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| [[File:Bingham_mayo.jpg|thumb|right|302px|[[Mayonnaise]] is a Bingham plastic. The surface has ridges and peaks because Bingham plastics mimic solids under low shear stresses.]]
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| A '''Bingham plastic''' is a [[viscoplastic]] material that behaves as a rigid body at low stresses but flows as a [[viscosity|viscous]] [[fluid]] at high stress. It is named after [[Eugene C. Bingham]] who proposed its mathematical form.<ref>E.C. Bingham,(1916) ''U.S. Bureau of Standards Bulletin'', 13, 309-353 "An Investigation of the Laws of Plastic Flow"</ref>
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| It is used as a common [[mathematical model]] of [[mud]] flow in [[drilling engineering]], and in the handling of [[slurry|slurries]]. A common example is [[toothpaste]],<ref name=Steffe>J. F. Steffe (1996) ''Rheological Methods in Food Process Engineering'' 2nd ed ISBN 0-9632036-1-4</ref> which will not be [[extruded]] until a certain [[pressure]] is applied to the tube. It then is pushed out as a solid plug.
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| ==Explanation==
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| [[File:Bingham1a.svg|thumb|left|302px|Figure 1. Bingham Plastic flow as described by Bingham]]
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| '''Figure 1''' shows a graph of the behaviour of an ordinary viscous (or Newtonian) fluid in red, for example in a pipe. If the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move (called the [[shear stress]]) and the volumetric flow rate increases proportionally. However for a Bingham Plastic fluid (in blue), stress can be applied but it will not flow until a certain value, the [[yield stress]], is reached. Beyond this point the flow rate increases steadily with increasing shear stress. This is roughly the way in which Bingham presented his observation, in an experimental study of paints.<ref>E. C. Bingham (1922) ''Fluidity and Plasticity'' McGraw-Hill (New York) page 219</ref> These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a Newtonian fluid. | |
| [[File:Bingham2a.svg|thumb|right|302px|Figure 2. Bingham Plastic flow as described currently]]
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| '''Figure 2''' shows the way in which it is normally presented currently.<ref name=Steffe/> The graph shows [[shear stress]] on the vertical axis and [[shear rate]] on the horizontal one. (Volumetric flow rate depends on the size of the pipe, shear rate is a measure of how the velocity changes with distance. It is proportional to flow rate, but does not depend on pipe size.) As before, the Newtonian fluid flows and gives a shear rate for any finite value of shear stress. However, the Bingham Plastic again does not exhibit any shear rate (no flow and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the slope of this line is the [[viscosity]], which is the only parameter needed to describe its flow. By contrast the Bingham Plastic requires two parameters, the '''yield stress''' and the slope of the line, known as the '''plastic viscosity'''.
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| The physical reason for this behaviour is that the liquid contains particles (e.g. clay) or large molecules (e.g. polymers) which have some kind of interaction, creating a weak solid structure, formerly known as a '''false body''', and a certain amount of stress is required to break this structure. Once the structure has been broken, the particles move with the liquid under viscous forces. If the stress is removed, the particles associate again.
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| ==Definition==
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| The material is an elastic solid for [[shear stress]] ''τ'', less than a critical value <math>\tau_0</math>. Once the critical shear [[shear stress|stress]] (or "[[yield (engineering)|yield stress]]") is exceeded, the material flows in such a way that the [[shear rate]], ∂''u''/∂''y'' (as defined in the article on [[viscosity]]), is directly proportional to the amount by which the applied shear stress exceeds the yield stress:
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| :<math>\frac {\partial u} {\partial y} = \left\{\begin{matrix} 0 &, \tau < \tau_0 \\ (\tau - \tau_0)/ {\mu_\infty} &, \tau \ge \tau_0 \end{matrix}\right.</math>
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| ==Friction Factor Formulae==
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| In fluid flow, it is a common problem to calculate the pressure drop in an established piping network.<ref>{{Cite book| title=Chemical Engineering Fluid Mechanics. | first1=Ron | last1=Darby | publisher=Marcel Dekker | year=1996 | isbn=0-8247-0444-4| postscript=<!--None--> }}. See Chapter 6.</ref> Once the friction factor, ''f'', is known, it becomes easier to handle different pipe-flow problems, viz. calculating the pressure drop for evaluating pumping costs or to find the flow-rate in a piping network for a given pressure drop. It is usually extremely difficult to arrive at exact analytical solution to calculate the friction factor associated with flow of non-Newtonian fluids and therefore explicit approximations are used to calculate it. Once the friction factor has been calculated the pressure drop can be easily determined for a given flow by the [[Darcy–Weisbach equation]]:
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| :<math> \ f = \ {2 h_f g D \over L V^2}</math>
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| where:
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| * <math>{\bold \ h_f}</math> is the frictional head loss ([[SI units]]: m)
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| * <math>{\bold \ f}</math> is the [[Darcy friction factor]] ([[SI units]]: Dimensionless)
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| * <math>{\bold \ L}</math> is the pipe length ([[SI units]]: m)
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| * <math>{\bold \ g}</math> is the gravitational acceleration ([[SI units]]: m/s²)
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| * <math>{\bold \ D}</math> is the pipe diameter ([[SI units]]: m)
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| * <math>{\bold \ V}</math> is the mean fluid velocity ([[SI units]]: m/s)
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| ===Laminar flow===
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| An exact description of friction loss for Bingham plastics in fully developed laminar pipe flow was first published by Buckingham.<ref>Buckingham, E. (1921). "on Plastic Flow through Capillary Tubes". ''ASTM Proceedings'' '''21''': 1154–1156.</ref> His expression, the ''Buckingham-Reiner'' equation, can be written in a dimensionless form as follows:
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| :<math> \ f_L = \ {64 \over Re}\left[1 + {He\over 6 Re} - {64\over3}\left({He^4\over {f}^3 Re^7}\right)\right]</math>
