write_up

Numerical Analysis of Foam Dynamics
Killian Tattan†
11316881, Manya Sahni
School of Physics, Trinity College Dublin
†Author
Supervisor: Professor Stefan Hutzler
January 12, 2015
A water and soap based vertical column of foam is modelled by applying a finite difference
method to the Foam Drainage Equation. Several physical systems are simulated. We find that
our simulations are in good agreement with theory and experimental data.
1 Introduction
The study of foam has many applications, from distillation in the beer industry to the engineering
of Beijing’s ”Bubble Cube” to healthcare products such as shampoo[1][2]. The properties and
structure of any one foam are different from any other foam depending on what the foam is made
of (air, liquid, plastic polymers, rock), how liquid flows through the foam, and local physics such
as thermodynamic properties and gravitational strength. This paper focuses on water and soap
based foams in a constant gravitational field.
Underpinning the dynamics of how liquid flows through a foam is the Foam Drainage Equation
(FDE). The FDE describes how the channels permeating a foam, known as Plateau Borders, vary
as a function of time and vertical position. The FDE has been studied extensively and applied
to many physical systems in works by Weaire, Hutzler, Verbist and Cox[1][2][3][4][5][6]. In this
paper we attempt to replicate some established results and conclude with a simulation of a rising
column of foam, a system which has been observed experimentally, but has yet to be modelled.
A finite difference scheme is employed to solve the FDE. We analyse several distinct physical
systems corresponding to unique boundary conditions of the FDE.
1

2 Foam Structure
Figure 1: Structure of a typical wa-
ter and soap based foam
To understand the FDE we must first study the structure
of a simple foam. A water and soap based foam has the
structure shown in Fig 1. Plateau Borders permeate the
foam between bubble films. We consider liquid flow domi-
nated through Plateau Borders rather than over films (see
Appendix 2). Four Plateau Borders meet at an angle of
109◦
at an intersection called a node [2][3]. We denote the
Plateau Border cross sectional area as α. The node is al-
ways a confluence for exactly four Plateau Borders. These
Plateau Borders have a concave triangular shape. Gravity
and diffusion provide the force that drives the liquid through
the Plateau Borders.
It is known that more liquid flowing through a foam results
in a larger Plateau Border cross sectional area. The reasons
why are not explored here, however the relation between
flow rate and α will be set out in section 4.
For simplicity we consider vertical columns of foams. This
results in an approximation whereby the FDE has one spa-
tial variable (one direction) in which liquid flows, namely
vertically downwards. In the lab, these vertical columns of foam are contained within a transpar-
ent hollow tube (see Appendix 1, Fig. 7).
3 The Foam Drainage Equation and Discretisation
We have the Foam Drainage Equation as derived in Appendix 2
∂α
∂τ
+
∂
∂ξ
α2
−
√
α
2
∂α
∂ξ
= 0 (1)
Recall that α, ξ and τ represent the dimensionless quantities of Plateau Border cross sectional
area, (downward) vertical distance and time respectively. Certain initial and boundary conditions
2

on the FDE correspond to specific physical systems as will be demonstrated.
To solve Eq. 1 numerically we must discretise it. We start by labelling α(τ, ξ) → αi,j, with i
the time index and j the spatial index. Now we have
∂α
∂τ
→
αi+1,j − αi,j
∆τ
∂α
∂ξ
→
αi,j+1 − αi,j
∆ξ
.
Implementing this discretisation, Eq. 1 becomes
αi+1,j = αi,j −
∆τ
∆ξ
α2
i,j+1 −
√
αi,j+1
2∆ξ
(αi,j+1 − αi,j) − α2
i,j −
√
αi,j
2∆ξ
(αi,j − αi,j−1) (2)
This is the discretised FDE which will be solved numerically and used in all simulations in this
paper. The difference in boxed parentheses arises from the first spatial derivative operator in
Eq. 1 and we now have a finite difference scheme. Eq. 2 gives α at time i + 1 in terms of α at
the previous time step i. We will set initial and boundary conditions for α so our program will
have αi,j and can therefore calculate αi,j+1 and αi+1,j. We have left the difference in ξ and τ as
an arbitrary small step, ∆, which is left to us to set in our model.
4 Liquid Fraction and Flow Rate
The wetness of a foam is measured by the liquid fraction[2]. The liquid fraction, φ, of a foam can
be thought of as the ratio of the volume of liquid to the volume of gas in a foam. In our case the
liquid is water and soap and the gas is air, so φ is less than 1. φ is related to geometric parameters
of the foam through[2]
φ =
5.35α
V
2/3
b x2
0
(3)
where Vb is the bubble volume and x0 is the length scale introduced in Appendix 2. As these
are both constants, we see that φ and α are essentially the same quantity but simply scaled
differently. Therefore the cross sectional area α of any given (or a collection of) Plateau Border(s)
can be thought of as a local measure of wetness. Note that in general φ = α. For convenience,
3

