## Chapter 16

### One-dimensional two-phase flow

The motivating problem of interest is the turbulent one-dimensional two-phase flow of steam and water in boilers, condensers, thermosyphons, etc. Firstly, the different flow régimes are described, and the simplest mass/momentum model for separated, adiabatic flow is given. The homogeneous and drift-flux models are briefly described. Then it is shown that the simple two fluid model is ill-posed (it has complex characteristics), and there follows a more detailed description of the process of averaging, whereby the basic governing equations are derived.

In exposition, the treatment of averaging follows Drew and Wood's 1985 treatise from the Gaithersburg workshop, which is unfortunately not readily available (not even from the authors), although Don Drew tells me an updated version is in preparation*.

Simplified models are then presented for the annular flow régime (which is the one most requiring a two fluid model), and for a homogeneous model of a thermosyphon. Although quite a bit of simplification is possible, the model in most basic form is still an integro-partial differential equation with delay. Analysis is possible for steady states, linear and nonlinear stability, much of this being consigned to the exercises.

The exercises (1-5) are based on a homogeneous model studied by me ( Fowler 1978); this paper demonstrates that negative slopes are not necessary for Ledinegg instability, and also shows how to do multiple scales in an integral delay system, something which is not perhaps totally obvious. For stability and nonlinear simulations of a two fluid model, see Chris Aldridge's D. Phil. thesis. A synopsis of this was published by Aldridge and Fowler (1996).

Exercise 6 concerns the waves you can see propagating downwards in a pint of guinness when it is first poured (and is settling). The question (and its answer) were written down in the Hyland Hotel in Ballyvaughan, Co. Clare, while consuming the product with Stephen O'Brien and Felix Ng. The illustration is by courtesy of Michael Manga. For behind the scenes stories about efforts to get this photo on the front cover, see the miscellany.

*A recent reference which may help is :
D. A. Drew and G. B. Wallis 1994 Fundamentals of Two-Phase Flow Modeling. In: Multiphase Science and Technology, Vol. 8, eds. G. F. Hewitt, J. H. Kim, R. T. Lahey, J. M. Delhaye, and N. Zuber, pp. 1-67; Begell House, New York.