Bingham critical shear stress

Since elements in Bingham s model are parallel to each other, their strains are identical, and the stresses are additive. The shear stress in Coulomb s element can not exceed the critical value, x Consequently, the strain rate generated by viscous element, should be proportional to the difference between the acting stress and the critical shear stress, namely ... 

Bingham s critical shear stress, tb, corresponding to the beginning of an intense fracture of the structure, may be regarded as the strength of the system (shear strength, to be exact). 

For a Bingham slurry, the yield stress is added to Equation 6-45 to express the critical shear stress at the point of the incipient motion ... 

The Bingham fluid has a linear shear stress-shear strain relationship, but a critical shear stress is required to initiate the flow. The shear stress-shear rate curve of a Bingham fluid does not pass through the origin, and it can be expressed as ... 

In other terms, above a critical shear stress, it flows as a Newtonian fluid of (constant) viscosity t). It follows that a fluid obeying the Herschel-Bulkley model is sometimes called a generalized Bingham fluid, since with n=1 and K=r in Equation 5.3, one obviously obtains Equation 5.4. The three fit parameters of the Herschel-Bulkley equation can be reduced to two, when considering that n=0.5. This was in fact the approach used by Casson in proposing the following model ... 

Newtonian fluids. Complex fluids, such as polymers, exhibit Newtonian behavior for low values of the shear rate until a value of k is reached above which the shear stress falls below the linear relationship of CTi2 versus K. Hence the apparent viscosity decreases as k increases, and these fluids exhibit non-Newtonian behavior. For a comparatively small number of fluids, the viscosity increases as the shear rate increases. Typical curves showing the dependence of the shear stress on the shear rate are shown in Figure 13.2. For some fluids, known as Bingham fluids, a critical stress is necessary for flow to occur. The flow behavior for different Bingham fluids is also shown in Figure 13.2. 

Laminar flow conditions cease to exist at Rcmod = 2100. The calculation of the critical velocity corresponding to Rcmod = 2100 requires an iterative procedure. For known rheology (p, m, n, Xq) and pipe diameter (D), a value of the wall shear stress is assumed which, in turn, allows the calculation of Rp, from equation (3.9), and Q and Qp from equations (3.14b) and (3.14a) respectively. Thus, all quanties are then known and the value of Rcmod can be calculated. The procedure is terminated when the value of x has been found which makes RCjnod = 2100, as illustrated in example 3.4 for the special case of n = 1, i.e., for the Bingham plastic model, and in example 3.5 for a Herschel-Bulkley fluid. Detailed comparisons between the predictions of equation (3.34) and experimental data reveal an improvement in the predictions, though the values of the critical velocity obtained using the criterion Rqmr = 2100 are only 20-25% lower than those predicted by equation (3.34). Furthermore, the two... 

Examples of non-Newtonian behavior are shown in Figure 13.3. Shear-thickening (dilatant) is observed when the resistance to deformation increases with the shear rate, whereas the opposite is true for a system undergoing shear-thinning pseudoplastic). In some cases, a minimum critical value of stress, x, is needed for flow to occur Bingham flow). This situation is characteristic of fluids where a structured arrangement of the molecules exists, and therefore a critical x is required to break down the structure. 

The shear viscosity of the suspension exhibits shear rate sensitivity above this critical stress value, This model resembles in form to the purely viscous approximation of the Bingham... 

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