Abstract
The steady propagation of an air finger into a fluid-filled tube of uniform, rectangular cross-section is investigated. This paper is primarily focused on the influence of the aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field are also considered. The three-dimensional interfacial problem is solved numerically using the object-oriented multi-physics finite-element library oomph-lib and the results agree with our previous experimental results (de Lózar et al. 2007 ) to within the ±1% experimental error.
At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure drops across the finger tip and relative finger widths decrease with increasing α. The dependence of the wet fraction m (the relative quantity of liquid that remains on the tube walls after the propagation of the finger) is more complicated: m decreases with increasing α for low Ca but it increases with α at high Ca. Our results also indicate that the system is approximately quasi-two-dimensional when α ≥ 8, when we obtain quantitative agreement with the two-dimensional model of McLean & Saffman (1981) for the relative finger width as a function of the governing parameter 1/B = 12α 2 Ca. The action of gravity causes an increase in the pressure drops, finger widths and wet fractions 2 A. de Lózar, A. Juel and A. Hazel at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca.
For α ≥ 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α 2 the capillary number at which global recirculation flows disappear Ca c has been observed to follow the simple empirical law: Ca 2/3 c α = 1.21.

at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca.
For α ≥ 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α 2 the capillary number at which global recirculation flows disappear Ca c has been observed to follow the simple empirical law: Ca 2/3 c α = 1.21.
Introduction
The displacement of one fluid by a second, immiscible, fluid is a phenomenon present in many applications including enhanced oil recovery, flow in porous media, thin-film coating and the biomechanics of the lung. The canonical model problem concerns a finger, or long bubble, of air that is driven through a tube of uniform cross section, initially filled by a wetting fluid.
The majority of previous studies of this model problem have considered two particular tube geometries: (i) a circular tube (Bretherton 1961) , and (ii) a Hele-Shaw cell, a rectangular tube whose width is much greater than its depth so that the system is approximately two-dimensional (Saffman & Taylor 1958) . In many applications, however, the tubes are neither circular nor quasi-two-dimensional. In particular, many pharmaceutical and food processing applications require an understanding of interfacial flows at microfluidic scales (Baroud & Willaime 2004) . In microfluidic devices the manufactured microchannels are usually of rectangular cross section, but with widths comparable to their depths.
Motivated by these applications, recent studies have investigated air-finger propagation
The steady propagation of an air finger into a rectangular tube. 3 in polygonal tubes, see the recent review by Ajaev & Homsy (2006) . The tubes are usually of rectangular cross-section, which can be characterised by a single aspect ratio α = w/b, defined to be the ratio of the tube width, w to the tube depth b. If the aspect ratio is large, the Hele-Shaw cell geometry is recovered, in which the displacement of viscous fluid by a propagating air finger is known as the Saffman-Taylor problem. The Saffman-Taylor problem is an archetype of pattern forming instabilities and has been extensively studied in the literature, see the review of Homsy (1987) . Fewer studies have been conducted at low aspect ratios, but for 1 ≤ α ≤ 2 steady air-finger propagation in fluid-filled rectangular tubes has been comprehensively described by the numerical simulations of Hazel & Heil (2002) . Not surprisingly, comparison of the available results suggests that the dynamics of the system are rather different at low and high aspect ratios. At intermediate aspect ratios, we are aware of only three previous studies: (i) a lubrication-theory-based approach valid only at very low finger-propagation speeds (Wong et al. 1995a,b) ; (ii) an investigation of the specific case of buoyancy-driven "free" rise of bubbles (Clanet et al. 2004) ; and (iii) our recent experimental results covering the range 1 ≤ α ≤ 15 (de Lózar et al. 2007) .
In this paper we extend the study of Hazel & Heil (2002) and complement the experimental results of de Lózar et al. (2007) by solving the three-dimensional, free-surface Stokes equations numerically in order to investigate the steady propagation of air fingers in rectangular tubes of intermediate aspect ratio (1 ≤ α ≤ 8). The chosen range of aspect ratios is the relevant one for microfluidics devices, in which microchannels are usually manufactured with 1 ≤ α ≤ 10 (Whitesides & Stroock 2001) . In the interests of simplicity, we neglect the effects of fluid inertia, but we do include the effects of a transverse gravitational body force for direct comparison with the experimental results.
