Simple shear flow start

We start with the ground state (°), fi(° defined by the simple shear flow y(°), Fig. 17. The principal effect is, as expected, the appearance of a small tilt of the director from the layer normal (flow alignment), predominantly in z direction (Fig. 18). Note that the configuration of layers is also modified by the shear (Figs. 19 and 20), i.e., the cylindrical symmetry is lost. This is analogous to the shear-flow-induced undulation instability of planar layers (wave vector of undulations in the... 

The stress in viscoelastic liquids at steady-state conditions is defined, in simple shear flow, by the shear rate and two normal stress differences. Chapter 13 reviews the evolution of both the normal stress differences and the viscosity with increasing shear rate for different geometries. Semiquantitative approaches are used in which the critical shear rate at which the viscosity starts to drop in non-Newtonian fluids is estimated. The effects of shear rate, concentration, and temperature on die swell are qualitatively analyzed, and some basic aspects of the elongational flow are discussed. This process is useful to understand, at least qualitatively, the rheological fundamentals of polymer processing. 

In this section, we consider one example of this type of problem, namely, the start-up from rest of the flow in a circular tube when a nonzero pressure gradient is suddenly imposed at some instant (which we shall denote as i = 0) and then held at the same constant value thereafter (t > 0). In Section G, we consider the start-up of simple shear flow between two infinite plane boundaries. Other relatively simple examples of start-up flows are left to the reader to solve (for example, see Problem 19 at the end of this chapter). 

Figure 3-12. The velocity profiles at different times t for start-up of simple shear flow between two infinite, parallel plane walls.

Sibillo V et al. Start-up and relraction dynamics of a Newtonian drop in a viscoelastic matrix under simple shear flow. J Non-Newtonian Fluid Mech 2006 134(l-3) 27-32. 

It seems, therefore, useful to take the numerically simple laminar shear flow as a starting point for the description. As recently published (Bouldin), an easily manageable model has been derived which enables an ad hoc prediction to be made of the critical sfteanate at which mechanical chain scission takes place (cf. Fig. 34). 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

In this section, we return to the analysis of simple, unidirectional shear flow that was considered in Section B of Chap. 3, but instead of neglecting viscous dissipation altogether, we consider its influence when the Brinkman number is small, but nonzero. The starting point is Eqs. (3-34) and (3-35), which are reproduced here for convenience ... 

In Section 10.8.1 it was noted that molecular orientation results in flow birefringence in a polarizable polymer, and if the melt is transparent, optical techniques can be used to determine the three components of the stress tensor in uniform, shear flows [91-93]. To determine the transient normal stress differences, the phase-modulated polarization technique was developed by Frattini and Fuller [ 126]. Kalogrianitis and van Egmond [ 127] used this technique to determine the shear stress and both the normal stress differences as functions of time in start-up of steady simple shear. Optical techniques are particularly attractive for measurements of normal stress differences, since such methods do not require the use of a mechanical transducer, whose compliance plagues measurements of normal stress differences by mechanical rheometry. 

When comparing extensional flow data with the LVE prediction, care must be taken in the calculation of the linear prediction. If the data used to establish the relaxation spectrum do not include very short-time (high-frequency) data, the initial portion of the curve will not be correct. It may thus be better to use data from start-up of steady simple shear to measure tfit) directly rather than inferring it from complex modulus data. 

If a sample shows elastic, solid-like deformation below a certain shear stress ay and starts flowing above this value, ay is called a yield stress value. This phenomenon can occur even in solutions with quite low viscosity. A practical indication for the existence of a yield stress value is the trapping of bubbles in the liquid Small air bubbles that are shaken into the sample do not rise for a long time whereas they climb up to the surface sooner or later in a liquid without yield stress even if their viscosity is much higher. A simple model for the description of a liquid with a yield stress is called Bingham s solid ... 

The structure of Immiscible blends Is seldom at equilibrium. In principle, the coarser the dispersion the less stable It Is. There are two aspects of stability Involved the coalescence In a static system and deformability due to flow. As discussed above the critical parameter for blend deformability Is the total strain In shear y = ty, or In extension, e = te. Provided t Is large enough In steady state the strains and deformations can be quite substantial one starts a test with one material and ends with another. This means that neither the steady state shearing nor elongatlonal flow can be used for characterization of materials with deformable structure. For these systems the only suitable method Is a low strain dynamic oscillatory test. The test Is simple and rapid, and a method of data evaluation leading to unambiguous determination of the state of miscibility is discussed in a later chapter. 

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