Solution in Terms of a Stream Function . . . . . . . . 115
7.2.3 Shape of the Bubble . . . . . . . . . . . . . . . . . . . 116
7.3 Translation of a Liquid Drop in a Quiescent Fluid . . . . . . 117
7.3.1 Hadamard–Rybczynski Drag on a Drop . . . . . . . . 117
7.3.2 Stability of the Stokes Solution . . . . . . . . . . . . . 119
7.3.3 Validity Limits of the Stokes Solution . . . . . . . . . 119
7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.4.1 Thermocapillary Motion of a Gas Bubble . . . . . . . 122
7.4.2 Flow in a Droplet Due to an Electric Field . . . . . . 123
8 General Solutions of the Stokes Equations 127
8.1 Flow Due to a Point Force . . . . . . . . . . . . . . . . . . . 128
8.1.1 Stokeslet . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.1.1.1 Demonstration . . . . . . . . . . . . . . . . . 129
8.1.2 Stokeslet Properties . . . . . . . . . . . . . . . . . . . 130
8.1.3 Correspondence with the Flow around a Sphere . . . . 131
8.1.4 Solutions Derived from a Stokeslet . . . . . . . . . . . 132
8.2 Irrotational Solutions . . . . . . . . . . . . . . . . . . . . . . 134
8.3 Series of Fundamental Solutions: Singularity Method . . . . 135
8.3.1 Singularities for External Flows . . . . . . . . . . . . . 136
8.3.2 Singularities for Internal Flows . . . . . . . . . . . . . 137
8.3.3 Example: Translating Gas Bubble . . . . . . . . . . . 140
8.3.4 Applications of the Singularity Method . . . . . . . . 141
8.4 Integral Form of the Stokes Equations . . . . . . . . . . . . . 142
8.4.1 Velocity Field in . . . . . . . . . . . . . . . . . . . . 143
D
8.4.2 Velocity Field on the Boundary ∂ . . . . . . . . . . . 144
D
8.4.2.1 Demonstration . . . . . . . . . . . . . . . . . 145
8.4.3 Boundary Integral Method . . . . . . . . . . . . . . . 145
8.4.4 Motion of a Solid Particle . . . . . . . . . . . . . . . . 146
8.4.5 Applications of the Boundary Integral Method . . . . 147
8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.5.1 Liquid Droplet Translating in a Quiescent Liquid . . . 150
8.5.2 Solid Sphere Freely Suspended in a Linear Shear Flow 150
8.5.3 Hydrodynamic Interaction between Three Spheres . . 151
8.5.4 Integral Equation for the Flow around a Solid Particle 153
8.5.5 Integral Equation for the Flow around a Liquid Drop 154
vii
9 Introduction to Suspension Mechanics 155
9.1 Homogenisation of a Suspension . . . . . . . . . . . . . . . . 156
9.2 Micro–Macro Relationship . . . . . . . . . . . . . . . . . . . 157
9.3 Dilute Suspension . . . . . . . . . . . . . . . . . . . . . . . . 160
9.3.1 Dilute Suspension of Identical Spheres . . . . . . . . . 160
9.3.2 Dilute Suspension of Anisotropic Particles . . . . . . . 161
9.3.3 Approximation O(c2) to the Viscosity of a Suspension
of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.4 Highly Concentrated Suspension of Spheres . . . . . . . . . . 165
9.5 Numerical Modelling of a Suspension . . . . . . . . . . . . . 166
9.5.1 Global Mobility and Resistance Tensors . . . . . . . . 167
9.5.2 Application to a Suspension of Spherical Particles . . 168
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.7.1 Constitutive Law of a Suspension of Spheres . . . . . 172
9.7.2 Intrinsic Convection in a Suspension . . . . . . . . . . 173
10 O(Re) Correction to Some Stokes Solutions 177
10.1 Translation of a Sphere: Oseen Correction . . . . . . . . . . . 177
10.2 Translation of a Cylinder: Stokes Paradox . . . . . . . . . . . 180
10.3 Validity Limits of the Stokes Approximation . . . . . . . . . 182
10.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.4.1 Evaluation of the Reynolds Number . . . . . . . . . . 183
11 Non-Newtonian Fluids 185
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.2 Non-Newtonian Fluid Mechanics . . . . . . . . . . . . . . . . 187
11.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . 187
11.2.2 Formulation of a Constitutive Law . . . . . . . . . . . 188
11.2.3 Viscometric Parameters in Simple Shear Flow . . . . . 190
11.3 Viscous Non-Newtonian Liquid . . . . . . . . . . . . . . . . . 191
