Flow of Foams
Katgert, G.
Citation
Katgert, G. (2008, December 11). Flow of Foams. Casimir PhD Series.
Retrieved from https://hdl.handle.net/1887/13329
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Flow of Foams
PROEFSCHRIFT
Ter verkrijging van
de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus
Prof. mr. P.F. van der Heijden,
volgens besluit van het College voor Promoties te verdedigen op donderdag 11 december 2008
klokke 15.00 uur door
Gijs Katgert geboren te Leiden
in 1981
Promotiecommissie:
Promotor: Prof. dr. M.L. van Hecke
Overige leden: Prof. dr. N.D. Denkov (Universiteit Sofia, Bulgarije) Dr. O. Dauchot (CEA-Saclay, Frankrijk)
Prof. dr. D. Bonn (UvA, Amsterdam) Prof. dr. ir. W. van Saarloos
Prof. dr. J.M. van Ruitenbeek Prof. dr. P. Kes
Casimir PhD Series, Delft-Leiden, 2008-08 ISBN 978-90-8593-045-7
Dit werk maakt deel uit van het onderzoeksprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), die financieel wordt gesteund door de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
"Yon foaming flood seems motionless as ice;
Its dizzy turbulence eludes the eye"
William Wordsworth, Address to Kilchurn Castle.
Voor mijn ouders
Contents
1 Introduction 1
1.1 Microscopics: Foam films, stability . . . 2
1.1.1 Surfactants and surface tension . . . 2
1.1.2 Microscopic nature of foam evolution . . . 5
1.2 Mesoscopics: Shape, Forces and Pressures . . . 7
1.2.1 Plateau rules . . . 9
1.2.2 Capillary and Disjoining pressure . . . 9
1.2.3 Bubble deformation . . . 12
1.2.4 Viscous drag forces . . . 16
1.2.5 Capillary forces on floating bubbles . . . 18
1.3 Structure, rearrangements and rheology . . . 22
1.3.1 Foam structure . . . 22
1.3.2 Structural rearrangements and visco-elastic behaviour 23 1.4 Experiments and numerics on foams and emulsions . . . . 26
1.4.1 Experiments . . . 26
1.4.2 Numerics . . . 27
1.4.3 Differences between foams and emulsions . . . 28
2 Linear shear of two dimensional foams 31 2.1 Overview of the field . . . 31
2.2 Linear shear of two dimensional foams . . . 36
2.2.1 Experimental details . . . 36
2.2.2 Results . . . 41
2.2.3 Model . . . 43 i
CONTENTS
2.2.4 Fits . . . 46
2.2.5 Continuum Limit . . . 48
2.3 Rheometry of viscous forces in 2D foams . . . 48
2.3.1 Bubble-wall drag . . . 49
2.3.2 Bubble-bubble drag . . . 51
2.4 Discussion . . . 53
2.5 Ordered foams . . . 55
3 Rheology of foams in a Couette geometry 59 3.1 Introduction . . . 60
3.2 Experiment . . . 62
3.2.1 Setup . . . 62
3.2.2 Imaging . . . 63
3.3 Results . . . 64
3.4 Model . . . 66
3.4.1 Continuum limit . . . 69
3.5 Fits . . . 70
3.5.1 Flows without a top plate . . . 70
3.5.2 Flows with a top plate . . . 71
3.6 Local rheology and power-law fluids . . . 72
3.6.1 Fit to a power-law fluid model . . . 73
3.6.2 Rheometry: an anomalous local flow rule . . . 74
3.7 Discussion . . . 77
3.8 Conclusion . . . 78
3.A Stress and strain rate in polar coordinates . . . 79
3.A.1 Stress equilibrium . . . 79
3.A.2 Strain and strain rate . . . 80
3.B Velocity profiles in a curvilinear geometry . . . 81
4 Packing fraction and Jamming 83 4.1 Varying and measuringφ . . . . 84
4.1.1 Direct measure ofφ from experimental images . . . 85
4.1.2 The contact numberZ and its scaling with φ . . . . 87
4.2 Scaling of the effective viscosity with φ . . . 89
4.2.1 φ-dependence of β . . . . 89
4.2.2 Scaling ofk with φ . . . . 92
4.3 Measures of jamming . . . 95
4.3.1 Voronoi area distribution . . . 96 ii
CONTENTS
4.3.2 The force distributionp(f) . . . . 103 4.3.3 The shear modulusG . . . . 107 4.4 Conclusion . . . 109
5 Outlook 111
5.1 Viscous drag, fluctuations an kymographs . . . 111 5.2 Foam as a granular material: jamming and flow . . . 113
Summary 115
Samenvatting 119
Publication List 123
Curriculum Vitae 125
Dankwoord 127
Bibliography 129
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