our graphs will alternate between α and φ and are related simply through Eq. 3. It is found
experimentally that a water based foam cannot have a liquid fraction greater than 0.36[2]. This
number is important as it is a boundary condition we use throughout this paper.
When liquid is pumped into a foam it has a flow rate, Q. Flow rate has units of [volume/time],
but for our purposes it can be related to α through[2]
Q = α2
(4)
Therefore the Plateau Border cross sectional area increases as the square root of the flow rate. A
larger flow rate also means that the liquids average flow velocity, v, will be larger. We can relate
Q to v by[2]
Q = v2
(5)
⇒ α = v (6)
This relation serves to be extremely useful when analysing the analytical solution to forced
drainage, as will be shown in section 6.
5 Free Drainage
The first physical system we will analyse is free drainage. In this case, the column of foam is
initially uniformly wet and is allowed drain under gravity. The column is in contact with liquid at
its base, which through capillary action ”sucks” some liquid up vertically into the foam, leaving
the base of the foam with a large liquid fraction. The boundary and initial conditions are as
follows. The foam is initially uniformly wet and set at an arbitrary value of φ = 0.15. The input
flow is set to zero so that there is no liquid input. The base (bottom) of the foam remains at a
constant wetness of φ = 0.36 for all times.
Running the model with these parameters, we obtain the graph shown below in Fig. 2. We observe
several interesting features in Fig. 2. Firstly, for small times φ decreases at the top while increasing
(due to capillary action) near the base. This means there must be a turning point in between top
and bottom. Indeed, we see that at ξ ≈ 7 the liquid fraction is initially constant at 0.15 for a
short time. Initially the liquid fraction increases near the base before decreasing at larger times.
4

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 10
liquidfractionφ
(downward) vertical distance ξ
wet base
progression through time
Top Bottom
equilibrium
Figure 2: Liquid fraction profiles at different times of a foam undergoing free drainage. The foam
has an initial wetness of φ = 0.15 (straight line) except at the base where the boundary condition
imposed. After sufficient time the foams liquid fraction tends to the equilibrium profile (dashed
line).
As time goes to infinity φ reaches an equilibrium profile. In approaching the equilibrium limit φ
decreases throughout the foam except at the base where it tends to 0.36.
Fig. 2 depicts what we would expect to measure experimentally. The foam dries as liquid is drained
out resulting in a decreasing liquid fraction. Capillary action results in a larger liquid fraction at
the base of the foam. The very bottom of the foam is in contact with liquid and remains at a
constant liquid fraction of 0.36.
6 Forced Drainage
In a forced drainage system, the foam is initially dry and therefore has a very small liquid fraction.
Liquid is then pumped in from the top of the foam at a constant rate. For this case, there is an
analytical solution to Eq. 1[2][3]
α(τ, ξ) =
v tanh2
√
v(ξ − vτ) for ξ ≤ vτ (7)
0 for ξ ≥ vτ (8)
where v is the liquids average flow velocity.
The boundary and initial conditions are as follows. The foam initially has a small liquid fraction
<< 0.1 (dry foam). Liquid is then pumped in at the top at a constant flow rate. With these
5

