^{1}, W. Drenckhan

^{1}and D. Weaire

^{1}

### Abstract

We present a computational analysis of the flow of liquidfoam along a smooth wall, as encountered in the transport of foams in vessels and pipes. We concentrate on the slip of the bubbles at the wall and present some novel finite element calculations of this motion for the case of fully mobile gas/liquid interfaces. Our two-dimensional simulations provide for the first time the bubble shapes and entire flow field, giving detailed insight into the distribution of stresses and dissipation in the system. In particular, we investigate the relationship between the drag force and the slip velocity of the bubble, which for small slip velocities obeys power laws, as predicted by previous semianalytical treatments.

The authors would like to thank N. Denkov and I. Cantat for stimulating discussions at the FRIT workshop (http://users.aber.ac.uk/sxc/frit.html) organized by S. J. Cox. The research was funded by the European Space Agency (MAP AO-99-108:C14914/02/NL/SH, MAP AO-99-075:C14308/00/NL/SH), Enterprise Ireland (BRG SC/2002/011), and the Science Foundation Ireland (RFP 05/RFP/PHY0016). One of the authors (A.S.) was supported by the Conseil Régional de Rhône-Alpes. Another author (W.D.) is an IRCSET Fellow funded by the Embark Initiative Ireland.

I. INTRODUCTION

II. DEFINITION OF THE MODEL

A. Bulk equations

B. Boundary equations

1. Liquid/wall interface

2. Liquid/liquid side boundaries

3. Film

4. Gas/liquid interface

C. Scaling

III. COMPUTATIONAL BACKGROUND

A. Reference domain and deformed domain

B. Weak formulation of the force balance on the interface

C. Projection of the fluid equations on the reference domain

D. Simulation

IV. RESULTS

A. General description

B. Shape properties

C. Force on the wall

D. Viscous dissipation

E. Influence of bubble size

V. FINITE INTERFACE MOBILITIES

VI. CONCLUSION

### Key Topics

- Wetting
- 29.0
- Gas liquid interfaces
- 28.0
- Foams
- 25.0
- Liquid thin films
- 16.0
- Viscosity
- 13.0

## Figures

Sketch of two typical experiments to study wall slip of a liquid foam. (a) Foam is sheared in a rheometer between a smooth and a rough plate moving at relative velocity . (b) A train of equal volume bubbles is pushed through a narrow channel at velocity by a pressure drop .

Sketch of two typical experiments to study wall slip of a liquid foam. (a) Foam is sheared in a rheometer between a smooth and a rough plate moving at relative velocity . (b) A train of equal volume bubbles is pushed through a narrow channel at velocity by a pressure drop .

Sketch of a sliding bubble. (a) Gas bubbles are immersed in a liquid of viscosity , flow field , and pressure field . They slide along a smooth wall at velocity and are separated from the wall and from each other by thin liquid films. In the latter we set a pressure reference, equivalent to a reservoir at constant pressure. The gas/liquid interface is assumed to be fully mobile. The bubbles are stationary while the wall moves at . (b) The stress balance on the gas/liquid interface takes into account the normal stress exerted by the liquid, the surface tension , the disjoining pressures in the thin films and the gas pressure . (c) Definition of the unit cell of length with periodic boundary conditions on either side . The gas/liquid interface separates the gas domain and liquid domain . On the liquid film boundary we apply a special point boundary condition (pressure reference) at [see (a)].

Sketch of a sliding bubble. (a) Gas bubbles are immersed in a liquid of viscosity , flow field , and pressure field . They slide along a smooth wall at velocity and are separated from the wall and from each other by thin liquid films. In the latter we set a pressure reference, equivalent to a reservoir at constant pressure. The gas/liquid interface is assumed to be fully mobile. The bubbles are stationary while the wall moves at . (b) The stress balance on the gas/liquid interface takes into account the normal stress exerted by the liquid, the surface tension , the disjoining pressures in the thin films and the gas pressure . (c) Definition of the unit cell of length with periodic boundary conditions on either side . The gas/liquid interface separates the gas domain and liquid domain . On the liquid film boundary we apply a special point boundary condition (pressure reference) at [see (a)].

A transformation is computed, which maps the reference domain onto the deformed physical domain . The flow equations are then mapped onto using and solved there.

A transformation is computed, which maps the reference domain onto the deformed physical domain . The flow equations are then mapped onto using and solved there.

The weak form can be viewed as the virtual work associated with an infinitely small displacement of an interface of length associated with a liquid domain of area .

The weak form can be viewed as the virtual work associated with an infinitely small displacement of an interface of length associated with a liquid domain of area .

Variation of the bubble shape (solid line) with the capillary number . The dashed line represents the reference domain .

Variation of the bubble shape (solid line) with the capillary number . The dashed line represents the reference domain .

