Shear rate/stress

Important for polymer processing is the fact that when the concentration of a hard filler is increased in the composite, the unsteady flow (in the sense of large-scale distortions) of the extrudate occurs at higher shear rates (stresses) than in the case of the base polymer [200, 201,206]. Moreover, the whirling of the melt flow is even suppressed by small additions of filler [207]. 

In this investigation, a set of six experimental heatset lithographic inks were subjected to a variety of rheological measurements both as a dry ink, as well as after being emulsified with a commercial fountain solution. Determinations of apparent viscosity at 2500 secs l and yield stress at 2.5 secs l were made 25°C from 5 point shear rate/stress curves, and inkometer tacks at 1200 RPM/90°F were also measured. 

With regard to thermal noise, only weak indications of an increased noise level when increasing the shear rate were found. These eflFects will thus not be discussed here. When recording the current noise, however, the noise level increased significantly with the shear rate (stress). Figure 6 provides a typical illustration (PEO solution in water). 

Fig. 3.3-36 Poly(oxymethylene), POM-H viscosity ver- Fig. 3.3-37 Poly(oxymethylene-co-ethylene), POM-R sus shear rate stress versus strain...

Fig. 3.3-t. 5 Polycarbonate, PC viscosity versus shear rate Stress a (MPa)... 

A well-known feature of yield stress fluids is the appearance of a low-shear-rate stress plateau which marks the yield stress. 

The major component of the synovial fluid, that is, hyaluronan, has two main functions namely lubricant and shock impulse damping [25]. Furthermore, hyaluronic acid is a major component of the extracellular matrix of the connective tissue. This kind of polymer should exhibit viscoelastic properties that directly depend on its microstructure and external parameters such as shear rate, stress and temperature. Knowledge of dependence of model synovial fluid viscosity on the shear rate, stress and temperature is very useful for biomedical applications, for example, in treatment of joint diseases. 

There are limited measurements of the viscoelastic properties of hard-sphere colloids at elevated shear rates, stresses, and frequencies, de Kruif, et al. report rir of silica spheres in the small and large shear-rate limits, as functions of (53). At smaller concentrations, rjr depends but weakly on /c. Above

shear thinning becomes apparent, de Kruif, et al. propose that r]r diverges 2 — 4>/4>m), with 4>m being 0.71 or 0.63 in the large and small shear limits. Jones, et al. also find weak shear thinning for (p > 0.395, the extent of shear thinning increases quite substantially for volume fractions between 0.59 and 0.60(55). 

Note that there are many (complex) fluids that do not exhibit a Newtonian plateau at low shear rate (stress) and whose shear viscosity function feeds the controversy on the existence of a yield stress. As noted by Barnes, when the flow is so slow than ages are necessary to detect it, at least one could consider that the yield stress is an engineering reality (H. Barnes. The yield stress—a review or "jtotvra pei" —everything flows /. Non-Newtonian Fluid Mech., 81,133-178,1999.)... 

Rios et al, (2007a) investigated the rheology of activated sludge from the aerobic zone of a lab-scale side-stream MBR with MLSS concentration of 8 g/L.The sludge was concentrated or diluted to obtain 11 different concentrations and correlated with shear rate-stress. The Herschel-Buckley rheology model with Papanastasiou s (H-B-P) adaption was found to exhibit the best fit to the experimental data for MLSS concentrations between 6 and 16 g/L and shear rates between 0.05 and 1600 s h... 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

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. 

Pseudoplastic fluids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Figure 1.1). 

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

Shaving products Shaw process Shear breeding Shear energy Shearlings Shearometer Shear plane Shear rate Shear stresses Shear test Shear thinning behavior Shear viscosity Sheath-core fiber... 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

Polyolefin melts have a high degree of viscoelastic memory or elasticity. First normal stress differences of polyolefins, a rheological measure of melt elasticity, are shown in Figure 9 (30). At a fixed molecular weight and shear rate, the first normal stress difference increases as MJM increases. The high shear rate obtained in fine capillaries, typically on the order of 10 , coupled with the viscoelastic memory, causes the filament to swell (die swell or... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Between 50 and 300°C, PTEE obeys the relationship between stress T and the apparent shear rate 7 r =. Melting of PTEE begins near... 

C. Above this temperature, the shear stress at constant shear rate increases and the rheological exponent rises from 0.25 toward 0.5 at the final melting point (68). 

Apparent viscosity of a grease at low shear rates, eg, below about 10, is approximately equal to the yield stress divided by the shear rate. This... 

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

For a Hquid under shear the rate of deformation or shear rate is a function of the shearing stress. The original exposition of this relationship is Newton s law, which states that the ratio of the stress to the shear rate is a constant, ie, the viscosity. Under Newton s law, viscosity is independent of shear rate. This is tme for ideal or Newtonian Hquids, but the viscosities of many Hquids, particularly a number of those of interest to industry, are not independent of shear rate. These non-Newtonian Hquids may be classified according to their viscosity behavior as a function of shear rate. Many exhibit shear thinning, whereas others give shear thickening. Some Hquids at rest appear to behave like soHds until the shear stress exceeds a certain value, called the yield stress, after which they flow readily. 

Fig. 2. Flow curves (shear stress vs shear rate) for different types of flow behavior.

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

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. 

The Williamson equation is useful for modeling shear-thinning fluids over a wide range of shear rates (15). It makes provision for limiting low and high shear Newtonian viscosity behavior (eq. 3), where T is the absolute value of the shear stress and is the shear stress at which the viscosity is the mean of the viscosity limits TIq and, ie, at r = -H... 

Time-dependent effects ate measured by determining the decay of shear stress as a function of time at one or more constant shear rates (Fig. 7)... 

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