The position of the foam front, h _f, during the vertical rise of the foam column as a function of time for different pipe diameters is shown in Figure 2. The error bars on each measurement in Figure 2A correspond to the standard error of the sample calculated as its standard deviation divided by the square root of its size (i.e., the square root of the number of measurements for each pipe). The standard errors vary between ∼ 5 % (for Pipe 1 with a confidence interval of 94.9%) and ∼ 3.8 % (for Pipe 5 with a confidence interval of 96.2%). As the pipe diameter is increased, it takes longer for the foam column to reach its maximum height. This is the obvious consequence of using the same airflow rate in all experiments. However, the foam column experiences a non-linear rise in pipes with h/D ≥ 30. A transition from non-linear to linear growth is evident in Figure 2A when the pipe diameter increases from 6.05 mm (Pipe 4) to 7.32 mm (Pipe 5). The transition from non-linear to linear behavior can be better observed from Figure 2B, where the height of the foam front normalized to the height of the vertical pipe, h _f/h, is plotted against the dimensionless time t/t _end, where t _end is the time of foam generation.

The log-log plot of Figure 3 shows that the growth of h _f obeys an approximate power-law behavior with time, i.e., h _f ∼ t ⁿ , where n depends on the specific pipe diameter. Pipes 1 and 2 with the smallest diameters exhibit a temporal evolution characterized by two different power-law dependencies. For times less than about 1.35 s (marked by the vertical arrow on the left), h _f evolves as t ^1.11 for Pipes 1 and 2, while for times greater than 1.35 s, the height of the foam front evolves as t ^0.69 for Pipe 1 and t ^0.53 for Pipe 2. Since in Pipes 3 and 4 with larger diameters, the foam column experiences a slower growth compared to Pipes 1 and 2, the threshold time has been shifted to approximately 1.5 s as marked by the vertical arrow on the right. In this case, h _f ∼ t ^0.97 in Pipe 3 and h _f ∼ t ^0.85 in Pipe 4 when t < 1.5 s, while h _f ∼ t ^0.81 in Pipe 3 and h _f ∼ t ^0.68 in Pipe 4 when t > 1.5 s. In Pipe 5 with the largest diameter, the log-log plot reveals an approximate linear behavior (h _f ∼ t) as was already noticed from Figure 2. According to Denkov et al. [19], as the foam front rises a capillary number, Ca = μv/σ, can be defined, where μ is the liquid viscosity, v is the relative velocity between the foam and the pipe wall (here the front rise velocity) and σ is the surface tension. They found that the wall stress is proportional to Ca^1/2. Although we have not measured the viscous stress in the foam/wall region, it has been verified experimentally that friction forces were relatively stronger in pipes with smaller h/D-ratios, thereby affecting the surface mobility of the bubbles [19, 20]. This was confirmed by the increased apparent foam viscosity, which slowed down the front velocity in pipes with decreasing h/D-ratios. This occurs because foams with larger bubble sizes were formed in those pipes, resulting in larger zones of viscous friction with the pipe wall and therefore in reduced surface mobility of the bubbles.

