The mathematical foundations of mixing

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 15
Acknowledgments......Page 22
1 Mixing: physical issues......Page 23
1.1 Length and time scales......Page 25
Example: flow in a small channel......Page 26
Example: more on channel flow......Page 27
1.2 Stretching and folding, chaotic mixing......Page 28
1.3 Reorientation......Page 33
1.4 Diffusion and scaling......Page 34
1.5 Examples......Page 35
1.5.2 Samelson\'s tidal vortex advection model......Page 36
1.5.3 Chaotic stirring in tidal systems......Page 37
1.5.4 Cavity flows......Page 39
1.5.5 An electro-osmotic driven micromixer blinking flow......Page 40
1.5.6 Egg beater flows......Page 41
1.5.7 A blinking flow model of mixing of granular materials......Page 44
1.5.8 Mixing in DNA microarrays......Page 48
1.6 Mixing at the microscale......Page 50
2.1 Introduction......Page 53
2.2 Linked twist maps on the torus......Page 54
2.2.1 Geometry of mixing for toral LTMs......Page 57
Superimposed shear flows......Page 59
2.3 Linked twist maps on the plane......Page 64
2.3.1 Geometry of mixing for LTMs on the plane......Page 68
2.4 Constructing a LTM from a blinking flow......Page 70
2.5 Constructing a LTM from a duct flow......Page 71
Construction of the first twist map......Page 72
Action-angle-axial transformation in the half cell......Page 73
2.6 More examples of mixers that can be analysed in the LTMs framework......Page 75
3.1 Introduction......Page 81
3.2 Mathematical ideas for describing and quantifying the flow domain, and a \'blob\' of dye in the flow......Page 82
Metric space......Page 83
Normed vector space......Page 84
Topological space......Page 85
3.2.2 Describing sets of points......Page 86
3.2.3 Compactness and connectedness......Page 88
3.2.4 Measuring the \'size\' of sets......Page 89
3.3 Mathematical ideas for describing the movement of blobs in the flow domain......Page 92
Measure-preserving transformations......Page 94
3.4.1 Terminology for general fluid kinematics......Page 95
3.4.2 Specific types of orbits......Page 96
3.4.3 Behaviour near a specific orbit......Page 97
3.4.4 Sets of fluid particles that give rise to \'flow structures\'......Page 98
3.5 Fundamental results for measure-preserving dynamical systems......Page 100
3.6 Ergodicity......Page 102
3.6.1 A typical scheme for proving ergodicity......Page 106
3.7 Mixing......Page 108
3.8 The K-property......Page 115
3.9 The Bernoulli property......Page 116
SIGMAN is a compact metric space......Page 117
SIGMAN is a measure space......Page 120
3.9.2 The shift map......Page 121
3.9.3 What it means for a map to have the Bernoulli property......Page 123
The Baker’s transformation has the Bernoulli property......Page 124
3.10 Summary......Page 126
4.1 Introduction......Page 127
4.2.1 The standard horseshoe......Page 128
4.2.2 Symbolic dynamics......Page 131
4.2.3 Generalized horseshoes......Page 133
4.2.4 The Conley–Moser conditions......Page 134
4.3 Horseshoes in fluids......Page 135
4.4.1 A twist map on the plane......Page 137
4.4.2 Linking a pair of twist maps......Page 138
4.5.1 Construction of the invariant set Lambada j,k......Page 139
4.5.2 The subshift of finite type......Page 145
4.5.4 Hyperbolicity of LAMBDAj,k......Page 146
4.6 Summary......Page 147
5.1 Introduction......Page 148
5.2 Hyperbolicity definitions......Page 150
5.2.1 Uniform hyperbolicity......Page 151
Stable and unstable manifolds......Page 154
Results for uniform hyperbolicity......Page 156
The Arnold Cat Map......Page 157
Uniformly hyperbolic trajectories......Page 158
5.2.2 Nonuniform hyperbolicity......Page 159
5.2.4 Other hyperbolicity definitions......Page 161
5.3.1 Lyapunov exponents......Page 162
5.3.2 Lyapunov exponents and hyperbolicity......Page 165
5.3.3 Stable and unstable manifolds......Page 166
5.3.4 Ergodic decomposition......Page 167
5.3.5 Ergodicity......Page 168
5.3.6 Bernoulli components......Page 170
5.4 Smooth maps with singularities......Page 171
5.4.1 Katok–Strelcyn conditions......Page 172
5.4.2 Ergodicity and the Bernoulli property......Page 173
5.5.1 Invariant cones......Page 176
5.6 Summary......Page 180
6.1 Introduction......Page 181
6.2 Toral linked twist maps......Page 182
6.2.1 Twist maps on the torus......Page 183
6.2.2 Linking the twist maps......Page 186
6.2.3 First return maps......Page 188
6.2.6 Smooth twists......Page 190
6.2.8 Linear twists......Page 191
6.2.9 More general twists......Page 192
6.3.1 Co-rotating smooth toral linked twist maps......Page 193
6.4 The ergodic partition for toral linked twist maps with singularities......Page 202
6.4.1 Co-rotating toral linked twist maps with singularities......Page 204
6.4.2 Counter-rotating toral linked twist maps with singularities......Page 207
6.5 Summary......Page 215
7.1 Introduction......Page 216
7.2 Properties of line segments......Page 217
7.2.1 Definition, iteration and orientation of line segments......Page 218
Toral linked twist maps......Page 219
7.2.2 Growth of line segments......Page 222
7.2.3 Nu-segments and h-segments......Page 223
7.3 Ergodicity for the Arnold Cat Map......Page 226
7.4 Ergodicity for co-rotating toral linked twist maps......Page 227
7.5 Ergodicity for counter-rotating toral linked twist maps......Page 229
7.6 The Bernoulli property for toral linked twist maps......Page 235
7.7 Summary......Page 238
8.1 Introduction......Page 239
8.2.1 The annuli......Page 240
8.2.2 The twist maps......Page 241
8.2.3 Linked twist maps......Page 243
8.3 The ergodic partition......Page 244
8.3.1 Counter-rotating planar linked twist maps......Page 247
8.3.2 Co-rotating planar linked twist maps......Page 255
8.4 Ergodicity and the Bernoulli property for planar linked twist maps......Page 260
8.5 Summary......Page 261
9.1 Introduction......Page 262
9.2 Optimizing mixing regions for linked twist maps......Page 263
Counter-rotating toral linked twist maps......Page 264
Area of intersection of a pair of annuli......Page 267
The cotangent of alpha......Page 270
The counter-rotating planar LTM......Page 272
9.3 Breakdown of transversality: effect and mechanisms......Page 273
9.3.1 Separatrices......Page 277
9.3.2 Planar linked twist maps with a single intersection component......Page 278
9.3.3 More than two annuli......Page 279
9.4 Monotonicity of the twist functions......Page 282
9.4.1 Lack of monotonicity in toral linked twist maps......Page 284
9.4.2 Non-slip boundary conditions with breakdown of monotonicity......Page 285
9.5 Final remarks......Page 289
References......Page 293
Index......Page 301