UNIVERSITY OF CAMBRIDGE Isaac Newton Institute for Mathematical Sciences Free Boundary Problems in Industry 17 July - 4 August 2000 Report ______________________________________________________________________ Introduction The charts that follow were drawn up at the end of the "food" week. We collected all the problems that had been discussed during the week and put them into various categories (there is some repetition). Each problem has been given a star rating as a measure of (a) its industrial interest and relevance, (b) its academic interest and relevance, (c) progress made during the week. The notes include names of people who were (or intend to be) active on that problem. The earlier categories for each week are broadly "industrial", and the later ones are more academic. ___________________________________________________________________________ FOOD ___________________________________________________________________________ 1 Eating (a) Bolus modelling, "rapid granulation" Industry Interest: ** Academic Interest: * Notes: J Ockendon, J Harper, J Melrose, P Lillford; Unilever. An interesting coagulation problem related to (b) below. See Glass 1(b). (b) Tongue/palate interaction, squeeze films (positive and negative) Industry Interest: ** Academic Interest: ** Notes: See 4(d), 7(a), Nottingham thesis, R Davies (c) Fracture of cellular and porous media. Modes I & II; inter/intra granular fracture Industry Interest: *** Academic Interest: **(v hard!) Notes: See 6(e) and Metals 1(h), (d) Fracture of gels and soft solids Industry Interest: * Academic Interest: * Notes: J. Melrose, R Davies; Unilever 2 Chocolate/Cocoa Butter (a) Phase separation/alloy solidification. Industry Interest: * Academic Interest: ** Notes: P Fryer, M Mackley. Progress by looking at rate equations, need mat. sci. input. See also Metals 1(a). (b) Rate equations (differential scanning calorimetry) Industry Interest: * Academic Interest: * Notes: A Lacey, W Rudnicki, Australian MISG, some progress in week. (c) Melting Industry Interest: * Academic Interest: * Notes: (d) Enrobing Industry Interest: * Academic Interest: ** Notes: Very large literature. Some progress in week. P Howell, S Howison, J-M Vanden Broeck. Need to model thin films on substrates with corners. 3 Extrusion (a) Die-swell Industry Interest: * Academic Interest: * Notes: Very classical problem. Relates to popping 8(a). (b) Air lubrication of chocolate extruders Industry Interest: * Academic Interest: ** Notes: A lot of progress made, analytical and numerical. N Barton, M Mackley, J Ockendon, S Howison. See 6(f). (c) Internal heating, isothermal/non-isothermal. Phase energy Industry Interest: * Academic Interest: ** Notes: Need to understand phase changes; see 2(a,b) and Metals (1(l)). Chemistry, Materials Science. S J Chapman, S. Luckhaus. (d) Plug formation Industry Interest: * Academic Interest: * Notes: R Hinch 4 Bread and Cake Processing (a) Kneading in mixes Industry Interest: * Academic Interest: * Notes: Level sets may be useful. C Please. Some progress during week. (b) Forming and rolling Industry Interest: *** Academic Interest: ** Notes: Many applications throughout industry. P Howell, R Davies, C Elliott, C Please, S Howison. Much progress during week in cases when arbitary initial shapes become circular and then elliptical. See also Metals 2(b). (c) Sticking, wetting, peeling Industry Interest: * Academic Interest: * Notes: See 6(c) and Glass 2(b), 2(d), E Tuck (d) Baking and Crusts Industry Interest: ** Academic Interest: ** Notes: A Lacey, P Howell, A Fasano, K Parrott, D Jenkins. Much progress during week on free boundary model for crust formation. (e) Foam and Sponge Transitions Industry Interest: ** Academic Interest: ** Notes: E Tuck, EJ Hinch, P Howell, S Howison. Some progress during week. (f) Bubble growth and Fracture Industry Interest: ** Academic Interest: ** Notes: Relates to 1(c). Not enough time to discuss properly. (g) Aeration of