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The Mathematics of Foam

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The Mathematics of Foam CORE Metadata, citation and similar papers at core.ac.uk Provided by Mathematical Institute Eprints Archive The Mathematics of Foam Christopher James William Breward St Anne's College University of Oxford A thesis submitted for the degree of Doctor of Philosophy June 1999 Contents 1 Introduction 1 1.1 What is a foam? . 1 1.2 How do foams form? . 5 1.3 Why does a foam persist after it has been formed? . 5 1.3.1 Surfactants . 5 1.3.2 How does a surface tension gradient affect a liquid flow? . 6 1.3.3 Volatile Components . 7 1.4 What happens to lamellae when they become thin? . 7 1.5 How does a lamella rupture? . 7 1.6 How are foams destroyed in industrial situations where they are not wanted? 8 1.7 Background material . 8 1.8 Thesis plan . 12 1.9 Statement of Originality . 13 2 Modelling of liquids, surfactants and volatile systems 14 2.1 Liquid modelling . 15 2.1.1 Field Equations . 15 2.1.2 Boundary conditions . 15 2.2 Surfactants . 16 2.2.1 Effect of surfactant on the surface tension . 18 2.3 Volatile and Inert systems . 19 2.3.1 Volatile component at the free surface . 19 2.3.2 Model for evaporation . 20 2.4 Summary . 20 3 Modelling expanding free surfaces 22 3.1 Introduction . 22 3.2 The overflowing cylinder . 22 3.3 Previous work . 24 i 3.3.1 Experimental observations . 25 3.3.2 Plan . 31 3.4 Model formulation: velocity distribution . 31 3.4.1 Typical non-dimensional parameter sizes based on flow at depth . 33 3.5 Model formulation: surfactant distribution . 34 3.5.1 Linearisation . 36 3.5.2 Nondimensional parameter sizes based on flow at depth . 37 3.6 Outer (Inviscid) Problem . 37 3.6.1 Inviscid solution . 38 3.7 Outer (surfactant) Problem . 42 3.7.1 Solution to the problem in the limit Ma 0 and S . 42 ! ! 1 3.8 Boundary layers at the free surface . 42 3.8.1 The hydrodynamic boundary layer . 44 3.8.2 The diffusion boundary layer . 45 3.8.3 Coupling between the hydrodynamic and diffusive problems . 45 3.9 Series solutions to both the liquid and surfactant problems . 47 3.9.1 The Solution for C∗ . 48 3.9.2 Lowest order liquid problem and solution . 49 3.9.3 Problem and solution for C2 . 52 3.10 The lack of closure . 53 3.10.1 Termination of the series . 54 3.10.2 Neglected physical effects . 56 3.10.3 Edge Effects . 58 3.11 Conclusions . 58 4 Models for Marangoni flows in thin films with two free surfaces 61 4.1 Introduction . 61 4.1.1 Thin films with two free boundaries . 61 4.1.2 Films on substrates . 62 4.1.3 Plan . 63 4.2 Thin film equations for the liquid . 63 4.3 Reduction to thin film equations . 67 4.3.1 Distinguished Limits: Three forces balancing . 67 4.3.1.1 Viscous, capillary and Marangoni effects . 67 4.3.1.2 Capillary, Marangoni and lubrication effects . 69 4.3.2 Intermediate regimes where two forces balance . 69 4.3.2.1 Capillary and Marangoni . 69 ii 4.3.2.2 Capillary and viscous effects . 70 4.3.2.3 Marangoni and viscous effects . 70 4.3.2.4 Marangoni and lubrication effects . 71 4.3.3 Intermediate regimes where one force dominates . 71 4.3.3.1 Capillary effects . 71 4.3.3.2 Viscous effects . 71 4.3.3.3 Marangoni effects . 72 4.3.4 Summary . 73 4.4 The effects of gravity, pressure drops across the film, disjoining pressure, and surface viscosity . 73 4.4.1 Gravity . 76 4.4.2 Pressure drops across the film . 76 4.4.3 Van der Waals forces and disjoining pressure . 77 4.4.4 Closure of the models . 77 4.5 Thin film equations: presence of a surfactant . 78 4.5.1 Distinguished Limits: three mechanisms balancing . 80 4.5.1.1 Longitudinal diffusion, bulk convection and surface con- vection . 80 4.5.1.2 Diffusion across the film, bulk convection and surface con- vection . 81 4.5.2 Intermediate Regimes: Two mechanisms balancing . 82 4.5.2.1 Bulk convection and surface convection . 82 4.5.2.2 Diffusion across the film and bulk convection . 82 4.5.2.3 Longitudinal diffusion and surface convection . 82 4.5.2.4 Diffusion across the film and surface convection . 82 4.5.3 Regimes where one mechanism dominates . 83 4.5.3.1 Surface convection . 83 4.5.3.2 Bulk convection . 83 4.5.3.3 Longitudinal diffusion . 83 4.5.3.4 Convection dominated flows . 84 4.5.4 Summary . 85 4.5.5 The no-slip condition . 85 4.5.6 Insoluble surfactants . 85 4.6 Thin film equations: presence of a miscible, volatile component . 88 4.6.1 Distinguished limits: three mechanisms balancing . 89 4.6.1.1 Convection, longitudinal diffusion and evaporation . 89 4.6.1.2 Convection, diffusion across the film and evaporation . 90 iii 4.6.2 Intermediate regimes where two mechanisms balance . 90 4.6.2.1 Convection and longitudinal diffusion . 90 4.6.2.2 Convection and evaporation . 91 4.6.2.3 Convection and diffusion across the film . 91 4.6.2.4 Convection and diffusion across the film (and evaporation) 91 4.6.3 Regimes where one mechanism is dominant . 91 4.6.3.1 Diffusion . 91 4.6.3.2 Convection (well-mixed) . 92 4.6.3.3 Convection dominated flow . 92 4.6.3.4 Evaporation . 92 4.6.4 Summary . 93 4.7 Mathematical Structure of a typical coupled problem . 93 4.8 Conclusion . 96 5 Foam films 99 5.1 Introduction . 99 5.1.1 Relevant parameter sizes . 101 5.1.2 Plan . 102 5.2 Evolution of a thin aqueous film in the absence of surfactant . 103 5.2.1 Introduction and modelling ideas . 103 5.2.2 Model for a film acting under viscosity and capillarity . 104 5.2.3 Lengthscale of the transition region and velocity scalings . 105 5.2.4 Transition region . 106 5.2.5 Lamella model . 107 5.2.6 Implications . 109 5.3 Evolution of films stabilised by a well-mixed soluble surfactant . 109 5.3.1 Selection of velocity scale . 110 5.3.1.1 Parameter sizes and plan for the rest of this section . 111 5.3.2 Plateau border . 112 5.3.3 Lamella model . 112 5.3.4 Transition region . 114 5.3.4.1 Asymptotic behaviour of solutions at . 115 ∞ 5.3.5 Summary . 117 5.3.6 Results and discussion . 118 5.3.6.1 Unexpected results . 127 5.3.7 Including the `lubrication' term . 133 5.3.7.1 Behaviour as ξ . 133 ! ∞ iv 5.3.7.2 Surface and centre-line velocities as ξ . 134 ! 1 5.3.7.3 Summary . 135 5.3.7.4 Results and discussion . ..
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