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The Viscous Froth Model: steady states and the high-velocity

limit

BY S.J. COX1, D. WEAIRE2, G. MISHURIS1

1Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth SY23 3BZ, UK

2School of Physics, Trinity College, Dublin 2, Ireland

Thesteady-statesolutionsof theViscous FrothModelforfoamdynamicsareanalyzed,andshownto beoffinite extent

or to asymptote to straight lines. In the high-velocity limit the solutions consist of straight lines with isolated points

of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile

two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.

Keywords: foams, discrete microfluidics, interface motion

1. Introduction

Foams are widely used in traditional applications such as fire-fighting and froth flotation [1, 2]. More recently they

have begun to find new applications in the emerging field of discrete microfluidics [3, 4], in which small volumes of

gas (or, equivalently, liquid) can be manipulated within narrow channels.

When monodisperse bubbles are confined in narrow channels with a low liquid fraction, they generally form

ordered structures [3, 5] – see figure 1. They may be readily manipulated in networks of such channels. Such a system

is suggestive of a practical system of microfluidics [6], so the motion of the bubbles, when driven by a pressure, is

of practical as well as basic interest. The desire to predict the structure and dynamics of such of a foam arises from

the requirement to design both chemical formulation and container geometry with maximum efficiency, to perform

specific functions.

In a rectangular channel whose width is much greater than its depth, the structure in question may be regarded as

essentially two-dimensional. That is, each bubble touches both of the bounding plates and the soap films that span

the gap are perpendicular to those plates. Accordingly a 2D model, previously developed in the physics of foams [7]

has been used in this area. This “viscous froth” model is only a provisional or skeletal one (including in particular

linear relationships when power laws may be more realistic), but it has succeeded in shedding light on some observed

phenomena that are seen when the flow velocity is large enough to depart from the quasistatic condition [7, 8, 9, 10].

However, it then appeared that significant qualitative details of observed structures were apparently not compatible

with the model and it was not clear what needed to be added to it. The present study, which explores the high velocity

limit, was stimulated in part by these anomalous structures.

We review the model in §2 and find its steady-state solutions for single lines in §3. Even this apparently straight-

forward problem presents quite a wealth of mathematical detail. Simplicity is restored in the high velocity limit [10],

which is described in §4. Any of these steady-state solutions may be used to provide pieces of a more general solution

in which the lines meet at 120◦, allowing comparison with experiment, as discussed in §5.

2. The viscous froth model

A static dry foam can be representedby the ideal 2D soap froth [11], in which each film is a circular arc, and three arcs

meet at a vertex at equal angles of 120◦. The elastic force on each film, which is the productof curvatureκ and surface

(or line) tension γ, balances the pressure difference ∆p across it according to the Laplace law, ∆p = γκ. The Viscous

Froth Model (VFM) extends the ideal soap froth to predict the dynamics of such structures at finite deformation rate.

See figure 2. In the VFM three forces act on each element and sum to zero, since inertia is neglected. They are due to

surface tension, pressure difference and a drag force which resists motion normal to the film.

Consider a point on a soap film: a force balance in the direction of the normal leads to the following evolution

equation:

∆p − γκ = λvβ

n

(2.1)

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Cox, Weaire, Mishuris

Figure 1. A 211 (or staircase or hex-2) structure in static equilibrium.

120o

120o

120o

120o

120o

120o

120o

120o

1

2

2

1

Motion

γ

γ

γ

γ

λ

(a)(b)

n

v

p

p

p

p

Figure 2. Factors determining the local balance of forces in the static case and in the VFM. In the VFM, a viscous drag force is

included. The pressure difference is ∆p = p2− p1

where λ is the drag coefficient and vnthe normal velocity. The exponent β is usually set to unity (the linear VFM, as

considered here) to speed up the numerical algorithm and to simplify analytical theory [7].

Note that the 120◦rule for internal vertex angles is an essential feature of the model, since three equal surface

tension forces, and no others, act on a point.

3. Steady-state motion

This analysis is concerned with steady-state motion with constant velocity. It generalizes earlier treatments of steady-

state solutions of curvature-driven growth (the VFM with zero pressure difference), reviewed in [12], and soap film

motion in channels [9]. Here we state the key results, relegating derivations and details to the Appendix.

