## Abstract

A model system for theory and experiment that is relevant to foam fractionation consists of a column of foam moving through an inverted U-tube between two pools of surfactant solution. The foam drainage equation and its variants are used for a theoretical analysis of this process. In the limit in which the lengths of the two legs is large , exact analytic formulae may be derived for the key properties of the system. Numerical computations and experiments support these results.

## 1. Introduction

Foam fractionation is the process in which a foam rises in a column and overflows at the top. The liquid collected from the overflowing foam, when it is collapsed, is relatively rich in the surface-active components of the liquid. The process is of widespread practical importance in chemical engineering and biotechnology [1]. It has, therefore, been the subject of many analyses of a largely empirical nature [2]. Attempts to place them on a firmer basis have been frustrated by the difficulty of capturing a reliable theoretical description of the overflow, in which drainage, bubble rupture and rheological properties may all be implicated.

It is, therefore, desirable to explore a model system that is representative of the fractionation process, but involves boundary conditions that are realistic but less challenging to theory than the more typical overflow from an open column.

This is the motivation for the study of the inverted U-tube system shown in figure 1. Foam generated by a sparger rises in the left-hand leg and falls on the right-hand side. It is in contact with the surface of a liquid reservoir on both sides. As a result, we have well-defined boundary conditions at each end of the tube, allowing us to sidestep the challenging theoretical conditions of an open overflow. The attractive nature of the U-tube has also been noted for experimental work, used for example by Martin *et al.* [4] (who compared their results with previous models by Stevenson [5]).

There is a steady flow of both gas and liquid from left to right. The gas flow rate is constant. The key question in the present context is: *what is the liquid flow rate that is delivered?* This must vary with the gas flow rate, and other parameters such as bubble size.

A proportion of surface active molecules is trapped at the film surfaces and hence delivered at a rate determined by the gas flow. The interstitial liquid does not, however, travel at the same velocity. Ideally, we would seek to minimize the rate of delivery of this dilute solution.

We shall address this problem in terms of the steady-state version of the so-called foam drainage equation with a linear dissipation term, in the first instance. The modelling of foam drainage (including the case of fractionation) goes back to the work of Leonard & Lemlich [6], who had all the elements of the model that prevails today, but did not conduct the mathematical analyses that would have exposed its rich variety of non-uniform and time-dependent solutions. Only uniform profiles of steady drainage were considered. The later contributions of Gol’dfarb *et al.* [7], Verbist *et al.* [8], Koehler *et al.* [9], Cox *et al.* [10] and Saint-Jalmes & Langevin [11] developed the field in its full generality. This formulation has been applied with qualitative and semi-quantitative success to a range of drainage experiments, including ‘free drainage’ and ‘forced drainage’ [12,13], and foamability tests [14] in a single column. (For a more thorough summary of early foam drainage theory, see the work of Weaire *et al.* [13].)

The great advantage of the elementary foam drainage equation is that many problems may be treated *analytically* and this turns out to hold again here (at least within a limited regime), although the mathematical details are quite subtle. With a full analytical theory, we can hope to understand the process in greater detail. Having done so, we will take a step towards practical reality by generalizing the power-law at the heart of the equation, which has often been empirically adjusted to describe particular surfactant systems [15]. Much of the analysis can be extended to the more general case.

The key result of the more rudimentary theory states that liquid flow rate varies *quadratically* with gas velocity, in a certain limit that is of practical relevance. Other interesting relations are derived for the variation of liquid fraction in the two tube legs.

We will also present numerical solutions that corroborate the findings of this analytic theory and preliminary experiments to test it. We defer to a subsequent paper the details of the dependence of the results on the length of the two legs [16].

There are several points of contact and consistency with the previous work of Neethling *et al.* [17], who set out to analyse and model an overflowing fractionation set-up (based on the foam drainage equation), using numerical solutions for validation. We also comment on the work of the Stevenson group [2,5,15], which has significant points in common with our analysis.

## 2. Basis of the theory

The flow of gas and liquid will be treated as one-dimensional: that is, we do not allow for any variation across the finite cross section of the tube. Such variations are certainly detectable in experimental systems, but will be neglected here to allow the derivation of analytical results (and the numerical verification of those results).

We use, in the first instance, the foam drainage equation in its simplest form. It is a partial differential equation for the spatial and temporal variation of the liquid fraction *ϕ*(*x*,*t*).

