Stress coefficients, normal

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

Figure 10 Flow curves of a family of polyacrylamide (PAA)/water-solutions of different concentrations and viscosities. Left side normalized viscosity curves id/po = f y/yo), right side normalized stress coefficients ( i + 2 2) ol o =f2 Source From Ref. 13.

The symbols Nt and N2 denote the normal stress functions in steady state shear flow. Symmetry arguments show that the viscosity function t](y) and the first and second normal stress coefficients P1(y) and W2(y) are even functions of y. In the... 

Second normal stress coefficient, N2/y2-Number of distinguishable configurations. 

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

Here we have three parameters r/o the zero-shear-rate viscosity, Ai the relaxation time and A2 the retardation time. In the case of A2 = 0 the model reduces to the convected Maxwell model, for Ai = 0 the model simplifies to a second-order fluid with a vanishing second normal stress coefficient [6], and for Ai = A2 the model reduces to a Newtonian fluid with viscosity r/o. If we impose a shear flow,... 

Indicating that the convected Jeffreys model gives a constant viscosity and first normal stress coefficient, while the second normal stress coefficient is zero. 

The coefficients used to fit the data are summarized in Table 2.11 [43], The viscosity and first normal stress coefficient data presented in Figs. 2.30 and 2.31 where fitted with the Wagner form of the K-BKZ equation [41],... 

The cone-plate rheometer. The cone-plate rheometer is often used when measuring the viscosity and the primary and secondary normal stress coefficient functions as a function of shear rate and temperature. The geometry of a cone-plate rheometer is shown in Fig. 2.47. Since the angle Oo is very small, typically < 5°, the shear rate can be considered constant throughout the material confined within the cone and plate. Although it is also possible to determine the secondary stress coefficient function from the normal stress distribution across the plate, it is very difficult to get accurate data. 

Concentrated emulsions can exhibit viscoelasticity, as can gelled foams and some suspensions. Compared with the previous equations presented, additional coefficients (including primary and secondary normal stress coefficients) are needed to characterize the rheology of viscoelastic fluids [376,382]. 

We note that the primary normal stress coefficient P 1 is positive, whereas the secondary normal stress coefficient P2 is negative, but with a lot of scatter in the data. It is difficult to measure (r22 — T33) and its value is in doubt, but the ratio — (tn — X22)/ x22 — T33) appears to be about 0.1. 

If the normal stress coefficient functions Mr and T2 are ignored, the CEF equation reduces to the GNF equation... 

Fig. E3.5 Steady-state shear viscosity rj and first normal stress coefficient i, obtained from dynamic measurements versus shear rate for a low-density polyethylene melt, melt I. [H. M. Laun, Rheol. Ada, 17, 1 (1978).]...

N2 values are always lower than Nj values, see e.g. [40]. Therefore for many processes taking into consideration only Nj will suffice. The normal stress differences are independent of the direction of flow and, in laminar flow (low y), are proportional to y2. In following p = x/y for a Newtonian fluid, normal stress coefficients ipi = Nj/y2 and ip2 = N2/y2 are occasionally used. Their dependence on the shear rate i j(y) describes the non-linear viscoelastic behavior of the fluid. 

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

The principal quantities determining the rheological behaviour of polymer melts are the viscosity and normal stress coefficients in shear and extensional... 

For common liquids, the viscosity is a material constant which is only dependent on temperature and pressure but not on rate of deformation and time. For polymeric liquids, the situation is much more complicated viscosities and normal stress coefficients differ with deformation conditions. Because polymer melts are viscoelastic their flow is accompanied by elastic effects, due to which part of the energy exerted on the system is stored in the form of recoverable energy. For this reason the viscosities are time and rate dependent polymer melts are viscoelastic. 

MODES OF DEFORMATION AND DEFINITION OF VISCOSITY AND NORMAL STRESS COEFFICIENTS... 

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