## Industrial applied mathematics

### Two phase flow

Steam-water flows are of importance in boilers and
condensers. Typically a turbulent inlet water flow is heated till it
begins to boil, passing through a succession of flow régimes
(bubbly, slug, annular) before dryout occurs. The flow can exhibit
multiple steady states (the unstable intermediate state exhibits
*Ledinegg instability*), and is also prone to an oscillatory
instability (called a density wave oscillation) which is mediated by
the delay in the system due to the fluid transit time. The 1978 paper
below did linear and nonlinear stability analysis for a homogeneous
model, incidentally showing how to do multiple time scale analysis for
a delay integro-differential equation. Chris Aldridge's 1995 thesis
does this as well as a numerical simulation of a reduced two fluid
model, and more or less wraps up the problem.
### Alloy solidification

The main problem of interest to me is the formation of freckles in
castings. These are formed during solidification by a convective flow
within the porous dendritic mush, and an analysis of the instability
leading to such convection has been done by me, together with Paul
Emms, and also by Grae Worster in Cambridge, who is the leading
exponent of these studies. Paul Emms's 1993 thesis did both linear
stability analysis and also a numerical simulation of the problem. Of
interest is the way in which one treats instability of a time
dependent state. Most simply, one uses the frozen time hypothesis, but
something better ought to be done. The other main problems in the
modelling are (i) how does one choose the boundary condition at the
mush/liquid interface, and (ii) how does one model the evolution of
the dendritic spacing, i.e. the geometry of the microstructure.
### Miscellaneous

I am now a sporadic attender of the Study Groups for Industry, though
they were very formative on my outlook on applied mathematics. A
couple of industrial problems have led to publications: the
Søderberg paste electrode problem, and the thermistor
problem. The study groups have gone Hollywood style, with a glittering
array of stars to smash the problems to pieces. One of the irritating
things is that even if you solve the problem in a week, it will come
back again the next year, when a new set of people will re-do all your
previous efforts.
### Relevant publications

- Fowler, A.C. 1978 Linear and nonlinear stability of
heat exchangers. J. Inst.
Maths. Applics.
**22**, 361-382.

- Fowler, A.C. 1985 The formation of freckles in binary
alloys. IMA J. Appl.
Math.
**35**, 159-174.

- Bergstrom, T., S. Cowley, A.C. Fowler and P.E. Seward
1989 Segregation of carbon paste in a smelting electrode. IMA
J. Appl. Math.
**43**, 83-99.

- Fowler, A.C. and S.D. Howison 1990 Temperature surges
in thermistors. Proc.
3rd Conf. on Mathematics in Industry, eds. J. Manley, S. McKee, and
D. Owens, Kluwer, pp. 197-204.

- Fowler, A.C., I. Frigaard and S. Howison 1992
Temperature surges in current-limiting circuit devices. SIAM
J. Appl. Math.
**52**, 998-1011.

- Fowler, A.C. and P.E. Lisseter 1992 Flooding and flow
reversal in annular two-phase flows. SIAM J. Appl. Math.
**52**,
15-33.

- P.E. Lisseter and A.C. Fowler 1992 Bubbly flow.
Part I: a simplified mode. Int. J. Multiphase Flow
**18**,
195-204.

- P.E. Lisseter and A.C. Fowler 1992 Bubbly flow.
Part II: modelling void fraction waves. Int. J. Multiphase Flow
**18**, 205-215.

- Aldridge, C.J. and A.C. Fowler 1992 Mathematical
modelling of thermosyphons in cryogenic air
separation plants. Proc. 6th European Conference on
Mathematics in Industry, ed. F. Hodnett; B.G. Teubner, Leipzig, pp. 75-78.

- Emms, P. and A.C. Fowler 1992 The onset of freckling
in the solidification of binary alloys. In: Interactive dynamics of
convection and solidification, eds. S.H.
Davis, H.E. Huppert, U. Müller and M.G. Worster. NATO ASI Series E, Vol.
219, pp. 147-150.

- Aldridge, C.J. and A.C. Fowler 1993 Dynamics of a
reduced model of two-phase flow in a boiling channel.
Proc. Int. Symp. Instabilities in Multiphase Flows, eds. G.
Gouesbet and A. Berlemont, Plenum, New York, pp. 137-148.

- Emms, P. and A.C. Fowler 1994 Compositional
convection and freckle formation in the
solidification of binary alloys. J. Fluid Mech.
**262**,
111-139.

- Aldridge, C.J. and A.C. Fowler 1996
Stability and instability in evaporating two-phase flows (with C.
Aldridge). Surveys Math. Indus.
**6**, 75-107.

- Fowler, A.C. 2001
Analysis of an endothermic reaction in a packed
column. In: Mathematical modelling: case studies in industry,
eds. E. Cumberbatch and A.D. Fitt, pp. 160-180, C.U.P., Cambridge.