Viscosity Bingham fluids

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) ... 

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

Both polymeric and some biological reactors often contain non-Newtonian liquids in which viscosity is a function of shear rate. Basically, three types of non-Newtonian liquids are encountered power-law fluids, which consist of pseudoplastic and dilatant fluids viscoplastic (Bingham plastic) fluids and viscoelastic fluids with time-dependent viscosity. Viscoelastic fluids are encountered in bread dough and fluids containing long-chain polymers such as polyamide and polyacrylonitrite that exhibit coelastic flow behavior. These... 

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 ... 

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. 

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. 

A similar problem on a nonisothermal rectilinear flow of a viscoplastic Shvedov-Bingham fluid in a circular tube for the case in which the yield stress and the plastic viscosity are inversely proportional to temperature was studied in [298],... 

Figure 10. Pressure dependence of parameters from various models of the rheology of invert emulsion oil-based drilling fluids at various temperatures. Casson high shear viscosity Bingham plastic viscosity consistency, power law exponent, and yield stress from Herschel-Bulkley model. (Reproduced with permission from reference 69. Copyright 1986 Society of Petroleum Engineers.)...

Chemical potential Viscosity or viscosity coefficient Internal viscosity of fluid drop Bingham plastic viscosity Kinematic viscosity... 

The pseudoplastic fluids can be consider as the Bingham fluid with the yield stress value and plastic viscosity equal to tg a and it is the reason of their name (see Fig. 5.4). 

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 ... 

Figure 3 shows that, at 30° C, foam fracturing fluid still retained higher viscosity retention after lOmin at 170s shear. Foam fluid was Bingham fluid. 

Comparison of the two-viscosity models and Pa-panastasiou s modification for a Bingham fluid. Larger values of parameter a in Pa-panastasiou s modification (eq. 2.S.7) permit a better approximation to the Bingham model. 

At higher shear rates, three types of deviations are observable when compared to ideal Newtonian flow (see Fig. 2.1).The first kind of deviation relates to the existence of a flow threshold (yield point). In the case of a Bingham fluid, flow occurs only when the yield stress is exceeded. The second type of deviation is shear thickening, observed where the viscosity increases with shear rate. This is the case for a dilating fluid, behaviour which is seldom apparent in polymers. Last, where viscosity decreases with increase in shear rate, fluxing is observed and such fluids are usually referred to as pseudoplastic fluids. This last phenomenon is a general characteristic of thermoplastic polymers. Flow effects may also be time dependent. Where viscosity does not depend only on the shear rate, but also on the duration of the applied stress, fluids are thixotropic. Polymers in a molten state thus behave as pseudoplastic fluids having thixotropic characteristics. 

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]. 

Secreted mucus of a snaU is a kind of non-Newtonian fluid that has the characteristics of a visco-plasitic (Bingham) fluid because the viscosity... 

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 ... 

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... 

Power consumption for impellers in pseudoplastic, Bingham plastic, and dilatant nonnewtonian fluids may be calculated by using the correlating lines of Fig. 18-17 if viscosity is obtained from viscosity-shear rate cuiwes as described here. For a pseudoplastic fluid, viscosity decreases as shear rate increases. A Bingham plastic is similar to a pseudoplastic fluid but requires that a minimum shear stress be exceeded for any flow to occur. For a dilatant fluid, viscosity increases as shear rate increases. 

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). 

For a Bingham plastic fluid flow in a circular pipe and annular space, the effective viscosities are given as [61]. 

Thus, equation 3.127, which includes three parameters, is effectively a combination of equations 3.121 and 3.125. It is sometimes called the generalised Bingham equation or Herschel -Bulkley equation, and the fluids are sometimes referred to as having/n/re body. Figures 3.30 and 3.31 show shear stress and apparent viscosity, respectively, for Bingham plastic and false body fluids, using linear coordinates. 

Figure 3.31, Apparent viscosity for Bingham-plastic and false-body fluids using linear axes...

As in the case of Newtonian fluids, one of the most important practical problems involving non-Newtonian fluids is the calculation of the pressure drop for flow in pipelines. The flow is much more likely to be streamline, or laminar, because non-Newtonian fluids usually have very much higher apparent viscosities than most simple Newtonian fluids. Furthermore, the difference in behaviour is much greater for laminar flow where viscosity plays such an important role than for turbulent flow. Attention will initially be focused on laminar-flow, with particular reference to the flow of power-law and Bingham-plastic fluids. 

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