Chemical Engineering Science 58 (2003) 3765–3776 www.elsevier.com/locate/ces Analytical and numerical evaluation of two-uid model solutions for laminar fully developed bubbly two-phase ows Osvaldo E. Azpitarte, Gustavo C. Buscaglia * Centro At omico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina Received 15 July 2002; received in revised form 22 February 2003; accepted 4 March 2003 Abstract An analysis of the two-uid model in the case of vertical fully developed laminar bubbly ows is conducted. Firstly the phase distribution in the central region of the pipe (where wall eects vanish) is considered. From the model equations an intrinsic length scale L is deduced such that the scaled system reduces to a single equation without parameters. With the aid of this equation some generic properties of the solutions of the model for pipes with diameter greater than about 20L (the usual case, since L is of the order of the bubble radius) are found. We prove that in all physically meaningful solutions an (almost) exact compensation of the applied pressure gradient with the hydrostatic force e g occurs (with e the eective density and g the gravity). This compensation implies at void fraction and velocity proles in the central region not aected by the wall, even when no turbulence eects are accounted for. We then turn to consider the complete problem with a numerical approach, with the eect of the wall dealt via wall forces. The previous mathematical results are conrmed and the near-wall phase distributions and velocity proles 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 ow of bubbles). Finally, the same code is used to study the eect of reducing the gravity. As g decreases, so does the relative velocity between the phases, making the lift force increasingly dominant. This produces, in upward bubbly ows, narrower and sharper void fraction peaks that also appear closer to the wall. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Fluid mechanics; Multiphase ow; Laminar ow; Bubbly ow; Modelling; Two-uid model 1. Introduction The two-uid model (Ishii, 1975) is, at present, widely used in the simulation of two-phase ows. It emerges from the exact conservation equations after ensemble averaging, introducing unknowns such as the phase fraction k (x;t ), which is the probability of nding phase k at the position x at time t . The ensemble averaging process needs to be com- plemented with several closure relations which are far from established, either conceptually or quantitatively. For exam- ple, the lift term modeling transversal forces that act upon bubbles or particles in the presence of a velocity gradient, is not completely understood. The eective lift coecient for bubbles that ts pipe-ow data (C L ≃ 0:05; Lahey Jr., 1990) is much smaller than that deduced from potential ow ∗ Corresponding author. E-mail address: gustavo@cab.cnea.gov.ar (G. C. Buscaglia). calculations (C L = 1 2 ; Drew & Lahey Jr., 1987), while C L for particles depends on the rotational motion of the particle around its center. Some authors, because of this uncertainty, simply omit the lift term (Uchiyama, 1999). The eect that solid walls exert on its surrounding bubbles is included by means of a so-called wall force. Antal, Lahey Jr., and Flaherty (1991) modeled this force considering spherical rigid bubbles, but a recent model (Larreteguy, Drew, & Lahey Jr., 2002) assumes deforming and elastic bubbles. In this state of aairs it is interesting to go back to very simple ows and try to extract as much information as possible about the behavior of the model, starting with a model that satises physical constitutive principles (Drew & Lahey Jr., 2001). Perhaps the simplest ow is laminar fully developed ow in a circular pipe. A decade ago Antal et al. (1991) reported on fully developed solutions of the two-uid model equations. They solved the equations numerically for some specic data so as to reproduce experimental data from Nakoryakov et al. (1986).Ourobjectivehereistogobeyond 0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00239-2