The *steady-state* form of the equation relates local values of liquid flux (volume flow rate per unit cross-sectional area) *J* to liquid fraction *ϕ*. In the present context, we introduce a gas velocity *V* , and a variable *g*(*x*) term to allow for the orientation of the flow with respect to gravity.

In principle *V* is the quotient of the constant gas flux and the gas fraction, 1−*ϕ*, and hence a function of position *x*. In the spirit of the drainage equation, which is formulated for relatively dry foam (*ϕ*≪1), this velocity is treated as *constant*, neglecting its small variation owing to that of the gas fraction. (Here, as elsewhere, experiments lead us to test the validity of the theory somewhat beyond the regime in which this is a clearly reliable approximation. Stevenson [5] has proposed the inclusion of a correction for this effect in the drainage equation, but we will not consider it here in order to simplify analytical calculations.)

Note that in what follows *x* is the *upward* vertical coordinate for both the left (upward flow) and right (downward flow) tube, with *x*=0 at the liquid surfaces and *x*=*L* at the top (i.e. *x* refers to a height above the surface of the liquid reservoirs).

For the *left-hand side* (liquid flux and gas velocity in positive *x*-direction), we thus obtain the standard form of the steady-state equation, with an additional term arising from finite gas velocity,
2.1aand
2.1bwhereas for the *right-hand side*, where both liquid flux and gas velocity are now in the negative *x*-direction, we have
2.2aand
2.2b

The constants *c*_{1} and *c*_{2} are defined in appendix A. They contain the physical and geometrical parameters of the system, including surface tension, liquid viscosity and density, and average bubble diameter.

In the analytic theory, we neglect the effect of a finite bend, instead assuming that liquid fraction may be equated at the top of the two columns. This approximation will be examined by simulation in an appropriate regime of flow parameters (see §5). A brief analysis of the expected error from this assumption can be found in appendix C.

The boundary conditions are, therefore, taken to be 2.3and 2.4the value commonly taken for the liquid fraction at the liquid interface. The key results are insensitive to this precise value.

## 3. Analysis for

We seek solutions of the problem posed in the previous sections, in the limit of infinite leg length . We may think of the gas velocity *V* as fixed at the outset, but the liquid flux *J* is not. Indeed, a main objective is to determine *J*(*V*). Note that *J* must also depend on *L*, and it is the limiting case that we seek here.

We shall show that in this limit, with the above boundary conditions,
3.1which is the value for which the expression in parentheses in equation (2.1b) has two coincident roots for *ϕ*_{l}.

The roots in question (for equation (2.1b)) are 3.2whereas the corresponding roots for equation (2.2b) are 3.3All of these roots, when real, have the significance that they are constant-profile solutions. As these do not fit the boundary conditions, such solutions are not used directly here.

On the right-hand side, only *ϕ*_{3} is positive and *ϕ*_{4} is largely irrelevant. As , it is evident from equation (2.2b) that . One may, for example, integrate equation (2.2b) to obtain
3.4The integral must diverge as , and this requires . This further implies (from equation (2.3)) that . In this way, the right leg sets approximate boundary condition for consideration of the left-hand leg.

We have for the integrated form of equation (2.1b),
3.5Consider the case *J*<*J*_{0}, for which the roots *ϕ*_{1} and *ϕ*_{2} are real. By the same argument as given above, we require . But there is no finite value of *J* in this range for which *ϕ*_{1}=*ϕ*_{3}, as required by equation (2.3), hence no such solution is possible.

We now turn to the case *J*>*J*_{0}, for which *ϕ*_{1} and *ϕ*_{2} are complex. For any given *J* in this range, the denominator in equation (3.5) has a minimum (finite) value, and hence the integral in equation (3.5) cannot diverge.

The remaining possibility is that as . The integral 3.6is indeed divergent, as required. Thus, equation (3.1) must hold in the limiting case.

All of this may be seen more clearly by examining the numerical solutions (see §5) but a formal derivation, such as the above, is desirable. It becomes clearer when the nature of the approach to the limit of infinite *L* is analysed, as we will do in a subsequent paper [16], finding the appropriate asymptotic form for *J*(*L*) using the above integrals.

Alternatively, there exist analytic solutions of equations (3.5) and (3.4). Clearly, all of the above is contained in or implied by those solutions, but they are quite clumsy. For completeness, we provide them in appendix B.

