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A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form. It is used as a common mathematical model of mud flow in
drilling engineering Drilling engineering is a subset of petroleum engineering. Drilling engineers design and implement procedures to drill wells as safely and economically as possible. They work closely with the drilling contractor, service contractors, and complianc ...
, and in the handling of
slurries A slurry is a mixture of denser solids suspended in liquid, usually water. The most common use of slurry is as a means of transporting solids or separating minerals, the liquid being a carrier that is pumped on a device such as a centrifugal pu ...
. A common example is
toothpaste Toothpaste is a paste or gel dentifrice used with a toothbrush to clean and maintain the aesthetics and health of teeth. Toothpaste is used to promote oral hygiene: it is an abrasive that aids in removing dental plaque and food from the teeth, ...
, which will not be
extruded Extrusion is a process used to create objects of a fixed cross-sectional profile by pushing material through a die of the desired cross-section. Its two main advantages over other manufacturing processes are its ability to create very complex c ...
until a certain
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
is applied to the tube. It is then pushed out as a relatively coherent plug.


Explanation

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 Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
) 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. These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
. Figure 2 shows the way in which it is normally presented currently. The graph shows
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
on the vertical axis and
shear rate In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
, 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. The physical reason for this behaviour is that the liquid contains particles (such as clay) or large molecules (such as 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.


Definition

The material is an elastic solid for
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
\tau, less than a critical value \tau_0. Once the critical
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
(or " yield stress") is exceeded, the material flows in such a way that the
shear rate In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
, ∂''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: :\frac = \begin 0, & \tau < \tau_0 \\ \frac, & \tau \ge \tau_0 \end


Friction factor formulae

In fluid flow, it is a common problem to calculate the pressure drop in an established piping network. 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 In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation ...
: :f = where: * f is the
Darcy friction factor Darcy, Darci or Darcey may refer to: Science * Darcy's law, which describes the flow of a fluid through porous material * Darcy (unit), a unit of permeability of fluids in porous material * Darcy friction factor in the field of fluid mechanics ...
(SI units: dimensionless) * h_\text is the frictional head loss ( SI units: m) * g is the gravitational acceleration (SI units: m/s²) * D is the pipe diameter (SI units: m) * L is the pipe length (SI units: m) * V is the mean fluid velocity (SI units: m/s)


Laminar flow

An exact description of friction loss for Bingham plastics in fully developed laminar pipe flow was first published by Buckingham. His expression, the ''Buckingham–Reiner'' equation, can be written in a dimensionless form as follows: :f_\text = \left + - \left(\right)\right/math> where: * f_\text is the laminar flow Darcy friction factor (SI units: dimensionless) * \operatorname is the Reynolds number (SI units: dimensionless) * \operatorname is the Hedstrom number (SI units: dimensionless) The Reynolds number and the Hedstrom number are respectively defined as: :\operatorname = , and :\operatorname = where: * \rho is the mass density of fluid (SI units: kg/m3) * \mu is the dynamic viscosity of fluid (SI units: kg/m s) * \tau_o is the yield point (yield strength) of fluid (SI units: Pa)


Turbulent flow

Darby and Melson developed an empirical expressionDarby, R. and Melson J.(1981). "How to predict the friction factor for flow of Bingham plastics". ''Chemical Engineering'' 28: 59–61. that was then refined, and is given by: :f_\text = 4 \times 10^a \operatorname^ where: * f_\text is the turbulent flow friction factor (SI units: dimensionless) * a = -1.47\left + 0.146 e^\right/math> Note: Darby and Melson's expression is for a Fanning friction factor, and needs to be multiplied by 4 to be used in the friction loss equations located elsewhere on this page.


Approximations of the Buckingham–Reiner equation

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.


Swamee–Aggarwal equation

The Swamee–Aggarwal equation is used to solve directly for the Darcy–Weisbach friction factor ''f'' for laminar flow of Bingham plastic fluids. It is an approximation of the implicit ''Buckingham–Reiner'' equation, but the discrepancy from experimental data is well within the accuracy of the data. The Swamee–Aggarwal equation is given by: : f_L = + \left( \right)^


Danish–Kumar solution

Danish ''et al.'' have provided an explicit procedure to calculate the friction factor ''f'' by using the Adomian decomposition method. The friction factor containing two terms through this method is given as: : f_L = \frac where : K_1 = + , and : K_2 = - .


Combined equation for friction factor for all flow regimes


Darby–Melson equation

In 1981, Darby and Melson, using the approach of Churchill and of Churchill and Usagi, developed an expression to get a single friction factor equation valid for all flow regimes: :f = \left m + ^m \right\frac where: :m = 1.7 + 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.


See also

*
Bagnold number The Bagnold number (Ba) is the ratio of grain collision stresses to viscous fluid stresses in a granular flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold. The Bagnold number is defined by : \mathrm=\frac, where \ ...
*
Bernoulli's principle In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
*
Bingham-Papanastasiou model An important class of non-Newtonian fluids presents a yield stress limit which must be exceeded before significant deformation can occur – the so-called viscoplastic fluids or Bingham plastic A Bingham plastic is a viscoplastic material that be ...
*
Rheology Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with Plasticity (physics), plastic flow rather than deforming Elasticity (phy ...
*
Shear thinning In rheology, shear thinning is the non-Newtonian behavior of fluids whose viscosity decreases under shear strain. It is sometimes considered synonymous for pseudo-plastic behaviour, and is usually defined as excluding time-dependent effects, s ...


References

{{DEFAULTSORT:Bingham Plastic Materials Non-Newtonian fluids Viscosity Offshore engineering