Secondary normal stress difference

Furthermore, when the cone-and-plate rheometer is outfitted with pressure taps at various radial positions, the experimentally obtained pressure distribution is found to be increasing with decreasing radial distance. This, as we will see later, enables us to compute the secondary normal stress difference, namely, x22 — T33, where direction 3 is the third neutral spatial direction. 

Upon rearrangement and integration, and taking into account that the negative of the secondary normal stress difference, — ngg, is a constant (since jg is constant), and that ng at 0 — n/2 (the plate) is a function of the radius, we have... 

Figure E3.2b presents the primary normal stress difference data for LDPE, and Fig. E3.2c presents the primary and secondary normal stress-difference data for a 2.5% polyacrylamide solution, again using a cone-and-plate rheometer.

We therefore observe that unlike in the Power Law model solution with a single shear stress component, xn, in the case of a CEF model, we obtain, in addition, two nonvanishing normal stress components. Adopting the sign convention for viscometric flow, where the direction of flow z is denoted as 1, the direction into which the velocity changes r, is denoted as 2, and the neutral direction 8 is denoted as direction 3, we get the expressions for the shear stress in terms of the shear rate, the primary, and secondary normal stress differences (see Eqs. 3.1-10 and 3.1-11) ... 

We find the maximum pressure rise at the center of the disk to be proportional to the square of flR/H, which is the shear rate at r = R. Moreover, by comparing Eq. 6.5-18 to Eqs. 6.5-10 and 6.5-11, we find that this pressure rise is the sum of the primary and secondary normal stress-difference functions —[(tn — T22) + (J22 — T33)] at r = R, less centrifugal forces. Since lL is probably negative, it opposes pressurization hence, the source of the pressurization in the normal stress extruder is the primary normal stress difference function ffq. 

Similar definitions can be made for p and p1 related to the oscillating secondary normal stress difference. The quantities 0 and p1 are real, whereas tf, 8, and (t are complex. It is customary to write these complex quantities thus12 ... 

Equations (10.7), (10.8), and (10.10) show that (xyy — t2z)+ = —(2/7) (xxx — xyy)+ for small Xk0. This result and the result in Eq. (6.3) suggest that the secondary normal stress difference might be easier to observe in an unsteady-state experiment than in a steady-state experiment. Note that, from Eqs. (10.6) and (10.10), the relation... 

The absolute value of any particular component of normal stress is of no rheological relevance, whereas the values of the normal stress differences tn — T22 and T22 — T33 do have considerable rheological significance. The first is termed file primary normal stress difference while the latter is termed the secondary normal stress difference. Thus,... 

The viscoelastic forces that produce the remarkable manifestations illustrated above are properly characterised in shear flow by the so-called first and second normal-stress differences, Ni and N2, which occur in addition to the shear stress CT (with which we are already familiar) —note that occasionally Ni and N2 are called the primary and secondary normal stress differences. The complete stress distribution in a flowing viscoelastic Hquid may be written down formally as follows,... 

The normal stresses c,/ cannot be specified on an absolute basis because of the arbitrary hydrostatic pressure in equation 59, but their differences are predicted by continuum mechanics and several molecular theories and can be measured. The primary and secondary normal stress differences are defined as (Th — an and 022 — an respectively. Data are often expressed in terms of the primary normal stress coefficient... 

The secondary normal stress difference in steady shear flow, aii o ss, has been studied much less extensively. It has a similar dependence on 721 but is smaller than [Pg.51]

Thus oscillatory measurements of the primary normal stress difference give the same information as oscillatory shear stress measurements. Oscillatory measurements of the secondary normal stress difference, however, would provide additional... 

For parallel plate geometry, the vertical thrust depends on both primary and secondary normal stress differences. A device for providing this measurement at very high shear rates has been described by Walters. ... 

The form of this equation implies that P is a linear function of In r, as is found experimentally. Apparatus for this purpose has been described by several authors. " By combining equations 14 and 15, the secondary normal stress difference, 0-22 ff33, can be obtained. 

A combination of these geometries is provided by the plate and truncated cone apparatus of Lxxige, which provides the radial gradient of pressure in the outer cone region as well as the pressure at the center of the truncated (parallel plate) area. From these measurements, both primary and secondary normal stress differences can be obtained in principle preliminary work suggests, however, that the accuracy obtainable for [Pg.107]

Non-linearity shows up in the second finding. We observe that simple shear is accompanied by the development of a non-vanishing normal stress difference < xx — The effect is non-linear since it is proportional to the square of 7. The result tells us that in order to establish a simple shear deformation in a rubber, application of shear stress alone is insufficient. One has to apply in addition either pressure onto the shear plane gzz < 0) or a tensile force onto the plane normal to the x-axis a x > 0), or an appropriate combination of both. The difference Gxx zz is called the primary normal stress difference likewise Oyy — Ozz is commonly designated as the secondary normal stress difference. As we can see, the latter vanishes for an ideal rubber. 

For simple shear, we have thus obtained again linearity for as for ideal rubbers, but non-vanishing values now for both, the primary and the secondary normal stress difference. 

Rheological measurements are performed so as to obtain a test fluid s material functions. Under viscometric flows we have seen that the shear viscosity and the primary and secondary normal stress differences suffice to rheologically characterize the fluid. If the flow field is extensional and the material is able to attain a state of dynamic equilibrium, then one measures the extensional viscosity otherwise, we measure the extensional viscosity growth or decay functions. In this section, we will examine steady and dynamic shear plus uniaxial extensional tests, since these make up the majority of routine rheological characterization. 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

Figure 6.11 Cone-plate viscometer allowing measurement of the primary and secondary normal stress differences. Normal force (F) and pressure transducers recording hydrostatic pressures (p,) at different radial positions are shown.

The secondary normal stress difference vanishes, as was the case for an ideal rubber... 

For most fluids, Ni N2 and, thus, the latter is often excluded in rheological discussions. Attempts to determine the value of secondary normal stress difference experimentally have been made by several rheologists, but without success. It is still a challenge to quantitatively determine this material function. Nevertheless, it is not very important in most hydrodynamic calculations baning, of course, wire coating [S] wherein the secondary normal stress difference helps provide the necessary restoring force for stabilizing the wire position whenever it becomes off-centered. 

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