Stress coefficient

In addition to the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary nomial stress coefficients expressed, respectively, as... 

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.

Here, Q are the elastic stiffness constants, are the thermal stress coefficients, and gkj and are the direct and converse piezoelectic stress coefficients, respectively. The superscript , on Pk, p k, and Xki indicates that these quantities are now defined under the conditions of constant strain. 

The total pyroelectric response at constant stress, p , is the sum of the primary pyroelectic response, given by p, and the secondary pyroelectric response, which is the product of the direct piezoelectric stress coefficient gs, and the thermal expansion coefficients ... 

In the experiments used to draw Fig. 20.2, wind speeds were measured at different heights above the water surface. Since wind speed generally decreases when approaching the water surface, these experiments can be compared only if we find a means to transform the wind speeds to a standard height (usually 10 m). Mackay and Yeun (1983) use the standard boundary layer theory with a roughness height of 0.03 cm and a wind stress coefficient of 1.5 x 10 3 to get ... 

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. 

The set of constitutive parameters contains the (drained) elastic volumetric compliance C and two poroelastic constants the Biot stress coefficient b, and the unconstrained storage coefficient Sa = d(/dp a which can be expressed as So- = bB 1C ([13]), where B is the Skempton pore pressure coefficient. The other three parameters, a, f3, and 7 quantify the physico-chemical interactions. Both a and (3 are constrained to vary from 0 when there is no chemical interaction to 1 when the salt ions are trapped in the pore network (this limiting case is referred to as the perfect ion exclusion membrane model ). The coefficient 7 can simply be approximated by 7 x0/n, where n is the porosity of the shale. 

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).]...

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