Shear flow oscillatory

As a result of the above scaling properties, the shear flow problem outlined previously in subsection 5.5.4 can be reduced from one involving the gap width d and the relative shear velocity to a problem with unit gap width and relative velocity V where 

As mentioned by Leslie [168, 171], if there is a unique solution n to this problem (via the solution for 0) then it follows that 

We know from the result in (5.150) that the apparent viscosity rj depends on the product Vd. It is also the case, following on from (5.146), (5.147), (5.152) and (5.156), that with d=2h and q = (P, as selected above, we have 

In this Section an investigation will be made of the response of a nematic liquid crystal to an induced oscillatory shear flow. After deriving some possible solutions 

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Typical for the spectroscopic character of the measurement is the rapid development of a quasi-steady state stress. In the actual experiment, the sample is at rest (equilibrated) until, at t = 0, oscillatory shear flow is started. The shear stress response may be calculated with the general equation of linear viscoelasticity [10] (introducing Eqs. 4-3 and 4-9 into Eq. 3-2)... 

Makinen R, Ruokolainen J, Ikkala O, De Moel K, ten Brinke G, De Odorico W, Stamm M. Orientation of supramolecular self-organized polymeric nanostructures by oscillatory shear flow. Macromolecules 2000 33 3441-3446. 

Wiesner U (1997) Lamellar diblock copolymers under large amplitude oscillatory shear flow order and dynamics. Macromol Chem Phys 198 3319-3352... 

For a small amplitude oscillatory shearing flow, the strain is defined as,... 

Fig. 21 SAXS of the self-assembled lamellar-wfr/iin-lamellar structures of PS-1)-P4VP (TSA)i.o(PDP)i.o macroscopically oriented by oscillatory shear flow [155]...

A dimensionless quantity called the Deborah number, De, is defined as the fluid s characteristic relaxation time t divided by a time constant tf characterizing the flow (Reiner 1964). Thus, De = t///. In an oscillatory shearing flow, for example, we might take tf to be the inverse of the oscillation frequency (o, and then De = xo). At high Deborah number, the flow is fast compared to the fluid s ability to relax, and the fluid will respond like a solid, to some extent. Thus, in an oscillatory shearing flow, when De = cur 3> 1 the complex modulus is solid-like, while when De = 1 a liquid-like terminal behavior is... 

All the above theories are derived for rigid rod nematic liquid crystal systems. The rheological behavior of chiral nematic liquid crystals is more complex and less understood than that of nematic systems. Rey introduced a model based on rigid rod chiral nematic liquid crystals to describe permeation shear flow and small amplitude oscillatory shear flow. The model can predict some common phenomena of chiral nematic liquid crystals, e.g., the three-region... 

There have been fewer studies in electrochemistry where the flow is known but the boundary-layer approach is inapplicable. One example has been recently analyzed and compared with experiment. In this case, mass transfer to a line electrode or an array of line electrodes in the presence of an oscillatory shear flow was treated. A finite-volume approach was used for the numerical analysis and a ferri/ferrocyanide redox couple was used to measure the mass-transfer rate. The studies show that boundary-... 

Following Doi and Ohta s work, a more general theory was derived for immiscible polymer blends by Lee and Park [1994]. A constitutive equation for immiscible blends was proposed. The model and the implied blending laws were verified by comparison with dynamic shear data of PS/LLDPE blends in oscillatory shear flow. 

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

Lee et al. [2007] studied the rheological behavior of poly(ethylene-co-vinyl acetate) (EVAc 40 wt% VAc) and its CPNC with < 10 wt% C30B the tests were conducted under steady-state and small oscillatory shear flow. The samples were prepared by melt compounding at 110 C for 25 min, which resulted in a high degree of dispersion. The flow behavior was quite regular, well described by the Carreau-Yasuda equation [Carreau, 1968,1972 Yasuda, 1979] ... 

RHEOLOGY OF POLYMERS WITH NANOFILLERS 16.2.3 Small-Amplitude Oscillatory Shear Flow... 

Because of the complications caused by the stress-induced orientation of clay platelets resulting in different rheological responses, the studies of CPNC flow focus on smaU-amplitude oscillatory shear flow (SAGS). As the discussion on the steady-state flow indicates, there is a great diversity of structures within the CPNC family. Whereas some nanocomposites form strong three-dimensional structures, others do not thus while nonlinear viscoelastic behavior is observed for most CPNCs, some systems can be smdied within the linear regime. 