The agreement between our simulations and experiments is within the ±1% experimental Figure 1 . Sketch of the problem: an air finger propagates at a constant speed UF into a tube of rectangular cross-section with height 2b and width 2w. The tube is initially filled by a wetting Newtonian fluid of viscosity µ and density ρ and the surface tension at the air-liquid interface is σ. A gravitational body force of magnitude g acts normally to the axis of the tube.
error demonstrating the accuracy of the simulations and the quality of the experimental data.
The simulations provide detailed information about the influence of gravity and aspect ratio on the system. Furthermore, we uncover the physical origin of the empirical scaling found by de Lózar et al. (2007) relating the wet fraction (the relative quantity of liquid that remains on the walls after the propagation of the air finger) to a modified capillary number that depends on the aspect ratio. We show that, although not exact, this relation is valid to within experimental error over a wide range of parameter values.
The present study also represents a significant step towards the direct numerical study of the Saffman-Taylor instability; and our simulations indicate that the Saffman-Taylor, quasi-two-dimensional, approximation is valid for α ≥ 8.
The steady propagation of an air finger into a rectangular tube. 
The model
A finger of air, assumed to be of negligible viscosity and density, propagates at a constant speed U F under an internal pressure p * F into a tube of uniform rectangular cross-section with aspect ratio α = w/b, where 2b is the depth of the cross-section and 2w is the width. The wetting fluid originally contained in the tube is assumed to be an incompressible Newtonian fluid of dynamic viscosity µ and density ρ and the liquidair interfacial tension is given by σ. The entire system is subject to a gravitational body force of magnitude g directed normally to both the direction of finger propagation and the wider plane of the tube's cross-section exactly as in de Lózar et al. (2007) 's experimental setup, see Figure 1 . For simplicity, we neglect inertial effects in the fluid.
Governing equations
The problem is formulated in dimensionless Cartesian coordinates x = (x 1 , x 2 , x 3 ) = x * /b and the finger propagates in the negative x 2 -direction. Throughout this paper we use an asterisk to denote dimensional quantities as opposed to their dimensionless equivalents.
We non-dimensionalise the fluid velocity on the finger velocity u = u * /U F and the fluid pressure on the viscous scale p = p * /(µU F /b). In an inertial frame moving with the constant velocity of the finger the motion of the fluid is governed by the steady Stokes equations:
and the equation of continuity
The Einstein summation convention is used and the indices take the value i = 1, 2, 3.
The vector k = (0, 0, −1) indicates the direction of the gravitational body force. Two dimensionless parameters govern the behaviour of the system: the capillary number, Ca = µU F /σ (the ratio of viscous to surface-tension forces) and the Bond number Bo = ρgb 2 /σ (the ratio of buoyancy to surface tension forces). Hence, the factor Bo/Ca appearing in equation (2.1) represents the ratio of buoyancy to viscous forces, sometimes called a Galileo or Archimedes number.
Boundary conditions
On the free surface we impose a non-penetration condition
and a dynamic boundary condition for the jump in stress across the interface
where n is the unit vector normal to the free surface (directed out of the fluid) and κ = κ * b is the dimensionless mean curvature of the surface. On the tube walls we impose the non-slip boundary condition and in the moving frame of reference the wall velocity is u = (0, 1, 0). (2.5)
We must truncate our computational domain and require boundary conditions at the domain ends. Far behind the tip (at the outlet in figure 2), the variations in the axial direction are gentle enough that a long-wavelength approximation is valid. The usual assumption (see Reinelt & Saffmann (1985) ) is to neglect all axial variations at the end of the domain. Using this condition in equation (2.1), yields a Laplace equation for the axial velocity. Instead of this condition we impose a weaker condition: we assume that only the axial variations of the pressure and axial velocity are zero: ∂p/∂x 2 = ∂u 2 /∂x 2 = 0.
