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.


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

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

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

  3. 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.

  4. 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.

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

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

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

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

  9. 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.

  10. 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.

  11. 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.

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

  13. 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.

  14. 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.