Bretherton theory

By asserting that the film thickness remains proportional to the 2/3 power of the capillary number, they establish that the dynamic pressure drop for surfactant-laden bubbles also varies with the capillary number to the 2/3 power but with an unknown constant of proportionality. New pressure-drop data for a 1 wt% commercial surfactant, sodium dodecyl benzene sulfonate (Siponate DS-10), in water, after correction for the liquid indices between the bubbles, confirmed the 2/3 power dependence on Ca and revealed significant increases over the Bretherton theory due to the soluble surfactant. 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

Figure 8. Experimental data of the dimensionless pressure drop per bubble as a function of capillary number for 1 and 2 mm diameter glass capillaries. The solid line denoted by E - 0 gives the theory of Bretherton.

The theory of Bretherton agrees nicely with experimental data for low Ca. For other geometries, the values oftheproportionaUty constant changes [97-99,101,152-154], but the scaling rules remain valid. 

As the Re number approaches 1, Equation 9.22 follows Bretherton s theory. In a rectangular channel, the film thickness at the lateral walls differs from that above and below the bubble [67,68], and these, in turn, differ from the diagonal film thickness in the channel corners. At small capillary numbers, the liquid film formed on the channel center becomes very thin. However, as capillary number increases, the interface shape becomes axisymmetric. As the Reynolds number increases, transition from nonaxisymmetric to axisymmetric flow pattern occurs. Based on their experimental work in capillaries of 0.3, 0.5, and 1.0 mm, Han and Shikazono [69] provided a correlation which calculates the film thickness in microchannels with a square cross section with an accuracy within 5% ... 

The Uquid fihn thickness surrounding the Taylor bubble in a microchannel can be precisely predicted with a number of correlations derived from Bretherton s theory. In the closed micro-channels, the pressure drop in Taylor flow can be correctly estimated by the Wamier model and in the annular regime by the Lockhart-Martinelli model. A number of correlations for the C-factor are available for the Lockhart-MartinelU model... 

The theoretical difficulties in calculating the lift force are significant. It was proved by Bretherton that no lift force is exerted on a body of revolution by a parallel flow if we remain within the scope of the theory of creeping flows (removing nonlinear terms from the Navier-Stokes equations). It is therefore necessary to take into account the inertial (nonlinear) terms. This difficulty had been identified in 1910 by Oseen, who had indicated that the nonlinear terms cannot be... 

Figure 5.49 shows the comparison of film thickness from Bertherton s above equation with the experimental data of Taylor (1961) and Bretherton (1961). There is a good agreement between the theory and experimental data for 10 capillary number may be attributed to (a) Marangoni... 

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