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Journal ArticleDOI

Free drainage of aqueous foams: Container shape effects on capillarity and vertical gradients

01 Jun 2000-EPL (IOP Publishing)-Vol. 50, Iss: 5, pp 695-701
TL;DR: In this paper, the standard drainage equation applies only to foam columns of constant cross-sectional area and generalizes to include the effects of arbitrary container shape and develops an exact solution for an exponential, "Eiffel Tower", sample.
Abstract: The standard drainage equation applies only to foam columns of constant cross-sectional area. Here, we generalize to include the effects of arbitrary container shape and develop an exact solution for an exponential, "Eiffel Tower", sample. This geometry largely eliminates vertical wetness gradients, and hence capillary effects, and should permit a clean test of dissipation mechanisms. Agreement with experiment is not achieved at late times, however, highlighting the importance of both boundary conditions and coarsening.

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Summary

  • The standard drainage equation applies only to foam columns of constant cross-sectional area.
  • This geometry largely eliminates vertical wetness gradients, and hence capillary effects, and should permit a clean test of dissipation mechanisms.
  • Agreement with experiment is not achieved at late times, however, highlighting the importance of both boundary conditions and coarsening.
  • Disciplines Physical Sciences and Mathematics | Physics Comments.
  • The standard drainage equation applies only to foam columns of constant crosssectional area.
  • Here, the authors generalize to include the effects of arbitrary container shape and develop an exact solution for an exponential, “Eiffel Tower”, sample.
  • Because of the density mismatch between gas and liquid, the bubbles rise and collect at the top, while the liquid falls through the random network of plateau borders and accumulates at the bottom.
  • This behavior can be understood in general terms via a nonlinear partial differential “drainage equation” that expresses liquid conservation as flow proceeds in response to gravity, capillary, and viscous forces [3–5].
  • In spite of its obvious importance for applications, and its apparent simplicity, the problem of free drainage in foams is not well understood yet [6].
  • Even then it was not possible to isolate and identify the individual effects of capillarity, dissipation, and potentially coarsening, because of their strong nonlinear couplings.
  • The authors then develop an exact solution for special containers that flare out exponentially towards the bottom, much like the Eiffel tower.

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Europhys. Lett., 50 (5), pp. 695–701 (2000)
EUROPHYSICS LETTERS 1 June 2000
Free drainage of aqueous foams: Container shape effects
on capillarity and vertical gradients
A. Saint-Jalmes, M. U. Vera and D. J. Durian
UCLA Department of Physics and Astronomy - Los Angeles, CA 90095-1547, USA
(received 9 August 1999; accepted in final form 7 April 2000)
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.
Abstract. The standard drainage equation applies only to foam columns of constant cross-
sectional area. Here, we generalize to include the effects of arbitrary container shape and
develop an exact solution for an exponential, “Eiffel Tower”, sample. This geometry largely
eliminates vertical wetness gradients, and hence capillary effects, and should permit a clean test
of dissipation mechanisms. Agreement with experiment is not achieved at late times, however,
highlighting the importance of both boundary conditions and coarsening.
Introduction. Free drainage is the unavoidable fate of aqueous foams under Earth’s
gravity [1, 2]. Because of the density mismatch between gas and liquid, the bubbles rise
and collect at the top, while the liquid falls through the random network of plateau borders
and accumulates at the bottom. This behavior can be understood in general terms via a
nonlinear partial differential “drainage equation” that expresses liquid conservation as flow
proceeds in response to gravity, capillary, and viscous forces [3–5]. However, in spite of its
obvious importance for applications, and its apparent simplicity, the problem of free drainage
in foams is not well understood yet [6]. One reason is that analytic predictions of the full
time dependence cannot be achieved unless capillary terms, the most egregious nonlinearity,
are dropped from the drainage equation [7]. This cannot be a valid approximation for short
columns, at late times, or even at early times for dry foams [6]. Another reason is that
the mechanism of viscous dissipation is not clearly established; shear flow may occur within
the plateau borders at which three films meet [3], or only within the vertices at which four
plateau borders meet [8]. Only recently have experiments been carried out systematically
as a function of uniform initial liquid fraction and sample height [6]. Even then it was not
possible to isolate and identify the individual effects of capillarity, dissipation, and potentially
coarsening, because of their strong nonlinear couplings.
In this paper we attack the free-drainage problem with an experiment designed to min-
imize capillary effects and, thereby, to isolate and identify dissipation mechanisms. This is
accomplished by altering the sample geometry in such a way that drainage may proceed with-
out vertical gradients in liquid fraction. Since the standard drainage equation applies only to
straight columns of constant cross-section, we begin by generalizing to account for arbitrary
container shape. We then develop an exact solution for special containers that flare out ex-
ponentially towards the bottom, much like the Eiffel tower. At each height, we predict, more
liquid can be passed down than received from above, so that the foam becomes uniformly
drier with time. Finally, we compare with experiments using specially constructed “Eiffel
c
EDP Sciences

