Characteristic shear rate

Here r is the center-to-center distance, and m is a unit vector along the line joining the centers of the two fragments, so that r = rm. Distances are made dimensionless with respect to the radius of a fragment (a), shear rates and time with respect to the characteristic shear rate y = V2D D, and Fe = FC with respect to H a. These equations are identical in form to... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

Fig. 8.14. Product of reduced compliance eR and characteristic shear rate po for narrow distribution systems of linear polymers. The dashed lines indicate the range of / 0JeR for...

Characteristic shear rate locating the onset of shear rate dependence in the viscosity. 

This characteristic shear rate is equal to the reciprocal average entanglement time Tn at equilibrium. The solid curve is from Graessley s model, which reads... 

A characteristic shear rate /cha depends on the mean specific power input e ... 

Based on the characteristic shear rate, the Deborah number can be expressed as... 

The characteristic shear rate may be expressed as 7char = ( )/( c/2) = 2v/dc, where A is the relaxation time of the viscoelastic fluid measured in shear. 

In the commercial formulations the presence of particles at =0.20 merely increases the level of the viscosity and shifts the characteristic shear rates and frequencies. Jenkins data (48) conforms at 0.10, but at higher volume fractions the Newtonian plateau disappears. This raises the question of whether a reversible polymer network persists or the particles simply interact as fuzzy and, perhaps, sticky spheres with much slower dynamics that control the shear rate. Additional (kta exists beyond that cited here but d s not seem to resolve this issue. 

It is customary to account for the non-Newtonian fluid behavior by introducing the so called effective viscosity to define various dimensionless groups. Unlike its constant value for Newtonian liquids, the effective viscosity of non-Newtonian pseudoplastic type fluids depends upon the operating conditions (e.g., gas and liquid velocities) as well as on the geometrical details of the system. Indeed, the lack of a rationed definition of the apparent viscosity or characteristic shear rate appears to be the main impediment in extending the well established predictive correlations for Newtonian media to non-Newtonian media. When we develop correlations for design parameters in bubble columns with non-Newtonian media in an analogous manner to the case of Newtonian media, Newtonian viscosity p is simply replaced by an apparent viscosity for non-Newtonian media. 

This relation indicates that the apparent viscosity varies depending on the shear rate y. unlike the Newtonian viscosity. Therefore, the definition of the appropriate shear rate characterizing hydrodynamics is required to estimate the effective apparent viscosity for non-Newtonian fluids in bubble columns. In bubble columns, the shear rate is not uniform and unknown. The motion of bubbles results in the wide variation of the shear rate, which is hopelessly complicated and cannot be analyzed. Therefore, the characteristic shear rate should be evaluated on the basis of a simplified physical picture of hydrodynamics in bubble columns and aided by experimental observations. 

A relation for the effective characteristic shear rate ( y ) proposed by Nishikawa et al. has been commonly used in the literature [7,8]. 

It should be emphasized here that in the derivation of theoretical correlations for design parameters the introduction of the appeu ent viscosity concept is not necessary, and, as a result, the use of questionable definitions of characteristic shear rate such as Equation 3 is not required, as discussed in the ensuing sections. 

Another important dimensionless group, the Weissenberg number (White 1964), is deflned as the product of the characteristic time of the fluid and the characteristic shear rate of the flow, i.e.. 

The Weissenberg number compares the elastic forces to the viscous effects. It is usually used in steady flows. One can have a flow with a small Wi number and a large De number, and vice versa. Sometimes the characteristic time of the flow in the deflnition of the Deborah number has been taken to be the reciprocal of a characteristic shear rate of the flow in these cases, the Deborah number and the Weissenberg number have the same definition. Pipkin s diagram (see Fig. 3.9 in Tanner 2000) classifies shearing flow behavior in terms of De and Wi, and provides a useful guide for the choice of constitutive equations. 

These equations can be solved to predict the pressure drop, AP, as a function of the volumetric flow rate, Q, if the viscosity is known as a function of pressure, temperature and shear rate. Two rheological models are employed in this study for the functional relationship between viscosity euid shear rate. In the data analysis procedure, calculations are performed for Newtonian fluids in which the viscosity is Independent of shear rate. In the consistency check procedure, the experimental results are correlated with a truncated power-law model in which the fluid behaves as a Newtonian fluid below a characteristic shear rate, Yq, and exhibits shear-thinning behavior above this shear rate ... 

The role of bubble size is much more involved in viscoelastic fluids than that in Newtonian media. It is customary to use the bubble radius (equivalent volume for nonspherical bubbles) as the characteristic linear dimension in the scaling of the governing equations and the pertinent boundary conditions. The characteristic shear rate associated with a rising bubble (low Reynolds number region) is 0 V/R) and, hence, the radius enters in the calculations of the magnitudes of the viscous and elastic forces in addition to the linear dimension in the Reynolds and Weber numbers. It was conceivable that under appropriate conditions, that is, levels of surface and viscoelastic forces, the no-slip-no-shear changeover may occur as a smooth transition as reported by De Kee et al. (1990) or as an abrupt jump as suggested by the qualitative analysis of Leal et al. (1971). 

Some generally useful facts are known about these properties for melts and concentrated solutions of non-associating polymers. Both t](y) and 0i(y) decrease with increasing shear rate, and both begin to depart from t]o and 2J°rjl near the same shear rate fo- Moreover, this characteristic shear rate is closely related to the characteristic time of the liquid ... 

Many authors have described structural transitions in dilute lamellar phases under the influence of shear. For example, Roux and co-workers (19) studied different dilute lamellar phases which were stabilized by undulation forces and contained flat bilayers with defects, at rest. With increasing shear rates, these bilayers undergo a transition into relatively monodisperse multilamellar vesicles above a characteristic shear rate. The size of the formed vesicles is indirectly proportional to the shear rate. Beyond a second characteristic shear rate, the vesicles are again transformed into flat oriented bilayers. These results were explained in terms of a balance between shear stress and elastic forces which come from the bending and the Gaussian moduli of the bilayers. The same authors observed a similar sequence with increasing shear rate for other lamellar phases. It was found... 

In shear measurements one expects the described solutions behave like normal Newtonian aqueous solutions. This is in fact the case for small shear rates (Fig. 11.32). In Fig. 11.32 the shear viscosity, which was measured in a capillary viscometer, is plotted vs.the shear rate. One observes a sudden rise of the viscosity at a characteristic shear rate and for y> % the solutions show some shear thickening behaviour. Obviously something dramatic has happened to the micelles in the solutions. Some conclusions about what has happened can be drawn from flow birefringence measurements. Some typical results of flow measurements from a Couette system are shown in Fig. 11.33. We note a sudden increase of the flow birefringence at a critical shear rate. For y < yc no flow birefringence could be detected. 

Thus, each point on the viscosity curve rj y) corresponds to a value ofM. Figure 8.1 illustrates this. The continuous lines are viscosity curves for several of the hypothetical monodisperse polymers with molecular weights within the range of the polydisperse material, whose viscosity curve is shown by a dashed line. Each monodisperse material has a characteristic shear rate y M) and a zero-shear viscosity t]q M) that correspond to the molecular weight M. Thus, each shear rate between the Newtonian plateau and power law region of the polydisperse material will be the critical shear rate y M) of one of the monodisperse materials of which it is comprised. 

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