Shear rate values

The power law developed above uses the ratio of the two different shear rates as the variable in terms of which changes in 17 are expressed. Suppose that instead of some reference shear rate, values of 7 were expressed relative to some other rate, something characteristic of the flow process itself. In that case Eq. (2.14) or its equivalent would take on a more fundamental significance. In the model we shall examine, the rate of flow is compared to the rate of a chemical reaction. The latter is characterized by a specific rate constant we shall see that such a constant can also be visualized for the flow process. Accordingly, we anticipate that the molecular theory we develop will replace the variable 7/7. by a similar variable 7/kj, where kj is the rate constant for the flow process. 

Mendelson (169) studied the effect of LCB on the flow properties of polyethylene melts, using two LDPE samples of closely similar M and Mw plus two blends of these. Both zero-shear viscosity and melt elasticity (elastic storage modulus and recoverable shear strain) decreased with increasing LCB, in this series. Non-Newtonian behaviour was studied and the shear rate at which the viscosity falls to 95% of the zero shear-rate value is given this increases with LCB from 0.3 sec"1 for the least branched to 20 sec"1 for the most branched (the text says that shear sensitivity increases with branching, but the numerical data show that it is this shear rate that increases). This comparison, unlike that made by Guillet, is at constant Mw, not at constant low shear-rate viscosity. 

The raw data from the capillary rheometer measurements are subject to four important corrections to obtain true viscosity and shear rate values. These corrections, which are fully described in Kwag (1998) and Kwag et al. 

Capillary Flow Rheometry Next we examine the experimentally obtained results with the capillary flow rheometer shown in Fig. 3.1, which are directly relevant to polymer processing flows, since the attainable shear rate values are in the range encountered in polymer processing. The required pressure drop AP does not increase linearly with increases in the volumetric flow rate Q, as is the case with Newtonian fluids. Rather, increasingly smaller increments of AP are needed for the same increases in Q. The Newtonian Poiseuille equation, relating flow rate to pressure drop in a tube, is linear and given by... 

Experimental measurements of the normal stress differences A i and N2 for suspensions are rare, especially for hard spheres. According to a recent theory by Brady and Vicic (1995), in the dilute regime the low shear-rate values of the normal stress differences are M,o/hoK = O.8996 7r0 Pe and Af2,o/r/oy = —O.78867T0 Pe. These values are quite small at concentrations and shear rates where the theory might apply (0 0.15 Pe 0.1). The... 

A schematic representation showing the intervals of shear rates at which different geometries are used is given in Figure 13.4. In this figure, the interval of shear rates at which polymeric materials are processed is also included. It should be noted that injection molding tends to the upper shear rate values while compression molding tends to the lower shear rates. 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

Often, one finds a linear relation at a very small shear rate, a shear thinning behavior at intermediate shear rate, and a linear relation at high shear rate values (see, e.g., Van Diemen and Stein [18] and Hunter [19]). In this respect, the Meter model may be of special interest if a system behaves like Newtonian fluid at both low and high shear rates ... 

One of the unique rheological features of emulsions is that the apparent viscosity of the emulsion can drop below the viscosity of the continuous phase when the concentration of the dispersed phase is low, normally below 0.1 in volume fraction (194). When solids are added to the emulsion, the apparent viscosity can decrease even further and the volume fraction of the dispersed phase at which minimum viscosity occurs increases with increasing solids content. Figure 30 shows the apparent viscosity of water-and-sand-in-bitumen, pwsh, variation with the solid-free water volume fraction, j8w, for two shear rate values. The experimental data were provided by Yan (private communication), where the system consists of 52 pm sand particles treated with hexadecyltri-methylammonium bromide (HAB) and water droplets of a Sauter mean diameter of 9 pm dispersed in bitumen at 60 °C. The sand particle volume fraction on water-free basis is j8s = 0.193. The range of the water droplet volume fraction, on a solid-free basis, jfrw is between 0 and 0.4. It can be observed that a minimum viscosity is present at a solid-free water droplet volume fraction of about 0.1. For a lower solid concentration, Ps = 0.113, the minimum apparent viscosity is found at /3W = 0.05... 

