Taylor capillary number

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

To quantify the increase of a due to pressure, a mean bubble diameter has been estimated using Taylor s stability theory [7] on bubble deformation and break-up in sheared emulsions. According to this theory, bubble size in a sheared emulsion results from a balance between viscosity and surface tension forces. The dimensionless number that describes the ratio of these forces is called the capillary number Q. For large bubble deformations, the maximum stable bubble diameter in a shear flow is expressed as [8] ... 

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres ... 

The capillary pressure will have an effect only at high capillary numbers when the curvatures of the front and rear ends of the Taylor bubble are not symmetrical. At low velocity and in narrow channels, the frictional pressure drop is viscosity-dominant and can be calculated using the Hagen-Poisseuille equation... 

Figure 3. A generalized capillary number correlation, (Courtesy of K. Taylor, Petroleum Recovery Institute, Calgary.)...

The study of emulsion rheology was pioneered by Geoffr Taylor (1,2), who not only experimentally identified e dimensionless groups (capillary number and viscosity ratio) that control droplet deformation in an emulsion in simple shear and hyperboUc flow fields, but also proposed a linear theory for droplet deformation in flow. The droplet Cs illary number is defined as Ca f -... 

Qualitative sketches of the flow streamlines in the hquid slug ahead of the bubble have been presented by Taylor [3] (see Fig. 2). These were related to the capillary number, Ca (Ca = where p is the liquid viscosity and U, is the bubble velocity), and to the dimensionless number m, that gives the relative velocity between the bubble and the liquid ... 

The thickness of the wall film in Taylor flow in capillaries is mainly dependent on the ratio of viscous to interfacial forces, which is given by the capillary number, Ca. 

Taylor and Acrivos (1964) found an approximate expression for applicable for small valnes of the Reynolds nnmbCT (= l/ap /fi ) and capillary number (= U]4gly). They first obtained the creeping flow solution that satisfied all boundary conditions, those at the drop-flnid intaface being satisfied at r = a. The normal stress balance (Equation 7.19), which was not used in this initial proce-dnre, was then applied, with evaluated at r = a to obtain a first approximation... 

K capillary (or Taylor) number Kcrit critical capillary number A distortion wavelength X. viscosity ratio... 

Figure 8. Correlation between residual oil saturation reduction and the capillary number. (From Taylor and Hawkins [135]. Copyright 1990 Petroleum Recovery Institute, Calgary, AB.)...

In cylindrical capillaries where the effects of gravity can be neglected, the liquid film around the bubble has a constant thickness, which increases with the capillary number. Under inertia-dominated flow conditions, the liquid film thickness decreases and then increases with increasing Reynolds number [65, 66]. Han and Shikazono [66] studied hydrodynamics of Taylor flow in circular tubes with different diameters of 0.3, 0.5, 0.7, 1.0, and 1.3 mm, and they proposed empirical correlations for the dimensionless film thickness for Re < 2000 ... 

Taylor [64] found that in simple shear flow, a dispersed drop with viscosity ratio p = 1 breaks up when the Ca > 0.5. Breakup seems to occur when the shear stress and the interfacial stress are of the same order of magnitude. The critical Capillary number depends on the type of flow and on the viscosity ratio. In the mixing process two regimes can typically be distinguished ... 

Figure 3.13 Critical capillary number as a function of viscosity ratio in Newtonian systems. Reproduced with permission from [79] 1982, Taylor. Francis.

To the lowest order approximation, necessary conditions for the onset of instability are obtained in the form of Taylor expansion in the capillary number. For instance, using the Marango-ni number the motionless steady state of the fluid layer is stable provided that its value remains below... 

The pioneering work of Taylor on drop break-up in Newtonian systems in a simple shear field has been the basis of investigations on more complex systems and flow fields [40, 41]. The analysis of Taylor considers how the balance of applied shear forces and counteracting interfacial forces affects drop dimensions and stability. The results have been expressed in terms of the so-called capillary number,... 

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... 

Figure 5.49 Comparison of film thickness (S/d) between theoretical prediction of Bertherton and experimental results of Taylor and Bretherton as a function of capillary number...

Theory The release under shear of an active molecule that is initially encapsulated in the aqueous phase of a W/OAV multiple emulsion is a very promising phenomenon for applications in cosmetics or pharmaceuticals. Taylor (Taylor, 1932, 1934) was first to study the deformation of molecules under shear and their bursting in a simple, dilute emulsion. He considered that breakup occurred when shear stress exceeds cohesion stress. He defined this breakup by way of a capillary number, Ca ... 

As was expected, the distributions were observed to shift toward the lower diameter as the shear was increased. It was possible to compare quantitatively the experimental diameters 43 (diameter moment/volume) of the globules after shear with the theoretical diameters determined from the expression of the capillary number given previously. The correlation between experiment and theory was satisfactory. So we could conclude that the Taylor theoretical model does apply to the breakup of the multiple emulsions globules, at least at the first approximation. 

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