Cox and Weaire [1] rightly emphasize that our solution of the drainage equation for the “Eiffel Tower” geometry does not treat the boundary conditions. There should be a no- flow condition at the top, and, after leakage begins, the liquid fraction should be pegged to εc ≈ 0.36 at the bottom. They then show how approximating the no-flow conditions at the top can improve agreement with numerical solution. But as argued in [2], we maintain that the neglect of capillarity coming from boundary conditions at the bottom dominates, and that this cannot explain our measurements. At short times, capillarity can delay the onset of leakage, and at long times it can counter gravity and retain liquid in the foam indefinitely; in either case, leakage is slower than our approximate solution, contrary to experiment. Therefore, we speculated that the discrepancy arose from neglect of coarsening, whereby the average bubble size increases via gas diffusion from smaller to larger bubbles. This is an important puzzle because, while the drainage equation successfully predicts forced-drainage experiments, it fails dramatically for free-drainage experiment

Durian, Douglas J

Saint-Jalmes, A.

English

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsDepartment of Physics Papers Department of Physics2001Reply to Comment by S. J. Cox and D. Weaire on"Free Drainage of Aqueous Foams: ContainerShape Effects on Capillarity and Vertical Gradients"A. Saint-JalmesUniversity of California, Los AngelesDouglas J. DurianUniversity of Pennsylvania, djdurian@physics.upenn.eduFollow this and additional works at: https://repository.upenn.edu/physics_papersPart of the Physics CommonsAt the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles. Currently, he is a faculty member at thePhysics Department at the University of Pennsylvania.This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/600For more information, please contact repository@pobox.upenn.edu.Recommended CitationSaint-Jalmes, A., & Durian, D. J. (2001). Reply to Comment by S. J. Cox and D. Weaire on "Free Drainage of Aqueous Foams:Container Shape Effects on Capillarity and Vertical Gradients". Europhysics Letters, 55 (3), 447-448. http://dx.doi.org/10.1209/epl/i2001-00435-1Reply to Comment by S. J. Cox and D. Weaire on "Free Drainage ofAqueous Foams: Container Shape Effects on Capillarity and VerticalGradients"AbstractCox and Weaire [1] rightly emphasize that our solution of the drainage equation for the “Eiffel Tower”geometry does not treat the boundary conditions. There should be a no- flow condition at the top, and, afterleakage begins, the liquid fraction should be pegged to εc ≈ 0.36 at the bottom. They then show howapproximating the no-flow conditions at the top can improve agreement with numerical solution. But asargued in [2], we maintain that the neglect of capillarity coming from boundary conditions at the bottomdominates, and that this cannot explain our measurements. At short times, capillarity can delay the onset ofleakage, and at long times it can counter gravity and retain liquid in the foam indefinitely; in either case,leakage is slower than our approximate solution, contrary to experiment. Therefore, we speculated that thediscrepancy arose from neglect of coarsening, whereby the average bubble size increases via gas diffusion fromsmaller to larger bubbles. This is an important puzzle because, while the drainage equation successfullypredicts forced-drainage experiments, it fails dramatically for free-drainage experimentsDisciplinesPhysical Sciences and Mathematics | PhysicsCommentsAt the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles.Currently, he is a faculty member at the Physics Department at the University of Pennsylvania.This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/600Europhys. Lett., 55 (3), pp. 447–448 (2001)ReplyEUROPHYSICS LETTERS 1 August 2001Reply to the Comment by S. J. Cox and D. Weaire on“Free drainage of aqueous foams: Container shapeeﬀects on capillarity and vertical gradients”A. Saint-Jalmes(∗) and D. J. DurianUCLA Department of Physics and Astronomy - Los Angeles, CA 90095-1547, USA(received 14 February 2001; accepted 25 May 2001)PACS. 82.70.Rr – Aerosols and foams.PACS. 47.60.