Crumpets Industry Interest: * Academic Interest: *** Notes: Very nice travelling wave problem for columnar bubbles growing from hot base. J. Ockendon, J Harper, L Pyle 5 Spoilage/Storage (These areas were not discussed during the week but will almost certainly lead to interesting free boundary problems.) (a) Shelf-life, Packaging (b) Bruising, Damage (c) Rotting (d) Refrigeration, Sterilisation 6 Contact Problems (a) Bolus shattering, particle deformation Industry Interest: ** Academic Interest: ** Notes: See 1(a) (b) Dough sticking Industry Interest: * Academic Interest: * Notes: See 4(c) and Glass 2(d). (c) Bread slumping on meshes and rough surfaces Industry Interest: ** Academic Interest: ** Notes: Y Stokes. Quite specific to bread but see Glass 2(f). (d) Squeeze films & Hele-Shaw flows, non-Newtonian flows. Industry Interest: ** Academic Interest: ** Notes: See 1(b), 4(b). C Elliott, S Howison. Some progress during week. (e) Grape pressing Industry Interest: ** Academic Interest: ** Notes: P Howell. Relations to 1(c) and models of cells. Some progress during week. (f) Stick-slip Industry Interest: ** Academic Interest: ** Notes: See 3(b), 6(b), 4(c) and Glass 2(b). Some progress during week. 7 Rheology (a) Stress localisation in two-phase flow Industry Interest: ** Academic Interest: ** Notes: See 1(b) (b) Contrasts between normal and shear stresses, effects on structure, cavitation Industry Interest: * Academic Interest: ** Notes: (c) Multiphase polymeric systems Industry Interest: * Academic Interest: * Notes: 8 Mechanics and thermodynamics of porous media (a) Rupture (internal and external); (Mediation). Popping and Puffing Industry Interest: *** Academic Interest: ** Notes: See 1(c); E Tuck interested in popping and puffing. Some progress during week. (b) Mathematical homogenisation of deforming bubbles Industry Interest: * Academic Interest: ** Notes: Relevant to all of category 4 above. (c) Phase changes in porous media Industry Interest: * Academic Interest: ** Notes: Large academic literature, eg, on frost-heave, packed beds. Possibility of technology transfer. (d) Bubble morphology, foams Industry Interest: ** Academic Interest: * Notes: Possibility of technology transfer (esp. rheology) (e) Coupled oil/moisture transport Industry Interest: * Academic Interest: * Notes: P Howell, S Howison, L Pyle. Some progress during week on potato chips. (f) Scraped surface heat exchangers Industry Interest: ** Academic Interest: ** Notes: C Please has industrial support for a 3-year postdoc, names to him! 9 Numerical (a) Level sets methods in the food industry. Mesh generation. Industry Interest: ** Academic Interest: * Notes: A big issue with any numerical method is how it is incorporated into commercial packages. No level set methods in current commercial packages. (b) Finite element techniques Industry Interest: ** Academic Interest: * Notes: (c) Direct numerical simulation, particle following. Industry Interest: ** Academic Interest: * Notes: See Metals 5(g). ___________________________________________________________________________ GLASS ___________________________________________________________________________ 1 Viscous flow (a) Tin turbulence in float glass Industry Interest: *** Academic Interest: * Notes: No time for discussion. EJ Hinch. (b) Sintering Industry Interest: * Academic Interest: * Notes: See also Food 1(a). (c) Lens slumping. Industry Interest: * Academic Interest: * Notes: Y. Stokes has done Mechanics. Heat transfer still to be done. (d) Gob forming in air-lubricated moulds Industry Interest: ** Academic Interest: *** Notes: Hinch "floating discs" are relevant. Model formulated and some numerical comments made. EJ Hinch, V Entov; S Howison; N Barton. (e) Bubbling Industry Interest: * Academic Interest: *** Notes: See 5(a). (f) Murrino glass drawing Industry Interest: ** Academic Interest: * Notes: Numbers needed. Some academic literature, interesting "canonical" problems. P