Steady simple shear

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Note 3 fy t)=yt, where is a constant, then the flow has a constant shear rate and is known as steady (simple) shear flow. 

Quotient of shear stress (an) and shear rate (y) in steady, simple shear flow... 

First normal stress function, pt t — p22 at steady state in steady simple shear flow. 

To successfully measure non-Newtonian fluids, a known shear field (preferably constant) must be generated in the instrument. Generally, this situation is known as steady simple shear. This precludes the use of most single-point viscometers and leaves only rotational and capillary devices. Of these, rotational devices are most commonly used. To meet the criterion of steady simple shear, cone and plate, parallel plates, or concentric cylinders are used (Figure HI. 1.1). 

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

Rheological Response of Polymer Melts in Steady Simple Shear-Flow Rheometers... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

In the previous sections, the non-Newtonian viscosity rj) was used to characterize the rheology of the fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by... 

Steady simple shear of a liquid is accomplished by confining the liquid... 

In steady simple shear, the top plate in Fig. 7.23 is moved at a constant velocity v. The shear rate 7 = v]//t is a time-independent constant that can be pulled out of the Boltzmann superposition integral ... 

The last relation was obtained using the variable transformation s=t — t, which implies ds= — dt The integration limits change with this transformation because when F = —00, s = oc, and at t — t, s — 0. For any liquid, the relaxation modulus G(t) eventually decays to zero fast enough that the integral in the above equation is simply a number with units of stress time. Thus, the stress at long times in the steady simple shear experiment is constant, and proportional to the shear rate 7. Newton s law of viscosity [Eq. (7.100)] already defined the viscosity in steady shear as the ratio of shear stress and shear rate. Therefore, the viscosity of any liquid is the time integral of its stress relaxation modulus ... 

The mechanical properties of a liquid are fundamentally different from the solids discussed in Chapter 7. Solids have stress proportional to deformation (for small deformations). However, the stress in liquids depends only on the rate of deformation, not the total amount of deformation. If we pour water from one bucket into another bucket, there is only resistance during the flow, but there is no shear stress in the water in either bucket at rest. We describe the deformation rate of a liquid in shear by the shear rate 7 = d /dt [Eq. (7.99)]. For the steady simple shear flow of Fig. 7.23, the shear rate is the same everywhere, equal to the way in which velocity changes with vertical position. The stress a in a Newtonian liquid is proportional to this shear rate [Newton s law of viscosity Eq. (7.100), cr = 777], with the viscosity rj being the coefficient of -proportionality. 

Manneville, P. Dubois-Violette, E. Shear flow instability in sheared nematic liquids theory steady simple shear flows. J. Phys. Paris 1976,37,285-296. 

Following a procedure similar to that of Tokita [1977], for equilibrium drop diameter in steady simple shear flow, the following dependence was proposed [Fortelny et al., 1988, 1990] ... 

Cho and Kamal (2002) derived equations for the affine deformation of the dispersed phase, using a stratified, steady, simple shear flow model. It includes the effects of viscosity ratio and volume fraction. According to the equation, for viscosity ratio > 1, the deformation of the dispersed phase increases with the increase of the dispersed phase fraction. For compatibiUzed PE/PA-6 blends at high RPM (i.e., 100, 150, and 200 RPM) in the Haake mixer, the particle size decreases with concentration of the dispersed phase up to 20 wt%. This occurs because the total deformation of the dispersed phase before breakup increases as the volume fraction increases, and coalescence is suppressed. The increase of the particle sizes between 20 and 30 wt% results from the increase of coalescence due to the high dispersed phase fractions. The data for 1 wt% blends suggest that mixing in the Haake mixer follows the transient deformation and breakup mechanism, and that shear flow is dominant in the mixer. 

In a steady simple shear flow, the maximum connector force develops when the dumbbell is oriented at a 45° angle to the direction of shear this force is ... 

Consider a steady simple shear flow with the kinematics given by u = yi3JC3, U2 = 0, Ms = 0 where the shear rate yj3 is a constant. This flow has the rate of deformation... 

The generalized strain rate is y = yi3. There is a class of restricted flows called viscometric flows, which are motions equivalent to steady simple shearing. Tanner (2000) has show various viscometric kinematic fields where each fluid element is undergoing a steady simple shearing motion, with streamlines that are straight, circular, or helical. Each flow can be viewed as a relative sliding motion of a shear of inextensible material surfaces, which are called slip surfaces. 

Now, a remaining problem is how to determine the value of the interaction coefficient. An approach was presented by Bay (1991) based on experimental values of an in steady simple shear flows for different concentrations. These experimental results were fitted to the numerically calculated an and the value of Ci that best fits the experimental results was obtained. The polymer matrices used in the experiments were nylon, polycarbonate and polybutylene terephthalate. The data of Cl were then plotted against a cp to give an empirical relationship... 

Rheology studies the relationship between force and deformation in a material. To investigate this phenomenon we must be able to measure both force and deformation quantitatively. Steady simple shear is the simplest mode of deforming a fluid. It allows simple definitions of stress, strain, and strain rate, and a simple measurement of viscosity. With this as a basis, we will then examine the pressure flow used in capillary rheometers. 

The flow in Fig. 3 is called a drag flow, the top plate is dragging the material across the stationary plate to create the velocity profile that is shearing the fluid. In contrast, flow in a capillary rheometer is pressure-driven flow. All of the wall area inside the capillary is stationary so that the material has zero velocity at the walls and a maximum velocity along the centerline. Calculating the shear rate in a capillary is not as straightforward as with steady simple shear. Each fluid element still sees steady simple shear, but the shear rate is no longer constant it varies across the radius of the die. It runs... 

The steady simple shear flow in Fig. 3 is fully controllable, because the velocity profile of the flow depends only on the geometry and motion of the plates, regardless of the rheological properties of the fluid. Fully developed flow in a capillary tube is only partially controllable. The material properties can affect the velocity proflle, which must be known in order to calculate the shear rate that the sample is experiencing. If a Newtonian fluid... 

Fluid deformation under steady simple shear flow can be aptly described by considering the situation in Figure 2.1 wherein the fluid is held between two large parallel plates separated by a small gap d 2 and sheared as shown. 

Small-amplitude oscillatory flow is often referred to as dynamic shear flow. Fluid deformation under d)mamic simple shear flow can be described by considering ttie fluid wiflun a small gap dX2 between two large parallel plates of which the upper one undergoes small amplitude oscillations in its own plane with a frequency velocity field within the gap can be given by d , = ydxj but y is not a constant as in steady simple shear. Instead it varies sinusoidally and is given by... 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

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