Shear viscosities example problem

By contrast, quite different results have been obtained with dilute polymer solutions. Here the extensional viscosity may be as much as thousand times the shear viscosity. Measurement of extensional viscosity of such mobile liquids is far more difficult than shear viscosity, or even impossible. According to Barnes et al. (General references, 1993) "The most that one can hope for is to generate flow which is dominated by extension and then to address the problem of how best to interpret the data in terms of material functions that are Theologically meaningful". An example of the difficulties that arise with the measurement of extensional viscosity is shown In Fig. 16.21 for a Round Robin test... 

Problem 3.4(b) (Worked Example) Suppose you now mold a 1-cm-diameter spherical ball of 1,4-polyisoprene and place it on a flat surface at 25 °C and find that the time for it to sag to 90% of its original height is 10 minutes. Now you place the polyisoprene in a rheometer at 25°C to measure its viscosity, but the viscosity is too high to measure accurately, so you raise the temperature to lOO C and measure a zero-shear viscosity of 10 P. Use this information and that in Problem 3.4(a) to determine the viscosity of the gel in Problem 3.4(a), given that the gel density is = 3g/cm ... 

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). 

Problem 6.1 (Worked Example) Estimate the zero-shear viscosity of a suspension of hard spheres 100 nm in diameter at a volume fraction of [Pg.318]

Problem 10.3(b) (Worked Example) Derive Eq. (10-31), the time- or strain-dependent shear viscosity in the absence of Frank elastic stresses. 

As a result of the secular growth of the /-body collision integrals with time, we are compelled to conclude that, although the cluster expansion method can be used successfully to derive the Boltzmann equation from the liouville equation and to obtain corrections to the Boltzmann equation, there are serious difficulties in trying to represent these corrections as a power series in the density. An example of the difficulties that appear if one attempts to apply the generalized Boltzmann equation as it stands now to a problem of some interest is provided by the calculation of the density expansion of the coefficient of shear viscosity. By constructing normal solutions to the generalized Boltzmann equation, one finds that the viscosity 17 has the expansion of the form mentioned in Eq. (224),... 

Since it is the long-time behavior that is closely related to molecular structure, this is the information that is most interesting in the present context. For example, the zero-shear viscosity describes behavior in the limit of zero frequency and is very sensitive to molecular weight. However, for a material whose longest relaxation time is quite large, neither step-strain nor oscillatory shear experiments are useful to probe the behavior at very long times or very low frequencies. The main problem is that the stress is so small that it is not possible to measure it precisely. It is in this region that creep measurements are most useful. This is because it is possible to make a very precise measurement of a displacement, and it is also possible to apply a very small controlled stress. Controlled-torque (controlled-stress) rheometers are available from several manufacturers. 

The second approach to the problem of inferring a spectrum function from dynamic data is directed at establishing as accurately as possible the spectrum that is a material property of the material under study rather than simply fitting experimental data. To accomplish this, it is necessary to overcome the ill-posedness of the problem by providing information in addition to the experimental data. For example, Mead [35] proposed the addition of the zero-shear viscosity and the steady-state compliance to the data set. We can readily see how this constrains the discrete spectrum by noting that ... 

Predicting pressure profiles in a disc-shaped mold using a shear thinning power law model [1]. We can solve the problem presented in example 5.3 for a shear thinning polymer with power law viscosity model. We will choose the same viscosity used in the previous example as the consistency index, m = 6,400 Pa-sn, in the power law model, with a power law index n = 0.39. With a constant volumetric flow rate, Q, we get the same flow front location in time as in the previous problem, and we can use eqns. (6.239) to (6.241) to predict the required gate pressure and pressure profile throughout the disc. 

Problem 6.7(a) (Worked Example) Estimate the first normal stress difference Ni for a suspension of long, thin particles (approximated as spheroids) with p = 100 and L = 0.1/ m, if the solvent viscosity is 1 P, the shear rate y is 100 sec, and the particle concentration is 0 = 0.001, which is in the dilute regime. 

Other errors, which could influence the results obtained, are, for example, wall effects ( slipping ), the dissipation of heat, and the increase in temperature due to shear. In a tube, the viscosity of a flowing medium is less near the tube walls compared to the center. This is due to the occurrence of shear stress and wall friction and has to be minimized by the correct choice of the tube diameter. In most cases, an increase in tube diameter reduces the influence of wall slip on the flow rate measured, but for Newtonian materials of low viscosity, a large tube diameter could be the cause of turbulent flow. ° When investigating suspensions with tube viscometers, constrictions can lead to inhomogeneous particle distributions and blockage. Due to the influence of temperature on viscosity (see Section Influence Factors on the Viscosity ), heat dissipated must be removed instantaneously, and temperature increase due to shear must be prevented under all circumstances. This is mainly a constructional problem of rheometers. Technically, the problem is easier to control in tube rheometers than in rotating instruments, in particular, the concentric cylinder viscometers. ... 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

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