Wall normal stress

Near the die exit, the wall normal stress rapidly increased. Relationships were derived for the wall shear stress, normal stress difference at the channel wall, and the wall normal stress gradient in the flow through a converging channel and through a conical duct. Contrary to viscometric flows, in the converging flow field the normal stress alone does not permit to determine the pressure gradient of viscoelastic fluids. 

Let us consider the situation where pressure transducers are mounted at the tube wall along the axis of a cylindrical tube, including the reservoir section, as schematically shown in Figure 5.10. Under such a situation, pressure transducers measure the outward acting total normal stress at the tube wall (i.e., at r = R), which will be referred to as wall normal stress, denoted by T (R, z), and which consists of two parts, as defined by... 

Figure 5.11 Wall normal stress profiles of a polybutene (Indopol H1900) at 25 °C in a capillary die at various shear rates (s ) (A) 50.1, (O) 40.4, and ( ) 31.1. (Reprinted from Han, Rheology in Polymer Processing, Chapter 5. Copyright 1976, with permission from Elsevier.)...

Another experimental method that is as important as continuous-flow capillary rheometry is slit rheometry. The basic idea of slit rheometry is the same as that of continuous-flow capillary rheometry insofar as the measurement of wall normal stress along the die axis is concerned. But, there is a significant theoretical difference between the two methods, as we will make clear, and also in the die design. The slit rheometer has some advantages over the continuous-flow capillary rheometer in the way that transducers can be mounted on the die wall, but there are also some disadvantages. 

We now present the theory (Davis et al. 1973 Han 1974) that allows one to determine shear stress and first normal stress difference in steady-state shear flow using wall normal stress measurements along the axis of a slit die. Consider a fluid flowing through a slit die having the height h and the width w, and assume that flow has become fully developed. Then, for steady-state fully developed flow, the equations of motion... 

Therefore, measurements of wall normal stress along the die axis, T ib, z) in the fully developed region allows one to calculate dp/dz (see Eq. (5.55)), and thus cr and rj. Similar to the analysis of the capillary flow presented above, we have the following expression for y in slit flow (Han 1974, 1976) ... 

Equation (5.75) is a rather interesting result in that the pressure at the center of the slit at the exit plane is equal to the total outward acting wall normal stress at the die exit (at z = L). 

When using a continuous-flow capillary or slit rheometer, one must first make certain that the pressure gradients are constant (i.e., -dT ldz = dp/dz = constant in a capillary die or -dT ldz = dp/dz = constant in a slit die) in the region where wall normal stresses are measured. Nonlinear wall normal stress profiles of Tyy(b, z) may be observed in a slit die when pressure transducers are mounted in the die area that includes the entrance region (Eswaran et al. 1963, Leblanc 1976, Rauwendaal and Fernandez 1985). Nonlinear profiles of Tyy(b, z) in a slit die may also be observed... 

When a very viscous molten polymer is forced to flow through a slit or capillary die, viscous shear heating can become significant above a certain critical value of y or ct. Under such situations, nonlinear profiles of wall normal stress in a slit or capillary die may be observed, as described in the preceding section. Therefore, continuous-flow capillary/slit rheometry is limited to y or a, below which viscous shear heating can be neglected. 

What is the alternative The answer clearly lies in the use of a continuous-flow capillary (or slit) rheometer, which makes use of wall normal stress measurements in the fully developed region of a capillary (or slit) die (see Chapter 5). That is, as long as the wall normal stresses along the axis of a die are linear (i.e., in the fully developed region), a can be calculated from... 

Note in Eq. (13.1) that the total stress T (r, z) consists of two terms, pressure p(r, z) and deviatoric stress a (r,z), and all three quantities vary with the radial (r) and axial (z) directions. When pressure transducers are mounted on the die wall along the die axis, as schematically shown in Figure 5.10, the pressure transducers measure total normal stress T iR, z) at the die wall (r = R) and at position z (hereafter referred to as wall normal stress) ... 

Figure 13.2 Wall normal stress profiles along the axis of a capillary die (diameter of 0.3175 cm) for an 80/20 Rexene 143/(FC-114) mixture at 110 °C for two different shear rates ( ) 156.8 s l and (O) 260.3 s l. (Reprinted from Han and Ma, Journal of Applied Polymer Science 28 831. Copyright 1983, with permission from John Wiley Sons.)...

Figure 13.4 Schematic showing the wall normal stress profile along the axis of a capillary (or slit) die for a molten polymer with dissolved gaseous component that undergoes bubble nucleation inside the die. It is assumed in the schematic that a deviation of wall normal stress profile from linearity begins at a position where gas bubbles nucleate from the mixture.

Using two pressure transducers, Lee et al. (1999) measured wall normal stress T ri z) of molten PS with solubilized CO2 along the axis of a capillary die and... 

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