Normal stress difference From model

Figure 9.41 presents the predicted secondary flow patterns that result from the vicoelastic flow effects. The Giesekus model with one relaxation time was used for the solution presented in the figure. For the simulation, a relaxation time, A, of 0.06 seconds was used along with a viscosity, r], of 8,000 Pa-s and a constant a of 0.80. Similar results were achieved using the Phan-Thien Tanner-1 model. As expected, when the White-Metzner model was used, a flow without secondary patterns was predicted. This is due to the fact that the White-Metzner model has a second normal stress difference, N2 of zero. 

Fig. 12.37 First normal stress difference vs. shear stress, as predicted by the Leonov model. [Reprinted by permission from H. Mavridis and R. N. Shroff, Multilayer Extrusion Experiments and Computer Simulation, Polym. Eng. Sci., 34, 559 (1994).]...

Fig. 15.9 Primary normal stress differences in the calender gap calculated with the K-BKZ model for different relaxation times. [Reprinted by permission from D. Mewes, S. Luther, and K. Riest, Simultaneous Calculation of Roll Deformation and Polymer Flow in the Calendering Process, Int. Polym. Process., 17, 339-346 (2002).]...

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

The shear relaxation modulus Gs t) and the first normal-stress difference function G i(t), both normalized on a per-segment basis and with kT set to 1, are obtained from the constitutive equation of the Rouse model (Eq. (7.55) with Sp replaced by Tp) as... 

An important quantity for the interaction between polymer solutions and the turbulent flow is the retaxation-time behaviour of the polymer solution. The experimental results of MICHELE [l5] demonstrate that in the upper (Ostwald-de-Waele-)power-law region the measured relaxation times from start up tests are identical with those calculated from the measurements of the first normal-stress difference and the shear stress with a Maxwell model. Therefore effective relaxation times A were calculated by... 

Fig. 9.17 Nonlinear stress relaxation of the transient network model with a quadratic chain dissociation rate under a constant shear deformation for y = 0.5. The decay rate is fixed as (a) /3q = 0 and (b) /3q = 1. The total number Ve of active chains and the number Vg of chains that remain active from the initial state are shown on a logarithmic scale. These are normalized by the stationary value of Ve. The shear stress hxy, the first normal stress difference N, and the second normal stress difference N2 are shown in the unit of Ve B T. (Reprinted with permission from Ref. [19].)...

The Phan-Thien/Tanner (PTT) model is one of many generalizations that have been introduced to deal with the deficiencies of the basic Maxwell model constant viscosity, quadratic first normal stress difference, zero second normal stress difference, and infinite tensile stress at a finite extension rate. Many of these models, including the PTT, are derived from microstructural models that attempt to account for aspects of chain morphology and interactions. PTT is a network model, in which the chains are assumed to interact at entanglement points. There are kinetic expressions... 

A sinusoidally varying shear strain rate with small amplitude, such that 721 = 721 cos cor, evokes a sinusoidally varying normal stress difference symmetry considerations because of the proportionality to 721, is predicted by the phenomenological models previously quoted.5 -54 jj,g oscillatory stress difference is superposed on a constant stress and both are proportional to 721 if 721 is small. The coefficient is now defined as the ratio (ai i — ff22)/(72i) - It is the sum of a constant term and two oscillating terms ... 

It can be seen from Table 6.10 that the damping constant m lies between 0.13 and 0.2, in line with the findings of Wagner [15] for polymer melts. This fact can then be used to generate the unified normal stress difference curves for other polymers [16-18] whose parameter values for the modified Carreau model fit of unified viscosity data are given in Table 6.1. Figures 6.2-6.13 show die predicted normal stress difference curves for different polymer types. The solid line shows the plot generated for a median value of m = 0.16 and the band... 

It is evident that the Giesekus model can quantitatively describe the shear thinning behavior of the entangled solutions of rod-shaped micelles. The decrease of the viscous resistance is caused by the alignment of the anisometric aggregates in the streaming solutions. Similar conclusions can be drawn from measurements of the first normal stress difference. This parameter is often represented in terms of the first normal stress coefficient ... 

The rubberlike liquid model is able to predict, qualitatively, certain nonlinear viscoelastic phenomena. In particular, some effects arising from the finite orientation of chain segments are predicted, for example a nonzero first normal stress difference. However, it fails to describe many other nonlinear effects. For example, it predicts that the viscosity is constant with shear rate and the second normal stress difference is zero. In fact, all its predictions for the shear stress in simple shear are the same as those of the Boltzmann superposition principle. We can gain some insight into the origins of nonlinearity by examining the features of the rubberlike liquid model that limit its predictive ability. 

Figure 11.8 Comparison of the predictions of the MLD modei (soiid iines) and DEMG model (broken iines) with experimentai data (symbois) for the viscosity, shear stress and first normal stress difference of a 7 wt% soiution of neariy monodisperse polystyrene of molecular weight 2.89 million in tricresylphosphate at 40 °C.The open circles are the dynamic viscosity if oi. The parameter values for the MLD theory are G 5 = 3000 dyn/cm Tj, = 3.06 s, and = 0.13 s. From Pattamaprom and Larson [36].

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

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