## 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.