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. 14 Typical flow curves for a concentrated emulsion sheared between two rough surfaces closed circles the line is a fit to the Herschel-Bulkley equation), and between rough and smooth surfaces open circles hydrophilic glass surface open squares hydrophobic polymer surface). 7a denotes the apparent shear rate, and is the value of the apparent shear rate at the yield point (cr = CJy). CJs is the sticking yield stress below which the emulsion adheres to the surface...

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)...

Fig. 19 (a) Shear stress cr from simulations the lines are the best fits to the Herschel-Bulkley equation. The inset shows the elastic inverted triangles with open lower half) and hydrodynamic component inverted triangles with open upper half) of the stress for = 0.85. (b) Collapse of flow curves the line is of the form (26) with m = 0.52 and K = 320... 

Many systems show a dynamic yield value followed by a shear thinning behaviour [9]. The flow curve can be analysed using the Herschel-Bulkley equation ... 

Yield Stress Measurement. The foundations of the rheological treatment to fluids exhibiting a yield stress are due to Bingham (5). Under steady flow conditions, it is common to neglect the contribution from elastic deformation and to use the term Bingham fluid response. Normally, the Herschel-Bulkley equation 9 is used to characterize the flow. 

The power law, Herschel-Bulkley equation, and Casson model are simple and easy to use. However, these equations only work for modeling steady shear flows rather than transient or elongational flows. Thus, many other models have been proposed to fit experimental data more closely for food materials. Among these, it is worth mentioning the Ree-Eyring equation which has three constants... 

With increasing interparticle collisions the probability of formation of floes from dispersed (nonflocculated) particles increases. Thus the horizontal axis can also be interpreted to mean a change from weakly flocculated particles on the left to increasingly flocculated particles toward the right. An outcome is that, irrespective of the degree of interparticle interaction, at low values of cp the viscosity rises slowly, but tends to increase rapidly when particle packing becomes dense.For randomly packed spheres this change occurs at about 95 = 0.60. A simple viscoplastic model is the Herschel-Bulkley equation... 

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 ... 

When Op = 0, equation (4.27) reduces to the power fluid model. The Herschel-Bulkley equation Ats most flow curves with a good correlation coefficient and hence it is the most widely used model. 

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 ... 

Even if the White-Wang model remains open to discussion, the yield stress data that can be derived from experimental flow curves by fitting Equation 5.6 are likely to be similar to values that would be obtained with other models, for instance the Herschel-Bulkley equation. It is consequently interesting to pay attention to yield stress data that were obtained at the Polymer Engineering Institute, using this model. 

If there is no intermediate plateau (corresponding to TI02), one obviously recovers the Herschel-Bulkley equation, i.e. ... 

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