conditions we run the model where the output is shown in the graphs below. Fig. 3a depicts
a foam not in contact with liquid at its base (bottom) where the base liquid fraction is set to
be the same as the initial liquid fraction. While Fig. 3b depicts a foam in contact with liquid
at its base (bottom) resulting in capillary action and a larger liquid fraction near the base. We
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12
plateauborderareaα
τ=4 τ=6 τ=8 τ=10 v=1
Top Bottom
numerical data
tanh2
(ξ-10), ξ<10
(a) Forced drainage of a dry foam
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20
liquidfractionφ
Top Bottom
wet base
equilibrium
v
(b) Forced drainage of a dry foam with wet base
Figure 3: Forced drainage of dry foams. We have set the flow rate to 1 in Fig.(a) for convenience
such that Q = α2
= v2
= 1. With v = 1 the analytical solution in Eq. 7 takes the simple form
tanh2
(ξ − τ) for ξ ≤ vτ. We compare this analytical solution to the numerical data and find that
they are in good agreement. In Fig.(b) the foam is in contact with liquid at its base. As a result
the base remains at a constant liquid fraction of φ = 0.36. After enough time the foam reaches
an equilibrium profile (dashed line) where the foam is almost uniformly wet apart from the base
where the foam remains wetter.
see the numerical data is in good agreement with the analytical solution. The small discrepancy
is due to the numerical accuracy of our finite difference scheme. The stability of our code is
extremely sensitive on the steps we set, ∆ξ and ∆τ. If the step values are set below some
unknown small threshold the output diverges and the computation fails. However if the steps
are too large we do not obtain a high enough resolution. The task is to find values that give
high enough resolution but do not diverge. In finding optimum step values we attempted Von
Neumann stability analysis on Eq. 2 which should give constraints on the step values. However
the resulting equation was unsolvable, meaning we could not calculate optimum values for ∆τ and
∆ξ. A range of values between 0.00001 and 0.01 for ∆τ and ∆ξ were experimented with. When
the best possible resolution was obtained that did not diverge, these step values were recorded
and used for all other simulations.
From Fig. 3a and the analytical solution Eq. 7 we see that forced drainage with a constant
6

input flow rate behaves as a solitary wave moving through the foam expanding Plateau Borders
as it progresses. The solitary wave keeps the same profile as it moves through the foam. An
important feature of Fig. 3a is that the interface between dry and wet foam has a finite width,
rather than a sharp step function. This finite width interface between the dry and wet sections
of foam tells us that the Plateau Borders cross sectional area expands in a gradual and smooth
process, albeit an exponential one.
7 Coalescing Waves
This system is similar to forced drainage with a solitary wave front as in the previous section,
however now at some arbitrary time we increase the flow rate Q. We expect α to increase accord-
ingly. Indeed, this is what is observed in our numerical outputs and it is also what is observed
experimentally[2][3].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25 30 35 40
plateauborderareaα
v1 v1
v=v1+v2
Top Bottom
v2
flow rate = 1
flow rate = 0.49
Figure 4: Two solitary wave fronts of different velocities coalescing in a foam. The final velocity v2
is larger than the initial velocity of the first wave v1, but smaller than the coalescing wave velocity
v.
At first there is a single solitary wave front moving with velocity v1 =
√
Q1. The flow rate is
then increased to Q2 resulting in a new solitary wave behind (on top of) the original wave. This
new wave moves at velocity v > v1. The faster wave will eventually catch up to the first wave
where the two wave fronts will coalesce. The final wave front is solitary and moves with velocity
v2 =
√
Q2. It is found that as the waves coalesce the liquids average flow velocity is v = v1 +v2[3].
The largest of the three velocities is v due to the controlled increase in flow rate. At later times
7

after the two waves have coalesced the wave fronts’ velocity is reduced from v to v2 due to the
resistance of the first wave moving at v1 < v and conservation of momentum.
8 Pulse wetting of a Dry Foam
Having studied forced drainage at a constant rate, intuition tells us that if a short pulse of liquid
is injected into the top of a dry column of foam, we expect to observe some varying form of a wave
moving through the foam. Indeed, a wave front is observed, but it is not solitary as in the case of
forced drainage at a constant rate. As the wave is not solitary, we expect to observe diffusion of
the wave.
We have a column of foam that is initially dry. A short pulse of liquid is injected into the top.
Theory tells us that the liquid should diffuse in the foam through drainage and through capillary
action[2]. The resulting profile of the Plateau Border cross sectional area is very different to
that of forced drainage in Fig 3a. The numerical data produced is shown below in Fig 5. Our
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10 12
plateauborderareaα
τ=2
τ=14
Top Bottom
pulse direction
Figure 5: A sharp pulse of liquid injected into the top of a dry foam. The pulses undergoes
diffusion and drainage.
simulation agrees with experimental data produced by Weaire and Hutzler in [2]. We see that
initially the pulse is concentrated near the top of the foam for small times. As time progresses, the
pulse diffuses and drains through the foam. As the pulse diffuses through dry sections of foam,
it wets the foam (increases the liquid fraction) and thus increases α. As the pulse moves through
the foam the peak height decreases. This tells us that the largest value of α (the wettest the
8