Finally, we note the most subtle feature of these solutions that may cause concern. For *J*=*J*_{0}, as . How can this be compatible with the boundary condition (equation (2.3)), as *ϕ*_{1}≠*ϕ*_{3}?

The resolution of this paradox lies in the fact that for any finite large *L*, the liquid flux is given by *J*=*J*_{0}+*ϵ*^{2}, where *ϵ* is small and there is only an *apparent* asymptote at *ϕ*_{1}(=*J*_{0}), eventually crossed by the solution which then decreases to the required value, close to *ϕ*_{3}. Figure 2 illustrates this behaviour clearly. The existence of this inflection was also discussed by Neethling *et al.* [17].

For large *L*, *ϕ* is close to *ϕ*_{3} in most of the right-hand tube, and we may define the corresponding liquid velocity there as
3.7This enables us to arrive at the simple result
3.8relating gas and liquid velocities in the right-hand leg.

This would seem to be a surprising result, in that the two velocities are related by a numerical constant. It is not inevitable on purely dimensional grounds, because the theory contains a parameter (*c*_{1}) that has the dimension of a velocity.

## 4. Generalization

Examination of the essentials of this mathematical argument indicates that the final results can be generalized to the case where the steady-state equation is based on a power law for the viscous drag associated with liquid flow (different from the linear form implicit in equations (2.1) and (2.2)), as often assumed in empirical analysis.

The modified form of the equation for the left leg is
4.1where *b*_{1} and *b*_{2} are constants. A similar modification holds for the right leg.

The value of the exponent *n* may be related to the details of the dissipation mechanism for the flow of liquid through the foam. In the drainage model considered in the previous sections the Plateau border interfaces are treated as rigid, the dominant dissipation term is then owing to the resulting Poiseuille flow in the Plateau borders, leading to *n*=2 for this ‘channel-dominated’ drainage model [8]. Fully mobile interfaces result in plug flow in the Plateau borders. The dominant dissipation mechanism is then associated with the flow through the junction of Plateau borders, leading to *n*=3/2 for this ‘node-dominated’ drainage model [9]. It is important to note that both these models relate to dry foams; corrections are required for liquid fractions exceeding a few percent.

In practice, the value of the exponent *n* extracted from experiments has been found to depend on the surfactant. When used as an empirical parameter in drainage experiments, *n* takes values varying from 1.92 to 2.29 for surfactants giving rise to more or less rigid interfaces [15,18], and 1.56 to 1.64 for surfactants with more or less mobile interfaces [9,18].

As before, we begin by identifying constant profile solutions of the equation for the left-hand column, that is,
4.2where *c* is a constant containing physical parameters and *n*>1 (our previous discussion relates to *n*=2).

The same argument as before demands coincident roots of this equation (hence *dJ*/*dϕ*=0) for *ϕ*_{l}, as a condition for a matched solution in the limit . (Note however, that a full analytical formulation for the solution itself is not available in the general case, but is unnecessary in what follows). The generalized results for *J* and *v* obtained from this condition are
4.3and
4.4where again *k* is a numerical constant.

Thus, in the general case, the quadratic form for *J*(*V*) is lost but *v* remains proportional to *V* . The constant of proportionality *k*(*n*) is given by
4.5where the constant *d*(*n*) is the real positive root of
4.6The form of *k*(*n*) can be seen in figure 3.

Qualitatively, this generalization can also be further extended to the form of drainage equation used by Stevenson [5].

## 5. Numerical illustration

In order to validate and illustrate the analytical results discussed above, we will compare them with numerical simulations.

The foam drainage equation in its appropriate form is now solved for the entire U-tube, including a finite semicircular bend (a case for which there is no analytical equivalent).

The U-tube can then be thought of as a one-dimensional system, with the relevant component of gravity acting downwards in the left (input) leg of length *L*, upwards in the right leg (also of length *L*), and varying in the bend of length *B*. We thus write the (stationary) drainage equation for liquid fraction *ϕ* as a function of the new position variable *z* as
5.1The function represents the variation of gravity and is defined piecewise as
The boundary conditions are set as before, i.e. both ends of the tube are in contact with a liquid reservoir, *ϕ*(*z*=0)=*ϕ*(*z*=2*L*+*B*)=*ϕ*_{0}=0.36.

