## Abstract

The viscous froth model is used to study the evolution of a long and initially straight soap film which is sheared by moving its endpoint at a constant velocity in a direction perpendicular to the initial film orientation. Film elements are thereby set into motion as a result of the shear, and the film curves. The simple scenario described here enables an analysis of the transport of curvature along the film, which is important in foam rheology, in particular for energy-relaxing ‘topological transformations’. Curvature is shown to be transported diffusively along films, with an effective diffusivity scaling as the ratio of film tension to the viscous froth drag coefficient. Computed (finite-length) film shapes at different times are found to approximate well to the semi-infinite film and are observed to collapse with distances rescaled by the square root of time. The tangent to the film at the endpoint reorients so as to make a very small angle with the line along which the film endpoint is dragged, and this angle decays roughly exponentially in time. The computed results are described in terms of a simple asymptotic solution corresponding to an infinite film that initially contains a right-angled corner.

## 1. Introduction

Flowing foams occur in a variety of applications such as oil recovery, froth flotation and microfluidics, making them of research interest [1–3]. A detailed understanding of foam rheology would entail a description of foam structure on the bubble scale [4]. This has been achieved in the quasi-static slow flowing limit, for structures in mechanical equilibrium (at a minimal energy state [2]). In faster foam flows however, the structure departs from quasi-static mechanical equilibrium and dissipative processes become significant [5].

Various models have been developed to simulate the rheology of foams, such as the bubble model [6], the vertex model [7] and the viscous froth model [5,8]. Among such models, the viscous froth model has proved to be a powerful tool to simulate the rheology of dry foams, predicting realistic bubble shapes and agreeing with experimental results for ‘two-dimensional’ foam flows in channels, i.e. foam monolayers confined between upper and lower plates [8–10]. In this geometry, a foam film can be represented mathematically by a curve in the ‘two-dimensional’ plane. The viscous froth model [3,8–14] applied to such ‘two-dimensional’ systems is based on a force balance between the pressure difference across the two sides of the film (*ΔP*), the film tensions (*γ*) and the viscous drag on each film element (the drag being associated with moving the films over the confining upper and lower plates):
1.1where *λ* is the viscous drag coefficient per unit length of film, ** v** is the velocity of a film element,

**is the unit vector normal to the film element and**

*n**c*is the ‘two-dimensional’ film curvature.

One particularly attractive feature of the viscous froth model over others is its capability to account for film curvatures which are not simply arcs of circles. This feature is not strictly necessary when simulating foams which are slowly sheared, i.e. in systems with low capillary number (the ratio of the imposed foam deformation rate and the relaxation rate), as the viscous term in equation (1.1) would then be negligibly small, reducing the equation to the Young–Laplace law, and thus the curvature of each film would simply be well represented by a constant *ΔP*/*γ*. As velocities increase however, curvature deviates from this value and tends to vary with position.

We motivate this work by considering a periodic regular hexagonal array, the hexagonal honeycomb structure. The pressures in every bubble are identical, and thus films in the slowly sheared limit do not have any curvature. However, upon fast shear (high capillary number), the viscosity is important and curvature becomes spatially non-uniform, complicating the shape of the films even in the case of the hexagonal honeycomb structure [5,14,15].

Elongated bubbles then arise [16] which tend to have high surface energies. This situation comes about because bubble neighbour exchange transformations, so-called T1 topological transformations, which tend to relax film energy and reduce bubble elongation, are suppressed [5]. Films are required to meet threefold at vertices, and a T1 is produced by moving two vertices together until the point at which they collide. The velocity of a vertex is determined by the curvatures of the adjoining films [14]. In the hexagonal honeycomb case (with equal pressures in all bubbles), this can be expressed as:^{1}
1.2where *v*_{v} is the velocity of the vertex, *c*_{i}(0) is the curvature of film *i* at the vertex (*i*=1,2,3) and *n*_{i} is the normal to the film *i*.

In the hexagonal honeycomb case, for any finite *λ*, it is clearly necessary to have film curvature in order to set vertices in motion (and ultimately to induce a T1). Hence, a relevant question to ask is how rapidly curvature can be transported along foam films in a sheared foam, as curvature is ‘injected’ into the film by the shear (owing to film endpoints being moved perpendicular to the initial orientation of the film), but the T1 relies on transport of the curvature along the film. The purpose of this paper is therefore to consider transport of curvature for a highly simplified yet generic case, the shearing of a single semi-infinite film (see §2). We will compare computed simulation data (§3) with a late time asymptotic solution (§4). The asymptotic solution can be analysed further to gain insights into the behaviour of the system (§§5–6). Conclusions are offered in §7.