Figure 4 shows the foam front velocity, v _f = dh _f/dt, as a function of time for the five pipe diameters listed in Table 1. The velocity of the foam front was calculated by comparing successive measurements of the front height with time using one-sided finite differences. With the exception of Pipe 5, in all cases the front velocity decreases with time. However, during the first 50 s the decrease is considerably much steeper in Pipes 1 and 2 with the smallest diameters. In Pipe 2, with a diameter of 2.87 mm, the foam growth slows down after about 50 s and reaches its maximum height at a slower rate compared to Pipes 3, 4, and 5 with larger diameters. The same does not occur in Pipe 1 (with the smallest diameter) because there the foam achieves its maximum height within the first 50 s, i.e., faster than for the other pipe diameters. As the pipe diameter increases, the slope of the foam rise velocity decreases until it approximately vanishes for Pipe 5 with the largest diameter, where the foam grows at an almost constant rate. These differences in velocity are related to the movement of the bubbles. When the pipe diameter decreases faster rates occur due to direct pushing of bottom bubbles upward with little or no lateral movements, while when the pipe diameter is increased lateral motion of the bubbles slows down the rise of the foam. Gravity drainage of the liquid along the Plateau borders may also affect the front velocity. However, according to the experimental results of Enzendorfer et al. [8], who performed foam rheology measurements in long pipes of varying small diameters, the slower front velocity in pipes with smaller h/D ratios (i.e., with larger diameters) is due to the increasing apparent foam viscosity, which can be interpreted as a consequence of the increasing specific volume expansion ratio, defined as the ratio of the density of the base liquid to that of the foam [7], and not only as the consequence of the lateral bubble movement. An estimate of how the front velocity varies with time can also be obtained from the power-law scaling h f ∼ t n . (3) From Figure 4, the front velocity in Pipes 1 to 4 decays with time following approximately a power-law dependence of the form d h f d t ∼ t n − 1 . (4) Therefore, values of n < 1 implies that the front velocity decreases rapidly as depicted in Figure 4. However, as n → 1 as in Pipe 5, the front velocity is approximately constant as n−1 → 0. In this case, the formation of new bubbles due to gas injection and liquid drainage along the Plateau borders are no longer affecting the front velocity. According to Table 2 the Bikerman unit of foaminess increases with the pipe diameter, meaning that more stable foams are formed in pipes with larger diameters.

In a second set of experiments the pipe tops were sealed with a hermetic lid to impede the gas exit. Figure 5 shows the evolution of the foam front for the various pipe diameters when the pipe top is closed. The error bars correspond to standard errors, which vary from ∼ 4.8 % (for Pipe 1 with a confidence interval of 95.2%) to ∼ 2.5 % (for Pipe 5 with a confidence interval of 97.5%). Since the rising of the foam front is slower compared to the open top pipes, the errors carried by the height measurements were correspondingly smaller. The foam column rises much more slowly than in the open top pipes, with the rise exhibiting a much more linear behavior for h/D ≤ 63. This is a consequence of the increasing air pressure in the closed-top pipes, which in all cases slows down the rise of the foam front compared to the open top pipes. This becomes more prominent in closed-top pipes with lower h/D-ratios because of the combined effects of increasing apparent foam viscosity and lateral bubble movement. Only for Pipe 1 the rise of the foam column exhibits a non-linear behavior as shown in Figure 5B, where h _f/h is plotted against t/t _end.

The log-log plot of Figure 6 shows that the data for all pipe diameters follow approximately a power-law trend. For example, in Pipe 1 the foam height evolves as ∼ t ^0.70, while for Pipe 2 goes as ∼ t ^0.84 and for Pipe 3 as ∼ t ^0.95. For the case of the largest diameters, the foam grows as ∼ t ^0.89 (in Pipe 4) and ∼ t ^0.93 (in Pipe 5). In contrast to the case of open top pipes, here the exponents show a clear tendency to increase with the pipe diameter. As shown in Figure 6, the trends for Pipes 3 to 5 are very similar yielding on average a power-law dependence of ∼ t ^6/7. The foam rise velocity as a function of time is displayed in Figure 7. The foam column in all pipes, with the only exception of Pipe 1, grows with a nearly constant velocity. In these cases, n is close to unity and therefore n−1 → 0. However, for Pipe 1, n < 1 and the front velocity decays steepily during the first 30 s and then at a much slower rate until it reaches the maximum height after about 175 s. When the pipe top is closed, the pressure accumulated in its upper parts is correspondingly higher than in the open case. This explains why the rise of the foam columns is much slower in the former case. Also, in this case the bubbles tend to fill the foam volume almost completely. This has been evidenced by the high packing of bubbles which ultimately forms a foam with a well-marked honeycomb structure. The values of Σ listed in Table 2 are in all cases greater than those for the open-top pipes, implying that the growing foams are more stable when formed within closed-top pipes. As in the former experiments, the size of Σ also increases with the pipe diameter.

