Yield stress flow equations with

Some flow equations with yield stresses... 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Fluids that show viscosity variations with shear rates are called non-Newtonian fluids. Depending on how the shear stress varies with the shear rate, they are categorized into pseudoplastic, dilatant, and Bingham plastic fluids (Figure 2.2). The viscosity of pseudoplastic fluids decreases with increasing shear rate, whereas dilatant fluids show an increase in viscosity with shear rate. Bingham plastic fluids do not flow until a threshold stress called the yield stress is applied, after which the shear stress increases linearly with the shear rate. In general, the shear stress r can be represented by Equation 2.6 ... 

A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient —AP/l and volumetric flowrate Q is ... 

The flow curves for fluids with a yield stress are often fit by the constitutive equation of a Bingham plastic ... 

The apparent yield stress. Ihe complex viscosity n vs. oi for PP blends with LLDPE-B and LLDPE-C Is shown In Fig. 26. The plot clearly Indicates possible yield stress behavior especially for blends containing 50% PP. Ihe apparent yield stress In dynamic flow data was calculated using Equation 23, with F G or F G". The yield stress values as well as the assumed matrix material for calculating F are listed In Table V. For both systems the maximum value of the apparent yield stress occurred at 50% PP. In fact, there Is a direct correlation - In a given system the yielding Is primarily observed In blends having a co-contlnuous structure. As before (53 ) Gy > Gy... 

The pseudoplastic fluids do not show yield stress value. Their apparent viscosity decreases with the shear rate. The flow curve reveals linear character at very high shear rate. The logarithmic plot of shear rate as a function of shear stress of these fluids is often a straight line with the slope between 0 and 1. For the pseudoplastic behaviour description is hence frequently used the power-law equation ... 

It has been found that in the case of viscometers with co-axial cylinders, the flow is not homogenous if the distance between cylinders is too wide [1, 18]. The maximum distance should be not higher than 1 mm [ 1]. At higher distance the flow curves correspond to the pseudoplastic fluids with no clearly marked yield stress value. For low shear rate the flow of paste in the gap between the cylinders is not uniform as long as the stress does not exceed the yield stress value this can be derived from the Reiner-Rivhn equation ... 

This is the simplest equation describing the flow behaviom of a fluid with a yield stress and, in steady one dimensional shear, it is written as ... 

An examination of equation (2.14) shows that for any fluid with a finite yield point, the versus Xb curve approaches the Xb axis at zero slope, due to the requirement for such a system that the shear rate must become zero at finite Xb. This may lead to apparent shear-thinning characteristics being ascribed to systems, irrespective of the actual form of their flow curves above the yield point, i.e., whether Bingham plastic, shear-thickening (with a yield stress), or shear-thinning (with a yield stress). 

It is impossible to generalize the flow behavior of PP blends. The multitude of their types makes such an attempt impractical. However, the flow curves are expected to show a pseudoplastic behavior (equation (1)), with yield stress at higher loadings. When to a polymer-1, a polymer-2 is added, up to the percolation threshold volume fractions, ( ) 0.16, it forms a dispersed phase. When the concentration exceeds this limit, < ) > phase inversion concentration, 4> = <)>, where the distinction between the matrix and the dispersed phase disappears [6]. 

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