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| where:
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| * <math>{\bold \ f}</math> is the laminar flow friction factor ([[SI units]]: Dimensionless)
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| * <math>{\bold \ Re}</math> is the [[Reynolds number]] ([[SI units]]: Dimensionless)
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| * <math>{\bold \ He}</math> is the Hedstrom number ([[SI units]]: Dimensionless)
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| The [[Reynolds number]] and the Hedstrom number are respectively defined as:
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| :<math> \mathrm{Re} = { \rho {\ V} D \over {\mu}} </math>, and
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| :<math> \mathrm{He} = { \rho {\ D^2 } {\ \tau_o} \over {{\mu}^2}} </math>
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| where:
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| * <math>{\bold \rho}</math> is the mass density of fluid ([[SI units]]: kg/m<sup>3</sup>)
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| * <math>{\bold \ \mu}</math> is the dynamic [[viscosity]] of fluid ([[SI units]]: kg/m s)
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| ===Turbulent flow===
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| Darby and Melson developed an empirical expression <ref name=Darby>Darby, R. and Melson J.(1981). "How to predict the friction factor for flow of Bingham plastics". ''Chemical Engineering'' '''28''': 59–61.</ref>
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| that was then refined, and is given by:<ref name=Darbyetal>Darby, R. et al. (1992). "Prediction friction loss in slurry pipes." ''Chemical Engineering'' '''September''': .</ref>
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| :<math> \ f_T = \ {10^a} \ {Re^{-0.193}} </math>
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| where:
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| * <math>{\bold \ f_T}</math> is the turbulent flow friction factor ([[SI units]]: Dimensionless)
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| * <math> \ a = -1.47\left[1 + 0.146{\ e^{-2.9\times {10^{-5}}He}}\right] </math>
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| ==Approximations of the ''Buckingham-Reiner'' equation==
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| Although an exact analytical solution of the ''Buckingham-Reiner'' equation can be obtained because it is a fourth order polynomial equation in ''f'', due to complexity of the solution it is rarely employed. Therefore, researchers have tried to develop explicit approximations for the ''Buckingham-Reiner'' equation.
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| ===Swamee-Aggarwal Equation===
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| The ''Swamee Aggarwal'' equation is used to solve directly for the Darcy–Weisbach friction factor ''f'' for laminar flow of Bingham plastic fluids.<ref>Swamee, P.K. and Aggarwal, N.(2011). "Explicit equations for laminar flow of Bingham plastic fluids". ''Journal of Petroleum Science and Engineering''. {{doi|10.1016/j.petrol.2011.01.015}}.</ref> It is an approximation of the implicit ''Buckingham-Reiner'' equation, but the discrepancy from experimental data is well within the accuracy of the data.
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| The ''Swamee-Aggarwal'' equation is given by:
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| :<math> \ f_L = \ {64 \over Re} + {10.67 + 0.1414{({He\over Re})^{1.143}}\over {\left[1 + 0.0149{({He\over Re})^{1.16}}\right]Re }}\left({He\over Re}\right)</math>
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| ===Danish-Kumar Solution===
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| Danish ''et al.'' have provided an explicit procedure to calculate the friction factor ''f'' by using the Adomian decomposition method.<ref>Danish, M. ''et al.'' (1981). "Approximate explicit analytical expressions of friction factor for flow
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| of Bingham fluids in smooth pipes using Adomian decomposition method". ''Communications in Nonlinear Science and Numerical Simulation'' '''16''': 239–251.</ref> The friction factor containing two terms through this method is given as:
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| :<math> f_L = \frac{K_1 + \dfrac{4 K_2}{\left( K_1 + \frac{K_1 K_2}{K_1^4 + 3 K_2}\right)^3}}{1+ \dfrac{3 K_2}{\left(K_1 + \frac{K_1 K_2}{K_1^4 + 3 K_2}\right)^4}}</math>
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| where:
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| :<math> \ K_1 = \ {16 \over Re} + {16 He \over 6{Re^2}}</math>, and
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| :<math> \ K_2 = \ - {16 {He^4} \over 3{Re^8}}</math>
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| ==Combined Equation for friction factor for all flow regimes==
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| ===Darby-Melson Equation===
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| In 1981, Darby and Melson, using the approach of Churchill<ref>Churchill, S.W. (1977). "Friction factor equation spans all fluid-flow regimes". ''Chemical Engineering'' '''Nov. 7''': 91–92.</ref> and of Churchill and Usagi,<ref>Churchill, S.W. and Usagi, R.A. (1972). "A general expression for the correlation of rates of transfer and other phenomena". ''AIChE Journal'' '''18(6)''': 1121-1128.</ref> developed an expression to get a single friction factor equation valid for all flow regimes:<ref name=Darby/>
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| :<math> \ f = \ {\left[{f_L}^m + {f_T}^m\right]}^{1\over m}</math>
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| where:
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| :<math> \ m = \ 1.7 + {40000\over Re} </math>
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| Both ''Swamee-Aggarwal'' equation and the ''Darby-Melson'' equation can be combined to give an explicit equation for determining the friction factor of Bingham plastic fluids in any regime. Relative roughness is not a parameter in any of the equations because the friction factor of Bingham plastic fluids is not sensitive to pipe roughness.
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| ==See also==
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| * [[Bagnold number]]
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| * [[Bernoulli's principle]]
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| * [[Rheology]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Bingham Plastic}}
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| [[Category:Materials]]
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| [[Category:Non-Newtonian fluids]]
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| [[Category:Viscosity]]
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| [[Category:Offshore engineering]]
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