We look for solutions of the VFM for motion at a constant velocity of magnitude V . We use Cartesian axes and

seek curves that translate in the positive x-direction by writing eq. (2.1) in the following form [13]:

∆p + γ

xyy

(1 + x2

y)3/2=

λV

(1 + x2

y)1/2,

(3.1)

where xyis the derivative of x, etc. We normalize the x and y coordinates by the length-scale

L0=

γ

V λ

(3.2)

and write

α =∆p

λV

(3.3)

(cf. the mobility parameter a of [10, eq. (18)]) to give

α +

xyy

(1 + x2

y)3/2=

1

(1 + x2

y)1/2.

(3.4)

There are three classes of solutions, the details of which are given in the Appendix. Note that, given the form

of (3.1), we may choose each curve to originate at (0,0). They are classified according to their shape at the origin,

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The Viscous Froth Model: steady states and the high-velocity limit

3

y

x

(a)

y

x

(b)

y

x

(c)

Figure 3. Examples of steady-state solutions of the VFM for uniform translation in the x direction at velocity V . These consist of

either straight lines, curves that asymptote to straight lines or curves of finite extent, and have (a) linear (α = 0.5), (b) square-root

(α = -0.5 (short dashes), 0.5 (solid line), 1.5 (long dashes)) or (c) quadratic (α = -1.5 (dotted line), -0.5 (short dashes), 0.5 (solid

line), 1.5 (long dashes)) behaviour at the origin respectively. The parameter α, defined by (3.3) , represents the ratio of driving

pressure to the drag. For more detail see figure 4.

illustrated schematically in figure 3; for given α there may be more than one solution, distinguished by the tangent at

the origin.

The most straightforwardsteady states are simply straight lines (in which the middle term of eq. (3.4) is zero) with

slope

dy

dx= ±??α−2− 1??−1/2,

as in figure 3(a). These solutions are found only for |α| ≤ 1; for α = 0 (i.e. no pressure difference) they are parallel to

the x−axis while for α = ±1 they are parallel to the y−axis.

Inadditionthere arecurvesthat evolvefromthe originwith y ∼ ±√x, as x → 0. For 0 < α ≤ 1 theyasymptoteto

the above straight lines (eq. (3.5)). Otherwise they reach a region of zero slope at which we terminate our calculation

to leave curves of finite extent with accessible endpoints – see figure 4(a). The details of the envelopes, which are the

locus of the endpoints, are presented in the Appendix.

Finally, there are solutions with y ∼ ±x2at the origin, shown in figure 4(b). For positive α these lie above

the x-axis, and changing the sign of α results in a reflection in this axis. For |α| ≤ 1 they again asymptote to the

above straight lines (eq. (3.5)). As for the previous case, we follow the curves up to the point at which the respective

derivatives tend to infinity, and the correspondingpoints lie on essentially the same envelope.

The coincidence of the envelopes can be seen by appealing to figure 3: each curve of finite extent in figure 3(b),

with square-root behaviour at the origin, approaches the envelope as a parabola. Thus it is also represented in figure

3(c), i.e. it is one of the curves with quadratic behaviour at the origin, albeit with a shift and possible change of

orientation. For example, the dashed curve for α = −0.5 in the quadrant x > 0,y > 0 in figure 3(b) can be translated

to its (dashed) equivalent in the quadrant x < 0,y < 0 in figure 3(c).

Each curve of finite extent can also be extended with any other solution for the same value of α. For example,

the curve for α = 1.5 in the quadrant x > 0,y > 0 in figure 3(c) can be continued by the curve for α = 1.5 in the

quadrant x < 0,y > 0 in figure 3(b). This is elaborated upon below, in figure 5.

As the velocity increases, the extent of the finite, curved, solutions tends to zero, while the straight line solutions

remain invariant. The other solutions of infinite extent tend toward piece-wise straight-line forms, as discussed below.