696 EUROPHYSICS LETTERS
towers” of two different flaring lengths. At early times we find good agreement; at later times,
however, discrepancies arise due to neglect of both boundary conditions and coarsening.
Generalized drainage equation. The basic goal of all free-drainage models is to deter-
mine the time evolution of the liquid content vs. depth, z, into the foam. For fairly dry foams,
the liquid is distributed within three distinct structures: the flat soap films that separate two
neighboring bubbles; the long Plateau Borders (PBs), of scalloped-triangular cross-section,
at which three soap films meet; and the scalloped-tetrahedral vertices at which four PBs
meet [1–3]. Ignoring numerical factors, the liquid fraction is thus ε (R
2
l + Rr
2
+ r
3
)/R
3
,
where R is the sphere-equivalent bubble radius, r is the radius of curvature of both the PBs
and vertices, and l is the thickness of the soap films. Since l r R typically holds, one
gets ε (r/R)
2
, meaning that the liquid resides almost exclusively in the random network of
PBs. Within one PB, the flow speed may be deduced by the balance of gravity, viscosity and
capillarity:
u = u
0
ε
m
1
ε
c
ε
ξ
ε
∂ε
∂z
. (1)
The value of m is set by the nature of viscous dissipation, being 1 if shear flow is primarily
in the plateau borders [3] and 1/2 if primarily in the vertices [8]. The two quantities that
carry all dimensions are a maximum characteristic flow speed, u
0
, and a capillary rise scale, ξ,
set, respectively, by the competition between dissipation and capillary forces against gravity.
These must scale as u
0
ρgR
2
and ξ γ/ρgR; the precise numerical coefficients depend
on details of the film/PB/vertex geometry and the dissipation mechanism, but are not well
known [6]. And finally, ε
c
=1 0.635 is a critical liquid fraction where the bubbles become
randomly close-packed spheres. This way of writing the flow speed emphasizes the role of
capillarity through the gradient term. According to eq. (1), for any dissipation mechanism
and for any sample geometry of height h, free drainage eventually stops and u 0 when
the liquid fraction reaches a final capillary profile of ε(z)=ε
c
/[1 + (h z)/2ξ]
2
[9–12]; note
that at the bottom ε(h)=ε
c
and ε/(∂ε/∂z)|
h
= ξ.Thusξ may be defined operationally as
the extrapolation length of the equilibrium capillary profile. And similarly, if the boundary is
drier than ε
c
, eq. (1) implies a larger extrapolation length is required for a no-flow boundary
condition: ε/(∂ε/∂z)|
boundary
= ξ
ε
c
|
boundary
.
Assuming that the flow is only downwards (in the +z direction) according to eq. (1), it
is straighforward to write a drainage equation for the behavior of ε(z, t) applicable for any
container shape. We simply require conservation for the total amount of liquid at depth z,
which is proportional to the product of liquid fraction, ε, and the cross-sectional area, A(z),
of the container at z. After separating out the derivatives of A, then dividing by A,the
continuity equation becomes
∂ε
∂t
+
()
∂z
+
A
dA
dz
=0, (2)
where u is given by eq.( 1). The first term in brackets represents the usual drainage equation.
The dA/dz term represents the effects of container shape. It is straightforward to modify
eqs. (1)-(2) for three-dimensional problems, such as lateral flow in free drainage or radial flow
in a spinning container. To our knowledge, the only prior investigation into container shape
is ref. [10], where the total liquid content was compared with that in the remaining capillary
profile to determine whether or not leakage could occur; dynamics were not considered.
Uniform column predictions. For a column with constant cross-section, the complete
free-drainage problem may be solved only if the capillary term is dropped [13]. In this ap-

