Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach Book

Preface xi

1 Fluid Flows 1

1.1 Conservation laws 2

1.2 Laminar and turbulent flows 5

1.3 Turbulence 9

1.4 Kolmogorov's theory of turbulence 11

1.5 Two-dimensional flows 15

2 Mixing and Dispersion in Fluid Flows 20

2.1 Introduction 20

2.1.1 Advection 21

2.1.2 Diffusion 23

2.1.3 Advection and diffusion 30

2.2 Steady two-dimensional flows 31

2.2.1 Advection along streamlines 31

2.2.2 Dispersion of diffusive tracers in steady flows 35

2.3 Advection in weakly time-dependent two-dimensional flows 40

2.4 Chaotic advection in three dimensions 45

2.5 Dispersion by chaotic advection 48

2.5.1 The Lyapunov exponent 50

2.6 Chaotic advection in open flows 59

2.7 Chaotic advection and diffusion 64

2.7.1 The filament model 66

2.7.2 Asymptotic decay in chaotic flows 73

2.8 Mixing in turbulent flows 78

2.8.1 Relative dispersion in turbulence 78

2.8.2 Passive scalar in turbulent flows 81

2.9 Distribution of inertial particles in flows 85

3 Chemical and Ecological Models 89

3.1 Chemical dynamics 89

3.1.1 The Law of Mass Action 89

3.1.2 Binary, First-Order, and Zeroth-Order Reactions 92

3.1.3 Autocatalytic and Enzymatic Reactions: The adiabatic elimination 94

3.1.4 Oscillations and excitability 100

3.1.5 Multistability 108

3.2 Biological models 109

3.2.1 Simple birth, death and saturation 109

3.2.2 Predator-Prey models 111

3.2.3 Competition 117

3.3 Summary 123

4 Reaction-diffusion Dynamics 124

4.1 Diffusion and linear growth 124

4.1.1 Linear spreading of perturbations 125

4.1.2 The minimum habitat-size problem 128

4.1.3 Plankton filaments 129

4.2 Fisher waves 130

4.3 Multistability: Fronts advancing onmetastable states 137

4.4 Excitable waves 143

4.5 Turing diffusive instabilities 147

4.6 Oscillatory media and beyond 149

5 Fast Binary Reactions and the Lamellar Approach 151

5.1 Lamellar reacting models 151

5.2 Fast binary reactions in simple flows 152

5.3 The fast binary reaction in complex flows 158

6 Decay-type and Stable Reaction Dynamics in Flows 164

6.1 Stable reaction dynamics and its global steady state 165

6.2 The spectrum of decaying scalar in a flow 171

6.2.1 The inertial-convective range 171

6.2.2 The viscous-convective range 173

6.3 Smooth and filamental distributions 175

6.4 Structure functions, multifractality and intermittency 180

6.5 Two-dimensional turbulence with linear damping 189

7 Mixing in Autocatalytic-type Processes 192

7.1 Mixing in autocatalytic reactions 194

7.1.1 The closed-flow case 194

7.1.2 The open flow case 197

7.1.3 Results from the filament model 200

7.1.4 Front propagation in cellular flows 207

7.2 Mixing and bistable dynamics 209

7.3 Mixing in excitable dynamics 214

7.3.1 Excitable plankton dynamics 219

7.4 Competition dynamics 220

8 Mixing in Oscillatory Media 223

8.1 Synchronization of oscillatory dynamics by mixing 224

8.1.1 Persistent patterns in uniform medium 230

8.2 Synchronization in non-uniform medium 234

8.3 Noise induced oscillations in excitable media 238

8.4 The effect of chaotic dispersion on cyclic competition 241

9 Further Reading 247

9.1 Complex fluids and reactive flows 247

9.2 Self-propelled particles in prescribed flows 248

9.3 Bioconvection driven by swimming cells 250

9.4 Mesoscopic flows in active suspensions 251

Bibliography 253

Index 281