Before proceeding, we first illustrate the use of the steady state solutions with the example of a single bubble

confined within a channel and attached to a moving side wall, shown in figure 5. The ends of the interface enclosing

the bubble are fixed to the wall at y = 0 a distance 2L apart, and the wall is moved in the positive x direction with

speed V . We use a Surface Evolver-based implementation [8, 14] to solve the VFM to find the bubble shape, with

parameters γ = λ = 1, L = 1/√2, time step ∆t = 1 × 10−5and circa 60 line segments. The pressure p in the

bubble is varied to enforce a constraint of fixed area, Ab = πL2/2, although for the values of V considered here p

never differs greatly from γ/L. Figure 5(a) shows bubble shapes for different V , which cease to be physical when V

(3.5)

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Cox, Weaire, Mishuris

(a)

0

1

2

3

4

-2-1 0 1 2 3 4

α = 0 (Mullins)

Position x/L0

Position y/L0

α = 1

α = 2

α = −2

(b)

0

1

2

3

4

-2 -1 0 1 2 3 4

Position x/L0

Position y/L0

α = 1

α = 2

α = 0

Figure 4. Detailed results for steady-state solutions of the VFM for uniform translation with velocity V in the x direction and given

pressure difference ∆p, completing the sketch of figure 3. (a) Square-root behaviour at the origin: α = ∆p/(λV ) increases from

-2 to 2 in steps of 0.25. Mullins’ solution (the “grim reaper”) for grain-growth (i.e. the VFM without pressure differences) [15]

corresponds tothe case α = 0, whilethe case α = 1 isa vertical line. A second numerical calculation gives the shape of the limiting

boundary (envelope). The solutions are completed by reflecting in the line y = 0 and, in the case of solutions of infinite extent

(α ∈ [0,1], shown with thicker lines) taking the asymptotic slope. (b) Quadratic behaviour at the origin: α increases from 0 (the

x-axis) to 2 in steps of 0.25. Solutions for negative α can be found by reflecting in the line y = 0. Solutions of semi-infinite extent,

for α ∈ [0,1], are shown with thicker lines. Each curve (or part thereof) of finite extent can be obtained from (a) by reflection and

translation, and the limiting boundary is a reflection of the one in (a). The solutions may be normalized according to (3.2).

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The Viscous Froth Model: steady states and the high-velocity limit

5

is sufficiently large that the angle between the trailing edge and the wall goes to zero. We find this critical velocity to

be V ≈ 2.3 (figure 5(b)): the angles at the leading and trailing edges are equal only at low velocity, and as the velocity

increases, the angle at trailing edge decreases more quickly.

We show in figure 5(c) how the solutions of figure 4 can be used to predict the bubble shape. In the case V =

γ/L =√2, the parameter α is equal to one. Thus the shape of the leading edge is given by part of the curve for α = 1

in figure 4(b). Since the pressure difference acts in the opposite direction on the trailing edge, the latter’s shape is

given by part of the curve for α = −1 in figure 4(a). The two are joined at the point where the slope is vertical, and

truncated to give two endpoints the required distance apart. The simulation result is clearly consistent with the derived

steady-state solutions (figure 5(c)).

4. High velocity limit

Here we examine more carefully the forms that emerge as V tends to infinity, by reference to figures 3 and 4. It is

easiest to consider α to be constant which, in an experiment, would imply that ∆p is increased in proportion to V ,

according to (3.3), (i.e. fixing α whilst varying the pressure difference [9]) while λ is kept constant. This analysis

builds upon the results of Grassia et al. [10], who showed that the curvature is pushed towards the side walls and gave

approximate expressions for the film shape there.

The only effect of increasing V to a high value is then to uniformly shrink the curves progressively wherever they

have finite curvature. The normalized infinite (or semi-infinite) solutions have significant curvature only close to the

origin. The curvature here is of order 1/r ≈ α/L0(= ∆p/γ), and the curvature is concentrated within a distance of

the origin that is of order r. Now consider the evolution of the curve as the velocity goes to infinity. The curved parts

become ever sharper, as r goes to zero.

In the limit V → ∞, the solutions therefore consist entirely of straight lines. Consider, for example, the solution

that we have referred to as “square-root, α = 0.5” in figure 3(b). In this case, the limiting high-velocity solution

consists of a pair of straight lines inclined at ±30◦to the x-axis and joined at the origin, where the curvature is

infinite. The case that is referred to as “quadratic, α = 0.5” in figure 3(c) consists of just a single line, also inclined

at 30◦. All other solutions or parts of solutions vanish in the limit V → ∞, so we are left with essentially a single

possibility consisting of straight lines, arising from the three cases of figure 3.