We require to find pairs of (*V*,*J*) that lead to a liquid profile consistent with these boundary conditions. Fixing *V* and integrating from left to right, starting from *ϕ*_{0}=0.36, we use the ‘shooting method’ for determining the value of *J* for which the right-hand boundary condition is satisfied as well. This fitting of *J* is carried out using the MIGRAD minimizer from the CERN MINUIT software [19]. The integration is performed using Heun’s method (explicit improved Euler method [20]).

In this way we find the valid pairs of (*V*,*J*) for a given U-tube set-up (e.g. leg length, bend radius and physical parameters of the surfactant solution). Results from these simulations are presented in figures 4–6.

Figure 4 shows a numerical computation of a full liquid profile for a U-tube system with ratio of bend to leg length of 5/8, i.e. very similar to that used in some of our experiments (see §6). Note that the profile includes a full treatment of the bend, and that both legs and the bend have finite length. The analytic constant profile solutions for the left and right legs (equations (B3) and (B5), respectively) are also plotted, highlighting the key difference in the left leg—the presence of an inflection point.

Figures 5 and 6 show the variation of the liquid flux *J* with the gas velocity *V* and the variation of the liquid velocity *v* with the gas velocity *V* , respectively, for a U-tube system with ratio of bend length to leg length of 1/4. Despite the finite system dimensions in the simulations, the analytical predictions made by equations (3.1) and (3.8) for the limit of infinite legs and zero bend radius are found to be in excellent agreement with these.

A further paper addressed to the dependence of flux on leg length will determine the regime in which the infinite length is appropriate [16]. Appendix C gives a quantitative estimate of the effect of finite bend radius.

## 6. Experiments

Experiments were undertaken to test some of the key predictions made by the theory described above which can be summarized as follows:

— The ratio of the liquid fractions in the legs (measured sufficiently far from the liquid reservoirs) is approximately constant, and given by for rigid interfaces (by substituting equation (3.1) into equation (3.2) and (3.3)), and a different constant in other cases.

— Equation (4.3) predicts that the dependence of liquid flux

*J*on gas velocity*V*is a power law. Again the power law exponent*n*is dependent on the surfactant used (i.e. if the foam has mobile or rigid interfaces).— The dependence of the liquid velocity

*v*on the gas velocity*V*is predicted by equation (4.4) to be linear, with slope dependent on*n*.

The design of the experiments allowed us to check all three predictions. U-tube set-ups of the type shown in figure 1 were assembled, with internal tube diameters 5.8 and 15.7 mm. The lengths of the tube legs used were between 0.4 and 0.64 m (for a total system length of 1.0–1.7 m).

Foams were produced from aqueous solutions of sodium dodecyl sulfate (SDS) and Fairy Liquid (a commercial detergent), with concentrations above the critical micelle concentration. Gas was blown through a ceramic filter, resulting in a polydisperse foam flowing through the tube (average bubble diameter approximately 1.0 mm). Once the foam had filled the entire tube, and the gas velocity reached a steady state, experimental measurements were begun.

Gas velocity was measured by visually tracking individual bubbles in the foam. Liquid velocity was measured by adding fluorescein (a fluorescent dye) to the surfactant solution in the left reservoir (figure 1) and tracking the moving front that was visible under UV lighting. Liquid flux was measured by collecting the outflowing foam in a beaker, and measuring the mass of liquid collected over time to infer a liquid volume flow rate *Q*. The liquid flux is then simply *J*=*Q*/*A* where *A* is the cross-sectional area of the tube. Liquid fraction was estimated (to within a constant) from the thickness *d*_{PB} of Plateau borders at the tube surface. Liquid fraction should be proportional to the square of this quantity, at least in the dry limit. We simply measure *d*_{PB} in both legs, and take the squared ratio of these values as an approximate measure of the ratio of the corresponding liquid fractions.

Results from our experimental measurements with Fairy Liquid can be seen in figures 7 and 8. Figure 7 shows the relationship between the gas velocity *V* and the liquid flux *J*. The data are very well described by a power-law fit to equation (4.3), with exponent *n*/(*n*−1)=2.3±0.2. This corresponds to a value of *n*=1.8±0.2. Fairy Liquid has been associated with fairly rigid interfaces (i.e. *n*≈2) in previous foam drainage experiments [10,13], so the theoretical expectation is approximately realized. It would be interesting to repeat these experiments in combination with other drainage experiments that more directly determine *n*.

