Truncated power-law model

In the shear range of polymer processing the power law is a good approximation, but not in the low shear rate range. For this reason a better approximation is Spriggs truncated power law model law (Spriggs, 1965) ... 

The Carreau model [19] has the useful properties of the truncated power law model but avoids the discontinuity in the first derivative it can be written as ... 

These equations can be solved to predict the pressure drop, AP, as a function of the volumetric flow rate, Q, if the viscosity is known as a function of pressure, temperature and shear rate. Two rheological models are employed in this study for the functional relationship between viscosity euid shear rate. In the data analysis procedure, calculations are performed for Newtonian fluids in which the viscosity is Independent of shear rate. In the consistency check procedure, the experimental results are correlated with a truncated power-law model in which the fluid behaves as a Newtonian fluid below a characteristic shear rate, Yq, and exhibits shear-thinning behavior above this shear rate ... 

If it is assumed that the influence of pressure and temperature on the viscosity of the polymer solution is independent of shear rate, then this experimental technique results in values of the viscosity of the polymer solution as a function of pressure, temperature, and shear rate. Finally, the consistency of this overall data analysis technique can be checked. The computer simulation presented in the previous section was used to predict the flow of a non-Newtonian fluid which obeys the truncated power-law model. The parameters in this model were determined by coupling the high and low shear rate measurements of viscosity. Once the viscosity of the polymer solution has been determined as a... 

Table 6.1 Parameters appearing In the truncated power-law model for some molten polymers...

In order to solve Eqs. (11.6) and (11.7), one must specify a relationship between the stress tensor a and the rate-of-deformation tensor d. The most general situation would be many viscoelastic drops suspended within another viscoelastic medium, which is extremely complicated to handle, even when using the most sophisticated computational tools available. Therefore, let us consider a simpler model, the truncated power-law model (see Chapter 6), as schematically shown in Figure 11.35, describing the shear-rate dependence of viscosities, (k) and j()>), of the suspending medium and the drop phase, respectively. The truncated power-law model for the drop and the suspending medium in terms of dimensionless viscosities = t]a y)/rj and... 

Figure 11.35 Shear-rate dependence of viscosity in the truncated power-law model.

Table 11.1 Numerical values of the parameters appearing in the truncated power-law model for an aqueous solution of Separan and PIB solutions in decalin...

Plot these data in the form of t — / and yu, — / on logarithmic coordinates. Evaluate the power-law parameters for this fluid. Does the use of the Ellis fluid (equation 1.15) or of the truncated Carreau fluid (equation 1.14) model offer any improvement over the power-law model in representing these data What are the mean and maximum % deviations from the data for these three models ... 

In this spirit we investigated models where the dust density n fell off as r°, r, or r . The grain emission (and absorption) efficiency was taken as a three part power law in X. a for X < lOOoX e a X for lOOOA < X < 20y and e a X" for X > 20y. This crude form is consistent with expectations for H2O and silicate grains (Irvine and Pollack 1968 and Knacke and Thomson 1973). Once the dependence of dust opacity on radius from the star and on wavelength have been adopted, the only remaining critical parameter is the absolute value of the optical depth at lOOy, T] ooy measured from the outer radius of the dust shell through to the star. The adopted inner and outer radii at which one truncates the dust shell are easily seen to be unimportant provided the cloud is not optically thin (T] QQy O.l) and provided the outer layers do not contribute substantial optical depth respectively. 

Despite its simplicity, the model was applied with great success to, e.g., the STM-driven transfer of a xenon atom on a nickel surface. This pioneering experiment was the very first example of an STM-controlled atomic switch, where the xenon atom was moved from the nickel surface to the tungsten STM tip. Figure 6 shows the comparison between theoretical and experimental transfer rates for different values of the ratio The computed transfer rates are inferred from an implicit dynamics between truncated harmonic oscillators, with the initial conditions chosen as the fifth excited vibrational state located on the surface. This corresponds to a situation where above-threshold dynamics dominates. By construction, the rates exhibit the proper power-law dependence with increasing potential bias. It is found that, for this... 

---