### 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).

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).

#### Full Text:

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**