Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

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vi CONTENTS 2.3 Drag reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 Elastic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 Phase separation in binary fluids 51 3.1 Thermodynamics of binary mixtures . . . . . . . . . . . . . . . . 52 3.1.1 Spinodal decomposition . . . . . . . . . . . . . . . . . . 52 3.1.2 Surface tension . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Coarsening in a fluid at rest . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Domain growth . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Hydrodynamics and coarsening . . . . . . . . . . . . . . . . . . . 56 3.3.1 Growth laws . . . . . . . . . . . . . . . . . . . . . . . . 57 Part II 59 4 Small-scale statistics of viscoelastic turbulence 61 4.1 Uniaxial model . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1.1 Numerical settings . . . . . . . . . . . . . . . . . . . . . 64 4.2 Coil-stretch transition . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Energy cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Acceleration statistics . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.1 Fluctuations of viscoelastic accelerations . . . . . . . . . 68 4.4.2 Role of the back-reaction . . . . . . . . . . . . . . . . . . 69 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Elastic turbulence in two-dimensional flows 73 5.1 Viscoelastic model . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.1.1 Elastic instabilities . . . . . . . . . . . . . . . . . . . . . 74 5.1.2 Numerical setup . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Momentum budget . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Transition to turbulence . . . . . . . . . . . . . . . . . . . . . . . 77 5.3.1 Eulerian Lyapunov exponent . . . . . . . . . . . . . . . . 78 5.3.2 Intermediate regime: elastic waves . . . . . . . . . . . . . 79 5.3.3 Turbulent energy spectrum . . . . . . . . . . . . . . . . . 80 5.4 Mixing: Lagrangian Lyapunov exponent . . . . . . . . . . . . . . 81 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 Turbulence and coarsening in binary mixtures 85 6.1 Phase separation under stirring . . . . . . . . . . . . . . . . . . . 86 6.1.1 Evolution equations . . . . . . . . . . . . . . . . . . . . 86 6.1.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . 87 vi
CONTENTS vii 6.2 Active mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2.1 Unstirred case . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2.2 Stirred case . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Passive mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3.1 Role of Lagrangian chaos . . . . . . . . . . . . . . . . . 92 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Conclusions 95 Bibliography 97 Acknowledgements 103 vii
Page 1: UNIVERSITÀ DEGLI STUDI DI TORINO D
Page 5 and 6: Riassunto Questa tesi presenta uno
Page 7: Abstract This thesis presents a the
Page 13 and 14: Introduction This thesis presents a
Page 15: The results presented in the thesis
Page 19 and 20: Chapter 1 Newtonian turbulence Flui
Page 21 and 22: 1.1. Navier-Stokes equation 9 that
Page 23 and 24: 1.1. Navier-Stokes equation 11 1.1.
Page 25 and 26: 1.1. Navier-Stokes equation 13 wher
Page 27 and 28: 1.2. Phenomenology of turbulence 15
Page 37 and 38: 1.3. Two-dimensional turbulence 25
Page 39 and 40: 1.3. Two-dimensional turbulence 27
Page 41: 1.3. Two-dimensional turbulence 29
Page 44 and 45: 32 2. Polymer solutions Figure 2.1:
Page 46 and 47: 34 2. Polymer solutions Introducing
Page 48 and 49: 36 2. Polymer solutions 2.1.2 Weiss
Page 50 and 51: 38 2. Polymer solutions locity grad
Page 52 and 53: 40 2. Polymer solutions The convexi
Page 54 and 55: 42 2. Polymer solutions The stress
Page 56 and 57: 44 2. Polymer solutions which shows
Page 58 and 59: 46 2. Polymer solutions 2.3 Drag re
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48 2. Polymer solutions 2.4 Elastic
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Chapter 3 Phase separation in binar
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3.1. Thermodynamics of binary mixtu
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3.2. Coarsening in a fluid at rest
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3.3. Hydrodynamics and coarsening 5
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Part II 59
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62 4. Small-scale statistics of vis
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Chapter 5 Elastic turbulence in two
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5.1. Viscoelastic model 75 Figure 5
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5.3. Transition to turbulence 77 r
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5.3. Transition to turbulence 79 λ
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5.4. Mixing: Lagrangian Lyapunov ex
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5.5. Summary 83 We found evidence o
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86 6. Turbulence and coarsening in
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96 Conclusions studied by numerical
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98 BIBLIOGRAPHY [14] Parisi and U.
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100 BIBLIOGRAPHY [54] A. Groisman,
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102 BIBLIOGRAPHY [94] M. E. Cates,
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Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

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