foam becomes locally) obtained in the column of foam is at the top of the foam when the pulse is
initially injected.
We observe that the peak of the foam is continuously moving to the right (downward through the
foam). The relation between peak position is given by the relations[2]
αpeak ∝ τ−1/2
(9)
ξpeak ∝ τ1/2
(10)
These relations are verified in Appendix 3.
9 Rising Column of Foam
A rising column of foam has the following physical setup. Initially there is a small column of
foam inside the base of a hollow cylinder submerged in a liquid solution (Appendix 1, Fig. 7). By
pumping air through a nozzle submerged beneath the cylinder, air bubbles are created, rise and
merge with the foam column inside the cylinder. As bubbles are pumped in at the base the column
increases in height and liquid is carried up from the base while simultaneously being drained from
the foam due to gravity. The liquid fraction varies with position within the column; the base is in
contact with liquid and remains wet, while the top of the column becomes dry as liquid is drained
away. If the liquid fraction at the top of the column falls below some critical value, bubble films
begin to rupture and the column no longer grows. In this case the bubble input rate at the base
exactly matches the rate at which bubbles collapse at the top and the column has reached a steady
state (fixed height). However some theoretical models predict that at some critical bubble input
rate the foam height diverges where the column grows indefinitely[1].
We consider the former case where the column of foam reaches a steady state. In our model
we break the process of a rising column of foam into two processes: rising and drainage. For
every 2 × 103
discrete iterations that the foam column rises we allow one drainage iteration via
the FDE. This ratio corresponds to the bubble input rate and was chosen arbitrarily after running
the model for less rising iterations which produced non-physical outputs. We obtained the graphs
shown below in Fig. 6.
9

0
1
2
3
4
5
6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
foamheight(arbitraryunits)
liquid fraction φ
progression through time
(a) Foam height as a function of liquid fraction
4
4.2
4.4
4.6
4.8
5
5.2
5.4
0 1 2 3 4 5 6 7 8 9 10
dimensionless time τ
(b) Foam height as a function of time
Figure 6: Model of a rising column of foam reaching a steady state equilibrium. Fig.(a) shows
50 profiles of a rising column of foam taken at regularly spaced time intervals. We see that at
larger times the profiles become more compressed. Fig.(b) shows that the foam height reaches
equilibrium at 5.22 units in height.
In this scenario the column of foam reaches an equilibrium height of 5.22 units. Fig. 6a shows this
as the profiles become more compressed and converge at larger times. When we plot the foam
height as a function of time we see that the foam height initially increases and then levels out at
5.22 units in height.
10 Conclusions
We have seen the power and versatility of the Foam Drainage Equation to accurately model many
different physical systems. Our robust simulations have consistently been in good agreement with
experimentally derived results and theory. When comparing numerical data to analytical solutions
such as Eq. 7, there may be small differences between data and analytics. We can attribute this
to machine accuracy. Step sizes play an important role in obtaining higher resolution; made too
small and the simulation becomes unstable and diverges. If further analysis into the optimum step
size could have been conducted, we may have obtained more accurate numerical data.
If the project was to be extended, the author would investigate some properties of the FDE further,
for example the meaning of the constant of proportionality of a pulses peak position (Appendix
3). Another relation that could have been further explored is that of the maximum height of a
rising column of foam given the boundary conditions, such as the analysis conducted in [1].
10

Acknowledgements
We would like to thank both Professor Stefan Hutzler and Friedrich Dunne for guiding us through
this project. Both spent many hours for weeks with us, perfecting our simulations and explaining
the theory of foams. We are especially grateful to Friedrich who worked closely with us on the
numerical simulations.
References
[1] S. Hutzler et al, Evaluation of a steady-state test of foam stability, Philosophical Magazine 91
(2011)
[2] D. Weaire and S. Hutzler, The Physics of Foams, Clarendon Press, Oxford (1999)
[3] G. Verbist et al, The Foam Drainage Equation, J Phys. Condens. Matter 8 (1996)
[4] S. Cox, The Foam Drainage Equation, presentation, Institute of Mathematical and Physical
Sciences, University of Wales, Aberystwyth
[5] S. A. Koehler et al, Dynamics of foam drainage, Phys. Rev. E 51 (1998)
[6] A. Nikkar, Assessment of New Analytical Method for Solving the Foam Drainage Equation,
Hindawi (2014)
11

