Upper Newtonian region

In equation 1.75, pt0 and are the values of the apparent viscosity for the lower and upper Newtonian regions respectively. The constant ym is the shear rate evaluated at the mean apparent viscosity (po + p. )/2. 

In acknowledging that the upper Newtonian region is sometimes not observed due to experimental difficulties, the Cross and the Carreau models reduce to Eqs. (5) and (6), respectively ... 

Mathematical models, which can predict the shape of a flow curve of a shear thinning material including lower and upper Newtonian regions, require at least four parameters. The Cross model is one such model ... 

Considering that in many cases the viscosity measured is significantly smaller than that of the lower Newtonian region (i] i]oo), the Cross model can be further simplified ... 

The value of n describes the curvature of the flow curve. Values of n below unity are found for shear thinning non-Newtonian bodies, whereas a value of 1 would be found for Newtonian flow. It should also be noted that Eq. (9) only describes the shear thinning region of a flow curve without considering the possible existence of a lower or upper Newtonian region. 

An example of such a flow curve is presented in Fig. 6, and viscosity as a function of shear strain rate is depicted in Fig. 7. Many pharmaceutical ointments and creams show a similar shape of the flow curve with an extended upper Newtonian region. In these cases, an extrapolation of the linear portion of the flow curve to zero shear strain rate in order to obtain a dynamic yield stress is often utilized.l ... 

This behavior is shown graphically in Fig. 3.38. There is even an upper Newtonian region, where the polymer molecules are completely straightened out by the flow field. 

The shear-thinning region ends abruptly upon the onset of the upper Newtonian region, where the polymer mobility approaches the mobility of the solvent, as shown in Fig. 5.52. The limiting value of Xp is given by... 

The critical velocity for the onset of the upper Newtonian region is found by noting that, on the graph of Ap/L vs. u or v, the shearthinning and upper Newtonian regions intersect at the critical value of Ap/L and Thus, from Eqs. 5.16 and 5.20,... 

The existence of an upper Newtonian region can be determined by first finding the critical Darcy velocity where flow changes from shear-thinning to Newtonian flow. The viscosity of the water (solvent) in the polymer solution is 1.0 cp. When the upper Newtonian region is encountered, the polymer mobility is given by... 

The transition from the shear-thinning to the upper Newtonian region occurs when the Darcy velocity exceeds given by Eq. 5.44. Because there is a singjle non-Newtonian region, =n<. j and... 

Thus, as long as the injection rate is less than 1,644 B/D in the high-permeability zone, the upper Newtonian region will not be encountered. 

Eq. 5.52 (with the upper Newtonian region deleted) gives the pressure drop when there is Newtonian flow of polymer in the region between and r. ... 

Not long ago, an equation with four constants would have been deemed excessively complex for engineering calculations, but the computer does not mind a bit. Actually, in many practical applications, the shear rates don not get high enough to approach the upper-Newtonian region, and a truncated (three-parameter) form of the Carreau equation, with Voo = is adequate. 

Y. Ito andS. Shishido. Non-Newtonian behavior in steady flow over the range from the lower to the upper Newtonian region for polystyrene solution. J. Polym. Sci. Polym. Phys. Ed., 13 (1975), 35 1. 

At very high shear rates, an upper Newtonian region, with viscosity is attained. 

Add one constant to the modified Cross equation (15.9) so that it will reproduce an upper-Newtonian region. 

A typical shear stress-shear rate curve for a polymer can be divided into lower Newtonian, shear-thinning, and upper Newtonian regions if the shear rate range is very wide. Explain how the polymer chains behave under shear in these three regions. 

Sketch the shear stress-versus-shear rate behavior corresponding to the viscosity-versus-shear rate behavior depicted in Figure 14.1. Indicate the lower and upper Newtonian regions clearly on your sketch. 

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