Dynamic oscillatory flow

Fig. 2.5. Steady-state and dynamic oscillatory flow measurements on a 2 wt. per cent solution of polystyrene S 111 in Aroclor 1248 according to Philippoff (57). ( ) steady shear viscosity (a) dynamic viscosity tj, ( ) cot 1% from flow birefringence, (A) cot <5 from dynamic measurements, all at 25° C. (o) cot 8 from dynamic measurements at 5° C. Steady-state flow properties as functions of shear rate q, dynamic properties as functions of angular frequency m. Shift factor aT which is equal to unity for 25° C, is explained in the text, cot 2 % and cot 8 are expressed in terms of shear (see eqs. 2.11 and 2.22)...

FIGURE 6.4b Dynamic oscillatory flow measurements. Source Xia et al. (2001). 

Dynamic shear rheology involves measuring the resistance to dynamic oscillatory flows. Dynamic moduli such as the storage (or solid-like) modulus (G ), the loss (or fluid-like) modulus (G"), the loss tangent (tan 8 = G"IG ) and the complex viscosity ( / ) can all be used to characterize deformation resistance to dynamic oscillation of a sinusoidally imposed deformation with a characteristic frequency of oscillation (o). 

Our intent, as mentioned earlier, is not to review all the studies concerned with liquid crystalline fluids but to compare their properties with flexible chain polymers, interpret their properties in terms of the domain structure, and look for correlations between flow characteristics and processing conditions. We first examine the behavior of liquid crystalline copolyesters in steady shear flow and in small strain dynamic oscillatory flow. 

One of the most common transient shear flows used to analyze the rheological characteristics of macromolecular fluids is that of small strain dynamic oscillatory flow. Measurements are usually carried out at strain levels where the stresses are directly proportional to strain. In this experiment, components of stress in phase (G", which is called the loss modulus) with strain rate and those out of phase (G, which is called the storage modulus) are measured as a function of angular frequency (w). G represents... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

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. 

Steady shear flow measnrements, however, can measure only viscosity and the first normal stress difference, and it is difficult to derive information abont fluid structure from such measurements. Instead, dynamic oscillatory rheological measurements are nsed to characterize both enhanced oil recovery polymer solutions and polymer crosslinker gel systems (Prud Homme et al., 1983 Knoll and Pmd Homme, 1987). Dynamic oscillatory measurements differ from steady shear viscosity measnrements in that a sinusoidal movement is imposed on the fluid system rather than a continnons, nnidirectional movement. In other words, the following displacement is imposed ... 

It is clear that the peripheral resistance decreases, while compHance increases with mammalian body size. Thus, the dynamic features of blood pressure and flow pulse transmission can be scaled through this kind of modehng. The ratio of Zq/Hs corresponds to the ratio of pulsatile energy loss due to oscillatory flow to the energy dissipated due to steady flow (to overcome R ) and has been reported to be between 5 and 10% and is an invariant for the mammalian arterial circulation [Li, 1996,2004]. 

Acoustic streaming is a secondary steady flow generated from the primary oscillatory flow. It includes not only the Eulerian streaming flow caused by the fluid dynamical interaction but also the Stokes drift flow which arises from a purely kinematic viewpoint. 

As mentioned above, interfacial films exhibit non-Newtonian flow, which can be treated in the same manner as for dispersions and polymer solutions. The steady-state flow can be described using Bingham plastic models. The viscoelastic behavior can be treated using stress relaxation or strain relaxation (creep) models as well as dynamic (oscillatory) models. The Bingham-fluid model of interfacial rheological behavior (27) assumes the presence of a surface yield stress, cy, i.e.. 

Table 10.7 provides some of the current instrumentation for rheological measurements. Note that some of them are designed for flow, while others are designed for dynamic oscillatory measurements, while still others are basically uniaxial extension or creep instrumentation. 

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... 

Figure 3.9. A comparison of (a) steady shear and (b) dynamic oscillatory shear flows together with (c) the stress and strain response and the associated mathematical expressions (after Knoll and Prud homme, 1987).

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

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