Aqueous Foams in Space

1
Aqueous Foams in Space The Foam Drainage
Equation under Microgravity
Motivation
Simon Cox and Guy Verbist
Department of Physics, Trinity College, Dublin 2,
Ireland (e-mail Simon.Cox_at_tcd.ie)
and Shell Research and Technology Centre, PO Box
38000, 1030 BN Amsterdam, The Netherlands.
What happens to the liquid in a foam that is no
longer subject to a gravitational force? With the
advent of the International Space Station this is
a question that can now be experimentally
investigated in more detail than ever before. Our
goal is to provide a theoretical basis for this
work, based on an analysis of the foam drainage
equation 1, with modifications for the
reduction in gravity.
Consider a container of aqueous foam in
microgravity, in which the wetness is not
constant. It is straightforward to make a basic
prediction about the motion of the liquid rather
than a directed propagation of liquid there will
be only a general spreading in all possible
directions. This is reflected in the form of the
zero-gravity drainage equation, which is in
effect a diffusion equation.
Consider now the case of a constant input,
analogous to forced drainage, in which liquid is
pumped into a dry foam at the origin (r 0)
with constant flow rate Q0. We consider the
problem in d dimensions and first find the
self-similar scaling satisfied by the liquid
fraction in each case.
We treat two forms (limiting cases) of the
drainage equation the first corresponds to the
case in which the surface viscosity in the foam
is high, and is based upon an analysis of the
flow through the Plateau borders with a non-slip
velocity condition on their boundaries
(channel-dominated). The second was derived to
describe the limit in which the dissipation
occurs in the Plateau border junctions
(node-dominated). These equations describe the
variation of the liquid fraction in the foam with
position r and time t, represented here by a
dimensionless Plateau border cross-sectional area
?. Both are of the form
A similarity solution of each zero-g equation
exists, giving useful information such as the
position of the wetting fronts and the liquid
fraction at the origin. For the node-dominated
equation, the solutions are exponential in form,
so there is no sharp front as in the
channel-dominated case.
In the channel-dominated case, the similarity
variable is s x/t3/(d4) and ?
t2(2-d)/(d4)y(s). Note that in two dimensions
the liquid fraction at the point of input is
constant. We show below a numerical solution in
one dimension with Q00.01. The liquid fraction
at the point of input increases with t2/5 and the
wetting front spreads with t3/5.
where is the spatial derivative operator and
the relevant flow rate is
Channel-dominated
Node-dominated
Note that in deriving these equations, various
approximations have been made, including the
neglect of the liquid in the soap films. They
have, however, been found to show good agreement
with experiments on dry foams (low liquid
fraction) under terrestrial gravity.
Time
Input
One possibility has been well-studied 2,3,4.
This is the temporal evolution of a small pulse
of liquid placed at the origin in a dry foam.
Further possibilities are analysed in 5.
In the node-dominated case, the similarity
variable is s x/t1/2 and ?
t(2-d)/2y(s). In two dimensions the liquid
fraction at the point of input is also constant.
In one dimension the solution is
We consider the wetting of a dry foam in contact
with a liquid reservoir. We show solutions in
each case, with the critical liquid fraction set
equal to one. In both cases the wetting region
expands into the foam with t1/2, and the amount
of liquid in the foam also increases in
proportion to t1/2.
shown below with Q00.01. The liquid fraction at
the point of input increases with t1/2 and the
wet region spreads with t1/2.
Channel-dominated
Time
Node-dominated
Summary We can now solve the limiting cases of
the zero-gravity drainage equation to help
analyse various experiments of interest, to find
information such as the speed of the wetting
fronts. In the channel-dominated case these
fronts are sharp, and may therefore be easier to
measure in practice.
References 1 D. Weaire, S. Hutzler, G. Verbist
and E. Peters, Adv. Chem. Phys, 102, 315
(1997). 2 I.I. Goldfarb, K.B. Kann and I.R.
Schreiber, Fluid Dynamics, 23, 244 (1988). 3
S.A. Koehler, H.A. Stone, M.P. Brenner and J.
Eggers, PRE, 58, 2097 (1998). 4 S.A. Koehler,
S. Hilgenfeldt and H.A. Stone, Europhys. Lett.,
54, 335 (2001). 5 S.J. Cox and G. Verbist,
Euro. Phys. J. E, (submitted, 2002).
Acknowledgements This research was supported by
the Prodex programme of ESA, and is a
contribution to ESA contract C14308 / AO-075-99.