Rheology Bingham

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

The rheological characteristics of AB cements are complex. Mostly, the unset cement paste behaves as a plastic or plastoelastic body, rather than as a Newtonian or viscoelastic substance. In other words, it does not flow unless the applied stress exceeds a certain value known as the yield point. Below the yield point a plastoelastic body behaves as an elastic solid and above the yield point it behaves as a viscoelastic one (Andrade, 1947). This makes a mathematical treatment complicated, and although the theories of viscoelasticity are well developed, as are those of an ideal plastic (Bingham body), plastoelasticity has received much less attention. In many AB cements, yield stress appears to be more important than viscosity in determining the stiffness of a paste. 

Polymer rheology can respond nonllnearly to shear rates, as shown in Fig. 3.4. As discussed above, a Newtonian material has a linear relationship between shear stress and shear rate, and the slope of the response Is the shear viscosity. Many polymers at very low shear rates approach a Newtonian response. As the shear rate is increased most commercial polymers have a decrease in the rate of stress increase. That is, the extension of the shear stress function tends to have a lower local slope as the shear rate is increased. This Is an example of a pseudoplastic material, also known as a shear-thinning material. Pseudoplastic materials show a decrease in shear viscosity as the shear rate increases. Dilatant materials Increase in shear viscosity as the shear rate increases. Finally, a Bingham plastic requires an initial shear stress, to, before it will flow, and then it reacts to shear rate in the same manner as a Newtonian polymer. It thus appears as an elastic material until it begins to flow and then responds like a viscous fluid. All of these viscous responses may be observed when dealing with commercial and experimental polymers. 

The branch of science related to the study of deformation and flow of materials was given the name rheology by Bingham, whom some call the father of modern rheology. The prefix rheo is derived from the Greek rheos, meaning current of flow. The study of rheology includes two vastly different branches of mechanics—fluid and solid. The polymer scientist is usually concerned with viscoelastic materials that act as both solids and liquids. 

Buckingham (BIO) integrated the rheological equation [Eq. (2)] for the isothermal flow of Bingham plastics in round pipes. The familiar and important resulting equation ... 

That is to say, the same variables are relevant in both problems with the exception of the rheological properties of the fluid. For Newtonian fluids, the viscosity y. defines these adequately for Bingham plastics, the two parameters t and ij are required. 

Feed rheology (liquid) Newtonjan/pseudoplastic/dilatant/Bingham plastic/thixotropic/rheopectic/viscoelastic... 

While the Bingham plastic model is an adequate approximate description of foam rheology, it is by no means exact, especially at low strain rates. More detailed models attempl to relate the rheological properties of foams to the structure and behavior of the bubbles. 

Rheology has also been used to locate sol-gel transitions in concentrated block copolymer solutions, as described in Chapter 4. Gels exhibit a finite yield stress (i.e. they are Bingham fluids), which can be measured in steady shear experiments. 

The effect of pseudo-plasticity was reportedly known for different plastisols, in particular, for plastisols II described in Sect. 3. The rheological behaviour of these systems with an accuracy sufficient for the purposes of engineering is given by the Bingham-Shvedov law within the extended range of shear rates (10-2 < y < 102) ... 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

The Bingham Fluid. The Bingham fluid is an empirical model that represents the rheological behavior of materials that exhibit a no flow region below certain yield stresses, tv, such as polymer emulsions and slurries. Since the material flows like a Newtonian liquid above the yield stress, the Bingham model can be represented by... 

The Bingham fluid is a two-parameter, somewhat different model from the previous rheological models, in that it has a final yield stress below which there is no flow, whereas above it, the stress is a linear function of the rate of strain... 

A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient —AP/l and volumetric flowrate Q is ... 

It is easy to understand that these solutions must exhibit viscoelastic properties. Under shear flow the vesicles have to pass each other and, hence, they have to be deformed. On deformation, the distance of the lamellae is changed against the electrostatic forces between them and the lamellae leave their natural curvature. The macroscopic consequence is an elastic restoring force. If a small shear stress below the yield stress ery is applied, the vesicles cannot pass each other at all. The solution is only deformed elastically and behaves like Bingham s solid. This rheological behaviour is shown in Figure 3.35. which clearly reveals the yield stress value, beyond which the sample shows a quite low viscosity. 

II. The modified Bingham body - a useful rheological model. J. Text. Stud. 3, 194-205. Enjalbert, F., M. C., Nicot, C., Bayourthe, M., Vernay, Moncoulon, R. 1997. Effects of dietary calcium soaps of unsaturated fatty acids on digestion, milk composition, and physical properties of butter. J. Dairy Res. 64, 181-195. 

Princen and Kiss [13] have studied the rheological properties of emulsions in which the effects of drainage, coalescence and diffusion transfer is much less expressed than in foams. A concentric-cylinder viscometer was used. The slip was estimated by rheograms in the t vs. 0) lx form (t is the stress measured on the inner cylinder wall, co is the angular velocity of the outer cylinder). The Xq values obtained are conform well with those from Eq. (8.24) but Xo (y) function is not a Bingham one, i.e. does not obey Eq. (8.11) at rj = const. 

The influence of addition of sodium bentonite (a commonly used antisettling system) on the rheological behaviour of a pesticide suspension concentrate (250 g dm ) has been investigated. Steady state shear stress-shear rate curves were carried out to obtain the yield value and viscosity as a function of shear rate. The shear modulus was also measured using a pulse shearometer, and the residual viscosity was obtained in afew cases from creep measurements. The rheological parameters Tg (Bingham yield value),... 

Tg and . This is shown in Fig. 7 both for the extrapolated (Bingham) and Casson s yield value. In both cases the linear relationship is maintained indicating that such crude models may be applied to the rheology of the complex system of bentonite clay plus pesticide suspension. It should be mentioned, however, that the elastic floe model is a more realistic description of the system, since the assumption of a maximum of doublets in the floe rupture model is not justified with a concentrated suspension with many body interactions. 

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