Bubble displacement

Fig. 6.26 Velocity of bubble displacement. C/ls = 0.046 m/s, q = 80kW/m. Reprinted from Hetsroni et al. (2003) with permission...

The question then arises as to how foam moves through porous media. Foam injection and displacement is achieved in both the field and the laboratory. Obviously, a displacement mechanism other than single bubble displacement is operative. 

The above discussion is concerned with single bubble displacement. To obtain results analogous to Equations 6, 8, and 9 for bubble trains, it is necessary to account for changes in curvature at the bubble ends (in the Plateau border regions) due to the compression between adjacent bubbles. Referring to Figure 5(b), the contact radius Rc can be related to the capillary pressure Pc = Pi - Pi = P - P by... 

The above results are based upon the pressure drop-flow rate relations of Equations 16 and 17. Experimental studies for bubble displacement in round and square tubes (35) suggest a generalization of these relations. In particular, relations of the form... 

If the time difference between successive frames of the same bubble is known and the bubble displacement can be measured, then bubble rise velocity can also be measured. However, visual methods are limited to systems with optical access, so observations are limited to regions near the wall even under moderate gassing rates in gas-liquid systems. The wall and liquid must also be transparent. 

Collecting the gas involves careful co-ordination with the heating process. Have at least three water-filled test tubes lying in the trough, with corks nearby. When the gas needs to be collected, use one hand to hold the tube above the slit in the valve. The gas bubbles displace water from the tube. Try to keep an eye on the heating at the same time, but if necessary, stop heating and move the Bunsen away from the apparatus. Cork each tube as it fills with gas. It can be hard at first to tell which gas is air and which are alkanes/alkenes, so if in doubt have more tubes ready for gas collection rather than fewer. 

At the beginning of the 18th century S. Hales, an English scientist, invented a pneumatic bath. In this apparatus the vessel where a gas was formed (a retort with a reaction mixture) was separated from the collector for the liberated gas. The collector was a flask which was turned upside down and filled with water. Penetrating into the flask, the gas bubbles displaced water and the flask became filled with the gas under study. 

Initiate pumping at a slow flow rate (100 mL/min). Avoid surging. Observe air bubbles displaced from sample tube to assess progress of steady pumping until water arrives at the surface. 

The apparatus used in our laboratory to measure water vapor permeability is schematically shown in Figure 10-4 (Tohge, 1996 Tadanaga, 1996) as an example. The sample to be evaluated is placed between two parts of the cell and clamped. The temperature of the sample can be controlled by the thermostat, or an electronic heater. Both sides of the sample in the permeation cell are evacuated to 10 Torr and the sample is degassed in the permeation apparatus for several hours. A saturated water vapor of at 0°C (4.58 mmHg) is then introduced to the upstream side and the pressure change on the downstream side due to the permeated water vapor is measured with a pressure meter. The water permeation coefficient (P) is evaluated from the increasing rate of the pressure (dp/dt) at a steady state and normalized with the thickness of the sample. When the vacuum line is not used, a test gas is supplied to the upstream side of the sample. The pressure difference between the upstream side and down stream side is determined and the flow rate is also measured, by, for example, soap-bubble displacement (Klein, 1990). 

Our goal in the present section is to explore the energetics of quasistatic deformations. We will relate the energy of deformation, and hence the shear modulus, to the highly inhomogeneous spatial character of the bubbles displacement field (Figure 12.12). Later, we will build on these observations to describe viscoelastic response in the bubble model, which includes driving at finite frequency. 

The shear modulus relates stress and strain. These are macroscopic quantities used in coarse-grained or continuum descriptions of materials. The bubble model takes a mesoscopic perspective, viewing the material as a collection of individual bubbles the shear modulus is related to the bubbles displacements via Equation 12.15. To develop this relationship, the elastic energy stored by the deformation. 

Consider the bubble model under oscillatory driving (at finite frequency). When foams and emulsions are driven at a sufficiently high rate, the quasistatic limit breaks down. In qualitative terms, this happens when viscous forces are large enough that bubbles displacements no longer minimize the elastic potential energy AU. (Figure 12.15). 

In the low-frequency limit, the bubble displacements are, to very good approximation, the same as under quasistatic driving, with the magnitude of the typical... 

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