According to the Laplace equation, the pressure difference between the inside and outside of a gas bubble is given by [12, 21]. Δ p = 4 σ r , (5) where σ denotes surface tension and r is the bubble radius. Therefore, it follows that the smaller the radius, the greater the pressure inside. That is, the inner pressure of small bubbles is higher compared to that of larger bubbles. Hence as a foam collapses, the pressure above it increases. If the collapsing foam is contained within a sealed vessel, the rate of pressure increase inside the vessel will be directly proportional to the rate of foam collapse [12, 13]. In the present experiments foams formed in open-top pipes may collapse after a few hours (typically 3–4 h), while those formed in closed-top pipes were observed to collapse after more than 3 days.

We have performed measurements of the pressure close to the upper end of closed-top vertical pipes using a Dwyer pressure transducer D1635 with a pressure range between 0 and 103.24 kPa and operating temperature between 256.15 and 473.15 K, which is compatible with liquids and gases. For the data acquisition, it was necessary to develop a LabVIEW 2013 program and use a National Instruments SCB-68 data acquisition module. Figure 8 shows the measured pressure field as a function of time for all closed-top pipe diameters of Table 1. The pressure of the compressed air due to foam generation increases as the pipe diameter decreases. The dashed horizontal line marks the value of the atmospheric pressure (p = 77.98 kPa) in Mexico City. For all pipes, with the exception of Pipe 5, the pressure close to the top of the vertical pipes increases steeply during the first 75 s and then remains approximately constant. In particular, for Pipe 5 with the largest diameter the pressure at its top section remains slightly above the atmospheric pressure for all the evolution.

The size distribution of bubbles is an important factor for the stability and utility of foams. In particular, the mean bubble size in a flowing foam can be influenced by a number of factors, including the gas flow rate employed for foam generation, the liquid viscosity, and the foam generator method [22]. When measuring the size of foams there are at least four lengths of interest that characterize the properties of a dry foam, namely a) the length of the whole foam, with typical scales of ∼ 0.1 to 1 m; b) the length between opposite faces of bubbles in the dry part of the foam, with typical sizes of ∼ 1 0 − 4 to 0.1 m; c) the width of Plateau borders (liquid channels), with sizes from ∼ 1 0 − 6 to 10^–4 m; and d) the width of a soap film (bubble wall), with sizes ranging from ∼ 1 0 − 8 to 10^–6 m [23].

Figure 9A shows an amplified photograph of a small foam region inside Pipe 2 with top open. The picture shows that the bubbles that formed have a polyhedral shape and therefore their size cannot be characterized by a simple diameter. Instead their size can be defined by the length between two opposite faces as shown by the lengths l ₁, l ₂, and l ₃. In this case as well as for all other diameters the size of the forming bubbles is not uniform. For comparison Figure 9B shows an amplified view of a small region of the foam formed in Pipe 5 with top closed. In contrast to Figure 9A, the size of the bubbles looks much more uniform. Uniform sized bubbles were also produced for all other closed-top pipe diameters. The dry foam was cut at different pipe cross-sectional planes and the bubble size was measured on each section using 100 micro-photographs. The resulting pictures were analyzed using the commercial softwares AutoCad and ImageJ. The bubble size distribution influences the effective foam viscosity in the sense that it will be higher when the bubble size distribution is relatively homogeneous as in the case shown in Figure 9B [1].