(a) Applying the high-velocity solutions

In appropriate cases, we would hope to use these simple solutions directly. Note that we may cut any piece of the

infinite curves of figure 3 to fit certain boundary conditions. We may also use such pieces to construct high-velocity

solutions with vertices, for structures such as the one of figure 1.

The main subtlety in doing so relates to the infinite curvature at the kink of the solution in figures 3(b) and (c).

If a solution is excised that has this as an endpoint, the tangent at the rear end of this curve is not well-defined. By

reference to figures 3 and 4, we see that the direction of the tangent may be taken to lie anywhere in a broad range of

angles.

To demonstrate these results, we first perform representative VFM calculations on a single film.

(b) Single film with moving endpoints

Considerasinglefilm(orline)whichexperiencesapressuredifference∆p andwhoseendpoints,locateda distance

2Lapartinthey direction,moveat constantvelocityV in thex-direction.We solvetheVFM numerically,as described

above, to determine the steady-state shape of the film in this case, for fixed α and γ = λ = L = 1. Figure 6(a) shows

the convergence towards the predicted shape as the velocity (and pressure difference) increases for α = 0.5. In the

special case α = 0, the films consist of segments of Mullins’ solution.

(c) Single film between parallel walls

Now consider a single film which experiences a pressure difference ∆p that pushes it along a straight channel of

width 2. In addition to the drag on the bounding plates, with drag coefficient λ, we introduce a further drag on the

side walls of the channel proportional to velocity, with coefficient λsw. There is then a new evolution equation for the

points of the film at the side walls:

λswv = γ cosθ

(4.1)

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Cox, Weaire, Mishuris

(a)

V=0.0

V

V = 2.0

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5

Angle between film and wall

Velocity

Trailing edge

Leading edge

(c)

-0.8

-0.6

-0.4

-0.2

0

-0.6-0.4 -0.2 0 0.2

x position

0.4 0.6 0.8 1 1.2

Figure 4(b) (α=1)

Figure 4(a) (α=-1)

Simulation

y position

Figure 5. The shape of a bubble attached to a moving side wall. (a) Shapes for different velocities V , with the endpoints a distance

2L =

Comparison between the shape found in a numerical simulation of the VFM and the curves of figure 4 in the case V = 1.43.

√2 apart. V increases from 0 to 2 in steps of 0.5. (b) The angle between the film and the wall for different velocities. (c)

where θ is the angle between the film and the wall.

The numerical implementation of the VFM described above is used to find the shape and velocity of the film for

given pressure difference ∆p and drag coefficient λsw = 0.01. For these parameter values we find that the velocity

increasesindirectproportionto∆p, givingα ≈ 1.Figure6(b)showstheconvergencewithincreasingvelocitytowards

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The Viscous Froth Model: steady states and the high-velocity limit

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(a)

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

V = 1

V = 3

V = 10

V = 100

Position x

Position y

(b)

0

0.2

0.4

0.6

0.8

1

0 0.05

Position x

0.1

Centre-line of channel

Position y

∆p = 100

∆p = 50

∆p = 20

∆p = 5

Figure 6. (a) Examples of numerical solutions for the steady-state shape of a single film between two endpoints which move to

the right with velocity V . The applied pressure difference is chosen to give, in this case, α = 0.5. For the highest velocity shown

(V = 100), the curve consists essentially of a pair of inclined lines joined by a short arc, as predicted. Note then the difficulty

in defining the included angle at y = 0. (b) Examples of numerical solutions for the steady-state shape of a single film within a

channel with side wall drag and an imposed pressure gradient ∆p, following [10]. As ∆p increases, the steady velocity of the film

increases in proportion, so that the film converges to the predicted shape for α = 1 consisting of a straight line with curvature

concentrated at the wall.

the film shape expected from figure 4 for this value of α. That is, the film becomes vertical away from the wall and the

angle θ decreases to zero.

(d) Staircase structure

We next consider the extended array of bubbles depicted in figure 1; in order to mimic experiments [3] we choose

to find the steady-state shape of this 211 or staircase structure at finite flow-rate, with sidewall drag as in (4.1).

