Oscillatory Shear Flow Solutions

It is supposed that the director is strongly anchored parallel to the bounding plates and that the lower plate is fixed while the upper plate is subject to sinusoidal oscillations parallel to the initial alignment, as indicated in Fig. 5.7. As pointed out by Leslie [169, 171, 174], in a preliminary investigation it appears reasonable to suppose that both the surface and inertial effects are of secondary importance and concentrate on the influence of the oscillatory flow induced by the oscillating upper plate. This is equivalent to supposing that the associated Reynolds number for this problem, defined by 

In these circumstances, motivated by the set-up in Fig. 5.7, it is appropriate in this basic investigation to seek solutions to the Ericksen-Leslie dynamic equations of the form 

The constraints (4.118) are obviously satisfied. The governing dynamic equations (4.139) and (4.140), in the absence of any external body force Fi or generalised body force Gi are 

Following Clark et al [46], the fluid inertial contribution pvi = pdv/dt may be neglected because experimental sample depths are often very thin, or it may be omitted for the reasons suggested by Pieranski et al [220]. Therefore we can solve for the balance of linear momentum by setting the pressure p to be 

Multiplying equation (5.177) by sinP and equation (5.178) by cos and subtracting eliminates the Lagrange multiplier A, finally leaving the single governing dynamic equation 

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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],... 

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. 

Oscillatory shear flow has long been used to characterize the linear viscoelastic properties of polymer solutions and melts. In Chapter 5 we describe the basic principles of such experiments. In this section we present the material functions for small-amplitude oscillatory shear flow using the constitutive equations presented in the preceding section. 

Rouse (1953) also derived Eqs. (4.84) and (4.85) from the point of view of the work done on a polymer solution under oscillatory shear flow. 

One of the unique properties of soft matter is its viscoelastic behavior [13], Because of the long structural relaxation times, the internal degrees of freedom cannot relax sufficiently fast in an oscillatory shear flow already at moderate frequencies, so that there is an elastic restoring force which pushes the system back towards its previous state. Well-studied examples of viscoelastic fluids are polymer solutions and polymer melts [6,13],... 

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... 

Assuming that the flow is described by the set of equations (9.3) and (9.4), one can use equations (9.14) for arbitrary time dependence of velocity gradient, to obtain for oscillatory simple shear the solution in the form... 

Both strain- and stress-controlled rotational rheometers are widely employed to study the flow properties of non-Newtonian fluids. Different measuring geometries can be used, but coaxial cylinder, cone-plate and plate-plate are the most common choices. Using rotational rheometers, two experimental modes are mostly used to study the behavior of semi-dilute pectin solutions steady shear measurements and dynamic measurements. In the former, samples are sheared at a constant direction of shear, whereas in the latter, an oscillatory shear is used. 

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 ... 

If the particular product is not very non-Newtonian, a single-point measurement might be sufficient. Even then however we must be sure that the flow type is the same. For instance if our experience off-line is built up from steady-state, simple-shear flow curves, then a too-fast measurement might mean the measurement is not made under steady-state conditions. Equally if the on-hne flow has an appreciable extensional component, then problems can arise for some hquids, especially pol3nner solutions. Also if a vibrational mode is used, then it is probable that some non-hnear oscillatory function is being measured. Ah these facts could mean that we end up with no simple one-to-one correlation between on-hne and off-hne. Hence the safest way is to duphcate on-hne what is done off-hne. This is possible nowadays for most situations. 

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)...

Another parameter that can be related to molecular weight is the relative viscosity, defined as the ratio of the viscosity of a polymer solution rj to the viscosity of the solvent rjo (see Table 3.3). Reliable for molecular weights >10 g/mol, the viscosities can be determined by measuring flow times through capillary tubes (diameters 1 mm), usually with gravity as the driving force for the flow. Automated instrumentation is widely available. Rotational and oscillatory type viscometers are used where a uniform, well-defined, or low shear rate is required. The ratio of the two viscosities, is called the relative... 

The rheological behavior of the copolymers was measured with a DynAlyser 100 stress-control rheometer (EUieoLogica) equi] d with a cone and plate at 25 C. The radius of the cone is 40 mm, and the angle between the cone and plate is 4.0°. Steady shear and oscillatory flow measurements were conducted to obtain the steady shear viscosity and dynamic viscoelastic properties of polymer solutions. 

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