## 2. Description of the system to be studied

Consider a semi-infinite foam film that is initially straight, with no pressure drop across it. One of its endpoints is then moved perpendicular to the film orientation with a velocity *v*_{shear} (figure 1).

Then, because *γ*/*λ* has the units of a diffusivity *D*, it follows that *D*/*v*_{shear} is a length scale and *D*/*v*^{2}_{shear} is a time scale.^{2} We make lengths and times dimensionless on these scales and curvature dimensionless on the scale *v*_{shear}/*D*.

Because there is no imposed pressure difference, equation (1.1) then becomes
2.1where ** X** is the dimensionless location of a film element,

*t*is the dimensionless time,

*Θ*labels a material point,

*κ*is the dimensionless curvature and

**is the unit vector normal to the film surface. This then corresponds to motion by mean curvature [20], a model that has also been used in a different physical context to describe grain growth [14,21].**

*n*For dimensionless times *t*≪1, it turns out that the film turns through a very small angle (expected to scale as order *t*^{1/2}) over its entire length. However, for times *t*≥*O*(1), the film turns through an appreciable angle (i.e. a significant fraction of *π*/2) and consequently the endpoint of the film meets the horizontal boundary in figure 1 at an angle significantly different from *π*/2. In fact, the meeting angle decays very rapidly as *t* increases—a phenomenon which we will investigate later. One of the reasons for analysing the behaviour of this angle is that (in our dimensionless system) its cosine represents the force that must be applied to the film endpoint to drag the film along. This is also (again in our dimensionless system) the rate of working by the dragging force, some of that work being stored as an increased film length and some of it being dissipated viscously.

There is a well-known solution in the literature for a *finite* film dragged at both ends, known as the Mullins finger or the grim reaper [14,21]. Consider a film of initial length *L*, the endpoints of which are both pulled at velocity *v*_{pull} (perpendicular to the initial orientation of the film) such that the separation of the endpoints remains unchanged. Making the system dimensionless on scales as above, the dimensionless initial length is
2.2and the dimensionless velocity of the endpoints is unity. The final state depends on the initial length. If *l*<*π*, it is a finite segment of reaper propagating at unit velocity (figure 2). The angle through which the reaper turns between the symmetry point at its tail and the dragged endpoint is less than *π*/2. On the other hand, if *l*>*π*, then the solution is an infinitely long reaper propagating at velocity *π*/*l*<1 (again see figure 2). The reaper turns through *π*/2 between the symmetry point at its tail and the endpoint. The dragged endpoints move at unit velocity (hence the reaper becomes stretched infinitely long) in the direction of propagation.

The present case of a dragged *semi-infinite* film however is rather different. The curved region of the film is not confined to a fixed lateral distance but rather the curvature extends over longer and longer lateral distances as time evolves. This means the evolution slows down. The long-time asymptotic state of the semi-infinite film therefore cannot be mapped on to a reaper. There is however another solution to which it corresponds. This is the case of an infinite film that initially turns through a sharp right angle corner. This is sketched in figure 3 (and is analogous to some solutions considered by Weaire & McMurry [14], see fig. 5*b* in [14], albeit showing a case where turning angles on individual films are less than *π*/2). In our case (similar also to what is observed in [8]) surface tension acts to smooth out the corner and the film immediately evolves into a curve. This system (of a film that turns through a right angle) becomes the same as the sheared semi-infinite film provided that the extent of the region over which the curvature is distributed at any given time *t*, is considerably less than the distance that the endpoint of the sheared semi-infinite film has been moved; thus, the endpoint of the semi-infinite film is extremely far from the curved region. The endpoint's exact location (and the issue of whether or not it is moving) is no longer relevant to determining the evolution of that curved region—the endpoint has in effect moved an arbitrarily far distance away from that region. We shall return to consider this long-time asymptotic solution in due course, but first we consider some (numerical) simulation data at finite time.

## 3. Simulation of the dragged film

We have simulated the dragged film numerically using Surface Evolver [22]. Simulation data are provided in the electronic supplementary material. We cannot of course represent a semi-infinite film numerically, so we have chosen long finite films of length 40, 100 and 200 each discretized into segments (with individual segment lengths ranging between 0.01 and 0.05 units). *Both* ends of the film are displaced with unit velocity, and we used a time step of 10^{−4} units to evolve the film shape, using local film normals and local film curvatures which Surface Evolver readily determines from the discretized representation. If a segment shrinks below 0.01 units, it is combined with a neighbouring segment, whereas if it grows above 0.05 units, it is subdivided. Such subdivisions occur frequently adjacent to the film endpoints, because the film *endpoints* are moved at unit speed in the direction normal to the *initial* film orientation, whereas neighbouring adjacent film *material points* move in the direction of the instantaneous local film normal, i.e. nearby points in the discretized representation of the film are moving in different directions. The above numerical simulation procedure actually corresponds to the start up of the *reaper* problem discussed in §2.

