Dimensionless numbers Capillary number

There is a number of theoretical and experimental relations determining the dependence of the dynamic contact angle on flow velocity (Dussan 1979 Ngan and Dussan 1982 Cox 1986 Blake 1994 Kistler 1993). Hoffman (1975) expressed the dynamic contact angle as a function solely of dimensionless parameters capillary number Ca... 

Viscous and inertial forces are related to surface tension by the dimensionless Capillary and Weber numbers. Capillary number (Co), as shown in Equation 4.3, describes the relative importance ofviscosity and surface tension, where p represents the viscosity, u is the velocity and a is the surface tension. 

The fact that boiling heat transfer at the microscale is governed by the evaporation, hydrodynamics, capillary effects of the thin liquid film, has led to the introduction of some relevant dimensionless numbers. These numbers are used by many authors to analyse the behaviour of boiling heat transfer at the microscale (see Table 2). 

CCT, critical cracking thickness Boltzmann constant (1.381x10 local permeability [m ] fracture resistance [N m ] average permeability in/of compact [m ] particle shape factor compact thickness [m] initial particle number concentration [m refractive index of particle material refractive index of dispersion material number density of ion i dimensionless number dimensionless number Stokes number Peclet number capillary pressure [N-m ] dynamic pressure [N m ] local liquid pressure in the compact [N-m local solid pressure in the compact [N-m ] superficial fluid velocity [m-s q gas constant [J K ] centre to centre distance [m]... 

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 -... 

For other packings and for the case in which static holdup is changed by gas flowing through the bed, the method of Dombrowsld and Brownell [Jnd. Eng. Chem., 46, 1207 (1954)], which correlates static holdup with a dimensionless capillary number, should be used. 

We next determine the residual saturation, m , by first calculating the dimensionless capillary number ... 

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.

Another measure of wetting character in the field is the dimensionless capillary number N, which is the ratio of viscous to capillary forces. One expression for this number is (24)... 

Combining these dimensionless along with ng forms the well known capillary number as... 

Since the capillary number dominates the dispersion of the fluids, that dimensionless group must also be maintained constant. Since the rotational speed and the diameter have already been dealt with, the only remaining parameter in Ca is the channel depth, which must be maintained constant. Hence... 

Favelukis et al. (37,38) dealt with the problem of droplet deformation in exten-sional flow with both Newtonian and non-Newtonian Power Law model fluids, as wellas bubble breakup. For the Newtonian case, they find that as an inviscid droplet (or bubble) deforms, the dimensionless surface area is proportional to the capillary number... 

Vapor capacity factor, or C-factor, defined by Eq. (8.28), ft/e Capillary number of the liquid, Eq. (9.22), dimensionless Vapor flow rate, fta/s... 

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] ... 

Ca Capillary number Dimensionless Dimensionless N Number of data Dimensionless Dimensionless... 

The interplay between capillary and viscous forces determines the location of the two phases and the flow mechanism. Hence, their ratio, the capillary number Ca, is an important dimensionless parameter influencing flow. In this work it is defined simply as ... 

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

Figure 9.11 Dimensionless droplet size (capillary number) as a function of viscosity ratio M for various blends of 15 wt% ethylene-propylene rubbers in nylon or polyester produced by ex-trusion at an effective shear rate of around 100 sec The components are shear thinning, and the viscosity ratio M is evaluated at the effective shear rate. (From Wu 1987, reprinted with permission from the Society of Plastics Engineers.)...

Figure 9.33 Dimensionless shear stress a ill2T as a function of strain Y at zero capillary number for a hexagonal model foam, with initial orienta-tion (a) as shown in Fig. 9-32, and (b) rotated by jr/12. Here F is the surface tension and is the length of a side of the hexagonal foam. (From Kraynik and Hansen 1986, with permission from the Journal of Rheology.

In segmented flow in cylindrical capillaries, Taylor bubbles consist of a cylindrical part and two caps at the front and the rear menisci. The form of the caps may be axisymmetric or nonaxisymmetric. In a balance over a long inviscid bubble surrounded by a moving incompressible and viscous fluid, capillary, viscous, inertial, and gravity forces are taken into account. The latter three, relative to capillary force, are expressed in the following dimensionless numbers ... 

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