Dimensionless viscosity

From our earliest example, we saw that it was advantageous to use dimensionless variables and that the characteristic quantities should be capable of being held constant. In addition, if a parametric study on the effect of varying some input quantity is to be performed, that quantity should appear in only the distinguished parameter. This is no restriction, for the others are proportional to powers of the distinguished parameter, and the proportionality constants are themselves dimensionless numbers. For example, if the viscosity is to be varied, the Reynolds and the Schmidt numbers are both functions of v, but ReSc is not so, if Sc is chosen as the dimensionless viscosity, Re = Cl Sc, where C = ReSc is independent of v. 

The dimensionless number /3 determines the intensity of the coupling between the energy equation and the momentum balance. With the dimensionless viscosity, and assuming a characteristic pressure of p = fjuR,2/b2 (R2 was chosen as the characteristic r-dimension and b as the characteristic -dimension), the momentum balance can also be written in dimensionless form as... 

For low viscosity liquids when the dimensionless viscosity group [7], NJ, = /LtiAcrp,) is less than 1, a droplet wiU be stable below a maximum size, defined by the critical Weber number and the gas liquid contact time. For long gas—liquid contact times, two large droplets are produced. For short contact times, many smaU droplets are produced. This tjq) of droplet breakup yields a very broad droplet size distribution. 

Dimensionless temperature, T/Tq, dimensionless Viscosity, mass/length time Density, mass/volume... 

In Eq. 7.50 the interfacial viscosity, ry, is expressed in terms of the interfacial shear (subscript Si) and extensional (subscript Ei) components. The plot of emulsion viscosity as a function of the dimensionless viscosity ratio. A, is shown in Fig. 7.11. 

As mentioned in Section 1, the two main parameters determining the full-fluid-film lubrication regime for non-conformal contact are the combined elasticity of the tribopair and the viscosity of the lubricant. These parameters are often made dimensionless for simplification and generalization. For instance, according to Hamrock and Dowson [1], the dimensionless viscosity and elasticity parameters are defined as follows... 

Fig. 8 All MTM ( ) and pin-on-disk ( ) data at different concentrations of aqueous glycerol mixtures, plotted on a lubrication-tegime map, obtained from the Esfahanian-Hamrock-Dowson equations [28] for a circular contact (ellipticity parameter k = 1). The four different regimes in the dimensionless viscosity (gv) versus elastic (gD parameter plot are iso-viscous rigid (IR), iso-viscous elastic (IE), piezo-viscous rigid (VR), and piezo-viscous elastic (VE). All the values reported in this study lie in the iso-viscous elastic regime. It should be noted that while the equations in [28] apply to rolling contact, the model can also be used for sliding geometries at the low speeds used in our pin-on-disk experiments [35]...

Considerable measurements data can be converted into a prediction of the breakup length of threads from completely filled capillaries. The prediction (. pred) depends on the gas-Weber number, the dimensionless viscosity, and flow rate and is given in (22.8), proposed in [33]. 

The measurement of the drop size is compared for different process conditions and capillary configurations. For low gas-Weber number, the drop size increases slightly in all cases. The curves for the completely filled capillaries proceed nearly horizontally up to Weg = 2.3. Afterward, the drop size of fi = 0.9 and F = 1.5 increases and the drop size for fi = 0.33 and V = 3 stays constant. Higher dimensionless viscosities promote the increasing drop size. In case of the threads emerging from open channel flow, a similar trend is identified as the plots in Fig. 22.10 indicate. When comparing drop sizes from completely filled capillaries and open channel flow at similar process condition, it became obvious that the gas-Weber number of sudden drop size growth is considerably lower in case of open channel flow. 

The two photos in Fig. 22.11 illustrate the breakup of threads under the influence of cross-wind flow exemplary for threads emerging from open channel flow. The process conditions are the same except the increased dimensionless viscosity in Fig. 22.1 lb. In Fig. 22.1 la, there are Rayleigh waves on the thread, which grow due to surface tension. The cross-wind has no visible effect on the drop formation and the threads disintegrate into drops of quite uniform size. The higher viscous thread is longer as expected from the breakup length measurement. The thin liquid threads... 

Fig. 22.11 Two threads emerging from open channel flow with different dimensionless viscosities. In (a), the thread disintegrates due to surface-driven axisymmetric or Rayleigh wave breakup. The cross-wind flow leads to a slight increase in drop size, (b) Shows a different breakup mode, as the stronger cross-flow initiates a more stochastic-wind induced breakup...

Fig. 22.12 Span value for different dimensionless viscosities and flow rates depending on the gas-Weber number. The threads emerge from vertically orientated completely filled capillaries. The curves show similar tendency for all process conditions, as the span value increases considerably for low gas-Weber numbers. Afterward, it rises steadily. The increasing disturbance due to cross-wind flow leads to perturbation of the surface-driven breakup process [33]...

The span value for different process conditions is plotted in Fig. 22.12. The results from completely filled vertically oriented capillary are used for comparison at different dimensionless viscosities and flow rates. For thread breakup without cross-wind flow, the span values are very small (0.19 < span < 0.37). The course of the span value vs. gas-Weber number is equal for the given process conditions. For increasing gas-Weber number below Wgg<0.3, the span value increases substantially. Further on, the span value increases with a nearly constant rate [33],... 

For validation of the similarity trials, the drop size during spraying with the LamRot and the thread breakup experiments was compared. The prediction in (22.11) serves as comparison at similar gas-Weber number as well as the dimensionless viscosity and flow rate. In Fig. 22.13, the mean drop size is plotted. The prediction reproduces the mean drop size with sufficient agreement. 

In order to solve Eqs. (11.6) and (11.7), one must specify a relationship between the stress tensor a and the rate-of-deformation tensor d. The most general situation would be many viscoelastic drops suspended within another viscoelastic medium, which is extremely complicated to handle, even when using the most sophisticated computational tools available. Therefore, let us consider a simpler model, the truncated power-law model (see Chapter 6), as schematically shown in Figure 11.35, describing the shear-rate dependence of viscosities, (k) and j()>), of the suspending medium and the drop phase, respectively. The truncated power-law model for the drop and the suspending medium in terms of dimensionless viscosities = t]a y)/rj and... 

IF / / dimensionless elasticity parameter gv =G dimensionless viscosity parameter... 

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