At sufficiently early times (figure 4), there is a straight (vertical) central section of the film which has not moved significantly. At these early times, either end of the finite film will be a good representation of the semi-infinite case. The maximum amount of time for which we can use the finite film to represent the semi-infinite one increases as the film length increases. For instance, for a film of initial length 40, we could simulate out to time eight units before the film midpoint was moved by a distance of 10^{−6}. However, for a film of initial length 100, we could simulate out to time 45 before the film midpoint moved by the same amount, whereas for initial length 200, we could simulate all the way out to time 180. These simulations however also become much more expensive as the film length increases—for initial length 40 units, the run time was 15 min (on an i3 CPU, 3.10 GHz), whereas for initial length 200 units, the run time was 48 h.

Figure 5 shows data for the curved film shapes at times *t* equal to 40, 70 and 120 for the case of an initial film length of 200. The film is drawn on a Cartesian coordinate domain, and is assumed initially to be at a coordinate location *x*=0 while covering −200≤*y*≤0. On the figure, we plot only the domain −100≤*y*≤0 and *x*≥0, so we are plotting only half the film shape, it being symmetric about the line *y*=−100.

Our aim is to understand how far curvature is transported along the film as a function of time, because (as was mentioned in §1) film curvature leads (in the case of a foam, as opposed to that of just a single film) to vertex motion, and vertex motion is a prerequisite of any energy relaxing topological transformation in foam. Figure 5 shows (unsurprisingly) that at longer times curvature has been transported over a longer lateral distance of film (i.e. transported over a greater distance along the *y*-axis). There are however some additional interesting features in figure 5.

Even though the ultimate source of the curvature is the imposed motion of the top boundary (which at time *t* has displaced to *x*=*t*), the maximum of the curvature does not occur at that topmost point. Indeed, sections of the *t*=70 and *t*=120 curves are almost completely aligned with the *x*-axis from *x*=*t* (i.e. the film endpoint) down to about *x*∼40. The region of maximum curvature is much further back at smaller *x* values: figure 5 also shows a zoomed view of that region. In that zoomed view, we see at time *t*=40, that the shape of the film is qualitatively different either side of the quadrant ‘bisector’ *θ*=−*π*/4, i.e. there is asymmetry between polar coordinate angles −*π*/2≤*θ*≤−*π*/4 and angles −*π*/4≤*θ*≤0 (contrast figure 3). When *t*=40, the film tends to be more curved for −*π*/2≤*θ*≤−*π*/4, but exhibits a long rather flattened section for −*π*/4≤*θ*≤0. As curvature is required to drive motion here, the flattened section barely moves (e.g. it displaces comparatively little between times *t*=40 and *t*=70). By *t*=70, however, the flattened section has been eroded, and so these film points are now able to move again. The asymmetry between the polar angles −*π*/2≤*θ*≤−*π*/4 and −*π*/4≤*θ*≤0 is much reduced by *t*=70, a tendency which increases towards *t*=120.

Figure 6 shows data for rescaled film positions—rescaling distances by *t*^{1/2} (on the grounds that diffusive problems often exhibit this type of scaling). These collapse together imperfectly, but nevertheless reasonably well—although, if we subdivide the solution quadrant into two halves, this collapse tends to be much better for polar angles −*π*/2≤*θ*≤−*π*/4 than for polar angles −*π*/4≤*θ*≤0. Certainly, in the domain −*π*/2≤*θ*≤−*π*/4, the scaled curves in figure 6 are much closer together than their unscaled figure 5 counterparts. Given the (above noted) asymmetry of the *t*=40 film in these two angular domains, contrasted with the near symmetry^{3} of the *t*=120 film, we do not expect to obtain equally good collapse in both domains. Nevertheless, figure 6 is indicating that curvature is transported out to a lateral distance (i.e. out to a distance |*y*| in figure 6) of order *t*^{1/2}—this is a result we wish to understand (and we return to it shortly in §4).

Figure 7 shows data for the angle *ψ* at which the film meets the line along which its endpoint is dragged (see also the definition sketch figure 1). This is initially *π*/2—the film meets the line at right angles. Very rapidly, however, the film reorients—at its endpoint it becomes nearly parallel to the line along which its endpoint is dragged.