This assumption permits axial variations in the transverse velocity components at the end of the computational domain and, although not rational in an asymptotic sense, was found to provide accurate results in smaller domains than those required when applying the "full" long-wavelength approximation. Using this weaker condition in equation (2.1),
we obtain: 6) with β = 1, 3. The axial component of the momentum equation (2.1) is replaced by equation (2.6) at the outlet boundary subject to the boundary conditions of no-slip at the tube wall and the approximate dynamic boundary condition on the free surface ∂u 2 /∂n + ∂u n /∂x 2 = 0, where n is the vector perpendicular to the free boundary in the plane of the boundary and u n is the velocity component directed along the vector n.
In the transverse directions, tractions consistent with the prescribed axial velocity and hydrostatic pressure gradient are applied.
Far ahead of the tip (at the inlet in figure 2), we enforce parallel flow (u 1 = u 3 = 0) and the hydrostatic pressure gradient p = −x 3 Bo/Ca is applied as a traction boundary condition in the axial direction. By imposing the pressure far ahead of the finger tip, the pressure within the air finger is no longer independent and p F must be treated as an unknown. The required additional equation is obtained by fixing the tip of the finger to remove the translational invariance of the problem.
Numerical method
The governing equations are solved numerically using a finite element method via the object-oriented multi-physics library oomph-lib (Heil & Hazel 2006) . The fluid domain is divided into a number of volumetric elements, see figure 2. The symmetry of the problem allows us to restrict attention to half the fluid domain and symmetry boundary conditions are applied at x 1 = 0.
The deformation of the volume mesh in response to changes in the free surface is treated by the method of spines (Kistler & Scriven 1983) and is essentially the same as that described by Hazel & Heil (2002) . In brief, the mesh is divided into three subdomains: the canyon, tip and liquid subdomains, see Figure 2 . In the liquid domain, there is no free surface and so the mesh is fixed. In the canyon and tip regions, every node in the volume mesh is described by its fractional distance along a spine -a vector with a predetermined direction, but unknown length h. In the tip region the spines are all directed towards the point (0, 0, 0) and emanate from the tube walls or from the plane dividing the liquid and tip subdomains. In the canyon region, the spines lie in planes perpendicular to the axis of the tube and are directed from the tube walls to the centre of the tube. The spine lengths parameterise the position of the free surface and are determined during the solution procedure. As the free surface deforms, the nodal positions are updated in response to the changes in the spine lengths. A notable advantage of the scheme is that only a small number of nearby volume elements are remeshed in response to local changes in the free surface position. We note in passing that the translational invariance of the problem is actually enforced by fixing the length of the spine directed along the axis of the tube. The end of this spine corresponds to the finger tip in the absence of gravity.
The selection of the axial extent of the mesh is not obvious because there are generally two different length scales in the system: the height, 2, and width, 2α, of the tube.
However, for high aspect ratios the system behaves like a Saffman-Taylor system and α is the only important length scale (Reinelt 1987) . For low aspect ratios α is equal to or longer than the typical length scale in the system. Therefore, we adjust the mesh length based on the tube width. The length of the liquid, tip and canyon submeshes are chosen to be 2α, α and 3α respectively.
The fluid variables are discretized by using isoparametric Q 2 P −1 elements (Gresho & Sani 2000) , in which the quadratic fluid velocities are continuous, but the linear fluid pressures may be discontinuous between neighbouring elements. The spine lengths are discretized by isoparametric, two-dimensional quadratic elements. The discrete system of equations is constructed by introducing the dynamic boundary condition (2.4) into the weak form of the momentum equations (2.1) as explained by Ruschak (1980) ; and the system is completed by the weak forms of the continuity equation (2.2), non-penetration boundary condition (2.3) and outlet boundary condition (2.6) . The complete nonlinear system of equations is solved by Newton-Raphson iteration and the resulting linear system is assembled and solved by the HSLMA42 frontal solver (Duff & Scott 1996) .
In our standard resolution we deal with 60,000 degrees of freedom. An initial guess was generated by following the same procedure as Hazel & Heil (2002) and once a converged solution had been found other solutions were obtained by slowly varying α, Bo and Ca. 
