The Bingham equation

For most structured liquids at high shear rates, rjo rjoo and Ky l,and then it is easy to show that the Cross model simplifies to the Sisko equation 

For the most shear-thinning liquids, m is unity, and the equation above can easily be recast as the Bingham equation by multipl dng throughout by shear rate, so 

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Rheocalic V2.4. The Bingham mathematic model was used to determine viscosity. The Bingham equation is t = Tq + ly. Where t is the shear stress applied to the material, y is the shear strain rate (also called the strain gradient). To is the yield stress and p is the plastic viscosity. 

At high shear rates, when the gel network is broken down, the dominant viscoelastic contribution comes from floes that break apart and reform rapidly. For such dispersions, at modest particle volume fractions, a typical relationship between steady-state shear stress a and shear rate y is shown in Fig. 7-24 (Friend and Hunter 1971). Note that at the highest shear rates, the y-a relationship appears to be linear, but the extrapolation of this linear relationship to zero shear rate intersects the stress axis at a positive value, as, rather than zero. This intercept is called the Bingham yield stress, derived from the Bingham equation for shear stress (Friend and Hunter 1971) ... 

The adsorbed layer thickness of the graft copolymer on the latexes was determined using rheological measurements. Steady-state (shear stress a-y shear rate) measurements were carried out and the results were fltted to the Bingham equation to obtain the yield value and the high shear viscosity // of the suspension. 

The Bingham equation describes the shear stress/shear rate behaviour of many shear thiiming materials at low shear rates. Unfortunately, the value of obtained depends on the shear rate ranges used for the extrapolation procedure. 

The plastic and pseudo-plastic systems are described by the Bingham equation. 

The analysis of the results allows both constants in the Bingham equation to be determined. Different types of such computer-assisted equipment are available, but still the measurement techniques are not perfect and the results are not entirely independent of the measurement method. 

Figure 1 Hypothetical flow curve obeying the Bingham equation.

Let US suppose we have a set of shear-stress/shear-rate data (cr,y) which we want to fit to an appropriate equation. First we plot the data on a linear basis to see if it fits the Bingham equation. If it does, then aU we need to do is to perform a linear regression on the data, using a simple spread-sheet software package, or the software usually provided nowadays with the viscometer/rheometer we have used. Then the values of cto and % can be used to predict flows of the liquid in other geometries, see chapter 10. 

The experimental [44, 64, 651, and computer simulation [661 results suggest that the shear stress t of an ER fluid can be well expressed with the Bingham equation ... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

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. 

These equations can readily be solved by iteration, as follows. Assuming a value of / allows, VRe to be determined from Eq. (7-31). This is then used with Eq. (7-32) to find, Ylle. The friction factor is then calculated using these values of, VRe and AHe and the Bingham plastic pipe friction factor equation [Eq. (6-62)]. The result is compared with the assumed value, and the process is repeated until agreement is attained. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

The procedure for the Bingham plastic is identical to that for the power law fluid, except that Eq. (7-41) is used for the Reynolds number in the 3-K equation for fittings instead of Eq. (7-39), and the expression for the Bingham pipe friction factor is given by Eq. (6-62). 

Figure 1.5b shows the behaviour of a Bingham plastic and the fitting equation is ... 

Here the fitting parameters are the slope of the line (the plastic viscosity, rip) and the Bingham or dynamic yield stress (the intercept, constitutive equations will be introduced later in this volume as appropriate. 

So far we have restricted our discussion to Newtonian liquids, but the analysis will change somewhat if the systems are non-Newtonian. A useful illustration of the problems that arise is the case of a Bingham plastic. This gives us a linear response, as does a Newtonian liquid, but in this case there is an intercept or yield stress. The constitutive equation for a Bingham plastic is... 

We must use the constitutive equation for the Bingham plastic for the stress. This then gives the angular velocity of the outer cylinder from ... 

Clearly the Riener-Riwlin equation reduces to the Margules equation when the Bingham yield value is zero, but there is an important consequence in that it is assumed that all the material is flowing, i.e. the shear stress at the wall of the outer cylinder must be... 

Buckingham (BIO) integrated the rheological equation [Eq. (2)] for the isothermal flow of Bingham plastics in round pipes. The familiar and important resulting equation ... 

The Buckingham equation for Bingham flow in the laminar region is... 

Laminar Flow. Theoretically derived equations for volumetric flow rate and friction factor arc included for several models in Table 6.7. Each model employs a specially defined Reynolds number, and the Bingham models also involve the Hcdstrom number,... 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

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