How rapidly that angle decays to zero in the long-time limit is a behaviour that we would like to analyse. The data on figure 7 are plotted on a log-linear scale, and show a reasonably straight region on the plot. Thus, the angle *ψ* is decaying roughly exponentially to zero over time. By fitting the decay data over the (arbitrarily chosen) range 10^{−4}≤*ψ*≤10^{−1} to an exponential, the characteristic decay time turns out to be 2.65 units. The fit is poor at early times (because early time data were excluded from the fit) but is also poor in relative terms at very late times (an artefact of the fitting procedure which minimizes the sum of squares of absolute errors, and hence incurs little penalty from relative errors if *ψ* is small). Nevertheless, the notion of an exponential fit is a fair approximation to the data: over the range of *ψ* mentioned above, the fitting procedure reported r.m.s. errors in *ψ* between the data and the exponential fit of only 5×10^{−4}. The origin of this rapid near-exponential decay for *ψ* is another feature that we wish to understand.

To summarize, Surface Evolver furnishes us with useful simulation data for understanding the dragged film problem. However, to access very long times (and still have a reasonable representation of the semi-infinite film case) requires the use of very long but finite films in the simulation and this becomes expensive numerically. What we seek therefore is to understand the long-time behaviour of the dragged film using an asymptotic approach without relying on simulations—this is the topic of section 4.

## 4. A late time asymptotic solution

In what follows, we compute the similarity solution for the corner in figure 3, which corresponds to the long-time asymptotic behaviour of a sheared film. Such an asymptotic behaviour comes about because, at long times, the sheared film endpoint has already moved far outside the domain where significant curvature-driven dynamics is occurring. The motion of the sheared endpoint has ceased to be important *per se*, and what is relevant is the film shape which this motion has now produced.

In order to obtain this solution, there are various ways one can parametrize the film. For instance, one could parametrize based on material points or based on arc length along the film (see appendix). Here, however, a polar coordinate system versus *θ* will be chosen. The solution is clearly symmetric under reflection about the bisector of the original corner, meaning we need to solve only for half of the original corner domain.

Projecting the vector equation (2.1) onto the radial and angular directions in polar coordinates, one obtains
4.1and
4.2where is the radial position, *θ* is the angular position, and and *n*_{θ} are the radial and angular components of the normal vector ** n**, respectively. Furthermore, the radial position of a film element can be considered to be a function of time and angular position, which itself is a function of time and material point, i.e. . Then, the following differential equation holds
4.3

Assuming that , where *a*(*t*) is a scale factor to be determined and *r*(*θ*) is a rescaled radial position, we deduce
4.4and
4.5where *r*′ denotes the first derivative of *r* with respect to *θ*. Moreover, because the tangent vector is just a rotated version of the normal, and because the curvature is simply the derivative of the tangent with distance along the film projected back onto the normal, we obtain
4.6here^{4}
4.7where *r*′ is as above, and *r*′′ is the second derivative of *r* with respect to *θ*.

Substituting equations (4.1), (4.2) and (4.6) into equation (4.3) (and using to denote the derivative of *a* with respect to time),
4.8From equation (4.8), it is clear that we should choose (thereby confirming the diffusive nature of the system), from which it follows that
4.9Further substituting from equations (4.4), (4.5) and (4.7) and rearranging gives
4.10

A description *r*=*r*(*θ*) as in equation (4.10) is convenient near the symmetry line *θ*=−*π*/4. However, near the asymptote *θ*=0, the value of *r* should tend to infinity, and a more convenient description then is *θ*=*θ*(*r*); the chain rule gives trivially
4.11where *θ*′ is the derivative of *θ* with respect to *r*. Further differentiating with respect to *r* and substituting equations (4.10) and (4.11) where appropriate gives
4.12

Equation (4.10) can be numerically integrated given initial values of *r* and *r*′, and may be switched to equation (4.12) at any point to continue integrating. Assuming the shearing takes place in the fourth quadrant (i.e. for −*π*/2≤*θ*≤0), the obvious symmetry condition is that *r*′=0 at *θ*=−*π*/4. However, the value of *r* at *θ*=−*π*/4 (which we denote as *r*_{0}) must be found by a trial and error shooting procedure, on the basis that *θ* should tend to zero as *r* tends to infinity. Results of the numerical integration with varying values of *r*_{0} are given in figure 8. To avoid unnecessary numerical expense, the step size was increased adaptively.^{5} As a result, the curves shown in figure 8 could be obtained in under 100 steps.

Clearly, the asymptotic angle between the ends of the film is dependent on *r*_{0}. Too large a value of *r*_{0} leads to films which curve too sharply and turn through more than *π*/4 between the symmetry point and the asymptote. Conversely, too small a value of *r*_{0} leads to films which curve too little and turn through too small an angle. From the solutions obtained, *r*_{0}≈1.0445 gave a film curved by the correct amount, and thus was adopted as the right initial condition thereafter.

