Shear-Free Flow Material Functions

FIGURE 3.10 Time dependence of the elongational viscosity at the startup of simple elongational flow for two polystyrene samples (PS III and PS IV). (Reprinted by permission of the publisher from Miindstedt, 1980.) 

VISCOELASTIC RESPONSE OF POLYMERIC FLUIDS AND FIBER SUSPENSIONS 

FIGURE 3.12 Spring and dashpot analogs for rheological equations top, spring element middle, dashpot element bottom, spring and dashpot element in series. 

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Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

The flowability of a particulate material is determined by its shear properties. When internal shear deformation is just about to occur, the local shear stress is called the shear strength. The shear strength is a function of the normal stress this functional relationship is referred to as the yield locus (YL). For a free-flowing material, the yield locus under fully mobilized friction conditions is... 

There are two broad classes of rheometric experiments that have been developed shear flows and shear-free flows. Within each category one can speak of steady flows and various unsteady flows the latter can include step-function experiments, sinusoidal experiments and others. We now discuss these idealized flows and the material functions that are commonly defined. For a much more... 

The material functions tii and rji depend on both t and 8q, and of course on the parameter b that specifies the type of shear-free flow. For elongational flow, with b = 0 and Sq positive, rj becomes rj, the elongational stress growth function. This quantity has been measured for a number of polymer melts. Further information on elongational properties can be found in several extensive... 

It is not the intention here to give an in-depth description of the techniques used in measuring rheological properties of polymer melts as these details can be found elsewhere (Dealy, 1982 Macosko, 1994 Walters, 1975). The goal is to make sure that one is aware of at least the most common methods, how data is manipulated to obtain material functions, and the limitations of various techniques. Methods for measuring shear flow properties are discussed first followed by methods for measuring shear-free flow properties. 

The shear strength of non-free-flowing (cohesive) materials is not a unique function of the normal stress. The shear strength of these materials increases with pressure. The YL is a function of consolidation pressure and consolidation time. Thus, the shear strength has to be described by a series of yield loci, each curve representing a certain consolidation pressure and time. These curves can often be described by ... 

Enstad and Maltby (1992) concluded, from a series of shear tests carried out by five different laboratories on the certified material, CRM 116, that reproducible flow function results can be obtained only with skilled and experienced operators. Harwood (1971) applied the technique of Jenike to evaluate the flow of a range of pharmaceutical powders and found that as size of the powder increased the flowability became more free flowing (Table 1.12). 

The stress acting on an exposed surface is also the only non-zero principal stress, because the exposed surface is assumed self-supporting and traction-free (i.e., no shear stresses acting on the surface). The flow factor is determined by the geometry of the hopper and the properties of the bulk material. Another function used by Jenike is the flow function . This flow function is the ratio of the consolidating pressure CTi to the unconfined yield strength as defined in Section 6.1.2 ... 

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