Elasticity number

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... 

E is one of several elasticity numbers characterizing the stabilizing effect which adsorbed surfactant molecules have on an interface during mass-transfer processes (22). Note that E is inversely proportional to the capillary radius so that the effect of soluble surfactants on the bubble-flow resistance is larger for smaller capillary radii. 

Figure 6. The bubble shape at the front for the elasticity number equal to 0 and 1.

The first term in both Equations 17 and 18 is the constant surface-tension contribution and the second term gives the first-order contribution resulting from the presence of a soluble surfactant with finite sorption kinetics. A linear dependence on the surfactant elasticity number arises because only the first-order term in the regular perturbation expansion has been evaluated. The thin film thickness deviates negatively by only one percent from the constant-tension solution when E = 1, whereas the pressure drop across the bubble is significantly greater than the constant-tension value when E - 1. 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

R E, radius independent surfactant elasticity number, m f f., dimensionless shifted axial coordinate... 

Elasticity number El pv elastic force inertial force Viscoelastic flow... 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

Archimedes number Bingham number Bingham Reynolds number Blake number Bond number Capillary number Cauchy number Cavitation number Dean number Deborah number Drag coefficient Elasticity number Euler number Fanning friction factor Froude number Densometric Froude number Hedstrom number Hodgson number Mach number Newton number Ohnesorge number Peclet number Pipeline parameter... 

The surface tension gradient in the thinning film, which is created by the efflux of liquid from the film and the sweeping of surfactant along the film surfaces to the Plateau borders (Figure 5), can be characterized by the dimensionless elasticity number, Es, which is defined for one surface-active component (21) by... 

Figure 6. Interfacial mobility, or dimensionless drainage velocity, versus dimensionless film thickness, at three values of the dimensionless interfacial elasticity number. (Reproduced with permission from reference 21. Copyright 1991 Butterworth—Heinemann.)...

It is not easy to summarize the nonisothermal results. Generally, at some fixed value of an elasticity number, reduction in Fair/Tmeit leads to a stable region from an unstable one. Physically, the enhanced cooling causes thin regions in the spin line to toughen. 

However, in general the rheological properties of a viscoelastic fluid are shear rate (y) dependent. Thus the various dimensionless parameters are also shear rate dependent. For example, the Elasticity number El is now dependent on the change in shear rate y, which could be written as... 

A smaller channel has a smaller flow characteristic length and time. Thus, Re is smaller and it is difficult to have inertia/viscous flow instability. Conversely, De becomes larger and it is easier to have elastic/viscous instability. The relative dominance of elastic to inertial effects is typified by the Elasticity number, El, i.e., the ratio of fluid elasticity to fluid inertia. El is expressed as... 

In the second limiting case, the surface elasticity roj(-dY/dT5), or more precisely the dimensionless surface elasticity number [To5(-dY/dr5)//Lateral flow along the interface is virtually precluded and effects of surface deflection are important in this inextensible case which might be expected for a concentrated, nearly incompressible monolayer. 

Elasticity Number. A dimensionless quantity characterizing the surface-tension gradient in a thinning foam film. 

FIG. 15 Dimensionless drainage time for the film to drain from a dimensionless thickness, fi, to the thickness, hf, vs. Boussinesq number, at various values of the dimensionless interfacial elasticity number. (From Ref. 110.)... 

At such a large Elasticity number, the inertial effects were neghgible. Thus, the Reynolds number is no longer relevant in describing the flow behaviors. The flow dynamics of these dissimilar viscoelastic fluids were mainly governed by the competition of the dominant viscous and elastic forces in the flow field. As such, the Deborah number would be the dominant governing parameter. They observed experimentally that with increasing... 

For a given value of the temperature gradient the role of the impurity is rather clear. In accordance with the sign of E (solutal Marangoni or elasticity number) there is a dramatic lowering of the threshold for thermoconvective instability or the possibility of overstable modes (oscillations). When all buoyancy phenomena in the bulk are negligible (Ra = Rs = 0) transition to steady convection is expected above the line... 

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