+i – Flows in ducts, channels, nozzles, and conduits.PACS. 47.55.Mh – Flows through porous media.Cox and Weaire [1] rightly emphasize that our solution of the drainage equation for the“Eiﬀel Tower” geometry does not treat the boundary conditions. There should be a no-ﬂow condition at the top, and, after leakage begins, the liquid fraction should be pegged toεc ≈ 0.36 at the bottom. They then show how approximating the no-ﬂow conditions at the topcan improve agreement with numerical solution. But as argued in [2], we maintain that theneglect of capillarity coming from boundary conditions at the bottom dominates, and that thiscannot explain our measurements. At short times, capillarity can delay the onset of leakage,and at long times it can counter gravity and retain liquid in the foam indeﬁnitely; in eithercase, leakage is slower than our approximate solution, contrary to experiment. Therefore, wespeculated that the discrepancy arose from neglect of coarsening, whereby the average bubblesize increases via gas diﬀusion from smaller to larger bubbles. This is an important puzzlebecause, while the drainage equation successfully predicts forced-drainage experiments, it failsdramatically for free-drainage experiments [2].While Cox and Weaire [1] consider conditions at the top of the sample, we clarify here therole of boundary conditions at both top and bottom. Numerical results from the drainageequation are shown in ﬁgs. 1(a)-(b) for an initial liquid fraction of ε0 = 0.36, and in ﬁg. 1(c)for several diﬀerent ε0. In all cases the sample height is H = 70 cm, the ﬂaring length isz0 = 25 cm, the capillary rise scale is ξ = 5 cm, the characteristic ﬂow speed is u0 = 0.026 cm/s,and the dissipation exponent is m = 1. We predicted [2] that the liquid fraction proﬁleis ε(z, t) = ε0/(1 + t/t0), and that the normalized volume of drained liquid is V (t)/Vf =1/(1 + t0/t), where t0 = z0/(ε0u0), i.e. that drainage is uniform and all the liquid eventuallyleaks out. By contrast the full numerical solution for ε(z, t), in ﬁg. 1(a), shows that the samplebecomes drier at the top and remains wet at the bottom. Consequently, in ﬁg. 1(b), V (t)/Vfdoes not approach 1 at long times. Also in ﬁgs. 1(a)-(b), we see that the liquid fraction at thetop, ε(0, t), is not zero as assumed in [1].To see the role of boundary conditions, we disable them one at a time by taking ∂ε/∂z = 0.When only the bottom conditions are correct, we ﬁnd the dotted curve of ﬁg. 1(b), in whichthe volume of drained liquid is correct at short and long times. The ﬁnal approach is too(∗) Now at Laboratoire de Physique des Solides, Universite´ Paris-Sud - 91405 Orsay, France.c© EDP Sciences448 EUROPHYSICS LETTERS00.20.40.60.81102 103 104 105full solutionno BC'sBC's at topBC's at bottomt  (s)(b)V(t)/Vfε(0,t)00.10.20.30.40 10 20 30 40 50 60 70z  (cm)(a) ε(z,t)00.20.40.60.8110 102 103 104 105 106full solutionmodelt  (s)(c) εo=0.360.100.080.20V(t)/VfFig. 1 – (a) Liquid fraction vs. depth; from top to bottom the times shown follow a 1-2-5-10 sequencefrom 10 s to 105 s. The bottom-most curve is the equilibrium proﬁle. (b) Volume of drained liquidvs. time for various combinations of boundary conditions, and (c) for diﬀerent initial liquid fractions.slow because liquid enters via “ﬂow” conditions at the top. When only the top conditions arecorrect, we ﬁnd the long-dashed curve of ﬁg. 1(b), in which the volume of drained liquid iscorrect only at short times. For later times the drainage is too fast, and all the liquid eventuallyleaves the sample, because capillary forces do not act at the bottom boundary. When neitherboundary condition is correct, we recover our original approximation, the short-dashed curveof ﬁg. 1(b). The errors introduced at top and bottom act in opposite directions, extendingthe agreement with the full solution to later times. However, the error due to neglect of thebottom boundary conditions eventually dominates and all the liquid leaves the sample.From ﬁg. 1(b) we conclude that the bottom boundary conditions are most crucial. Thisis emphasized in ﬁg. 1(c), where numerical solutions show that, as the initial liquid fractiondeceases, the onset of leakage grows later and a smaller fraction of liquid eventually leaves.Inspired by ﬁg. 1(a), we model the liquid fraction proﬁle by our original approximation exceptnear the bottom, where it is taken according to the equilibrium proﬁle εc/[1 + (H − z)/2ξ]2.These two forms intersect at a point that travels upwards until equilibrium is reached. Theresulting volume of drained liquid, shown in ﬁg. 1(c) by the dashed curves, agrees very wellwith the crucial features of the full numerical solution. This gives signiﬁcant improvementover not only our original approximation, but over that of [1] as well. Note that our modelingapproach cannot be used in a rectangular column, where the liquid fraction in the centralportion remains constant until altered by boundary eﬀects that propagate inwards. Thisshows the advantage of the “Eiﬀel Tower” geometry introduced in [2].∗ ∗ ∗This work was supported by NASA through the microgravity ﬂuid physics program.REFERENCES[1] Cox S. J. and Weaire D., Europhys. Lett. this issue, p. 445.[2] Saint-Jalmes A., Vera M. U. and Durian D. J., Europhys. Lett., 50 (1999) 695.