Validation of the code
In figure 3 (a) we compare our results with the results of Hazel & Heil (2002) 
The limit of the simulations: contact lines
Far behind the finger tip, the fluid is at rest in the lab frame and the system is independent of the axial coordinate. Hence, the interface has a non-zero curvature in the transverse plane only and the curvature is equal to the pressure difference across the interface. In the absence of gravity, the curvature must be a constant and the only possible interface shapes are arcs of circles. If the radius of curvature of the interface is greater than one, then the interface must intersect the tube walls because the fluid is wetting, i.e. the contact angle is zero. The final equilibrium configuration will, therefore, consist of fluid lobes separated by dry spots. The evolution towards the equilibrium configuration is driven by transverse draining flows that persist over long axial distances, whereas the axial velocity approaches its undisturbed value a relatively short distance behind the tip.
The full details of the evolution dynamics were given by Wong et al. (1995a,b) in the limit of vanishing capillary number.
In Figure 4 we show a cross-section of the flow domain at a distance 3.92α behind the A. de Lózar, A. Juel and A. Hazel tip for Ca = 0.076, Bo = 0 and aspect ratio α = 3. The axial velocity is approximately uniform and identical to the wall velocity (maximum difference |δu y | = 0.001, 0.1%).
Nonetheless there are considerable pressure variations around the interface driving a transverse flow from the fluid films above and below the finger to the edges of the tube, which causes a decrease in the finger width with increasing axial distance from the tip.
The films thin with increasing axial distance and would eventually rupture to create the dry spots.
The accurate numerical resolution of the evolution to equilibrium far behind the finger tip is precluded by the extreme thinning of the films, which leads to local under resolution in our finite element method. At a fixed distance behind the finger tip the film thickness decreases with capillary number and the development of the thin films at the end of our computational domain sets the lower limit of capillary number in our simulations.
The axial extent of the domain was chosen to be short enough that the films were sufficiently thick to be well resolved, but yet long enough so that the approximation of negligible axial variations in axial velocity or pressure was valid. A number of simulations in longer domains, in which the length of the canyon subdomain was doubled to 6α, were performed and the bulk properties were found to be identical to those in our standard domain (differences of the order of 10 −5 ).
Increasing the aspect ratio leads to a decrease in finger pressure at a fixed capillary number (Hazel & Heil 2002) , see also figure 6(b), associated with thinner fluid films.
Hence, the range of capillary numbers that can be simulated decreases with increasing aspect ratio. The maximum value of the aspect ratio in the simulations was α = 8. For larger values of α, it was not possible to obtain results at sufficiently low values of Ca to explore the parameter regime associated with the single finger Saffman-Taylor regime. At Ca = 0.02 and α = 1, the cross-sectional flow domain consists of four corner regions of constant curvature, connected by thin films. As discussed in section 2.5, the difference in curvature between the corner regions and the thin films generates a pressure drop that drives transverse draining flows from the films into the corner regions. The volume of fluid transported, however, is so small that it does not measurably deform the interface, and thus, the air finger is in quasi-equilibrium. In fact, a fluid domain consisting of four corner regions of constant curvature is an equilibrium configuration at Ca = 0 for any value of α (Wong et al. 1995a) . Experimental evidence (Moore et al. 2002) suggests that these configurations may be unstable for very high aspect ratios α 250, however.
At Ca = 0.02 and α = 3 and 7, the quasi-equilibrium configuration is altered compared to α = 1. The interface consists of two end regions of constant curvature along the short semi-axes of the tube, connected by two thin liquid films along the longer semi-axes. The thin films ensure that the draining flows are weak, so that the variation of the finger width in the axial direction is negligible (in both cases, the values of dimensionless finger width λ, defined in section 3.2, measured at cross-sections located at 3.92α and 2.92α behind the finger tip differ by less than 8 × 10 −4 ). McLean & Saffman (1981) and Reinelt (1987) used this cross-sectional configuration to derive the boundary conditions for their two-dimensional simulations of the Saffman-Taylor instability.
As Ca increases, the fluid films thicken and in near-square tubes the ultimate finger shape becomes axisymmetric at sufficiently high Ca (Kolb & Cerro 1991) . Hazel & Heil (2002) showed that these axisymmetric configurations were possible for α 2.04, but for α 2.04 the axisymmetric configuration never occurs. Instead, the equilibrium finger shape consists of end regions of constant curvature connected by thick films along the longer semi-axes of the tube, as shown for Ca = 1.0 and α = 3, 7. The thick films enhance the effect of the draining flows, resulting in a stronger axial dependence of the finger width within our finite flow domain. (λ measured at cross-sections 3.92α and 2.92α
far from the tip differ by 8 × 10 −3 for α = 3 and by 3 × 10 −3 for α = 7.) The finger is still expected to relax into a quasi-equilibrium configuration similar to that shown at Ca = 0.02, but only at very long distances from the tip (not seen in our simulations).
