UNIVERSITY OF CAMBRIDGE
Isaac Newton Institute for Mathematical Sciences
Free Boundary Problems in Industry
17 July - 4 August 2000
Report
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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.
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FOOD
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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).
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GLASS
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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!
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METALS
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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).