Fluids Bingham

The Bingham model applied to pipe flow is one of the oldest mathematical flow solutions, and it is called the Buckingham-Reiner equation  

If the ratio of stresses Oo/ow is small (as when the yield stress is small anyway or the flow stress is quite high), we can ignore the third term in brackets. The formula is then simpler, and can be re-arranged to give Pas a function of Q hence 

In order to accommodate a Bingham fluid flowing down a pipe with a Newtonian slip layer (usually made up of the continuous phase), we need to add the 71 5 a P 

Newtonian slip layer whose viscosity is t]s In most practical cases where slip layers are observed, they are usually less than one micron in thickness, but because the viscosity is so low, they can still dominate the flow. 

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Binding Binding agents Binding mechanisms Bingham fluids... 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Many investigators beheve that the Bingham model accounts best for observations of electrorheological behavior (116,118), but other models have also been proposed (116,119). There is considerable evidence that ER materials behave as linear viscoelastic fluids while under the influence of electric field (120) thus it appears that these materials maybe thought of as elastic Bingham fluids. 

Determine the yield stress of a Bingham fluid of density 2000 kg/m3 which will just flow out of an open-ended vertical tube of diameter 300 mm under the influence of its own weight. 

Bingham fluid, annular region, pipe flow 126... 

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

Figure 6.6. Critical Reynolds number for transition from laminar to turbulent flow of Bingham fluids. The data also are represented by Eqs. (6.56) and (6.5T) (O) cement rock slurry (A) river mud slurries ( ) clay slurry (B) sewage sludge (A) Th02 slurries ( ) lime slurry. [Hanks and Prall, SPE Journal, 342-346 (Dec. 1967)].

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 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... 

Figure 6.50 presents the cumulative residence time distribution for a tube with a Newtonian model and for a shear thinning fluid with power law indices of 0.5 and 0.1. Plug flow, which represents the worst mixing scenario, is also presented in the graph. A Bingham fluid, with a power law index of 0, would result in plug flow. 

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... 

Fig. 13.28 Deformation of ABS at different piston displacements 0, 11, 22, 33, 44, and 55 mm. (a) Experimental, (b) Calculated using a Bingham fluid model (T = 230°C). [Reprinted by permission from E. Vos, H. E. H. Meijer, and G. W. M. Peters, Multilayer Injection Molding, Int. Polym. Process., 6, 42 (1991).]...

In the case of fluids with yield stress, viscoplastic fluids differ from elastoplastic fluids. With the application of a shear stress o above the yield strength o0. Bingham fluids show a linear dependence of shear stress on shear rate, whereas Casson and Herschel-Bulkley fluids show a nonlinear dependence on these parameters. 

Figure 3.3 illustrates the special cases of Eq. 3.6 used to describe Herschel-Bulkley fluids and, depending on the flow exponent and yield stress values, Newtonian fluids, shear thinning, shear thickening, and Bingham fluids. The values for Eq. 3.6 are given in Table 3.1. 

From the results in [35] it follows that the foam behaves as a simple Bingham fluid, characterised by... 

Fig. 8.1 Idealized plots of shear rate (y) against shear stress (x) for fluids of various types. (A) Newtonian fluid. (B) Bingham fluid. (C) Shear thinning, (D) Shear thickening. (E) Positive hysteresis 1, 2, 3A thixotropy 1,2, 3B. rhcodestruction, (F) Negative hysteresis with antithixotropy.

Toothpaste is a material that has to be solid at rest and liquid when sheared. We will assume that it behaves as a Bingham fluid, and is described by two parameters ... 

In concentrated suspensions, the particles touch each other. If there is also an attraction between the particles, the suspension may not flow when the shear stress is small it is a solid (Figure C4-14). The stress at which the liquid starts moving is known as the yield stress. Once the liquid yields, it often behaves like a Newtonian liquid with a constant differential viscosity. The behaviour of such Bingham fluids is similar to that of shear thinning fluids ... 

For Bingham fluids, these thickness fluctuations will self-level to a height, A8 , equivalent to the height of fluid supported by the yield stress or... 

A Bingham fluid has a yield stress of 1 kPa and a viscosity of 1 Pa - sec. It is used in extrusion with a die which forms a green body in the form of a pipe. The pressure used for extrusion is 100 kPa and the length of the die is 5 cm. Determine the velocity profile for a Bini am fluid during extrusion and the wall shear stress on both walls of the die. 

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