Bulk properties
In figure 6 we present results for tubes of aspect ratios α = 1, 2 . . . the finger tip, see figure 5 and section 5.1. In the limit of low Ca, and indeed for Ca = 0 (Wong et al. 1995a) , the final interface shape consists of four corner regions, whose constant curvature is not imposed by the geometry of the tube. Thus, the wetted area does not alter significantly with increasing α, resulting in decreasing values of m. For higher Ca, the fluid domain far behind the finger tip consists of two "side" regions, whose constant curvature is set by the shorter semi-axis of the tube's cross-section. The width of these regions grows more rapidly with α than the width of the finger, resulting in a net increase in m.
In figure 6 (b), we plot the pressure drop across the tip of the finger on the capillary scale (∆p Ca = ∆p/(γ/b)) as a function of the capillary number. The pressure drop decreases with the aspect ratio, reflecting a reduction in tip curvature that is associated with widening tubes, but the separation between the curves is also reduced as α increases. Our simulations suggest that the α dependence of ∆p Ca weakens further as Ca increases.
In the literature, the typical parameter used to characterize air fingers in rectangular tubes of high aspect ratio is the relative finger width, λ, the width of the air finger divided by the width of the tube's cross-section. This quantity appears naturally in the two-dimensional model and is easy to measure experimentally. The definition of λ in the three-dimensional system is not clear, however. Based on what would be observed in experiments we define the relative finger width to be the maximum x 1 -coordinate of the finger divided by the aspect ratio. The measurement is taken in the last x 1 x 3 −cross section of our computational domain, at a distance of 3.92α behind the tip. An example cross-section at Ca = 0.076 is shown in figure 4 for which the relative finger width is λ = 0.795. At this distance from the tip, λ is independent of the axial coordinate for low Ca. At high Ca, however, the finger width does not stabilize until much greater distances. This is because the relatively weak surface-tension forces require longer axial length-scales to restore the interface to an quasi-equilibrium state after the distortion caused by the non-axisymmetric passage of fluid around the tip, as explained by Hazel & Heil (2002) . Ca. The finger width decreases with Ca as expected, but for sufficiently high Ca, the width reaches a minimum and starts to increase; a consequence of measuring the width at a fixed axial distance behind the tip. At high values of the capillary number the finger's cross-section is no longer independent of the axial coordinate at our chosen distance. We expect that the finger width measured sufficiently far behind the tip does indeed decrease monotonically with increasing Ca, as found by Tabeling et al. (1987) in experiments at high aspect ratios. Increasing the aspect ratio leads to reduced relative finger widths for all Ca because the curvature and hence, the surface-tension forces, decrease as α is raised.
In figure 6 (d) we show the relative finger width plotted as a function of the SaffmanTaylor parameter 1/B = 12 α 2 Ca. This is the typical plot shown in the literature to characterize the Saffman-Taylor instability, see Homsy (1987) . Again, the separation between the curves decreases as the aspect ratio increases and the curves corresponding The values of Ca and 1/B for which λ reaches its minimum in figures 6(c) and 6(d),
respectively, give an indication of the parameter thresholds beyond which the interface does not reach a quasi-equilibrium configuration within our simulation domain. Whereas the threshold value of Ca decreases with α, the corresponding value of 1/B increases with α. This explains why λ has not been found to vary significantly with the distance from the finger tip in Saffman-Taylor experiments, which are performed at much higher aspect ratio than our simulations (α 20). However, in low aspect ratio tubes the relaxation of the interface must be taken into account when defining λ, which generally depends on the distance from the finger tip.