Prigogine, Trappeniers, and Mathot pressure-volume-temperature measurements lead zirconate titanate quaternary ammonium salts quasi-two-parameter theory rigid amorphous fraction Rheometrics extensional rheometer Rheometrics elongational rheometer for melts room temperature small-angle neutron scattering small-amplitude oscillatory shear flow small-angle x-ray scattering side-chain LCP... 

Zhang, R. Y., H. Cheng, C. G. Zhang, T. C. Sun, X. Dong, and C. C. Han. 2008. Phase separation mechanism of polybutadiene/polyisoprene blends under oscillatory shear flow. Macromolecules 41 6818-6829. 

We point out parenthetically that in the kinetic theory of dilute gases it is just the deviation from the Maxwellian velocity distribution that is of primary interest in the evaluation of the transport properties. In the kinetic theory of polymers, on the other hand, it has been assumed that the deviations from the Maxwellian distribution are of minor importance, and to date few calculations or estimations have been made of the errors introduced by this assumption [16-19], These exploratory efforts indicate that there may be a significant effect on the components of the complex viscosity in high-frequency oscillatory shearing flows. 

So, the mixtures show a viscoelastic behavior which is very different from neat PPS. Strong solid-like responses in the small amplitude oscillatory shear flow are observed. After quiescent annealing, a strainscaling transient stress behavior is observed in the mixtures in the reverse flow [61]. 

Rheological properties under steady state and oscillatory shear flow of isotropic and nematic solutions of PpPTA, PBT and PBO were studied by Baird [70] and Berry et al. [46]. Baird observed shear thinning for a series of PpPTA solutions in sulfuric acid (4-15%). These results also suggest that at higher shear rates very little difference exists between the anisotropic and isotropic phases. Steady-state viscosities as a function of the temperature observed for solutions of PBO in methane sulfonic acid showed a sharp increase near T,, a behavior which has also been reported for PpPTA and PpBA solutions [46],... 

Yu, W., Bousmina, M., Grmela, M., and Zhou, C. (2002) Modeling of oscillatory shear flow of emulsions under small and large deformation flelds./. Rhed., 46 (6), 1401-1418. 

Consider the linear response of the affine network (9.26) to a small oscillatory shear flow defined by the deformation tensor... 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

Therefore, it is more suitable to measure the shear sensitivity of polypropylene. Capillary rheometers or oscillatory shear flow rheometers are widely used for that purpose. Moreover, an investigation of elongational flow properties of molten PP can be used to check, for example, the presence of long-chain branching in some speciality grades of PP (to study the strain-hardening effect). 

Study of rheological properties and oscillatory shear flow... 

Hausnerova et al. [33] studied the rheological properties of carbon fiber-filled low-density polyethylene. These woikers investigated rheological properties of carbon-filled low-density polyethylene melts undergoing parallel superposed steady and oscillatory shear flows. It was also found that the critical angular frequency where the phase angle becomes 90° and storage modulus decreases sharply to zero. This was mainly dependent on the fiber volume fraction. 

The rheological behavior of a 1% w/w solution of hydrophobically modified (hydroxypropyl) guar (HMHPG) in water was investigated by Aubry and Moan (1996) in the presence of a nonionic surfactant. The response to steady and oscillatory shear flow, at different surfactant concentrations around the CMC, showed different behaviors below and above the CMC point. Below the CMC, a reinforcement of the intermolecular hydrophobic network occurs due to an increase in the number of intermolecular hydrophobic associations. Above the CMC, the intermolecular hydrophobic network is destroyed. 

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).

With the use of the emulsion time t = t]mR/r, where R is the radius, Eq. 10 is made dimensionless. In Eq. 10, I is the second rank unit tensor, D and are the deformation rate and the vorticity tensors, respectively, II is the second scalar invariant of tensor and the capillary number is Ca = yocor mRt f, see also [30]. At rest, the droplet is spherical with = I. Please note that droplet break-up will not occur under oscillatory shear flow as long as the ratio of the viscosities is greater than or equal to 2.5, i.e. that A = Tjm/rjd > 2.5 [15]. 

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