Flow at High Shear Rates

We have already discussed one aspect of non-linear behavior in polymer melts, namely shear thinning. A second aspect manifests itself when we examine the flow of polymer melts through small diameter tubes or capillaries. This is the phenomenon of jet or die swelling, where a polymer forced into a narrow tube, diameter d0, swells when 

FIGURE 13-69 Jet swelling [drawn schematically from the data of Burke and Weiss, Characterization of Materials in Research, Syracuse University Press (1975)J. 

Clearly, this is an elastic rather than a simple viscous type of response and simple fluids like water do not behave in this fashion. A qualitative understanding of this is relatively straightforward the chains in the melt are deformed as they are pushed into the capillary (or die, if we are talking about extrusion) and as they are sheared in the narrow channel. Normal stresses then develop in a direction perpendicular to the flo (i.e., axially in the capillary). These stresses are relieved and deformation is recovered when the polymer exits the die, so the polymer extrodate swells. (It s a bit more complicated than this, because some relaxations can also occur in the capillary, if it is long enough— but you get the idea.) 

This elastic property of polymer melts also 

FIGURE 13-70 Melt fracture [reproduced with permission from J. J. Benbow, R. N. Browne and E. R. Howells, Coll. Intern. Rheol., Paris, June-July I960)]. 

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Many fluids show a decrease in viscosity with increasing shear rate. This behavior is referred to as shear thinning, which means that the resistance of the material to flow decreases and the energy required to sustain flow at high shear rates is reduced. These materials are called pseudoplastic (Fig. 3a and b, curves B). At rest the material forms a network structure, which may be an agglomerate of many molecules attracted to each other or an entangled network of polymer chains. Under shear this structure is broken down, resulting in a shear... 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

Savage, S. B. (1983). Granular Flows at High Shear Rates. In Theory of Dispersed Multiphase Flow. Ed. R. E. Meyer. New York Academic Press. 

Savage, S. B. and Jeffery, D. J. (1981). The Stress Tensor in a Granular Flow at High Shear Rates. 

Up to now, two regions of shear flow have been discussed Newtonian flow at low shear rates and non-Newtonian flow at high shear rates. In the first region, the viscosity is independent of the shear rate, while in the second region the viscosity decreases with increasing shear rate. 

Savage SB, Jeffrey DJ The stress tensor in a granular flow at high shear rates, J Fluid Mech 110 255-272, 1981. 

A characteristic flow pattern at the capillary entrance develops when a polymer flows at high shear rates from a cylindrical reservoir through a capillary or die as shown in Figure 2.12. The qualitative difference between the capillary entry flows of linear and branched polyethylenes has been convincingly presented by Tordella [50] and discussed by others [67-70]. For linear polymers, the converging flow at the die entry fills the available space, while for branched polymers there is a large... 

The concept of the Graessley theory that the steady-state concentration of entanglements is diminished during flow at high shear rates implies that the steady-state compliance Ry observed in recovery after cessation of steady-state flow will be larger than J and given by the equation ... 

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