(a) Reduced friction force [Eq. (31)] exerted by the bubble on the wall as a function of its velocity, given by . The + symbols correspond to the six cases shown in Fig. 5. The line styles represent three distinct regimes: disjoining pressure (dotted line), interest in this article (full line), and thick film (dashed line). (b) Slope of (a) in a log-log plot, which represents in Eq. (1) whenever it is reasonably constant.

(a) Reduced friction force [Eq. (31)] exerted by the bubble on the wall as a function of its velocity, given by . The + symbols correspond to the six cases shown in Fig. 5. The line styles represent three distinct regimes: disjoining pressure (dotted line), interest in this article (full line), and thick film (dashed line). (b) Slope of (a) in a log-log plot, which represents in Eq. (1) whenever it is reasonably constant.

Same graphs as in Fig. 6, but here we show four computational cycles run for different ranges of with appropriately adjusted disjoining pressures. 1: , 2: , 3: , and 4: in Eq. (14). The symbols mark the beginning and the end of the scaling regime (solid line) of each cycle according to the definitions given in the text. All solutions in this regime collapse on a single curve.

Same graphs as in Fig. 6, but here we show four computational cycles run for different ranges of with appropriately adjusted disjoining pressures. 1: , 2: , 3: , and 4: in Eq. (14). The symbols mark the beginning and the end of the scaling regime (solid line) of each cycle according to the definitions given in the text. All solutions in this regime collapse on a single curve.

Sketch of the key features of the bubble shape and flow field obtained in the simulations. We obtain plug-like flow in the wetting film and a slow-moving vortex in the Plateau border. The downstream end of the wetting film forms a constriction, which is preceded by a small bulge.

Sketch of the key features of the bubble shape and flow field obtained in the simulations. We obtain plug-like flow in the wetting film and a slow-moving vortex in the Plateau border. The downstream end of the wetting film forms a constriction, which is preceded by a small bulge.

Equilibrium bubble shapes, velocity profiles and dynamic pressure fields for three different values of . The bubble is increasingly sheared and the wetting film thickens significantly toward higher .

Equilibrium bubble shapes, velocity profiles and dynamic pressure fields for three different values of . The bubble is increasingly sheared and the wetting film thickens significantly toward higher .

Curvature along the free interface for different values of . The distance on the free interface is measured from the upper left end of the gas/liquid interface.

Curvature along the free interface for different values of . The distance on the free interface is measured from the upper left end of the gas/liquid interface.

Variation of the wetting film thickness in the middle of the film and at the constriction with . Both conform very well to power laws [Eqs. (46) and (48)]; see the inserted log-log plot of the same data.

Variation of the wetting film thickness in the middle of the film and at the constriction with . Both conform very well to power laws [Eqs. (46) and (48)]; see the inserted log-log plot of the same data.

Increase of the liquid area with . The results are fitted well by a power law [Eq. (49)], which can be attributed to the fact that most of the area increase is related to the thickening of the wetting film, and hence to Eq. (46).

Increase of the liquid area with . The results are fitted well by a power law [Eq. (49)], which can be attributed to the fact that most of the area increase is related to the thickening of the wetting film, and hence to Eq. (46).

Distribution of the tangential viscous stress along the wall of the periodic cell for various values of (the same as in Fig. 9). The stress at the exit of the wetting film displays a pronounced peak of *opposite* sign.

Distribution of the tangential viscous stress along the wall of the periodic cell for various values of (the same as in Fig. 9). The stress at the exit of the wetting film displays a pronounced peak of *opposite* sign.

Total dissipative force [Eq. (31)] exerted on the wall as a function of . The insert shows the slope, which corresponds to the power in the relation [Eq. (1)] whenever it is reasonably constant. This is only the case for small .

Total dissipative force [Eq. (31)] exerted on the wall as a function of . The insert shows the slope, which corresponds to the power in the relation [Eq. (1)] whenever it is reasonably constant. This is only the case for small .

Viscous dissipation field for . It emphasizes the role of the entrance and the exit of the wetting film.

Viscous dissipation field for . It emphasizes the role of the entrance and the exit of the wetting film.

Variation of the integrated and normalized viscous dissipation along the wall of the periodic cell for the same as in Figs. 13 and 10. This emphasizes the dominance and similar magnitude of the dissipation at the entrance and exit of the wetting film.

Variation of the integrated and normalized viscous dissipation along the wall of the periodic cell for the same as in Figs. 13 and 10. This emphasizes the dominance and similar magnitude of the dissipation at the entrance and exit of the wetting film.

Investigation of the influence of the bubble length on the relationship between and . The film length is given by . We see that it has little influence. For small values of small deviations begin to show up in the slope at high .

Investigation of the influence of the bubble length on the relationship between and . The film length is given by . We see that it has little influence. For small values of small deviations begin to show up in the slope at high .

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