Solutions obtained using this *r*_{0} value have also been plotted in figure 6 and compared with the scaled simulation data. Agreement is imperfect (as might be expected, because the evolver data were obtained at finite times, rather than in the long-time asymptotic limit). Nevertheless, the evolver data do indicate a tendency to converge towards the asymptotic state.

Returning to the asymptotic solution, as the film approaches the coordinate axes (either the *x*-axis or the *y*-axis), the step-wise changes in angle become smaller. Because numerical calculations are limited by finite precision, it is preferable to switch eventually to an approximate analytical formula for *θ* versus *r*. As *r* increases and *θ*′ decreases, equation (4.12) becomes approximately
4.13Integrating with respect to *r*,
4.14where *A* is an integration constant.

In the limit as , *θ* should approach some ‘final’ value *θ*_{f} which should itself be close to zero provided the parameter *r*_{0} has been chosen correctly.^{6} At *θ* close enough to the final angle, *θ*_{f}, the value of *A* can be obtained by fitting the numerical *θ*′ versus *r* values to equation (4.14). Then, the final angle *θ*_{f} can be obtained by integrating equation (4.14) out to an arbitrarily large *r*. For consistency, we have checked that the *A* value is insensitive to the choice of angle at which we switch from the numerical integration to the analytical formula, and moreover that the predicted final angle *θ*_{f} is close to zero.

Table 1 shows the computed results varying the angle at which the switch to the approximate analytic solution takes place: convergence is obtained provided we switch around −0.001×*π*/2 or at any other angle closer to zero than that. The analytic solution (i.e. the decay of *θ* with *r*) can then be followed for several decades more, down to the point at which *θ* reaches the neighbourhood of *θ*_{f}. Once however *θ* attains a value close to *θ*_{f}, the solution becomes ‘unreliable’ being sensitive to the fact that our guess for the parameter *r*_{0} was not quite correct (i.e. the film turns through not quite the correct angle along its length, so *θ*_{f} is not exactly zero). However, the *θ*_{f} values reported in table 1 are so tiny that this discrepancy poses no real difficulty.

## 5. Reorientation of the film endpoint

In the computed results of §3 we saw that the angle *ψ* at which the film meets the line along which its endpoint is dragged decays very rapidly with time. Although the asymptotic results of §4 deal with an infinite film with an initial right angle corner (rather than a sheared semi-infinite one), we can nevertheless use the asymptotic results to gain insights into the behaviour observed in the semi-infinite case. The dragged endpoint of the semi-infinite film is displaced a distance (i.e. ) away from its original location. Hence, we can look at the film orientation at points on the asymptotic solution which are the same distance away from the origin. Unlike in the simulations, these will not be exactly at polar angle *θ*=0, but (provided the distance is large) the *θ* value will be very close to zero. The angle *ψ* can be evaluated as
5.1where *e*_{x} is the Cartesian unit vector, and ** t** is the tangent vector
5.2and

*e*_{r}and

*e*_{θ}are polar unit vectors. Here, and . Values for

*ψ*versus

*t*evaluated at

*r*=

*t*

^{1/2}according to the above formula and using the asymptotic analysis of §4 are plotted in figure 9. As in figure 7, we plot here on a log scale, so that it is clear that

*ψ*actually exhibits very rapid decay, with a near straight line on the plot being indicative of a near-exponential decay.

In the limit where *θ* is exceedingly close to zero, we have
5.3where *y*≈*rθ* and *x*≈*r*. Hence
5.4We write this as *rθ*′+*θ*_{f}+(*θ*−*θ*_{f}), where *θ*′ is given by equation (4.14) and hence
5.5with a dummy integration variable. Integrating by parts:
5.6where equation (4.14) was used. Substituting from equations (5.3) and (5.4) then gives
5.7and (as alluded to above) we are required to evaluate this at location *r*=*t*^{1/2} so that
5.8We are interested here in *r* and *t* values which are large compared with unity, but nevertheless such that the complementary error function term is considerably larger than *θ*_{f} (otherwise, the *ψ* value we compute is sensitive to the fact that our chosen *r*_{0}, as obtained via shooting, gives *θ*_{f} values that differ very slightly from zero).

Based on equation (5.5), we can estimate that *θ*_{f}−*θ* will be at most a number on the order of and hence is much smaller than *rθ*′ (found using equation (4.14)) when *r* is large. It follows that the two terms on the right-hand side of equation (5.6) nearly cancel (because the left-hand side of that equation is smaller than the first term on the right). If we are in a regime of *r* values (as alluded to above) where the complementary error function term is still much larger than *θ*_{f}, then it follows that
5.9Evaluated at *r*=*t*^{1/2}, this gives
5.10hence the near-exponential decay of *ψ* observed in figure 9.

