Bingham fluid model

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).]...

For the region Rp < r < R, the value of shear stress will be greater flian the yield stress of the fluid, and the Bingham fluid model for pipe flow is given by (equation (1.16) in Chapter 1) ... 

It is found that the shear stress and yield stress of the PAG suspension are higher than those of the pure PANI suspension at the equal electric field strength. Furthermore, under electric fields, the shear stress of the PAG suspension shows a decline as a function of shear rate to a minimum value after the appearance of yield stress. The widely accepted flow model for ER suspensions, i.e., the Bingham fluid model cannot fit well the flow curves of the PAG suspension, especially in the low shear rate region (see Figure 14.8a). However, the flow curves of the pure PANI suspension maintain a relatively stable level, which can be fitted by the Bingham fluid model (see Figure 14.8b). This different flow behavior reflects that the PAG sheets possess a different ER response from the pure PANI particles under the simultaneous effect of both electrical and mechanical fields. In addition. 

Another model in which the viscosity is described as a function of shear stress is the Bingham model [20]. This model is used for fluids with a yield stress Xq. Below this yield stress, the viscosity is infinite (no motion) above the yield stress, the viscosity is finite (motion occurs). The Bingham Fluid model is written as ... 

As mentioned above, interfacial films exhibit non-Newtonian flow, which can be treated in the same manner as for dispersions and polymer solutions. The steady-state flow can be described using Bingham plastic models. The viscoelastic behavior can be treated using stress relaxation or strain relaxation (creep) models as well as dynamic (oscillatory) models. The Bingham-fluid model of interfacial rheological behavior (27) assumes the presence of a surface yield stress, cy, i.e.. 

A behaviour of the dense fine grained slurries in laminar flow regime can be described by Bingham fluid model or the yield pseudoplastic rheological model, respectively... 

In more recent years. White and Suh [53] and White et al. [54] have developed three-dimensional models of compoimds with anisotropic disldike particles, which exhibit yield values. These produce direction dependent flow characteristics. Later Robinson et al. [55] described a transversely isotropic Bingham fluid model. 

This model accounts for a yield stress combined with power law behavior in stress as a function of shear rate. Besides, this model predicts a viscosity that diverges continuously at low shear rates and is infinite below the yield stress. When n = 1, the Herschel-Bulkley model reduces to the Bingham fluid model where the flow above the yield stress would be purely Newtonian and the constant k would represent the viscosity [28]. 

As early mentioned, the viscosity of PE dispersions is highly dependent on the concentration as it is shown in Fig. 14 for the system C-lidocaine. It was also observed that the elastic modulus of C varies from almost purely newtonian properties in diluted dispersion to the pseudoplastic behavior. At concentrations above 0.25 % C dispersion show a yield stress value with a plastic behavior which can be described by the Bingham fluid model [43, 44]. 

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... 

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. 

You must determine the horsepower required to pump a coal slurry through an 18 in. diameter pipeline, 300 mi long, at a rate of 5 million tons/yr. The slurry can be described by the Bingham plastic model, with a yield stress of 75 dyn/cm2, a limiting viscosity of 40 cP, and a density of 1.4 g/cm3. For non-Newtonian fluids, the flow is not sensitive to the wall roughness. 

The Bingham plastic model usually provides a good representation for the viscosity of concentrated slurries, suspensions, emulsions, foams, etc. Such materials often exhibit a yield stress that must be exceeded before the material will flow at a significant rate. Other examples include paint, shaving cream, and mayonnaise. There are also many fluids, such as blood, that may have a yield stress that is not as pronounced. 

A procedure analogous to the one followed can be used for non-Newtonian fluids that follow the power law or Bingham plastic models (Darby and Melson, 1982). 

Newtonian fluids can be correlated by this method that is, the same correlation applies to both Newtonian and non-Newtonian fluids when the Newtonian Reynolds number is replaced by either Eq. (7-40) for the power law fluid model or Eq. (7-41) for the Bingham plastic fluid model. As a first approximation, therefore, we may assume that the same result would apply to friction loss in valves and fittings as described by the 2-K or 3-K models [Eq. 7-38)]. 

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

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... 

As the power law model [Equation (20.3)] fits the experimental results for many non-Newtonian systems over two or three decades of shear rate, this model is more versatile than the Bingham model, although care should be taken when applying this model outside the range of data used to define it. In addition, the power law fluid model fails at high shear rates, whereby the viscosity must ultimately reach a constant value - that is, the value of n should approach unity. 

When (Tp = 0, Equation (20.14) reduces to the Power Fluid Model, but when n = 1, Equation (20.14) reduces to the Bingham model. When = 0 and n= 1, Equation... 

The Shvedov-Bingham fluid. For the Shvedov-Bingham fluid (the first model in Table 6.3), the dependence of the shear rate on the stress is given by (6.2.13), where... 

Shvedov-Bingham Fluids. In the special case of a viscoplastic Shvedov-Bingham medium (the first model in Table 6.3), we have the following expression for the function / in (6.4.9) ... 

In Equation (2), n is the flow behavior index (-),K is the consistency index (Pa secn), and the other terms have been defined before. For shear-thinning fluids, the magnitude of nshear-thickening fluids n>l, and for Newtonian fluids n=l. For PFDs that exhibit yield stresses, models that contain either (Jo or a term related to it have been defined. These models include, the Bingham Plastic model (Equation 3), the Herschel-Bulkley model (Equation 4), the Casson model (Equation 5), and the Mizrahi-Berk model (Equation 6). 

The Fann rheometer has been calibrated to give direct readings of plastic viscosity (PV) and yield point (YP) as given by the simple Bingham plastic fluid model relating shear stress (r) to shear rate (7) ... 

Figure 7. Determination of yield point and plastic viscosity of drilling fluid using the Bingham plastic fluid model.

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