Infiniti shear rate viscosity

The rheological properties of gum and carbon black compounds of an ethylene-propylene terpolymer elastomer have been investigated at very low shear stresses and shear rates, using a sandwich rheometer [50]. Emphasis was given to measurements of creep and strain recovery at low stresses, at carbon black filler contents ranging between 20 and 50% by volume. The EPDM-carbon black compounds did not exhibit a zero shear rate viscosity, which tended towards infinity at zero shear stress or at a finite shear stress (Fig. 13). This was explained... 

This fitted the data well up to volume fractions of 0.55 and was so successful that theoretical considerations were tested against it. However, as the volume fraction increased further, particle-particle contacts increased until the suspension became immobile, giving three-dimensional contact throughout the system flow became impossible and the viscosity tended to infinity (Fig. 2). The point at which this occurs is the maximum packing fraction, w, which varies according to the shear rate and the different types of packings. An empirical equation that takes the above situation into account is given by [23] ... 

A convenient form of 3-parameter equation which extrapolates to a constant limiting apparent viscosity (m0 or /jlqc) as the shear rate approaches both zero and infinity has been proposed by Cross(i3) ... 

For a shear-thickening fluid the same arguments can be applied, with the apparent viscosity rising from zero at zero shear rate to infinity at infinite shear rate, on application of the power law model. However, shear-thickening is generally observed over very much narrower ranges of shear rate and it is difficult to generalise on the type of curve which will be obtained in practice. 

Thus, the apparent viscosity falls from infinity at zero shear rate ( / r T Ry) to pp at infinite shear rate, i.e. the fluid shows shear-thinning characteristics. 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

From the above analysis it follows that the increase in Ti g at smaU v cannot be interpreted as shear thickening. Tribological driving (at positive temperatures) would result in a curve similar to that shown in Fig. 15, in which the linear-response viscosity is obtained at the smallest shear rates. However, as the two model systems mentioned above are athermal, their tribologically determined rjgff would tend to infinity at zero shear rate [216]. 

Isothermal steady time tests are used to determine the gel point of a thermoset system as the point at which the shear viscosity tends towards infinity. In these tests the viscosity is measured as a function of time at a constant shear rate. This method has the following major disadvantages. Firstly, the infinite viscosity can never be measured due to equipment limitations and thus the gel time must be obtained by extrapolation. Secondly, shear flow may destroy or delay network formation. Finally, gelation may be confused with vitrification or phase separation since both these processes lead to an infinite viscosity (St John et al., 1993). However, some work by Matejka (1991) and Halley et al. (1994) has shown that extrapolation to zero values of reciprocal viscosity or normal stress (i.e. extrapolation to infinite viscosity and normal stress) can be used with some success. 

Analyzing Fig. 1 we see that for shear stress values T reaching the plastic level, which is tantamount to shear rates y near zero, the shear viscosity 77 reaches infinity. This situation and the discontinuity of T in his first derivative at leads to remarkable difficulties in the course of numerically solving the conservation equations (l)-(3). Suggestions how these problems can be circumvented will be discussed later in chapter 3.1. 

One of the obvious disadvantages of the power law is that it fails to describe the low shear rate region. Since n is usually less than one, at low shear rate r] goes to infinity rather than to a constant %, as is usually observed experimentally (Figures 2.4.1 and 2.1.2a, b). It is also observed that viscosity b mes Newtonian at high shear rates for many suspensions and dilute polymer solutions (see Figure 2.1.2a). 

The lower limit on the integral is zero rather than minus infinity, since the sample is known to be in a stress-free state at t = 0. The ratio of the stress to the shear rate is called the shear stress growth coefficient and has units of viscosity ... 

Thus, a log-log plot of ri vs. y for a power-law fluid is linear with a slope of n — 1 (Fig. 15.7). This points up an interesting limitation of the power law. Where the shear rate goes to zero (e.g., for Poiseuille flow at the center line of a cylindrical tube) the viscosity approaches infinity for a pseudoplastic power-law fluid, for which n < 1. 

Although the indirect method is more difficult to implement, it renders better results because the rate of deformation and temperature fields are smoother and better bounded than the viscosity field. For example, when using the power law shear thinning model, the viscosity goes to infinity when the rate of deformation goes to zero. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

A step change of viscosity at critical conditions to infinity (3) is a convenient model for investigating the flow of a curing liquid. The simplest model employs the most important physical property of the process. In a number of papers [83-87], the dependence of the induction period t, the time during which a reactive substance retains the ability to flow, on the rate of shear y is studied. 

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