The time constant of the exponential decay for *ψ* for this asymptotic model according to equation (5.10) is not quite the same as that obtained from the Surface Evolver results (figure 7), but this is unsurprising given the different *θ* values at which the angle *ψ* is evaluated in the two models (at *θ*=0 for the numerical Surface Evolver data, but at a small non-zero *θ* for the asymptotic solution). Moreover, the *t*^{−1/2} prefactor in the formula (equation (5.10)) means that the decay is not quite a pure exponential in any case.

## 6. Movement of material points on a film

Having obtained the shape of the curved film in §4, the movement of material points can be tracked: following material points is of interest because, even though shearing a film (as in §3) adds new material to the film and increases its overall length, it turns out that film material elements once created, invariably *shrink*. In other words, any overall increase in film length here, is due to new film elements being extracted from a fluid reservoir (the reservoir corresponds physically to a so-called Plateau border located at the film endpoint that is being dragged along), but after an element is thereby extracted, it begins to shrink. Taking d** X** to represent an element of film with length d

*s*, and differentiating the relationship (d

*s*)

^{2}=d

**⋅d**

*X***with respect to time and substituting**

*X***=d**

*t***/d**

*X**s*and d

**/d**

*n**s*=

*κ*

**, yields the shrinkage rate over a material domain [9]: 6.1where is the derivative of arc length,**

*t**s*, with respect to time. Because the characteristic length is known to scale with

*t*

^{1/2}(see §4), it is appropriate to consider the rescaled arc length

*S*, defined by 6.2Substituting equation (6.2) into (6.1) and rearranging using

*κ*=

*K*/

*t*

^{1/2}from §4, gives 6.3The second term shows that even if the film element shrinkage stops, i.e. , then

*S*will continue to decrease because of the overall expansion of the structure (scaling with

*t*

^{1/2}) induced by the shearing.

Clearly (on geometrical grounds), both *S* and *K* can be expressed solely in terms of either *r* or *θ*. We choose to define *S*=0 to be the symmetry point at *θ*=−*π*/4, as identified in §4. Thus, the similarity solution obtained in §4 (via equation (4.10) or equivalently equation (4.12)), allows the movement of material points at any point on the film to be tracked by equation (6.3).

Because the film is being pulled at unit velocity, at time *t*=*t*_{0}, a material point is injected at a polar coordinate location . For our asymptotic solution (which for computational simplicity is what we choose to analyse here), this corresponds to a *θ* value very close to (albeit not exactly) zero.

The movement of material points injected at *t*_{0}= 4, 16 and 64 was calculated taking the asymptotic solution from §4, and considering points at the relevant initial location and tracking their motion in time. The results are shown in figures 10 and 11.

The behaviour of the angular location *θ* of material points as seen in figure 10 is straightforward. The period of time for which the angular position of a material point is nearly constant occurs when the point is on the near-straight section of the film (at |*θ*|≪1, adjacent to the *x*-axis). Hence, this behaviour is always observed for any material point with *t*_{0} large enough that its initial location is indeed on the straight section of the film, and the duration of the period of nearly constant *θ* becomes longer with larger *t*_{0}. Eventually, typically at some time *t* on the order of , the curved region of the film reaches the initially injected point and its polar angle *θ* begins to decrease. Eventually all material points approach −*π*/4 asymptotically at long times, at least in principle.^{7}

Figure 11 shows similar data but expressed in terms of the rescaled arc length coordinate *S*. This log–log plot suggests one straight line regime at early times, and a second straight line regime at long times.^{8} At early times and for large *S*, equation (6.3) is dominated by , because the integral remains finite. This gives a slope of on the log–log plot and (as noted above) corresponds to *S* decreasing due solely to the *t*^{1/2} overall length rescaling. At late times, i.e. small *S*, the slope is steeper. Equation (6.3) becomes
6.4where *K*_{0} is the curvature at *S*=0. Evaluating equation (4.9) at *θ*=−*π*/4 gives *K*_{0}=−*r*_{0}/2 where, as stated in §4, *r*_{0}≈1.0445. Integration of equation (6.3) reveals a straight line on a log–log plot but now with a larger slope, , agreeing with the second straight line regime seen in figure 11. Interestingly, for the smallest *t*_{0} value shown in figure 11 (i.e. *t*_{0}=4), the evolution follows the slope of this ‘second’ straight line region even at early times. The reason for this is made clear by comparing with figure 10: the *t*_{0}=4 material point never has exceedingly small values of |*θ*|, thus it never finds itself on any long straight section of film adjacent to the *x*-axis. Rather it is injected onto an already curved section of film and its *θ* and *S* values evolve accordingly.^{9}

