Newtonian behaviour, limit

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, 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. 

Many liquids display Newtonian behaviour under a wide range of shear rates, and many more show this behaviour within limited ranges. If we limit ourselves to everyday shear rates, then the liquids in table 1 show Newtonian behaviour and encompass a large range of viscosities, best shown—like the race distances of chapter 3 —on a logarithmic basis. 

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. 

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. 

Only a very limited amount of data is available on the motion of particles in non-Newtonian fluids and the following discussion is restricted to their behaviour in shear-thinning power-law fluids and in fluids exhibiting a yield-stress, both of which are discussed in Volume 1, Chapter 3. 

Note that in the limit of ti/2 oo, i.e. for Newtonian fluid behaviour, this equation reduces to the Hagen-Poiseuille equation. 

Thus, the index m is the slope of the log-log plots of the wall shear stress Xw versus (8V/D) in the laminar region (the limiting condition for laminar flow is discussed in Section 3.3). Plots of x versus (8V/D) thus describe the flow behaviour of time-independent non-Newtonian fluids and may be used directly for scale-up or process design calculations. 

Re = pVl x"jm where jc is the distance (from the leading edge) at which the flow ceases to be streamline. For Newtonian fluids (n = 1), B(n) = 0.323 and RCc = 10 which is the value at the transition point. For a Uquid of flow behaviour index, n = 0.5, the limiting value of the Reynolds mnnber is 1.14 X lO which is an order of magnitude smaller than the value for a Newtonian fluid. 

Drag reduction is also obtained in the case of laminar flow of polymer solution, in the limits of usual concentrations, accompanied by a slightly increase of viscosity without to modify the linear equation shear-stress/shear-flow rate. Figure 3.378, the behaviour being typical newtonian one. Figure 3.379. Based on the last Figure a small increase a friction coefficient could be expected but not an important drag reduction as was found. 

The Type 2 products comprising cross-linked polymers micro-particles, highly swollen by water, have a different profile. Figure 3.10 compares the profile of Polymer B (Type 1) with Polymer D (Type 2). The cross-linked polymer shows shear thinning behaviour across a very wide range of shear rates with little evidence of either Newtonian Plateau. Both these plateau regions may exist, but the extreme limits of instrument conditions maybe needed before these are apparent on the viscosity versus shear rate graph. 

For non-Newtonian media with a yield stress some particles do not settle in quiescent conditions, and this observation has led to the concept of stable slurries. However, an externally-applied shear in the medium may initiate particle settling, with possible deleterious effects for pipeline transport. It was decided to add to the rather limited existing data in this area by experimenting with a novel cup-and-bob apparatus in which shear rate decreases with depth. The experimental media were transparent, and approximated Bingham-plastic behaviour. The particles that were used did not settle under quiescent conditions. 

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