Generic Properties Of The Solutions Of The Two-Fluid Model In Laminar Fully Developed Two-Phase Flow.

Abstract

An analysis of the solutions of the two-fluid model for vertical fully-developed
flow is conducted. The resulting equation system is reduced to a single ordinary differential
equation (ODE). Introducing an intrinsic length scale L deduced form the system, which
value is approximately that of the bubble radius, the ODE is rendered non-dimensional.
With the aid of this equation, some generic properties of the solutions of the model for
pipes with diameter greater than about 20 L (the usual case)ar e found. Firstly the central
region of the pipe, where wall efects vanish, is considered. It is proved that an almost
exact compensation of the applied pressure gradient with the hydrostatic force ρeff g occurs
(with ρeff the effective density and g the gravity). This compensation implies that flat void
fraction and velocity profiles are the only possible solutions in the central region, and that
the void fraction at the center of the pipe only depends on the pressure gradient.
Finally, the complete problem is considered with a numerical approach, with the effect
of the wall dealt via wall forces. The previous mathematical results are confirmed and the
near-wall phase distributions and velocity profiles are found. With the numerical code it
is also possible to investigate the regime in which the pressure gradient is greater than the
weight of the pure liquid, in which case a region of strictly zero void fraction develops,
surrounding the axis of the pipe (in upward flow of bubbles), or at the wall of the pipe (in
downward flow of bubbles).