Figure 8 shows the dependence of the liquid velocity *v* on the gas velocity *V* . Again, the general theoretical prediction (*v*=*k*(*n*)*V*) is verified, with a linear fit describing the data well. The fitted slope is *k*=1.23±0.07. From figure 3 we can see that this slope corresponds to a value of *n*=2.0±0.4, consistent with the value from the *J*(*V*) relation.

Measurements for the Plateau border thickness were taken at approximately halfway up each tube leg (see figure 4 for a numerical calculation of a full liquid fraction profile, noting the non-constant *ϕ* in the left leg) for an SDS foam. Multiple measurements were taken and averaged, giving a *ϕ*_{1}/*ϕ*_{3}=2.5±0.5. The relatively large error in this measurement is larger than the variation in the left leg liquid fraction (figure 4), thus the precise point the measurements were taken is of limited influence. The theoretical prediction for this value is as outlined above, assuming *n*=2.

In summary, we find general agreement between theory and experiment on points 1–3 above. It may be interesting to repeat these experiments for surfactants with different interface type (and hence, different predicted values of *n*).

## 7. Conclusion

It has been gratifying to find that so much of the solution behaviour could be explored by purely analytic mathematics. In summary, for the foam drainage equation as presented above:

— the limiting value of the liquid flux

*J*as is quadratic in the gas flow velocity*V*(equation (3.1));— the ratio of liquid velocity

*v*to gas velocity*V*is linear (equation (3.8), generalized in equation (4.4)); and— such results may be generalized to other forms of the foam drainage equation, giving other power laws for

*J*, for example.

Preliminary experimental tests are in agreement with the above and will be pursued further in future, to, for example, test the theory with surfactants which have different values for *n*.

As a measure of the effectiveness of the process in concentrating the surfactant solution, one may calculate the ratio of the final concentration to that of the original solution. In terms of the present theory this ratio may be estimated as follows.

Let the original concentration be *ρ*_{1}. The rate of delivery of dissolved surfactant is *Jρ*_{1}, whereas that associated with the surface is *λV* , where *λ* is a constant involving the surface concentration etc. Hence, the required ratio is
7.1Using equation (3.1) for *J*, results in
7.2where *μ*=4*λc*_{1}/*ρ*_{1}.

Hence, as may be instinctively obvious, this measure of efficiency increases with diminishing gas velocity *V* .

This conclusion must, however, be qualified in the light of the model and approximations that were used. In particular, the limit of infinite leg length was taken: in practice this requires that the height of the columns is sufficient. As the required length for this to be the case increases.

Further consideration of this limitation is deferred to a subsequent paper in which length-dependence will be thoroughly investigated [16]. For the moment, let us suppose that the practical limitation of length entails a minimum value *V* _{0} of gas velocity. The maximum efficiency, according to equation (7.2) (and assuming that *R*≫1), hence is *R*≈*μ*/*V* _{0}.

It is possible to operate two (or more) systems in *tandem* [21], with a combined efficiency as follows. In the above approximation,
7.3and
7.4Hence, tandem operation with two columns doubles the maximum efficiency, and *N* columns increase it by a factor *N*, within the stated approximation.

A detailed numerical and analytical treatment of the length dependence of performance is deferred to a later paper. Most of the present results are for columns that are of effectively infinite length. This means they exceed some minimal length, beyond which limiting behaviour is approached. That length is, however, a function of gas flow rate *V* . Accordingly, consideration of this should enter the discussion of optimization and clearly answer the question: how short can the legs be made without significantly reducing the efficiency implied by equation (7.2)?

In developing a proposed theory of a single overflowing tube, Stevenson [5] has attempted to show that only a single value of *J* renders the constant profile solution *internally stable*. We do not believe this is the case for the foam drainage equation or any of its variants, including the node-dominated form used by Koehler *et al.* [9]. This may be checked by resorting to the time-dependent form of the equation and adding a local perturbation to a steady state (S. Cox 2012, private communication). Nevertheless, the value of *J* chosen as stable corresponded to the maximum of equation (2.1), so that analysis has, in the end, important points in common with this one.

