Shear-free flow

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

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. 

Extensional, elongational or shear-free flows play a dominant role in the post die-forming step, such as stretching of melt strands in spinning, uniaxial stretching of molten films... 

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 uniaxial extensional flow, with regard to describing both the deformation and the resulting stresses, is uniform shear free flow, in which the strain rate is the same for every material element, and there is no relative... 

Another class of flows, in which the fluid is not sheared, is known as shear-free flows (Bird et a/., 1987a). In this case all of the off-diagonal elements in the rate of strain tensor are zero and it has form ... 

The equations of motion for all of these shearing and shear-free flows are the so-called SLLOD equations. 

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... 

Several shear-free flows are shown in Figure 5. In each of these flows there are only normal stresses (and no shear stresses). These flows are much more difficult to maintain than shear flows and as a consequence they have not been so extensively studied. They are, however, extremely important, since the data obtained from them can provide crucial tests of continuum and molecular theories. 

The two functions and 2 depend on the parameter b as well as on the elongation rate e in steady shear-free flows. When b = 0, the function 2 is zero, and is replaced by the symbol tj, which is called the elongational viscosity . The elongational viscosity describes the resistance to elongational flow if 8 is positive and the resistance to biaxial stretching if e is negative (the terms extensional viscosity and Trouton viscosity have also been used for ). 

Because of the difficulty of attaining steady-state shear-free flows, it is thought that it may be preferable to study some of the unsteady-state flows experimentally. One can, of course, study the shear-free analogs of any of the unsteady-state experiments depicted in Figure 2 for shear flows. For example, one may define growth functions analogously to equations (5)-(7)... 

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... 

The invariant / is always zero when the assumption of incompressibility is made. For the shear flows discussed in Section 8.2.1, both / and III are zero and ll=yy = AvJAy). For the shear-free flows of Section 8.2.2, the invariant I is zero, II= 6 + 2b ) and III = 6—6b )e. Another quantity often used is the magnitude of the rate of strain tensor y=J(l/2)II which is defined to be a positive quantity for shear flows y = >>j,, which is called the shear rate . 

There are two basic flows used to characterize polymers shear and shear-free flows. (It so happens that processes are usually a combination of these flows or sometimes are dominated by one type or the other.) The velocity field for rectilinear shear flow is given below ... 

FIGURE 3.2 The deformation of (a) a unit cube of material from time ti to f2 (t2 > fi) in (b) steady simple shear flow and (c) three kinds of shear-free flow. The volume of material is preserved in all of these flows. (Reprinted by permission of the publisher from Bird et al., 1987a.)... 

It is noted that only the off-diagonal components of this tensor exist. For shear-free flow the rate of deformation tensor also takes on a distinct form. In particular, the components are... 

Here it is seen that only diagonal components exist. The physical significance of these matrices is that in shear flow the velocity gradient is transverse to the flow direction while in shear-free flow it is in the same direction as flow. 

For shear-free flows it can be shown using symmetry arguments again that the extra stress tensor is of the form... 

Similar flow histories for shear-free flows as described for shear flows in Figure 3.3 can also be used. Here we discuss only steady and stress growth shear-free flows. For steady simple (i.e., homogeneous deformation) shear-free flows two viscosity functions, 7Ji and 1)2, are defined based on the two normal stress differences given in Eq. 3.13 ... 

This constitutive equation is referred to as the upper converted Maxwell (UCM) model. In Example 3.1 this time derivative will be written out for simple shear and shear-free flows. The predictions of this model for steady shear... 

As there is no intention in this book to use the nonUnear constitutive equations in conjunction with the equations of motion to solve polymer processing problems, we at least show in the next several examples how one determines the predictions of a nonlinear model for flows in which the kinematics are known. In particular, we consider shear and shear-free flows. Furthermore, we show how one goes about finding the material parameters in a constitutive equation from rheological data for a polymer melt. 

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