## 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.