Reply to Comment by S. J. Cox and D. Weaire on  Free Drainage of Aqueous Foams: Container Shape Effects on Capillarity and Vertical Gradients

Kosmopolis

University of Pennsylvania
ScholarlyCommons
Department of Physics Papers Department of Physics
2001
Reply to Comment by S. J. Cox and D. Weaire on
"Free Drainage of Aqueous Foams: Container
Shape Effects on Capillarity and Vertical Gradients"
A. Saint-Jalmes
University of California, Los Angeles
Douglas J. Durian
University of Pennsylvania, djdurian@physics.upenn.edu
Follow this and additional works at: https://repository.upenn.edu/physics_papers
Part of the Physics Commons
At the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles. Currently, he is a faculty member at the
Physics Department at the University of Pennsylvania.
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/600
For more information, please contact repository@pobox.upenn.edu.
Recommended Citation
Saint-Jalmes, A., & Durian, D. J. (2001). Reply to Comment by S. J. Cox and D. Weaire on "Free Drainage of Aqueous Foams:
Container Shape Effects on Capillarity and Vertical Gradients". Europhysics Letters, 55 (3), 447-448. http://dx.doi.org/10.1209/epl/
i2001-00435-1
Reply to Comment by S. J. Cox and D. Weaire on "Free Drainage of
Aqueous Foams: Container Shape Effects on Capillarity and Vertical
Gradients"
Abstract
Cox and Weaire [1] rightly emphasize that our solution of the drainage equation for the “Eiffel Tower”
geometry does not treat the boundary conditions. There should be a no- flow condition at the top, and, after
leakage begins, the liquid fraction should be pegged to εc ≈ 0.36 at the bottom. They then show how
approximating the no-flow conditions at the top can improve agreement with numerical solution. But as
argued in [2], we maintain that the neglect of capillarity coming from boundary conditions at the bottom
dominates, and that this cannot explain our measurements. At short times, capillarity can delay the onset of
leakage, and at long times it can counter gravity and retain liquid in the foam indefinitely; in either case,
leakage is slower than our approximate solution, contrary to experiment. Therefore, we speculated that the
discrepancy arose from neglect of coarsening, whereby the average bubble size increases via gas diffusion from
smaller to larger bubbles. This is an important puzzle because, while the drainage equation successfully
predicts forced-drainage experiments, it fails dramatically for free-drainage experiments
Disciplines
Physical Sciences and Mathematics | Physics
Comments
At the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles.
Currently, he is a faculty member at the Physics Department at the University of Pennsylvania.
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/600
Europhys. Lett., 55 (3), pp. 447–448 (2001)
Reply
EUROPHYSICS LETTERS 1 August 2001
Reply to the Comment by S. J. Cox and D. Weaire on
“Free drainage of aqueous foams: Container shape
effects on capillarity and vertical gradients”
A. Saint-Jalmes(∗) and D. J. Durian
UCLA Department of Physics and Astronomy - Los Angeles, CA 90095-1547, USA
(received 14 February 2001; accepted 25 May 2001)
PACS. 82.70.Rr – Aerosols and foams.
PACS. 47.60.+i – Flows in ducts, channels, nozzles, and conduits.
PACS. 47.55.Mh – Flows through porous media.
Cox and Weaire [1] rightly emphasize that our solution of the drainage equation for the
“Eiffel Tower” geometry does not treat the boundary conditions. There should be a no-
flow condition at the top, and, after leakage begins, the liquid fraction should be pegged to
εc ≈ 0.36 at the bottom. They then show how approximating the no-flow conditions at the top
can improve agreement with numerical solution. But as argued in [2], we maintain that the
neglect of capillarity coming from boundary conditions at the bottom dominates, and that this
cannot explain our measurements. At short times, capillarity can delay the onset of leakage,
and at long times it can counter gravity and retain liquid in the foam indefinitely; in either
case, leakage is slower than our approximate solution, contrary to experiment. Therefore, we
speculated that the discrepancy arose from neglect of coarsening, whereby the average bubble
size increases via gas diffusion from smaller to larger bubbles. This is an important puzzle
because, while the drainage equation successfully predicts forced-drainage experiments, it fails