Flow fields
The flow field associated with the propagating air finger in rectangular tubes has important similarities with its analogue in cylindrical tubes. Taylor (1961) showed that, at sufficiently low flow rates, the velocity of the fluid near the centreline of a cylindrical tube is faster than the finger velocity, due to the existence of a recirculating flow in the moving frame of reference. The centreline velocity changes sign for increasing flow rates (i.e. increasing Ca), resulting in the disappearance of the recirculation region ahead of the finger. In the cylinder, the transition between these two flows involves two topological changes that were conjectured by Taylor (1961) and observed by Giavedoni & 
The effect of gravity
We focus on the influence of transverse gravity (along the x 3 direction) that would apply to experiments performed in a horizontal tube (de Lózar et al. 2007 ). The effect of gravity is most pronounced at low Ca, where viscous forces are relatively weak. For a fixed Bo, the buoyancy-force term in equation (2.1) decreases in magnitude with increasing
Ca and so the influence of gravity on the system is negligible at high Ca.
In figure 9 we show cross-sections of the flow domain at a distance of 3.92α behind the tip for two values of the Bond number (Bo = 1.0, 2.5) and three aspect ratios (α = 1, 3, 7)
A. de Lózar, A. Juel and A. Hazel at Ca = 0.2. In the presence of transverse gravity the hydrostatic pressure that increases linearly with distance from the the upper wall of the tube must be balanced by the pressure drop across the free surface, in order for the pressure inside the finger to remain constant. Thus, the curvature of the free surface decreases with distance from the upper wall. For α = 1, as the Bond number is raised from Bo = 1.0 to 2.5, the finger becomes increasingly distorted from the circular configuration adopted when Bo = 0. In tubes of higher aspect ratios (α = 3, 7), it is the end regions, of constant curvature when Bo = 0, that deform to balance the hydrostatic pressure. Jensen et al. (1987) described the resulting fingers as "banana shaped". Jensen et al. (1987) also demonstrated that for Bo > 1 static interface shapes were possible in which the air finger lifts off the bottom wall of the tube, because surfacetension forces cannot balance the hydrostatic pressure across the entire depth of the tube. As a result, in our simulations, the bottom film is much thicker than the top for Bo = 2.5, compared to the case when Bo = 1.0. In the bottom part of the tube, the effect of the hydrostatic pressure is to drive a flow toward the bottom film, thus opposing the draining flow due to curvature changes. As fluid is transported from the upper part of the flow domain toward the bottom, the finger lifts and widens with distance from the tip until a quasi-equilibrium configuration is reached. The enhancement of the transverse flows by the hydrostatic pressure gradient means that the finger widths stabilise over shorter axial distances, but that the upper film is thinner than the equivalent case in the absence of gravity. Hence, the range of Ca that can be accurately simulated is reduced as Bo increases.
The reduction in the curvature of the free surface in the presence of transverse gravity also means that the finger occupies a smaller proportion of the cross-section of the tube.
Hence, at fixed Ca, the wet fraction increases with Bo, as shown in figure 10 and α = 3, respectively. The considerable increase in the wet fraction between Bo = 1.0
and Bo = 2.5 is again a consequence of the change in equilibrium shape for Bo > 1, The presence of gravity breaks the reflection symmetry about x 3 = 0, and complicates the topology of the flow considerably. The flow field remains simple for large Ca, however, where it is characterised by a single stagnation point, as shown in figure 11 for Ca = 0.292
and Bo = 1.0. This flow regime is topologically analogous to that seen in the absence of gravity shown in figure 7(f), but the stagnation point is located above the finger tip rather than exactly at the tip as for Bo = 0. Increasing Ca causes the stagnation point to migrate closer to the tip as the influence of gravity decreases. Non-zero gravity also reduces the minimum value of the capillary number for which the flow regime defined by one stagnation point is observed. At Ca = 0.292, the flow field at Bo = 1.0 includes one stagnation point, figure 11 , whereas at Bo = 0, it has three, figure 7(e).
The empirical scaling of de Lózar et al. (2007)
One of the most surprising results of the experimental study of de Lózar et al. (2007) was the discovery that the wet fraction appeared to depend on a single modified capillary number Ca = Ca 1 + a 1 (α − 1) + a 2 (α − 1) 2 for all aspect ratios above Ca > Ca t ≈ 0.035. The scaling has obvious practical applications, but its origin is strictly empirical and it was determined from a set of experiments conducted at a single Bond number, Bo = 1.0.
