Viscosity infinite shear rate

Casson analysis of shear stress s. shear rate for a linear ramp of shear rate with time yields viscosity jvs. shear rate and the Casson "infinite shear rate" viscosity. 

Zero-shear-rate viscosity Infinite-shear-rate viscosity High shear relative viscosity Intrinsic viscosity, polymer rheology,... 

To numerically illustrate the effect of particle size on the dimensionless groups, we note that in water at standard temperature for a = 100 (im, a potential of 10 mV, and A = 10 °J, we have Pe = 10 y, N5R = 10 y, and = 10 y. Clearly for 100 /um particles, all of the dimensionless groups are very large compared to unity even at a shear rate of 1 s In the high shear limit, non-Newtonian behavior should vanish and the viscosity should attain a stationary value independent of the shear rate. We note here the analogue between the high shear limit for suspensions and the infinite-shear-rate-viscosity limit for polymers discussed above. 

A simple but versatile model that can cope with a yield stress, yet retain the proper extreme of a finite zero and infinite shear-rate viscosity is the Cross model with the exponent set to unity, so... 

V/mm. The infinite shear-rate viscosity, was measured as 0.45 Pa. Reproduced with permission from T. C. Halsey, J.E. Martin, and D. Adolf Phys. Rev. Lett., 68(1992)1519. 

Carreau (1972) llo-tloo r = infinite shear rate viscosity whole range ri/y fit for many... 

In the above equations, t]o is the zero shear rate viscosity, t)oo is the infinite shear rate viscosity, Xc is a time constant, and n is again the power-law index. The magnitude y of the rate-of-strain tensor is given by... 

The square root of viscosity is plotted against the reciprocal of the square root of shear rate (Fig. 3). The square of the slope is Tq, the yield stress the square of the intercept is, the viscosity at infinite shear rate. No material actually experiences an infinite shear rate, but is a good representation of the condition where all rheological stmcture has been broken down. The Casson yield stress Tq is somewhat different from the yield stress discussed earlier in that there may or may not be an intercept on the shear stress—shear rate curve for the material. If there is an intercept, then the Casson yield stress is quite close to that value. If there is no intercept, but the material is shear thinning, a Casson plot gives a value for Tq that is indicative of the degree of shear thinning. 

For a shear-thickening fluid the same arguments can be applied, with the apparent viscosity rising from zero at zero shear rate to infinity at infinite shear rate, on application of the power law model. However, shear-thickening is generally observed over very much narrower ranges of shear rate and it is difficult to generalise on the type of curve which will be obtained in practice. 

Thus, the apparent viscosity falls from infinity at zero shear rate ( / r T Ry) to pp at infinite shear rate, i.e. the fluid shows shear-thinning characteristics. 

It was pointed out in Section 1.12 that the coefficient of rigidity /3 is equal to the apparent viscosity at infinite shear rate. Govier also defined a dimensionless yield number Y by... 

The shear rates in the heat exchanger are believed to have been much higher than the maximum shear rates obtained in the viscometer hence constancy of the differential viscosities at these higher shear rates is indicated. They should not, however, be considered equal to the apparent viscosities at infinite shear rate, as they were termed by Miller, as no data are available to support such a statement. 

Apparent viscosity of a non-Newtonian fluid at some specified shear rate, lb.M/(sec.)(ft.) or lb.ii/(hr.)(ft.). mo and m refer to the apparent viscosities of non-Newtonian fluids at zero and infinite shear rates, respectively... 

The Casson plastic viscosity can be used as the infinite shear viscosity, t]oo, (Metz et al., 1979) of dispersions by considering the limiting viscosity at infinite shear rate ... 

Here, the constants, K (mPa s ) and n (dimensionless), are the consistency index and the exponent, respectively Up is the Darcy velocity (m/s) of the polymer-containing water phase k is the average permeability in m is the water phase relative permeability S is water satnration (fraction) is porosity (fraction) is the viscosity at infinite shear rate and C is an empirical constant. Note that Eq. 5.23 is made more general by including the nonunit water saturation, Sw, and the water relative permeability, k,w, as was done previously by Hirasaki and Pope (1974). To consider the polymer permeability reduction fector Ffa explicitly (to be discussed later), we should divide the permeability k by Ekr, and n, is substitnted for Up. Then Eq. 5.23 becomes... 

In Figure 3.28, the shear stress is shown as a function of shear rate for a typical shear-thinning fluid, using logarithmic coordinates. Over the shear rate range (ca 10 to lO-" s the fluid behaviour is described by the power-law equation with an index n of 0.6, that is the line CD has a slope of 0.6. If the power-law were followed at all shear rates, the extrapolated line CCDD would be applicable. Figure 3,29 shows the corresponding values of apparent viscosity and the line CCDD has a slope of n — I = —0.4. It is seen that it extrapolates to jXa = oo at zero shear rate and to /Zq = 0 at infinite shear rate. 

The four parameters of this model, namely, rjo, 7oo, 2, and n, are the apparent viscosity at zero shear rate, apparent viscosity at infinite shear rate, time shear relaxation constant, and the exponential index, respectively. The parameter 1 has a unit of time and can assume any value in the range (0, oo). The index n is dimensionless, with 0 < n < 1. Equation 8 is graphically represented in Fig. 1 for illustration. Various parameters assumed for obtaining the plots appearing in Fig. 1 are as follows rjo = 900, rj o = 0.1, and the nondimensional time shear relaxation... 

Limiting viscosity at infinite shear rate, i.e., at the lower Newtonian plateau... 

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