## 7. Conclusion

We have used the viscous froth model to describe a sheared foam film, with shear being introduced by dragging the film's endpoint at a constant velocity in a direction at right angles to the original film orientation. Although the system we consider is a highly idealized one, it nonetheless describes a generic process which is important in any sheared ‘two-dimensional’ foam (i.e. in any system where a layer of bubbles is confined between upper and lower plates, and subjected to shear). In the foam system containing many bubbles, it is necessary to transport curvature along the entire length of a film so as to set a vertex in motion, leading eventually to a collision between two adjacent vertices. This initiates a so-called topological transformation that relaxes the energy in the foam as bubbles exchange neighbours. The first step in this chain of events is transport of curvature.

Our data show that after a time *t*, curvature is transported over a distance scaling like *t*^{1/2}, which underlines the diffusive nature of the transport inherent in the viscous froth model. The film endpoint, however, is moved at a constant velocity, and hence its displacement grows linearly in time. It therefore migrates increasingly far away from the curved region as time evolves. The film endpoint becomes joined to the curved region of film by an almost straight segment which reorients itself (approximately exponentially in time) to align with the direction in which the film endpoint is being dragged. The film eventually approaches an asymptotic state corresponding to an infinite film containing an initial right-angled corner, a system which admits a similarity solution whereby distances indeed scale like *t*^{1/2} as expected.

## Acknowledgements

G.M. and S.C. acknowledge financial support from FP7 IAPP project HYDROFRAC (PIAP-GA-2009-251475). P.G. also acknowledges support from FP7 IAPP project HYDROFRAC (PIAP-GA-2009-251475) which funded a visit to Aberystwyth. Part of this work also was carried out while P.G. was a Royal Academy of Engineering/Leverhulme Trust Senior Research Fellow and funding from the fellowship is gratefully acknowledged. S.C. and P.G. acknowledge useful discussions with Prof. D. Weaire, and in particular Prof. Weaire's helpful insights at the ‘Infoamal’ Workshop on Foam Coarsening (Université Paris Diderot, 9–11 January 2013), partly supported by CFCAM Ile-de-France.

## Appendix A

We have stated in the main text that curvature ‘diffuses’ along films. The purpose of this section is to make this claim mathematically precise, and to indicate some of the physical consequences that this diffusion of curvature implies.

The idea of the diffusive nature of film curvature comes from the viscous froth model equation (2.1), ignoring pressure differences for simplicity. Using the notation from equation (2.1), and substituting *κ*** n**=−d

**/d**

*t**s*and

**=d**

*t***/d**

*X**s*, A1which is analogous to Fick's law. There is however a slight complication—the left-hand side of equation (A1) is most naturally written in terms of material elements labelled by

*Θ*, whereas the right-hand side of equation (A1) is most naturally written in terms of arc lengths

*s*. Material elements do not however correspond precisely to arc length elements—indeed, individual material elements tend to shrink as time goes on. Care must be exercised therefore, when manipulating equation (A1) to derive a diffusion equation for curvature.

We proceed as follows. Because ** X** can be considered as

**(**

*X**t*,

*s*(

*Θ*,

*t*)), we can derive a convective derivative as follows. First, we write A2Substituting equation (6.1) (and treating

*s*′ as a dummy integration variable), A3and hence in general A4

Applying ** n**⋅d

^{2}/d

*s*

^{2}to each term of equation (A3) and simplifying using equation (2.1) and the relationship d

^{2}

**/d**

*X**s*

^{2}=−

*κ*

**, gives A5where we have used the fact that**

*n***⋅**

*n***vanishes, we have also applied equation (A4) to the vector −**

*t**κ*

**. Note in the above that A6making it obvious that the operators d**

*n*^{2}/d

*s*

^{2}and (d/d

*t*)

_{Θ}do not commute. We can further simplify equation (A5) by recognizing that any derivative of

**is necessarily orthogonal to**

*n***itself and hence A7**

*n*Applying ** n**⋅d

^{2}/d

*s*

^{2}to the right-hand side of equation (A1), substituting d

^{2}

**/d**

*X**s*

^{2}=−

*κ*

**, and simplifying (using also the fact that d**

*n***/d**

*n**s*=

*κ*

**and hence is orthogonal to**

*t***), A8Equating (A7) and (A8) and simplifying gives A9which is the curvature diffusion equation that we seek.**

*n*Equation (A9), like equation (A1), is still parametrized in terms of material point and arc length coordinates, and takes an extremely compact and elegant form in those coordinates. When solving viscous froth problems however, one seldom parametrizes in terms of either material points or arc lengths. Indeed, the solution we obtained in §4 was described in terms of polar *r* versus *θ* or *θ* versus *r* coordinates. Once the system is solved in terms of polars, it is a straightforward geometry problem to obtain arc lengths. Meanwhile, the analysis of §6 has shown how to recover material point positions. Thus, arc lengths and material point locations are often determined after the film shape evolution has been solved, rather than as part of the technique for obtaining that evolution.