To avoid end-effects, we use a sample that is periodic in the x−direction, in a channel of width 2L = 7/4 with

bubblearea 7√3/4.A pressuregradientis appliedby invokinganadditionalcontributionof ∆p∗to thepressurediffer-

ence across each film. (This results in a drop of 2∆p∗between adjacent bubbles on the same side of the channel.) For

sufficiently high ∆p∗, the angle θ goes to zero and the system becomes unstable. In what follows we shall concentrate

on the configuration close to this unstable limit, denoting the critical values of pressure and velocity by ∆p∗

Vcritrespectively.

Applying the arguments of the previous sections leads us to expect the limiting structure at high velocity to take a

surprising form: it returns to the original static or low-velocity form of figure 1. This is consistent with what we find

(figure 7): note that curvature is here concentrated at the points of contact with the side walls, so that a contact angle

consistent with the drag at the side walls is maintained.

There is an apparent approximatescaling for the critical velocity and pressure difference in the high velocity limit:

critand

Vcrit≈ 0.1

γ

λsw,∆p∗

crit≈ 0.05γ

?

λ

λsw

?

,

(4.2)

which, with the exception of the pre-factors, can be derived from (2.1) and (4.1) in the limit in which θ → 0. This

scaling for Vcritis in agreement with the results in [10] for a single film (cf. §4.c).

5. Discussion

We are therefore able to distinguish two important cases: the case |α| ≤ 1 consists of solutions that asymptote to a

straight-line solution, while for |α| > 1 they do not. In the former case, we find the “inverted wing” and the “hockey

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Cox, Weaire, Mishuris

0.01

0.1

1

10

100

0.1 1 10 100 1000

Drag ratio λ/λsw

(a)

2∆p∗

Vcrit

crit

(b) (c)

Figure 7. (a) The critical velocity and twice the pressure difference of the 211 structure as functions of the ratio of drag coeffi-

cients λ/λsw. The ratio 2∆p∗

λsw = 0.002 and (b) ∆p∗

It is clear that the structure is returning to the low-velocity form in the limit of high velocity, which is consistent with the argu-

ments given here: those films separating bubbles on opposite sides of the channel have a slope of ±30◦(as predicted by (3.5) with

α = 0.5) while those separating neighbouring bubbles on the same side of the channel, which experience a pressure difference of

2∆p∗, become vertical (the case α = 1 in figure 4).

crit/Vcrit = 2α tends to one as the velocity tends to infinity. The 211 structure at steady-state with

crit= 11.0 and (c) ∆p∗

crit= 50.0, each superimposed on the static structure of figure 1 shown in grey.

stick” shown in figure 3. In the latter, all solutions are of finite extent. For any given system, with fixed drag coefficient

λ, V increases with ∆p (not necessarily in proportion) so that α does not vary much [10]. If α is sufficiently small in

magnitude, then as V or ∆p tends to infinity any curved region shrinks as its curvature too becomes infinite.

The analysis of the previous sections has proveduseful in understandingthe way in which the results of numerical

simulations vary with V . But can they, in the end, shed light on the existing experimental results which provoked this

study? These we show in figure 8; they may be summarized as follows:

• the lines are almost straight.

• internal vertex angles clearly deviate from 120◦.

The first result is entirely consistent with the kind of high-velocity solutions that we have described. At first sight, it

might seem that the existence of undefined tangents at endpoints is the source of the anomalous internal angles, but

a careful application of the rules that follow from §4 shows this to be impossible. That is, deviations from the 120◦

angle rule cannot be explained by the present model. We may ask what additional effects might be responsible for

them. Although we have no definite evidence as yet, a prime candidate is the effect of a drag force acting at the vertex,

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The Viscous Froth Model: steady states and the high-velocity limit

9

Figure 8. Snapshot of an experiment in which a 211 structure is propelled rapidly to the right along a rectangular channel (width

w ≈ 5mm, depth h ≈ 1mm) by an imposed pressure [3]. A dry foam in steady plug flow is observed, that is, all elements of the

structure move with the same velocity V of the order of centimetres per second.

which is in reality a junction of finite extent. One may envisage a vertex correction in the form of a force acting at the

vertex point. Such forces have been posited before, but without any experimental support.

If such a connection is considered, we find that the angles observed in figure 8 (notwithstanding the difficulty of

measuring such an angle from an image of a foam) are such that the required vertex force must act in the direction of

the velocity, rather than opposingit. This negativedrag on a vertexseems unphysicalat first. However,it may be taken

to represent the change in the local drag as a finite vertex (attached to a finite transverse Plateau border) is replaced

by the point vertex of the present model. There is no reason why the finite vertex should not suffer a lesser drag force.