Howell, V Entov. See 1(g). Some progress made: process is perfect in absence of surface tension. (g) Multifilament drawing Industry Interest: * Academic Interest: ** Notes: Generalisation of 1(f). V Entov. (h) Stretching of fibres in turbulent flows Industry Interest: ** Academic Interest: * Notes: See 5(c). EJ Hinch 2 Viscoelasticity/Elasticity (a) Buckling of drawn sheets Industry Interest: *** Academic Interest: ** Notes: Lively discussions. Considerable modelling difficulties (to put it mildly) but some progress during week. Key question is feedback mechanism from final elastic state to earlier viscoelastic state. (b) Stick-slip Industry Interest: * Academic Interest: *** Notes: Again, a tough modelling problem, generic. See Food 4(c), 6(f). (c) Rippling at moving contact lines Industry Interest: *** Academic Interest: ** Notes: See 5(b). Yet another modelling nightmare, but some progress during week. (d) Sticking Industry Interest: ** Academic Interest: * Notes: Not really considered in the week. See Food 4(c), 6(b). (e) Fracture of toughened glass Industry Interest: * Academic Interest: *** Notes: J Ockendon would like a mathematical theory of "hackle". Phenomenological theory needed for energy redistribution. (f) Windscreen sag on horizontal frames Industry Interest: * Academic Interest: ** Notes: D. Salazar has bibliography and some new variational inequalities. See Food 6(c). 3 Control & Optimisation (a) Inlet mould design for uniform outflow conditions Industry Interest: ** Academic Interest: *** Notes: P Howell, A Wheeler, S Howison, C Please, S Luckhaus. Still no consensus on well-posedness. EJ Hinch has preliminary solution. See Metals 5(b). (b) Windscreen sag (inverse problem) Industry Interest: *** Academic Interest: ** Notes: How to classify achievable shapes. P. Howell, D. Gelder have information. (c) Stirrer optimisation to maximise mixing of injected filaments Industry Interest: ** Academic Interest: * Notes: EJ Hinch; see 1(e). Some progress during week. See Metals 5(b). 4 Heat transfer & thermodynamics (a) Numerical approximations in radiative transfer Industry Interest: *** Academic Interest: *** Notes: N Siedow, R Mattheij. Not really a free boundary problem per se, but fundamental to many models. (b) Radiative transfer in foam Industry Interest: ** Academic Interest: *** Notes: C Please, R Mattheij. Some progress during week using 4(a). (c) Radiative transfer in fibres Industry Interest: ** Academic Interest: *** Notes: Further generalisation of 4(a), (b). (d) "Fictive temperature" Industry Interest: * Academic Interest: ** Notes: S Luckhaus. "Structural relaxation" might be a better term. Some progress made during week. Preprint available from P. Howell. 5 Coupled mechanics & heat transfer (a) Flux foam modelling for feed in furnaces Industry Interest: ** Academic Interest: ** Notes: EJ Hinch "moment of glory". See 4(b) (b) Necking corrosion at glass/refractory contact line Industry Interest: *** Academic Interest: ** Notes: Modelling needed, potentially very nice problems. See paper by Hrma, Chem. Eng. Sci. 1970. (c) Bubbling corrosion (vertical "drilling" of horizontal refractory by bubbles) Industry Interest: ***Academic Interest: ** Notes: Same as (c) (d) Weathering/condensation Industry Interest: Academic Interest: Notes: Helen Tucker's PhD problem, hence no rating! ___________________________________________________________________________ METALS ___________________________________________________________________________ 1 Solidification, casting (a) Fundamentals of microstructure in alloys, microsegregation. Industry Interest: ** Academic Interest: ** Notes: Too basic a problem for this week. See Food 2(a). (b) Macrosegregation Industry Interest: *** Academic Interest: ** Notes: Related to homogenisation 5(d). (c) Banded segregation at top of air-lubricated moulds Industry Interest: * Academic Interest: * Notes: Some progress (L. Katgerman). (d) Ternary systems