Shear rate tensor

Rate-of-deformation tensor Shear rate (a scalar)... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

A similar KR approach can be employed for the intrinsic viscosity. In this case it is necessary to assume the presence of a small amount of shear rate, which cancels out in the calculations. Now, the linear equations are used to evaluate an averaged crossed component of the stress tensor on unit i, .The final result for a long linear non-draining chain is... 

Both the Doll s and SLLOD algorithms are correct in the limit of zero-shear rate. However, for finite shear rates, the SLLOD equations are exact but Doll s tensor algorithm begins to yield incorrect results at quadratic order in the strain rate, since the former method has succeeded in transforming the boundary condition expressed in the form of the local distribution function into the form of a smooth mechanical force, which appears as a mechanical perturbation in the equation of motion (Equation (12)) (Evans and Morriss, 1990). To thermostat the... 

Figure 8.36 Unwrapped screw channel with conditions and dimensions [9]. where 7 is the magnitude of the shear rate tensor defined by...

Savage, S. B. and Jeffery, D. J. (1981). The Stress Tensor in a Granular Flow at High Shear Rates. 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

Note that shear rate y is the magnitude of the tensor y, and therefore it must always be positive. Thus we maintain the absolute-value sign over the term that reflects the shear dependence of the viscosity. 

Another method to calculate viscoelastic quantities uses measurements of flow birefringence (see Chap. 10). In these measurements, two quantities are determined as functions of the shear rate y the birefringence An and the extinction angle y. The following relationships exist with the stress tensor components ... 

Rheological properties of food materials over a wide range of phase behavior can be expressed in terms of viscous (viscometric), elastic and viscoelastic functions which relate some components of flie stress tensor to specific components of the strain or shear rate response. In terms of fluid and solid phases, viscometric... 

Figure 3.17 Birefringence as a function of the eigenvalue of the velocity gradient tensor, G, for planar flows generated in a four-roll mill, for dilute solutions of polystyrenes of three different molecular weights in polychlorinated biphenyl solvent. Here G is the strain rate and a the flow type parameter. For planar extension, a — 1 and G = is the extension rate for simple shear, a = 0 and G = y is the shear rate. The different symbols correspond to a values of 1.0 (0)> 0.8 (A), 0.5 (-1-), and 0.25 (diamonds). The curves are theoretical predictions from the FENE dumbbell model, including conformation-dependent drag (discussed in Section 3.6.2.2.2). (From Fuller and Leal 1980, reprinted with permission from Steinkopff Publishers.)...

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

The stress tensor for a semidilute solution of rods is given by Eq. (6-36), the formula for dilute solutions. However, if in a thought experiment one holds the shear rate fixed at a low value while increasing the concentration of rods from dilute to semidilute, the Brownian contribution to the stress will greatly increase, since the rotary diffusivity decreases according to (6-44). The viscous stress contribution, however, only increases in proportion to u. Thus, as Doi and Edwards (1986) argued, the ratio of viscous to Brownian stresses decreases as as the concentration increases in the semidilute regime. Hence, in the semidilute... 

When these, or other dimensionally correct, relaxation terms are added to the right sides of Eqs. (9-43) and (9-44), distinctive scaling relationships can be derived, which can be expressed as follows. Let a t, [y(t)]) be the stress tensor at time t during a shearing flow with shear history [y (f)]. Now choose a new shear history [y (t)] = c[y(cf)] in which the shear rate at each instant r is a constant c times the shear rate in the old history at time ct. Then, the stress a in the new shear history is given by... 

This scaling law, Eq. (9-48), implies that all components of the stress tensor are linear in the shear rate. Consider for example, a constant-shear-rate experiment. At steady state, not only is the shear stress predicted to be proportional to the shear rate, but so also is the first normal stress difference N This prediction has been nicely confirmed in recent experiments by Takahashi et al. (1994), who studied mixtures of silicon oil and hydrocarbon-formaldehyde resin. Both these fluids are Newtonian, and have the same viscosity, around 10 Pa s. Figure 9-18 shows that both the shear stress o and the first normal stress difference N = shear rate, so that the shear viscosity rj = aly and the so-called normal viscosity rjn = N /y are constants. The first normal stress difference in this mixture must be attributed entirely to the presence of interfaces, since the individual liquids in the mixture have no measurable normal stresses. A portion of the shear stress also comes from the interfacial stress. Figure 9-19 shows that the shear and normal viscosities are both maximized at a component ratio of roughly 50 50. At this component ratio, the interfacial term accounts for roughly half the total shear stress. 

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by... 

If we now generalize the definition of director to be the unit vector parallel to the major axis of the order parameter tensor, we find that at vanishingly low shear rates, where the order parameter tensor is nearly uniaxial, this definition reduces to the usual meaning of the term director. ... 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

Hongladarom and Burghardt (1994) measured the components of this tensor for a 13.5% solution of PBDG using polarimetry and found that, in accord with the mesoscopic theory, the birefringence is indeed independent of shear rate in Region II. The experimental birefringence tensor was found to be... 

For steady-state shearing flows, the relationship between the shear-stress tensor and the shear-rate tensor is given by Criminale-Ericksen-... 

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