The time dependency of the stress-strain rate relationship can be omitted for polymeric liquids in many practical situations. Now, let us consider Figure 22.6, which is a typical plot for viscosity in terms of shear rate for a polymer melt. Two different regions can be observed in the flgure. In the first region, which occurs at moderate low shear rate values, there is a smooth variation of polymer viscosity. In the second region, there is a more pronounced decrease of viscosity as shear rate is further increased. This section of the curve is often described mathematically by a power-law model that expresses the relationship between shear rate and the viscosity stated in Equation 22.9, as discussed in... 

Some modifications of the melt flow behavior of thermoplastics that can be observed depending on filler concentration are a yield-like behavior (i.e., in these cases, there is no flow until a finite value of the stress is reached), a reduction in die swell, a decrease of the shear rate value where nonlinear flow takes place, and wall slip or nearwall slip flow behavior [14, 27, 46]. Other reported effects of flllers on the rheology of molten polymers are an increase of both the shear thinning behavior and the zero-shear-rate viscosity with the filler loading and a decrease in the dependence of the filler on viscosity near the glass transition temperature [18, 47-49]. 

The following shear stress-shear rate values have been obtained for aqueous silica (bulk density = 800kg/m ) suspensions to elucidate the effect of concentration on the rheological behaviour of suspensions ... 

Figure 8.12 shows the total area of the individual particles of the SAN-rich region as a function of the shear rate calculated from the image analysis of the previous TEM images (Fig. 8.11). Obviously, the area of the dispersed domains remarkably decreased with the shear rate and leveled off at high shear rates. As mentioned above, this is due to a competition of particle breakup and coalescence which may occur at high shear rate values.

Melt viscosity, or flow, is typically measured using extrusion plas-tometers (or melt indexers), capillary rheometers, and parallel plate rheometers. The extrusion plastometer measures the flow of a polymer melt under conditions specified by ASTM standard D 1238. This test yields a single, low-shear-rate value which is typically used to specify resins. Capillary rheometers determine viscosity over a range of shear rates in channel flow. While they are subject to error, these rheometers are still the only means of measuring viscosity at high shear rates (typically -y > 1000 s i). Parallel-plate rheometers also measure viscosity over a range of shear rates, but the maximum allowable shear rate is about 100 s i. 

Diffusion and convection are then competitive processes, but according to the shear rate value, for large segregation scales, the diffusion process is slow compared with convection and mixing is almost controlled by stretching. At fine segregation scales, diffusion becomes the controlling step. 

Figure 11. The reduced rotational velocity, as inferred oni the angular moentum (black) and from the components of the gyration tensor (gray) as functions of the shear rate for calculations with a Nos Hoover thermostat. The horizontal dashed line marks the limiting smtdl shear rate value 0.5, the inclined dashed line shown for high shear rates corresponds to a power law exponent —6.

The Roll Technologies Company s (U.S.) STT reactor (see Fig. 6.34) consists of a tube inside another tube, where the inner tube spins inside the annular tube, whereby only a very small gap is maintained. The reactants All the slightly eccentric gap. Immediately upon entry the reactants encounter a very large interfacial contact area leading to extreme rates of surface renewal. Typical shear-rate values are in the range of 30,000/s to 70,000/s. Process rates in reactors are influenced by the minimum length of turbulent eddies and the molecular diffusive mixing time. This reactor... 

Table 3.2 Typical shear rate values in industrial end uses ...

Recalling the range of shear rate values (see Table 3.2) that a paint or cosmetic formulation may be exposed to during manufacture, storage and use, it is apparent that this sort of measurement should cover values ranging over several orders of magnitude or more for both viscosity and shear rate. It is common to see this information presented graphically as log viscosity versus log shear rate. 

Satisfactory film thickness predictions are also obtained with lubricants L4, L5 and L6 if the viscosity used in the formula is the viscosity value at Che high shear rates found in Che contact and not Che low shear rate value given by classical viscometer. Shear thinning effects observed with Che silicone oil L5 are less important Chan Chose found in elastohy-drodynamic line contacts for much higher viscous silicone fluids by Dyson and Wilson (13). 

More detailed information concerning permissible instantaneous and long-term shear rate values may be requested from the manufacturer of the plastic in question, together with a graph of viscosity q = f(7,T). 

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