Further experiments are required to test this suggestion and interpretation.

Experiments on single films in micro-channels [16, 17] show similar shapes to those found in figure 6(a), and

further analysis of those experiments may be useful in trying to observe the curvature of the films close to the wall. In

addition, the solutions derived here may be combined in different ways to shed light on, for example, the motion of

single bubbles in channels.

The present appreciation of the properties of the model is a step forward in modelling microfluidic networks, but

we would not wish to underestimate the further factors that may come into play in due course, particularly when

shearing states are considered. They include at least film stretching (Marangoni etc.) effects, finite Plateau borders,

flow in Plateau borders, surface viscosity and nonlinear drag laws. For example, the VFM as originally formulated

included a power-law form for the velocity, in order to accommodate the Bretherton-type analysis of bubble motion

in tubes [18], and Grassia et al. [10] find that for β < 1 there is an increase in the width of the region in which the

curvature is concentrated at steady-state. Nevertheless the present model provides an adequate platform to begin to

investigate designs of networks and their components, for experimental testing.

We thank W. Drenckhan for stimulating discussion and the provision of experimental results. Financial support from (DW) the

European Space Agency (MAP AO-99-108: C14914/02/NL/SH, MAP AO-99-075: C14308/00/NL/SH), (GSM) the Wales Institute

of Mathematical and Computational Sciences, and (SJC) EPSRC (EP/D048397/1, EP/D071127/1) is gratefully acknowledged.

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Appendix A. Derivation of steady-state solutions

It is apparent that the VFM is closely related to motion by local curvature [19]. Indeed, it was designed to be so in the

limit in which pressure differences between cells go to zero. Motion by mean curvature [20, 12], also known as the

curve-shorteningproblem and described by the eikonal-curvatureequation, is often invoked to describe the motion of

grain boundaries in crystals. Mullins [15] gave steady-state solutions in this limit (i.e. eq. (2.1) with ∆p set to zero)

for translation, rotation and expansion. Here, we generalize the solutions for translation to the case of the VFM.

We classify the solutions of (3.4) with respect to their behaviour at origin (since we can set x(0) = 0 without loss

of generality). We conclude that

x(y) = ay̟+ O(y̟+ǫ),y → 0+,

where the following three cases can be distinguished:

̟ = 2,a =1 − α

2

, −∞ < α < ∞,

̟ = 1,a = ±

√1 − α2

α

,−1 ≤ α ≤ 1,

̟ = 1/2,a =

?

2

α,

α > 0.

(A1)

Therearein factnoothercases. Afurtherconditiononthederivativeat theoriginis requiredto distinguishthedifferent

branches of the solution.

Solutions in the case ̟ = 1 are straight-lines given by (3.5), found only for |α| ≤ 1.

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Page 11

The Viscous Froth Model: steady states and the high-velocity limit

11

(a) Case ̟ = 2

To seek further, curved, steady-state solutions, we rewrite (3.1) with the substitution: tanζ = xy:

y =

?

cosζ

cosζ − αdζ.

(A2)

In the case ̟ = 2 we adopt the following boundary conditions:

x(0) = 0,xy(0) = 0.

(A3)

Depending on the value of the parameter α three different families of solutions with finite gradient are possible (see

figure 4(a)): finite length solutions for α < 0 and α > 1 and infinite length solutions for 0 ≤ α ≤ 1 that tend to the

respective straight-line solutions discussed above. Note that the limiting cases are the well-known Mullins solution

(α = 0) or a line parallel to the y axis when α = 1.

