Bingham fluids yield stress

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 ... 

Bingham y = 00 T < Tq Tq = fluid yield stress below which no (i) Used mainly for pastes, slurries and... 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

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) ... 

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... 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

The simplest type of behaviour for a fluid exhibiting a yield stress is known as Bingham-plastic. The shear rale is directly proportional to the amount by which the stress exceeds the yield stress. 

The pressure drop for a fluid exhibiting a yield stress, such as a Bingham plastic material, can be similarly reduced by air injection. 

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. 

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. 

It is recalled that a plastic is really two materials. At low stresses below the critical or yield stress (r0) the material behaves as a solid, whereas for stresses above the yield stress the material behaves as a fluid. The Bingham model for this behavior is... 

The slurry behaves as a non-Newtonian fluid, which can be described as a Bingham plastic with a yield stress of 40 dyn/cm2 and a limiting viscosity of 100 cP. Calculate the pressure gradient (in psi/ft) for this slurry flowing at a velocity of 8 ft/s in a 10 in. ID pipe. 

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

Shear stress (F/A), lb.p/sq. ft. t refers to the shear stress at the wall of a round pipe (DAP/ 4L) and r< to the shear stress at the wall of a viscometer bob Yield value or yield stress of a Bingham-plastic fluid, lb.F/sq. ft. Indicator of an unspecified functional relationship... 

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). 

Fluids that show viscosity variations with shear rates are called non-Newtonian fluids. Depending on how the shear stress varies with the shear rate, they are categorized into pseudoplastic, dilatant, and Bingham plastic fluids (Figure 2.2). The viscosity of pseudoplastic fluids decreases with increasing shear rate, whereas dilatant fluids show an increase in viscosity with shear rate. Bingham plastic fluids do not flow until a threshold stress called the yield stress is applied, after which the shear stress increases linearly with the shear rate. In general, the shear stress r can be represented by Equation 2.6 ... 

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... 

Non-Newtonian fluids include those for which a finite stress x is required before continuous deformation occurs these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material its rheogram has a constant slope p< called the infinite shear viscosity. 

Two of the most common empirical models used to describe the behaviour of pseudoplastic fluids with yield stresses are the Bingham plastic [377] model ... 

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... 

The shearing characteristics of non-Newtonian fluids are shown in Fig. 3.24 of Volume 1. This type of fluid remains rigid when the shear stress is less than the yield stress Ry and flows like a Newtonian fluid when the shear stress exceeds Ry. Examples of Bingham plastics are many fine suspensions and pastes including sewage sludge and toothpaste. The velocity profile in laminar flow is shown in Fig. 3c. 

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. 

In the case of the Bingham plastic type of fluids, if the yield stress is higher than the characteristic shear stress in the contactor, the gas bubbles may not have any relative motion with respect to the rest of the fluid. For viscoelastic... 

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. 

---