Steady-state shear-dependent behaviour

In this section, consideration will be given to the equilibrium relationships between shear stress and shear rate for fluids exhibiting non-Newtonian behaviour. Whenever the shear stress or the shear rate is altered, the fluid will gradually move towards its new equilibrium state and, for the present, the period of adjustment between the two equilibrium states will be ignored. 

The fluid may be either shear-thinning or, less often, shear-fliickening, and in either ease the shear stress and the apparent viscosity pa are functions of shear rate, or  

Here y is distance measured from a boundary surface. 

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter p, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

A useful two-parameter model is the power-law model, or Ostwald-de Waele law to identify its first proponents. The relation between shear stress and shear rate is given by  

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Steady state shear-dependent behaviour is discussed in Volume 1, Section 3.7.1. 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Two liquid crystalline polybenzylglutamate solutions, adjusted to the same Newtonian viscosity, have been investigated Theologically. The steady state shear properties and the transient behaviour are measured. For the same kind of polymer, the dynamic moduli upon cessation of flow can either increase or decrease with time. This change in dynamic moduli shows a similar dependency on shear rate as the final portion of the stress relaxation but no absolute correlation exists between them. By comparing the transient stress during a stepwise increase in shear rate with that during flow reversal the flow—induced anisotropy of the material is studied. 

Repeated shear cycles on the test sample will enable one to determine whether the sample exhibits time-dependent flow behaviour such as thixotropy. If the up and down curves for the first and successive cycles coincide, the sample is undergoing steady-state shear. However, if hysteresis loops between the up and down curves are observed for each successive cycle, the sample is exhibiting time-dependent flow behaviour. In such cases, it is advisable to repeat the experiment with the speed (or torque) held constant until the torque (or speed) attains a steady value before changing the speed (or torque) to the next value. This will yield an equilibrium flow curve in which the up and down curves coincide. 

Analysis of flow curves of these polymers has shown that for a nematic polymer XII in a LC state steady flow is observed in a broad temperature interval up to the glass transition temperature. A smectic polymer XI flows only in a very narrow temperature interval (118-121 °C) close to the Tcl. The difference in rheological behaviour of these polymers is most nearly disclosed when considering temperature dependences of their melt viscosities at various shear rates (Fig. 20). 

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