Yield-stress fluids Bingham material

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

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. 

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). 

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. 

Based on the magnitude of n and to, the non-Newtonian behavior can be classified as shear thinning, shear thickening, Bingham plastic, pseudoplastic with yield stress, or dilatant with yield stress (see Fig. 2 and Table I). The Herschel-Bulkley model is able to describe the general flow properties of fluid foods within a certain shear range. The discussion on this classiflcation and examples of food materials has been reviewed by Sherman (1970), DeMan (1976), Barbosa-Canovas and Peleg (1983), and Barbosa-Canovas et al. (1993). 

A fluid with a linear flow curve for Ty > ro is called a Bingham plastic fluid and is characterised by a constant plastic viscosity (the slope of the shear stress versus shear rate curve) and a yield stress. On the other hand, a substance possessing a yield stress as well as a non-linear flow curve on linear coordinates (for Xyx > ro ), is called a yield-pseudoplastic material. Figure 1.8 illustrates viscoplastic behaviour as observed in a meat extract and in a polymer solution. 

The rheology of yield-stress (or viscoplastic) fluids is complex and often time dependent. Considerable insight can be gained, however, by considering the simplest example, the Bingham material. The classical Bingham material is defined for a shear flow with a positive shear rate as... 

Consider a fluid placed between two parallel circular disks of radius R. The gap 2h between the disks is assumed to be narrow, i.e. to be much less than R at all times. A force F is applied to each disk so as to push them together. We wish to know what the force needs to be in order to maintain a prescribed motion of the disks when the fluid is Newtonian or PL. The case where the material is a Bingham fluid with a yield stress is complicated (see, however, Wilson [36]). 

Perhaps the best picture of a viscoplastic fluid is that of a very viscous, even solidlike, material at low stresses. Over a narrow stress range, which can often be modeled as a single yield stress, its viscosity drops dramatically. This is shown clearly in Figure 2.5.5b, where viscosity drops over five decades as shear stress increases from 1 to 3 Pa. (The drop is even more dramatic in Figure 10.7.2.) Above this yield stress the fluid flows like a relatively low viscosity, even Newtonian, liquid. Because of the different behaviors exhibited by these fluids, the model (Bingham, Casson, etc.) and the range of shear rates used to calculate the parameters must be chosen carefully. In Section 10.7 we will discuss microstructural bases for r. It is also important to note that experimental problems like wall slip are particularly prevelant with viscoplastic materials. Aspects of slip are discussed in Section 5.3. 

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... 

In studies of viscous fluids (such as paint), Bingham (1922) suggested that some fluids do indeed have a yield point and suggested the model shown in Fig. 11.10(a). In this model, a viscous element and a friction element are in parallel. Upon applying a stress, no movement occurs until the resistance of the friction element is overcome. The stress-strain response is that of a rigid-viscous material and is as shown in Fig. 11.10(b). 

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