dramatically for free-drainage experiments [2].
While Cox and Weaire [1] consider conditions at the top of the sample, we clarify here the
role of boundary conditions at both top and bottom. Numerical results from the drainage
equation are shown in figs. 1(a)-(b) for an initial liquid fraction of ε0 = 0.36, and in fig. 1(c)
for several different ε0. In all cases the sample height is H = 70 cm, the flaring length is
z0 = 25 cm, the capillary rise scale is ξ = 5 cm, the characteristic flow speed is u0 = 0.026 cm/s,
and the dissipation exponent is m = 1. We predicted [2] that the liquid fraction profile
is ε(z, t) = ε0/(1 + t/t0), and that the normalized volume of drained liquid is V (t)/Vf =
1/(1 + t0/t), where t0 = z0/(ε0u0), i.e. that drainage is uniform and all the liquid eventually
leaks out. By contrast the full numerical solution for ε(z, t), in fig. 1(a), shows that the sample
becomes drier at the top and remains wet at the bottom. Consequently, in fig. 1(b), V (t)/Vf
does not approach 1 at long times. Also in figs. 1(a)-(b), we see that the liquid fraction at the
top, ε(0, t), is not zero as assumed in [1].
To see the role of boundary conditions, we disable them one at a time by taking ∂ε/∂z = 0.
When only the bottom conditions are correct, we find the dotted curve of fig. 1(b), in which
the volume of drained liquid is correct at short and long times. The final approach is too
(∗) Now at Laboratoire de Physique des Solides, Université Paris-Sud - 91405 Orsay, France.
c© EDP Sciences
448 EUROPHYSICS LETTERS
0
0.2
0.4
0.6
0.8
1
102 103 104 105
full solution
no BC's
BC's at top
BC's at bottom
t  (s)
(b)
V(t)/V
f
ε(0,t)
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70
z  (cm)
(a) ε(z,t)
0
0.2
0.4
0.6
0.8
1
10 102 103 104 105 106
full solution
model
t  (s)
(c) εo=0.36
0.10
0.08
0.20
V(t)/Vf
Fig. 1 – (a) Liquid fraction vs. depth; from top to bottom the times shown follow a 1-2-5-10 sequence
from 10 s to 105 s. The bottom-most curve is the equilibrium profile. (b) Volume of drained liquid
vs. time for various combinations of boundary conditions, and (c) for different initial liquid fractions.
slow because liquid enters via “flow” conditions at the top. When only the top conditions are
correct, we find the long-dashed curve of fig. 1(b), in which the volume of drained liquid is
correct only at short times. For later times the drainage is too fast, and all the liquid eventually
leaves the sample, because capillary forces do not act at the bottom boundary. When neither
boundary condition is correct, we recover our original approximation, the short-dashed curve
of fig. 1(b). The errors introduced at top and bottom act in opposite directions, extending
the agreement with the full solution to later times. However, the error due to neglect of the
bottom boundary conditions eventually dominates and all the liquid leaves the sample.
From fig. 1(b) we conclude that the bottom boundary conditions are most crucial. This
is emphasized in fig. 1(c), where numerical solutions show that, as the initial liquid fraction
deceases, the onset of leakage grows later and a smaller fraction of liquid eventually leaves.
Inspired by fig. 1(a), we model the liquid fraction profile by our original approximation except
near the bottom, where it is taken according to the equilibrium profile εc/[1 + (H − z)/2ξ]2.
These two forms intersect at a point that travels upwards until equilibrium is reached. The
resulting volume of drained liquid, shown in fig. 1(c) by the dashed curves, agrees very well
with the crucial features of the full numerical solution. This gives significant improvement
over not only our original approximation, but over that of [1] as well. Note that our modeling
approach cannot be used in a rectangular column, where the liquid fraction in the central
portion remains constant until altered by boundary effects that propagate inwards. This
shows the advantage of the “Eiffel Tower” geometry introduced in [2].
∗ ∗ ∗
This work was supported by NASA through the microgravity fluid physics program.
REFERENCES
[1] Cox S. J. and Weaire D., Europhys. Lett. this issue, p. 445.
[2] Saint-Jalmes A., Vera M. U. and Durian D. J., Europhys. Lett., 50 (1999) 695.


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Reply to Comment by S. J. Cox and D. Weaire on Free Drainage of Aqueous Foams: Container Shape Effects on Capillarity and Vertical Gradients

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