Finally, we note that in this work we analysed what Shih & Lemlich [3] termed ‘simple mode’. In industry, foam fractionation is also carried out in the so-called ‘stripping mode’, where solution is added to the foam at a constant rate at some point along the left leg (above the reservoir surface). In the related ‘enriching mode’, this additional liquid is taken from the outflow reservoir of the right leg. Both modes could, in principle, be modelled using the analytical and numerical approach as described in this paper. Finally, it would be illuminating to change the boundary conditions (especially in the right or exit leg) to examine other experimental set-ups, such as the overflow model previously studied by Neethling *et al.* [17], or the ‘walking stick‘ fractionation column used by Martin *et al.* [4].

## Acknowledgements

Research supported by Science Foundation Ireland (grant no. 08-RFP-MTR1083), European Space Agency (MAP grants AO-99-108:C14914/02/NL/SH and no. AO-99-075:C14308/00/NL/SH) and COST Action MP1106 Smart and Green Interfaces. Our interest in problems related to fractionation was stimulated by discussions with Paul Stevenson, while D.W. held a visiting position at the University of Auckland. We thank V. Poulichet for experimental support. D.W. thanks Simon Cox for useful discussions. The authors thank the reviewers for their help in clarifying the manuscript.

## Appendix A: physical parameters, length and time scales

The constants *c*_{1} and *c*_{2} contain the physical parameters of the problem, i.e. the density difference of gas and liquid, *Δρ*; surface tension, *γ*; liquid viscosity, *η*; gravitational acceleration, *g* and the average bubble volume, *V* _{b}, together with numerical constants related to foam structure and geometry. These are a constant related to the cross section of Plateau borders, , the total length of Plateau borders per unit volume of foam and the effective viscosity *η**≃150*η* (see [8] for explanations of these factors). The constants *c*_{1} and *c*_{2} are thus given by *c*_{1}=*Δρg*/*l*_{V}*η** and with dimensions of length/time and length^{2}/time, respectively. These constants are used to non-dimensionalize lengths by dividing them by *c*_{2}/*c*_{1}, and velocities and flux by dividing them by *c*_{1}. See figures 5 and 6 for examples. The values for these constants for the surfactant mix used in our experiments are as follows: *η*=0.001 Pa s, *γ*=0.032 Nm, *ρ*=1000 kg m^{−3}, *V* _{b}≈0.54 mm^{3}. For these values we get *c*_{1}≈0.008 m s^{−1} and *c*_{2}=1.5×10^{−5} m^{2} s^{−1}.

## Appendix B: full analytic solutions for the case of infinite leg lengths

We may derive analytic solutions to the integrals that define the left- and right-leg liquid fraction profiles (as given in §3). The form of these integrals depends on whether the roots of the quadratic equations (3.2) (for the left leg) and equation (3.3) (for the right) are real or complex. We show here only solutions for the real roots; solutions for the complex case exist (although they are very cumbersome) and may be found, for example, in the tables of Petit Bois [22].

For the left-hand leg we must consider two cases, depending on whether the roots of equation (3.2), *ϕ*_{1} and *ϕ*_{2}, are distinct or coincident. For the case where *ϕ*_{1} and *ϕ*_{2} are real and distinct we integrate equation (3.5) to obtain:
B1where *f*_{l}(*ϕ*) is given by
B2In the case that the *ϕ*_{1} and *ϕ*_{2} are real and coincident, we instead consider equation (3.6):
B3In the limit , we obtain *ϕ*_{l}→*ϕ*_{1}.

For the right-hand leg, we must only integrate equation (3.4) (for real roots of equation (3.3)), arriving at
B4where *f*_{r}(*ϕ*) is given by
B5In the limit , we obtain *ϕ*_{r}→*ϕ*_{3} (figure 9).

Example solution profiles of equations (B3) and (B5) are shown in figure 4.

## Appendix C: the effect of finite bend radius

The presented analytical theory proceeded from assuming that the bend is short compared with the straight legs of the U-tube. The liquid fraction at the top of the two tubes was then equated and we showed that in the limit of infinite leg length *L*, their value is (see equations (3.7) and (3.8)).

We now consider the effect of finite bend radius *r* in the same limit and obtain for the derivative ∂*ϕ*_{l}/∂*x* for the left leg
C1whereas for the right leg, we obtain
C2Here, *c*_{1} and *c*_{2} are the constants as defined in appendix A, and .

The change in liquid fraction across a finite bend of length *B*=*πr* may thus be estimated as
C3which can be compared with the value for *r*=0 to predict the error in the liquid profile owing to finite bend length,
C4

- Received December 12, 2012.
- Accepted March 8, 2013.

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