Second normal stress coefficient

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. 

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. 

A number of experimental techniques have been developed for measuring the viscosity and the first (and second) normal stress coefficient. In Table 15.2 a survey of these methods has already been given. 

Integral of time-temperature history, t -First normal stress coefficient. Pa s Second normal stress coefficient. Pa s Dynamic frequency, rad s ... 

Problem 3.14 (Worked Example) Derive expressions for the shear viscosity and first and second normal stress coefficients in steady-state shearing of the Johnson-Segalman model, given by Eqs. (3-80) and (3-8 la). 

We use the definitions of the shear viscosity r , first normal stress coefficient I l, and second normal stress coefficient 1 2 [from Eq. (1-24)] to obtain... 

Doi and Edwards (1978) and Kuzuu and Doi (1980) have solved the Smoluchowski equation (6-47)-(6-48) for simple shearing and elongational flows, and they obtained predictions of rheological behavior that are similar to those of the reptation theory for concentrated flexible polymers discussed in Section 3.7.5.1. Figure 6-19, for example, shows the shear-rate-dependence of the shear viscosity and first and second normal stress coefficients predicted by the Doi-Edwards theory for semidilute rods these results are similar to those predicted by the Doi-Edwards theory for entangled flexible molecules. At... 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

Cone-and-Plate Instrument. The fluid to be tested is placed in the gap between the cone and plate. Three measurements are generally made the torque C on the plate, the total normal force F on the plate, and the pressure distribution P + Tee across the plate. Under the assumptions that (1) inertial effects are negligible, and (2) the angle between the cone and the plate is small (1 to 2°), the first and second normal stress coefficients can be evaluated from the following two equations ... 

Here, Ttee is the pressure, which may be measured by flush-mounted pressure transducers located on the plate, and F is the total force applied on the plate to keep the tip of the cone on the surface of the plate. However, evaluation of both the first and second normal stress coefficients requires the pressure distribution on the plate. Only a few instruments have the capacity to measure both F] and P2. 

Normal stress is an extra stress which is developed in viscoelastic material under shear, in directions normal to the plane of shear. The normal stress is expressed by the first and second normal stress coefficients (Bird et al, 1987)... 

Let ymaxin) and Kmax( I () be their peak positions, and let yo( 2) be the shear rate where sign inversion occurs in the second normal stress coefficient. Then, an inequality H)( I 2)sign inversion occurs at a larger shear rate than that of the viscosity peak. 

Similarly, we find for the second normal stress coefficient... 

Knowledge of the behavior of apparent viscosity and the normal stress coefficients is inversely proportional to the case of the experimental measurement needed. The most is known about the apparent viscosity and the least about the second normal stress coefficient (i /2). 

The viscoelastic equivalents to viscosity—the stress divided by the shear rate—are the so-called first and second normal-stress coefficients, F,and These are given by the first and second normal-stress differences divided by the shear rate squared, so... 

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