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 . ..