Appendix
Appendix 1. Foam column setup
The physical setup of the system we analyse in this paper is that of a vertical column of foam
contained in a clear cylinder as shown below in Fig. 7
Figure 7: A vertical column of foam contained within a clear perspex cylinder. All the simulations
in this paper are based on this physical setup. Gravity acts downwards in the same direction as ξ.
Note: in some cases, such as that of forced drainage where the column of foam is not in contact
with liquid at its base, the liquid solution is replaced by air. Image Source: G. Verbist, The Foam
Drainage Equation [2].
Appendix 2. Derivation of the Foam Drainage Equation
We derive the Foam Drainage Equation (FDE) from theory and first principles as set out by
Weaire and Hutzler in [2]. To do this we need to make some assumptions. Firstly, we consider
only Poiseuille flow as this yields a simpler form of the FDE making it analytically solvable.
Poiseuille flow imposes the constraint that liquid flow velocity is zero at the boundaries of the
Plateau Border (the interface between a Plateau Border and a bubble film) and maximum in the
centre, resulting in a parabolic velocity profile inside the Plateau Border. Next we assume only
liquid flow inside the Plateau Borders and no flow over the bubble films. Finally, we assume
12

that the liquid is incompressible such that the density, pressure and viscosity do not change as a
function of distance and time.
We begin with the one dimensional continuity equation
∂A(x, t)
∂t
+
∂
∂x
(A(x, t) · u(x, t)) = 0 (11)
where A(x, t) is the cross sectional area of a Plateau Border and is a function of vertical distance
x and time t, and u is the average ﬂow velocity of the liquid moving through the Plateau Border.
Recall the Laplace-Young law of pressure diﬀerence across a liquid surface[2]
Pl = Pg − γ/r (12)
with Pl and Pg the pressures of the liquid and gas respectively with units of [force/length]. γ is the
liquid surface tension and r is the radius of curvature of a Plateau Border. γ has units [mass/time2
]
and r has units of [length]. Using the geometric relation A = C2
r2
, C =
√
3 − π/2 and the
dissipative force −fηu/A we obtain
ρg −
∂Pl
∂x
−
ηfu
A
= 0 (13)
which is the dissipation balanced by gravity, ρg, and the pressure gradient ∂Pl
∂x
. The density of
the liquid, ρ, has units of [mass/length3
], while gravity, g, has units of [length/time2
]. f is a
numerical value which is determined by the shape of the cross sectional area, and in our case is
49. η is the viscosity of the liquid with units [mass]/[time ∗ length]. Combining Eq. 12 and
Eq. 13 we have
∂A
∂t
+
1
ηf
∂
∂x
ρgA2
−
√
ACγ
2
∂A
∂x
= 0. (14)
By a cleaver change of variable, we can make Eq. 14 dimensionless. Let ξ = x/x0 , τ = t/t0 and
α = A/x2
0 represent the dimensionless quantities of vertical distance, time and cross sectional area
respectively. By choosing the parameters
x0 = Cγ/ρg
t0 = η / Cγρg
13

where η = 3fη 150η is the effective viscosity, Eq. 14 reduces to
∂α
∂τ
+
∂
∂ξ
α2
−
√
α
2
∂α
∂ξ
= 0 (15)
This is the dimensionless Foam Drainage Equation with α = α(τ, ξ). A simple check will verify
that our scaling factors x0 and t0 do in fact posses the correct units. The equation describes how
the Plateau Border’s cross sectional area α varies as a function of downward vertical position ξ
and time τ. In going from Eq. 14 to Eq. 15 we average over all values of cosine of the angle that
a Plateau Border makes with the vertical. In other words, we make an approximation that the
Plateau Borders are, on average, vertical. This leads to the dimensionless quantity ξ which is the
downward vertical distance within the column of foam.
Appendix 3. Pulsed drainage relations
The relations of Eq. 9 and Eq. 10 are verified below in Fig. 8 using data from Fig. 5. As the
pulse moves through the foam in Fig. 5 the peak height decreases but the area under the curve
remains constant. Therefore we have conservation of wetness i.e. the total cross sectional area of
all the Plateau Borders throughout the foam remains constant. As a result, the total area under
the curve of the pulse in Fig. 5 remains constant.
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
plateauborderareaα
0.972*τ
-1/2
numerical data
(a) α as a function of time
0
2
4
6
8
10
0 2 4 6 8 10 12 14 16
(downward)verticaldistanceξ
2.125*τ
1/2
numerical data
(b) Position (ξ) as a function of time
Figure 8: Curve fitting to peak positions corresponding to pulsed drainage data in Fig. 5. Note the
τ−1/2
relation in (a) and the τ1/2
relation in (b). Given our boundary conditions, we see that the
constants of proportionality are 0.972 and 2.125 for Eq. 9 and Eq. 10 respectively. The meaning
of these constants is not investigated by the author.
14

write_up

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (16)

Similar to write_up

Similar to write_up (20)

write_up