Fluid shear thinning

The Williamson equation is useful for modeling shear-thinning fluids over a wide range of shear rates (15). It makes provision for limiting low and high shear Newtonian viscosity behavior (eq. 3), where T is the absolute value of the shear stress and is the shear stress at which the viscosity is the mean of the viscosity limits TIq and, ie, at r = -H... 

The Cross equation assumes that a shear-thinning fluid has high and low shear-limiting viscosity (16) (eq. 4), where a and n are constants. 

Some materials have the characteristics of both solids and liquids. For instance, tooth paste behaves as a solid in the tube, but when the tube is squeezed the paste flows as a plug. The essentia] characteristic of such a material is that it will not flow until a certain critical shear stress, known as the yield stress is exceeded. Thus, it behaves as a solid at low shear stresses and as a fluid at high shear stress. It is a further example of a shear-thinning fluid, with an infinite apparent viscosity at stress values below the yield value, and a falling finite value as the stress is progressively increased beyond this point. 

Figure 3.28. Behaviour of a typical shear-thinning fluid plotted logarithmically for several orders of shear...

The viscosities of most real shear-thinning fluids approach constant values both at very low shear rates and at very high shear rates that is, they tend to show Newtonian properties at the extremes of shear rates. The limiting viscosity at low shear rates mq is referred to as the lower-Newtonian (or zero-shear /x0) viscosity (see lines AB in Figures 3.28 and 3.29), and that at high shear rates Mo0 is the upper-Newtonian (or infinite-shear) viscosity (see lines EF in Figures 3.28 and 3.29). 

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. 

As indicated in Section 3.7.9, this definition of ReMR may be used to determine the limit of stable streamline flow. The transition value (R ur)c is approximately the same as for a Newtonian fluid, but there is some evidence that, for moderately shear-thinning fluids, streamline flow may persist to somewhat higher values. Putting n = 1 in equation 3,140 leads to the standard definition of the Reynolds number. 

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 ... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

Equation 5.2, with the modified parameter X used in place of X, may be used for laminar flow of shear-thinning fluids whose behaviour can be described by the power-taw model. 

For streamline flow of non-Newtonian liquids, the situation is completely different and the behaviour of two-phase mixtures in which the liquid is a shear-thinning fluid is now... 

Because n < 1 for a shear thinning fluid, < 7 will be Jess than unity and a reduction in pressure drop occurs. The lower the value of n and the larger the value of b, the greater the effect will be. It will be noted that the effects of expansion of the air as the pressure falls have not been taken into account. 

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. 

The y-velocities are all set to zero the problem is numerically underconstrained otherwise. Figure 2 also shows the finite-element prediction of this velocity profile for two cases a Newtonian fluid (power-law exponent = 1) and a shear-thinning fluid (power-law... 

Rheology concerns the study of the deformation and flow of soft materials when they respond to external stress or strain. If the ratio of its shear stress and shear rate is a straight line, the material is termed Newtonian otherwise, it is termed non-Newtonian (Figure 4.3.2(a)). As the slope of the curve is the viscosity rj, a shear-thinning fluid exhibits a reduced viscosity as the shear stress increases, whereas a shear-... 

Niederkom, T. C., and Ottino, J. M., Mixing of shear thinning fluids in time-periodic flows. AlChE J. 40, 1782-1793 (1994). 

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. 

Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have been correlated by Dodge and Metzner (1959) as a generalized form of the von Karman equation ... 

Because most shear-thinning fluids, particularly polymer solutions and flocculated suspensions, have high apparent viscosities, even relatively coarse particles may have velocities in the creeping-flow of Stokes law regime. Chhabra(35,36) has proposed that both theoretical and experimental results for the drag force F on an isolated spherical particle of diameter d moving at a velocity u may be expressed as a modified form of Stokes law ... 

From equation 3.59, it is readily seen that in a shear-thinning fluid (n < 1) the terminal velocity is more strongly dependent on d, g and ps — p than in a Newtonian fluid and a small change in any of these variables produces a larger change in no. 

The effect of particle shape on the forces acting when the particle is moving in a shear-thinning fluid has been investigated by Tripathi et alP7>, and by Venumadhav and Ciiiiaisra 41 1. In addition, some information is available on the effects of viscoelasticity of the fl uid135. ... 

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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