Shear viscosity Herschel-Bulkley equation

Thus, equation 3.127, which includes three parameters, is effectively a combination of equations 3.121 and 3.125. It is sometimes called the generalised Bingham equation or Herschel -Bulkley equation, and the fluids are sometimes referred to as having/n/re body. Figures 3.30 and 3.31 show shear stress and apparent viscosity, respectively, for Bingham plastic and false body fluids, using linear coordinates. 

Fig. 17 Flow curves measured at steady state for microgel pastes (a) and concentrated emulsions of silicone oil in water (b). In (a) the data for microgel pastes are presented for varying particle concentration (wt%), crosslink density, salt concentration, and solvent viscosity. Symbols are the same as used in (c). Rl and R5 refer to two different crosslink densities, A x = 128 and A x = 28, where A x is the average number of monomers between two crosslinks. In (b), data for emulsions are presented for varying packing fractions. Symbols are the same as used in (d). The solid lines in (a) and (b) are the best fits to the Herschel-Bulkley equation. Plots (c) and (d) show collapse of the different data sets when the shear stress is scaled by <7y and the shear rate by tis/Gq. The equations of the solid lines in (c) and (d) are of the form (26), where m = 0.47 and K = 280 for microgel pastes (c) and m = 0.50 and K = 160 for emulsions (d)...

Experiments show that in steadily sheared foams and concentrated emulsions, the viscosity coefficient n depends on the rate of shear strain, and in most cases the Herschel-Bulkley equation [931] is applicable ... 

For a non-Newtonian system, as is the case with most food colloids, the stress-shear rate gives a pseudoplastic curve and the system is shear thinning, i.e. the viscosity decreases with increasing sheeu rate. In most cases the shear stress-shear rate curve can be fitted with the Herschel-Bulkley equation. 

Flow behavior was modeled through Ostwald and Herschel-Bulkley equations. Experimental data fit adequately to the first one, showing no tq as well as apparent viscosity (r]a) (shear rate 20 s ) values ranging from 0.018 to 0.027 Pa.s, trend that showed a weak thickening effect of these fractions. Aqueous systems showed pseudoplastic behavior with values of exponential index of 0.85 and 0.73 for R 2-2 and R 2-3, respectively (Table 6). 

In other terms, above a critical shear stress, it flows as a Newtonian fluid of (constant) viscosity t). It follows that a fluid obeying the Herschel-Bulkley model is sometimes called a generalized Bingham fluid, since with n=1 and K=r in Equation 5.3, one obviously obtains Equation 5.4. The three fit parameters of the Herschel-Bulkley equation can be reduced to two, when considering that n=0.5. This was in fact the approach used by Casson in proposing the following model ... 

Using a critical shear rate rather than shear stress as a yield criteria makes application to numerical calculations much easier (Beverly and Tanner, 1989). Equation 2.5.6 with stress yield criteria (eq. 2.5.3) is known as Herschel-Bulkley model (Herschel and Bulkley, 1926 Bird et al., 1982). From Figure 2.5.5 we see that the two-viscosity models will better describe the iron oxide suspension data illustrated here. 

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