In Fig. 12 and a 2 = 0.019 ± 0.001, providing further evidence that the effective scaling is a good approximation when Bo ≤ 1. The separation between the curves increases with the Bond number, however, indicating that the physics underlying the scaling no longer applies for large Bo. The most likely explanation for the increased error is the dramatic change in the equilibrium configurations of the finger when Bo > 1. 
Interpretation
The wet fraction is a measure of the cross-sectional area occupied by the fluid far behind the finger tip and, in the absence of gravity, there are only three possible interface shapes in this region: (i) four quarter-circles, (ii) two semi-circles or (iii) a complete circle, see section 3.1, in particular Figure 5 . We believe that the observed scaling of the wet fraction in rectangular tubes expresses a relationship between situations in which the interface approaches case (ii) far behind the finger tip.
An alternative scaling has been observed in case (iii), a configuration that is only possible at high capillary numbers in tubes of low aspect ratio (1 ≤ α 2). In these cases, the wet fraction appears to be independent of the aspect ratio, as predicted by Hazel & Heil (2002) where h is the film thickness, see figure 13 . Assuming that the variations in curvature occur on an axial length-scale α, an axial force balance gives
The axial gradients of W are observed to be negligible at very short distances behind finger tip, leading to a two-dimensional Poisson equation for W . If we wish to relate configurations in which the wet fractions are the same, we must choose h ∝ α and we note that h ∝ α also implies that the axial pressure gradient remains approximately constant. The fluid in the thin films above and below the air finger reaches the velocity of the wall in very short axial distances and so the viscous term can be approximated by variations in axial velocity only at the side of the finger, see figure 13. If h ≈ Kα then
suggesting the putative scaling parameter
The two components of the viscous term balance at some transitional aspect ratio, α t when h = 1 ≈ Kα t and then K ≈ 1/α t , in which case our scaling becomes
which differs from the previous empirical scaling by the disappearance of the linear term.
On performing a least-squares fit to find α t the collapse of the experimental data is comparable to that of the three-term relationship proposed by de Lózar et al. The breakdown of the scaling at low Ca is a consequence of the fact that the interface approaches case (i) and the change in curvature can no longer be estimated from the finger width. The behaviour in this regime can be approximated by the lubrication theory approach of Wong et al. (1995a,b) , in which case the functional dependence on α follows from the static wet fraction. For increasing Bond numbers, the widths of the equilibrium configurations increase at fixed Ca, leading to a change in the transitional aspect ratio and consequent increase in the error of the scaling.
Conclusions
In this paper we have investigated the propagation of long bubbles into tubes of rect- In general, at a fixed capillary number the relative finger width and pressure drop across the finger tip decrease with increasing aspect ratio. For sufficiently large Ca, the fraction of the cross-section occupied by the fluid after passage of the air finger increases with aspect ratio, but at low Ca, this so-called wet fraction decreases with increasing aspect ratio. The pressure drop changes little between α = 7 and α = 8 and for these aspect ratios the relative finger width depends only on the Saffman-Taylor parameter 1/B = 12 α 2 Ca, suggesting that the system is approximately two-dimensional for α ≥ 8.
The main influence of a transverse gravitational body force is to cause an increase in pressure drop, finger width and wet fraction at fixed capillary number and aspect ratio.
A dramatic increase in these quantities is observed from Bo = 1.0 to Bo = 2.5, as a consequence of the fact that at Bo > 1 it is not possible to find equilibrium interfaces that touch the bottom wall of the tube and therefore the finger lifts, as predicted by Jensen et al. (1987) . The increase in finger width is driven by the enhanced draining flows from the film above the finger to that below it, causing the buoyant finger to be "squashed" up towards the upper wall of the tube, which also results in an increase in the wet fraction.
The topology of the flow in the high-α tubes presents a wider variety of states than in the low-α ones. For α ≥ 3 a new flow regime occurs at intermediate capillary numbers, in which a local counterflow develops ahead of the finger tip, keeping it clean. We also find that that the capillary number at which the flow velocity far from the tip is always slower than the bubble, Ca c , follows the simple empirical law: Ca 

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