Despite the fact that we seldom solve problems using equation (A9) directly, the equation remains useful conceptually. As a diffusion equation, it explains why curvature spreads over a distance of order square root of time as we showed in the analysis in the main text.

In a different physical context, it also explains why so-called grain growth problems behave differently from coarsening in a soap froth [14]. In both froth coarsening and grain growth—see reference [14] for details—certain domains grow while others shrink and disappear. Soap froth coarsening demands that domain boundaries are uniformly curved, according to the difference in gas pressure across the boundary. A shrinking bubble on the point of disappearance has a much higher pressure than any of its neighbours with the result that curvature is nearly the same along each and every edge. A shrinking soap bubble therefore adopts a very regular shape. Grain growth models (which correspond to equation (A9)) permit much less regular shapes in shrinking domains. The time scale to diffuse curvature uniformly along a film (on the order of *s*^{2} for a domain of length scale *s*) is similar to the time scale (again order *s*^{2}) for the domain to disappear altogether. Thus, irregularities in shrinking domains persist in grain growth models all the way up to the disappearance of the grain. Grains are therefore far less regular than their soap froth counterparts.

## Footnotes

↵1 Note that according to equation (1.2), only in the limit as

*λ*→0 does it become possible to displace the vertex without curvature. For a hexagonal foam, this corresponds to the ‘slow displacement’ quasi-static limit of Princen [17].↵2 Values of

*λ*can be quite difficult to determine and are sensitive to details of a particular experiment [18,19], but reference [9] has quoted a value 290 kg m^{−2}s^{−1}. Taking a film tension*γ*of say 29×10^{−3}N m^{−1}, gives diffusivity*D*of the order of 10^{−4}m^{2}s^{−1}. If the film end point is dragged at a velocity*v*_{shear}of 0.1 m s^{−1}, the characteristic length scale becomes 10^{−3}m and and the characteristic time scale would be 10^{−2}s.↵3 One quantitative measure of the degree of symmetry in the

*y*versus*x*plot (figure 5) is to consider the ratio between the value of (in the domain above the line*y*=−*x*), and that of (in the domain below the line*y*=*x*). For our data, this ratio turns out to be 4.1 for*t*=40, falling to 2.1 for*t*=70 and to 1.5 for*t*=120, and would asymptote to unity if the plot became perfectly symmetric in the limit.↵4 Note that

*κ*and*K*turn out to be negative quantities in the particular sign convention we adopt here.↵5 Specifically, while integrating for

*θ*as a function of*r*, we considered the discrepancy between doing two steps with an integration step size*δr*and doing one step with an integration step size 2*δr*: if the discrepancy was exceedingly small compared with the change in*θ*during the step, then we considered it safe to increase the step size.↵6 For any given

*r*_{0}, the value of*θ*_{f}is the angle that the curve in figure 8 makes with the*x*-axis, as*x*tends to infinity.↵7 Note however that the system may require exceedingly long times for a material point to approach close to that final angle −

*π*/4. Based on estimates given earlier, the time value of 10 000 units required in figure 10 for the points labelled as*t*_{0}=16 to migrate to within a few percentage of −*π*/4 corresponds to 100 s of physical time. For curvature with a ‘diffusivity’*D*of 10^{−4}m^{2}s^{−1}(given earlier), this corresponds to an initially sharp-cornered film becoming rounded over a curvature radius 0.1 m, which is longer than a typical film in a foam. In practice, then, material points injected at either*t*_{0}=16 or*t*_{0}=64 case will not reach the neighbourhood of*θ*∼−*π*/4 on time or distance scales likely to be of interest in a real foam, although such material points could still see significant evolution of*θ*away from the initial |*θ*|≪1 value.↵8 We note a similar caveat as previously, i.e. for larger

*t*_{0}values, the times required to approach the second straight line region may be prohibitively large.↵9 The supposition that the

*t*_{0}=4 material point is injected onto an already curved section of film is based here on the analysis of the long-time asymptotic solution of §4. Figure 5, however, showed that at finite times, a flattened section of film can occur in the angular domain −*π*/4≤*θ*≤0*before*the long-time asymptotic film shape is actually achieved, and such a flattened section could affect the time evolution of*θ*or*S*as experienced by a material point. This, however, is beyond the scope of the analysis presented here.

- Received May 31, 2013.
- Accepted August 15, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.