This behaviour, as well as the shape of the envelope, can be confirmed by an analysis of the corresponding differ-

ential equation (A2). Under the conditions (A3), this takes the form

y = G(tan−1(xy)),G(t) =

?t

0

cosζ

cosζ − αdζ

(A4)

with argumentrestricted to the interval (0,cos−1(α)). G(t) itself takes the following form [9], dependingon the value

of α:

where β =

?(1 + α)/(1 − α). For any α < 0, G(t) is a positive increasing and bounded function in the interval

[0,π/2]. Then the solution to (3.4) under the conditions (A3) takes the form:

G(t) =

t +

2α

√α2− 1tan−1??α + 1

α

√1 − α2log

α − 1tan

?t

2

??

|α| > 1,

t −

???1 − β tan(t/2)

1 + β tan(t/2)

???

|α| < 1,

(A5)

x(y) =

?y

0

tan?G−1(t)?dt,0 ≤ y ≤ y∗(α) = G(π/2),

(A6)

where G−1is the inverse to the function G. In the case α > 1, the function G(t) decreases in the interval [0,π/2],

and the solution takes the same form (A6) but now x(y) < 0 for y > 0. In both of these cases the curves are of

finite extent and we can compute their envelope, consisting of the points (y∗,x(y∗)), from (A6). On the other hand,

it follows from (A4) that xy(y) → ∞ as y → y∗for any fixed α < 0 or α > 1. As the point (y∗,x(y∗)) can be

always shifted to the origin of a new coordinate system, this means that the end of any curve for a given value of α

(satisfying conditions (A3)) can be continued by another curve from the remaining class of solutions, which satisfy

slightly different conditions (A7), described below. Note that the solutions obtained can be extended to negative y, as

x ∈ C2[0,y∗). In fact, due to the symmetry in (3.4), x(−y) = x(y).

(b) Case ̟ = 1/2

The third case of (A1), ̟ = 1/2, cannot satisfy the conditions(A3) since xy(y) is in the interval (cos−1(α),π/2]

(cf. (A2) and (A4)). This solution has infinite derivative at the origin, and we therefore apply the conditions

y(0) = 0,yx(0) = 0.

(A7)

The corresponding equation for the derivative yxcan be evaluated in a similar way to (3.4), to give:

α −

yxx

(1 + y2

x)3/2=

yx

(1 + y2

x)1/2.

(A8)

Note that the case α = 0 gives the trivial solution y(x) ≡ 0 and is excluded from the following analysis. The solutions

to (A8) are invariant with respect to changing the signs of both y and α together; thus solutions for negative α may be

found by reflecting those for the corresponding positive value of α in the x axis.

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Page 12

12

Cox, Weaire, Mishuris

We now have tanζ = yxand hence

x = −

?tan−1(yx)

0

cosζ

sinζ − αdζ = −log|sin(tan−1yx) − α|

|α|

.

(A9)

This can be re-written in the form

yx(x) =

α(1 − e−x)

?|1 − α2(1 − e−x)2|,

(A10)

with the following solution for |α| < 1 and x such that |e−x− 1| < |α|−1:

y(x) = sin−1?α(e−x− 1)?+

α

√1 − α2log

?????

1 + α2(e−x− 1) +√1 − α2?1 − α2(e−x− 1)2

e−x(1 +√1 − α2)

?????.

(A11)

Then, from (A10), the following asymptotic formula holds for any |α| ≤ 1:

lim

x→+∞yx=

α

?|1 − α2|,

(A12)

That is, as x → ∞ all curves again asymptote to straight lines, with the same slope as before (eq. (3.5)).

(c) Envelope

Finally, we note that the non-zero points where the gradient yxis zero (̟ = 2) or infinite (̟ = 1/2) coincide,

since they both correspond to a limit of integration in (A2), but from different sides (i.e. one can either integrate from

the point at which yx= 0 to the point at which yx→ ∞, or alternatively from yx→ ∞ to yx= 0). Thus, we can give

an expression for the envelope. In parametric form, for the case ̟ = 1/2 (figure 4(b)), it is

y∗(x∗)=π

2−

α

2√1 − α2log

α

2√α2− 1

2+

?????

1 +√1 − α2

1 −√1 − α2

2− sin−11

?????,x∗= log

α

1 + α,

0 < α < 1,

y∗(x∗)=π

2−

?π

α

?

,x∗= log

α

1 + α< 0, α > 1,

y∗(x∗)= −π

α

2√α2− 1

?

π + 2tan−1

1

√α2− 1

?

,x∗= log

α

α − 1> 0, α > 1.

(A13)

For negative α one needs to change the sign of y∗. It is possible to eliminate the parameter α to have explicit repre-

sentations in cartesian coordinates, but we deliberately stay with the parametric form since it shows clearly the value

of the physical parameter α for every point on the curve.

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