Industry Interest: ** Academic Interest: *** Notes: S Luckhaus: see 5(g), 5(f). Some progress during week but no weak formulation; free energy ideas may be the only possibility. (e) Exudation (solute rejection resulting from shrinkage near top of mould) Industry Interest: ** Academic Interest: *** Notes: V Voller; see 1(h). Some progress during week: physical picture needs mathematical description. (f) Freckles Industry Interest: ** Academic Interest: ** Notes: See 1(b). V. Voller. Some progress during week. (g) Heterogeneous nucleation Industry Interest: *** Academic Interest: * Notes: Very hard, may need molecular simulations. (h) "Hot tearing" in mushy regions Industry Interest: *** Academic Interest: *** Notes: Really good problem. EJ Hinch, some progress during week. Still need to explain the observed "frozen curtain" appearance of the tear. (i) Coherency (the onset of a rigid solid matrix in a mould) Industry Interest: *** Academic Interest: ** Notes: No time to formulate as a free boundary problem. (j) Butt curl (void formation at bottom corners of ingots) Industry Interest: ** Academic Interest: ** Notes: P Quintela has numerical solution. Model relies on suitable assumption about shrinkage through the mushy region. (h) Friction welding Industry Interest: * Academic Interest: * Notes: Has been discussed at a Southampton Study Group (EJ Hinch has report). See Food 3(c). 2 Forming, rolling (a) Spray forming Industry Interest: * Academic Interest: ** Notes: K Parrott. Levels sets being used by Z. Djuric. (b) Hot rolling Industry Interest: *** Academic Interest: *** Notes: H Engl, C Please, A Movchan. S. McElwain (Brisbane) has done some asymptotics (C. Please has report). See Food 4(b). 3 Hydrodynamics (a) Sloshing (Al reduction) Industry Interest: *** Academic Interest: *** Notes: EJ Hinch. Very lively discussion. $70 million per millimetre at stake, some progress made concerning aluminium flow away from electrodes. (b) Large-scale thermal convection Industry Interest: ** Academic Interest: ** Notes: See 5(f). Some progress during week towards resolving question of localisation of vorticity (P. Davidson). (c) Large-scale MHD flows Industry Interest: ** Academic Interest: * Notes: (d) MHD Starfish Industry Interest: * Academic Interest: ** Notes: Y. Fautrelle, J. Etay proposed this problem. (e) Flow/mush interaction Industry Interest: *** Academic Interest: *** Notes: See 1(b); 1(e); 1(f); 1(h). A fundamental problem. 4 Fracture, furnaces (a) Damage and fracture in coke ovens. Industry Interest: ** Academic Interest: ** Notes: D Jenkins proposed this problem and has analysed plastic zone. Fracturing remains an issue. (b) Damage and fracture in carbon electrodes. Industry Interest: *** Academic Interest: ** Notes: AB Movchan, O Selsil have papers on traditional thermoelasticity for this problem. A. Lacey has ideas for a damage model for microfracture. See 2(b). (c) Blast furnace modelling. Industry Interest: ** Academic Interest: * Notes: N Barton, V Voller have ideas for numerical modelling. 5 Mathematical methodology (a) Level sets. Industry Interest: * Academic Interest: ** Notes: K. Parrott. See 2(a). (b) Inverse problems, control. Industry Interest: *** Academic Interest: *** Notes: H. Engl. See Glass 3(a), 3(c). (c) Phase-field, free energy modelling. Industry Interest: ** Academic Interest: ** Notes: S. Luckhaus. Lively discussion on physical relevance of phase field. (d) Homogenisation. Industry Interest: * Academic Interest: ** Notes: This methodology was implicit in almost all this discussion in all three industries. (e) Enthalpy methods, weak formulations. Industry Interest: ** Academic Interest: ** Notes: S. Luckhaus. See (c) above. (f) Code comparisons/verification and benchmarking. Industry Interest: **** Academic Interest: **** Notes: A universal problem. (g) Smooth particle hydrodynamics Industry Interest: * Academic Interest: ** Notes: Applicability to granular flow. See Food 9(c).