Figure 10 depicts typical cumulative bubble size distributions (CDs) for the case when the top of the vertical pipe is open (Figure 10A, corresponding to Pipe 2) and closed (Figure 10B, corresponding to Pipe 5). In both plots the cumulative size percentage (open circles) and frequency (crosses) are plotted against the bubble size. Here the cumulative size distribution was calculated by dividing the cumulative frequency by the total number of observations ( ≈ 130 for all experiments) and multiplying it by 100. In the case of pipes with top open, the cumulative size percentage is indicative of a wider dispersion in the bubble size distribution compared to the cases when the top of the pipe is closed, where the dispersion looks much narrower. The intersection between the cumulative size percentage and the frequency curves indicates the mean bubble size (vertical dot-dashed line). Similar plots were obtained for all other pipe diameters. The narrower dispersion in the bubble size distributions resulting in the foams produced in pipes with closed tops are indicative of foams with more homogeneous bubble sizes. The bubble sizes in terms of the average length l between opposite faces are listed in Table 3 for both top open and top-closed pipes. As the pipe diameter is increased, the size of the produced bubbles also increases. Therefore it follows from Table 3 that the same is also true for the bubble cross-sectional areas, A _b, and perimeters, P _b. In addition, the bubbles generated tend to have greater cross-sectional areas and perimeters when they are produced in pipes with closed tops. However, this is not entirely true for the resulting lengths l as can be seen from Table 3. In fact only for Pipes 1 and 2 are the bubble lengths l longer in closed-top pipes, while for larger diameters (Pipes 3 to 5) the opposite occurs.

A foam is said to be draining under the influence of gravity when there is liquid flowing along the Plateau borders (or channels intersecting three liquid films), which ultimately accumulates at the bottom and results in foams with non-uniform liquid content, that is, in foams that are wet at the bottom and drier at the top [24]. Even on the scale of a single bubble, the dynamics of foam drainage is intricate because the flow of liquid that converges in a node from different channels merges and then separates into flows through other channels. The situation complicates even more when the gas-liquid interfaces are coated with surfactants to prevent the liquid films from rupture by just keeping their opposite faces from merging [25]. A generalized foam drainage equation describing the spatial and temporal dynamics of the liquid volume fraction was derived by Koehler et al. [26], which takes into account the effects of viscous damping from flow in the nodes as well as in the liquid-carrying Plateau borders. However, the foam drainage equations found in the literature apply mostly to polydisperse foams [27].

For the dry foams formed in the present set of experiments, the flow velocity as measured along a Plateau border shows a marked difference from the traditional drainage model based on Poiseuille-type flow characteristic of channels with rigid gas-liquid interfaces, in which case viscous damping of the liquid flow is dominated by the channels. Figure 11A shows the velocity of a polyamide tracer particle along a Plateau border of the dry foams produced in Pipes 1 (open squares) and 2 (open circles) with top closed as a function of time. Figure 12 shows a photograph of the Plateau border analyzed, corresponding to the case of Pipe 2 in Figure 11A. The photograph captures an instant when the tracer particle (enclosed by the black circle) was flowing across the channel. In both cases the velocity of the tracer particle is rather erratic, showing a zig-zag behavior with time. The zig-zagging dynamics is due to mobile gas-liquid interface across the channels. The amplitude of the oscillations looks a bit larger in Pipe 1 than in Pipe 2. Pipe 3 with a larger diameter displayed on average smaller amplitudes compared to Pipe 2, confirming a dependence of the zig-zag motion on the pipe diameter. For both pipes in Figure 11A, the velocity of the particle zig-zags about a mean value, which is approximately uniform in time. This appears to be more indicative of a plug-like flow across a channel with fully mobile gas-liquid interfaces, in which case viscous dissipation takes place in the nodes where the converging flow is diverted into three different directions [26, 28]. According to Koehler et al. [26] two different drainage regimes can be identified, which are characterized by the power law ɛ ∼ t ^−η , where η > 1 indicates a mobile gas-liquid interface corresponding to a plug-like flow and η < 1 indicates a more rigid interface with a Poiseuille flow through the Plateau borders. However, the value of η for the foams produced in the present experiments could not be determined because it would have required to measure the decay time of the foams, which for the case of the closed-top pipes usually it took more than 3 days. Figure 11B depicts the instantaneous positions of the tracer particle during its passage across the full length of the Plateau border. Independently of the pipe type, the instantaneous positions varies linearly with time as the tracer particle crosses the entire channel.