Shear-thickening/-thinning

For a Hquid under shear the rate of deformation or shear rate is a function of the shearing stress. The original exposition of this relationship is Newton s law, which states that the ratio of the stress to the shear rate is a constant, ie, the viscosity. Under Newton s law, viscosity is independent of shear rate. This is tme for ideal or Newtonian Hquids, but the viscosities of many Hquids, particularly a number of those of interest to industry, are not independent of shear rate. These non-Newtonian Hquids may be classified according to their viscosity behavior as a function of shear rate. Many exhibit shear thinning, whereas others give shear thickening. Some Hquids at rest appear to behave like soHds until the shear stress exceeds a certain value, called the yield stress, after which they flow readily. 

The fluid may be either shear-thinning or, less often, shear-thickening, and in either case the shear- stress and the apparent viscosity fia are functions of shear rate, or ... 

Thus, by selecting an appropriate value of n, both shear-thinning and shear-thickening behaviour can be represented, with n = 1 representing Newtonian behaviour which essentially marks the transition from shear-thinning to shear-thickening characteristics. 

In general, for shear-thinning pseudoplastic fluids the apparent viscosity will gradually decrease with time if there is a step increase in its rate of shear. This phenomenon is known as thixotropy. Similarly, with a shear-thickening fluid the apparent viscosity increases under these circumstances and the fluid exhibits rheopexy or negative-thixotropy. 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

Compared with the parabolic profile for a Newtonian fluid (n = 1), the profile is flatter for a shear-thinning fluid ( < 1) and sharper for a shear-thickening fluid (n > l). The ratio of the centre line (uCl) to mean (k) velocity, calculated from equation 3.133, is ... 

Show how. by suitable selection of the index n, the power law may be used to describe the behaviour of both shear-thinning and shear-thickening non-Newtonian fluids over a limited range of shear rates. What are the main objections to the use of the power law Give some examples of different types of shear-thinning fluids. 

Shearing stress velocity 704, 715 Shear-thickening 106, 111, 121 Shear-thinning 106, 196... 

It should be noted that for shear thinning and shear thickening behaviour the shear stress-shear rate curve passes through the origin. This type of behaviour is often approximated by the power law and such materials are called power law fluids . Using the negative sign convention for stress components, the power law is usually written as... 

Clearly, shear thinning behaviour corresponds to nshear thickening behaviour to n> 1. The special case, n = 1, is that of Newtonian behaviour and in this case the consistency coefficient K is identical to the viscosity fx. Values of n for shear thinning fluids often extend to 0.5 but less commonly can be as low as 0.3 or even 0.2, while values of n for shear thickening behaviour usually extend to 1.2 or 1.3. 

Equation 3.29 is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, n = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate yw is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, yw is of a... 

Nienow and Elson (1988) have reviewed work done mainly by them and their co-workers on the mixing of non-Newtonian liquids in tanks. The above approach for inelastic, shearing thinning liquids has been largely substantiated but considerable doubt has been cast over using this method for dilatant, shear thickening materials. 

The flow behaviour of aqueous coating dispersions, because of their high pigment and binder content, is often complex. They have viscosities which are not independent of the shear rate and are therefore non-Newtonian. Shear thickening (when the viscosity of the dispersion increases with shear rate) and shear thinning or pseudoplastic behaviour (when the viscosity decreases with shear rate), may... 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n ... 

Bingham fluids that are either shear-thinning or shear-thickening above their yield stresses have corresponding power-law expressions incorporated into their viscosity models. 

Fluids with shear stresses that at any point depend on the shear rates only and are independent of time. These include (a) what are known as Bingham plastics, materials that require a minimum amount of stress known as yield stress before deformation, (b) pseudoplastic (or shear-thinning) fluids, namely, those in which the shear stress decreases with the shear rate (these are usually described by power-law expressions for the shear stress i.e., the rate of strain on the right-hand-side of Equation (1) is raised to a suitable power), and (c) dilatant (or shear-thickening) fluids, in which the stress increases with the shear rate (see Fig. 4.2). 

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