Bingham solids

Oldroyd, J. G., 1947. A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Philos. Soc. 43, 100-105. 

From eqn. (95) it follows that sintering of a Bingham solid should take place only if the product Jf(g) is smaller than unity. If A is small (low yield point), this condition is met for all densities and sintering to unit relative density occurs within a finite period of time. If A > 0.89, the rate of sintering decreases to zero as the material density increases towards a value given by equation f(e) = i/A. 

Fluids in which no deformation occurs until a certain threshold shear stress is applied, in which upon the shear stress x becomes a linear function of shear rate y. The characteristics of the function are the slope (viscosity) and the shear stress intercept (yield value) Xy. The rheological expression for this type of material, known as a Bingham solid, is... 

H) Bingham solid. This type of system responds to stress with a restoring force and cannot flow as long as the stress remains below a critical value, known as the yield stress value beyond this value, the sample flows like a liquid. This behaviour is shown by various polymer and surfactant solutions ... 

Bingham-plastic slurries require a shear stress diagram showing shear rate vs. shear stress for the slurry in order to determine the coefficient of rigidity, T], which is the slope of the plot at a particular concentration. This is laboratory data requiring a rheometer. These are usually fine solids at high concentrations. 

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. 

Plastic fluids are Newtonian or pseudoplastic liquids that exhibit a yield value (Fig. 3a and b, curves C). At rest they behave like a solid due to their interparticle association. The external force has to overcome these attractive forces between the particles and disrupt the structure. Beyond this point, the material changes its behavior from that of a solid to that of a liquid. The viscosity can then either be a constant (ideal Bingham liquid) or a function of the shear rate. In the latter case, the viscosity can initially decrease and then become a constant (real Bingham liquid) or continuously decrease, as in the case of a pseudoplastic liquid (Casson liquid). Plastic flow is often observed in flocculated suspensions. 

Adsorption on Kaolinite. For kaolinite, the polymer adsorption density is strongly dependent on the solid/liquid ratio, S/L, of the clay suspension. As S/L increases, adsorption decreases. This S/L dependence cannot be due totally to autocoagulation of the clay particles since this dependence is observed even in the absence of Ca2+ at pH 7 and at low ionic strength where auto-coagulation as measured by the Bingham yield stress is relatively weak (21). Furthermore, complete dispersion of the particles in solvent by ultra-sonication before addition of... 

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... 

A slurry pump operating at 1 atm must be selected to transport a coal slurry from an open storage tank to a rotary drum filter, at a rate of 250 gpm. The slurry is 40% solids by volume and has an SG of 1.2. The level in the filter is 10 ft above that in the tank, and the line contains 400 ft of 3 in. sch 40 pipe, two gate valves, and six 90° elbows. A lab test shows that the slurry can be described as a Bingham plastic with v = 50 cP and t0 = 80dyn/cm2. 

Example 15-1 Determine the pressure gradient (in psi/ft) required to transport a slurry at 300 gpm through a 4 in. sch 40 pipeline. The slurry contains 50% (by weight) solids (SG = 2.5) in water. The slurry contains a bimodal particle size distribution, with half the particles below 100 pm and the other half about 2000 gm. The suspension of fines is stable and constitutes a pseudohomogeneous non-Newtonian vehicle in which the larger particles are suspended. The vehicle can be described as a Bingham plastic with a limiting viscosity of 30 cP and a yield stress of 55 dyn/cm2. 

As the shear stress for flow in a pipe varies from zero at the centre-line to a maximum at the wall, genuine flow, ie deformation, of a Bingham plastic occurs only in that part of the cross section where the shear stress is greater than the yield stress ry. In the part where r< rv the material remains as a solid plug and is transported by the genuinely flowing outer material. 

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. 

While polymer melts and non-cross-linked elastomers flow readily when stress is applied, structural plastics must resist irreversible deformation and behave as elastic solids when relatively small stresses are applied. These plastics are called ideal or Bingham plastics with their behavior described mathematically by... 

In laminar flow of Bingham-plastic types of materials the kinetic energy of the stream would be expected to vary from V2/2gc at very low flow rates (when the fluid over the entire cross section of the pipe moves as a solid plug) to V2/gc at high flow rates when the plug-flow zone is of negligible breadth and the velocity profile parabolic as for the flow of Newtonian fluids. McMillen (M5) has solved the problem for intermediate flow rates, and for practical purposes one may conclude... 

One simple rheological model that is often used to describe the behavior of foams is thai of a Bingham plastic. This applies for hows over length scales sufficient ly 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 solid with the viscous flow of a liquid. 

If a sample shows elastic, solid-like deformation below a certain shear stress ay and starts flowing above this value, ay is called a yield stress value. This phenomenon can occur even in solutions with quite low viscosity. A practical indication for the existence of a yield stress value is the trapping of bubbles in the liquid Small air bubbles that are shaken into the sample do not rise for a long time whereas they climb up to the surface sooner or later in a liquid without yield stress even if their viscosity is much higher. A simple model for the description of a liquid with a yield stress is called Bingham s solid ... 

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. 

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 ... 

Kuhnle, G. Dell Aquila, C. Low, Y. Kussmaul, M. Bingham, S. 2007. Extraction and quantification of phytoestrogens in foods using automated solid-phase extraction and LC/MS/MS. Anal. Chem. 79 9234-9239. 

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