A. Saint-Jalmes et al.: Eiffel tower drainage 697
proximation, the liquid fraction decreases from an initial value of ε
0
according to [6]
ε(z,t)=ε
0
(z/vt)
1/m
,z vt
1,z vt
with v = u
0
(m +1)ε
m
0
. (3)
There is thus a drying front that travels from the top of the sample downwards at constant
speed v. Below it, the foam maintains the same initial wetness as liquid leaks out at a constant
rate. Once the front hits the bottom of the sample, the rate of drainage progressively decreases
until no liquid remains. The amount of drained liquid vs. time may be found by integration:
V (t)/V
f
=
1
m+1
(vt/h),vt h,
1
m
m+1
(h/vt)
1/m
,vt h,
(4)
where the final value V
f
equals the total amount of liquid within the sample at time zero.
Eiffel tower predictions. Even though approximation is necessary for a uniform column,
exact analytic solution of the drainage equation can be achieved if the container shape flares
out exponentially towards the bottom, like the Eiffel tower. For A(z)=A
0
exp[z/z
0
], it is
simple to verify that a solution of eq. (2) exists where the capillary term, ∂ε/∂z, vanishes and
where the liquid fraction thus decreases uniformly across the entire height of the sample:
ε(z,t)=ε
0
(1 + t/t
0
)
1/m
, with t
0
= z
0
/ (u
0
m
0
). (5)
The corresponding volume of drained liquid is simply
V (t)/V
f
=1 (1 + t/t
0
)
1/m
. (6)
At short times, both ε(z,t)andV (t) vary linearly with t. At long times, they exhibit power
law asymptotes with an exponent that depends on dissipation mechanism. Note that neither
eq. (3) nor eq. (5) satisfies an extrapolation or ε ε
c
boundary condition at the bottom,
z = h. Therefore, these predictions neglect a possible delay in onset of leakage at short times
and the final wetness profile at long times. Such capillary effects may be especially important
for dry foams and short columns, but can be handled only by numerical solution.
Experiments. Foams are produced using a turbulent mixing technique described earlier
[6]. It rapidly provides large volumes of foam, enough to fill our tanks in less than 1 minute,
with any initial liquid fraction desired in the range 3%
0
< 45%. Here the gas is N
2
, and the
liquid is water with 0.4% α-olefinsulfonate by weight. The bubble size distribution is slightly
polydisperse with an average diameter of about 110 µm, independent of ε
0
. The polydisersity
does not noticeably change with either drainage or coarsening. Also as before [6], the viscous
flow speed and capillary length scales are, respectively, of order u
0
0.05 cm/s and ξ 4cm.
At time zero such foams are flowed directly into one of three different containers, all with
the same thickness of 1.25 cm and the same height of h = 70 cm. The first is rectangular,
for reference, with a constant width of 25 cm. The other two have the same 25 cm width at
the bottom, but flare out exponentially from a smaller width at the top according to either
z
0
= 25 or 46 cm. These “Eiffel towers” can also be inverted for additional shape variation
(though eqs. (5)-(6) apply only for z
0
> 0).
The importance of container shape may be demonstrated by differences in drainage curve
data, V (t)/V
f
vs. t, for foams with the same initial liquid fraction of ε
0
=0.36 (close to
ε
c
). The results in fig. 1(a) show that drainage is initially most rapid in the Eiffel tower with
z
0
= 25 cm; it is initially slowest when this tank is inverted. For z
0
= 46 cm the shape variation

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Q1. What contributions have the authors mentioned in the paper "Free drainage of aqueous foams: container shape effects on capillarity and vertical gradients" ?

At the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles. This journal article is available at ScholarlyCommons: https: //repository. In this paper the authors attack the free-drainage problem with an experiment designed to minimize capillary effects and, thereby, to isolate and identify dissipation mechanisms. Even then it was not possible to isolate and identify the individual effects of capillarity, dissipation, and potentially coarsening, because of their strong nonlinear couplings. At each height, the authors predict, more liquid can be passed down than received from above, so that the foam becomes uniformly drier with time.