11.3.1 Reiner–Rivlin Fluid . . . . . . . . . . . . . . . . . . . 191
11.3.2 Fluid with a Yield Stress . . . . . . . . . . . . . . . . 193
11.4 Viscoelastic fluid . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.4.1 Relative Deformation . . . . . . . . . . . . . . . . . . 195
11.4.2 General Constitutive Law . . . . . . . . . . . . . . . . 197
11.5 Linear Viscoelastic Laws . . . . . . . . . . . . . . . . . . . . 197
11.5.1 Maxwell Fluid . . . . . . . . . . . . . . . . . . . . . . 197
11.5.2 Generalised Maxwell Fluid . . . . . . . . . . . . . . . 199
11.5.2.1 Proof of Equation (11.39) . . . . . . . . . . . 200
11.5.3 Linear Viscoelastic Law: Integral Form . . . . . . . . 201
11.6 Non-Linear Viscoelastic Laws . . . . . . . . . . . . . . . . . . 202
viii
11.6.1 Non-Linear Integral Laws . . . . . . . . . . . . . . . . 202
11.6.2 Non-Linear Differential Laws . . . . . . . . . . . . . . 203
11.7 Non-Newtonian Flow Examples . . . . . . . . . . . . . . . . 204
11.7.1 Stationary 2D Flow between Two ParallelPlates . . . 204
11.7.1.1 Example (a): Power Law Fluid . . . . . . . . 205
11.7.1.2 Example (b): Bingham Fluid . . . . . . . . . 207
11.7.1.3 Example (c): Unknown Constitutive Law . . 208
11.7.2 Oscillatory Flow of a Viscoelastic Liquid . . . . . . . . 209
11.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.9.1 Flow of a Bingham Fluid in a Cylindrical Tube . . . . 212
11.9.2 Weissenberg Effect . . . . . . . . . . . . . . . . . . . . 213
Appendix A Notations 217
A.1 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . 217
A.2 Einstein Summation Convention . . . . . . . . . . . . . . . . 218
A.3 Integration on a Sphere . . . . . . . . . . . . . . . . . . . . . 219
Appendix B Curvilinear Coordinates 221
B.1 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . 221
B.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . 224
Bibliography 227
Index 231
List of Figures
1.1 Definition of a fluid domain bounded by ∂ . . . . . . . . 5
D D
2.1 Stokesflowinthefluiddomain boundedby∂ ∂ ∂ . 10
0 1 2
D D ∪ D ∪ D
2.2 Definition of the fluid domain. The velocity is prescribed on
the boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Stokes flow around a solid particle with a symmetry plane. . 19
3.1 2D Stokes flow in the xy-plane in terms of Cartesian(x,y) or
polar (r,θ) coordinates.. . . . . . . . . . . . . . . . . . . . . 22
3.2 Stokes flow near a wall with longitudinal obstacles. . . . . . 24
3.3 Definition of the Taylor paint-scraper. . . . . . . . . . . . . 25
3.4 Sketch of the streamlines for different values of Ψ and com-
parison with an exact solution for α=π/2. . . . . . . . . . 27
3.5 Forcesonthescraperandonthefluid.Thestreamlinepattern
is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Belt plunging into a quiescent liquid. . . . . . . . . . . . . . 30
3.7 Examples of flow near 2D grooves or protrusions. . . . . . . 30
3.8 Flow in a fixed wedge of angle 2α.. . . . . . . . . . . . . . . 31
3.9 Graphical representation of Equation (3.55). . . . . . . . . . 33
3.10 Streamlines forα>73o:(a)antisymmetricflow;(b) symmet-
ric flow; (c) arbitrary flow. . . . . . . . . . . . . . . . . . . . 33
3.11 Streamlines forα<73o:(a)antisymmetricflow;(b) symmet-
ric flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.12 Experimental observation of corner eddies for 2α=28.5o . . 36
3.13 Corner eddies in cavities with different width to height ratios 36
3.14 Closing of a wedge with angle 2α at time t. . . . . . . . . . 37
3.15 Flow in an unbounded cavity with parallel walls. . . . . . . 38
3.16 Flow near a stagnation point on a flat plate. . . . . . . . . . 39
4.1 Two-dimensional lubrication flow between two solid walls. . 42
4.2 Mass balance over a small film element. . . . . . . . . . . . 45
4.3 Two-dimensional slider bearing. . . . . . . . . . . . . . . . . 46
4.4 2D slider bearing: (a) velocity profile in the gap; (b) pressure
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Sector thrust bearings used on rotor shafts: example of a
Michell thrust bearing with patented pads anchor system. . 49
ix