thumb|302px|[[Mayonnaise is a Bingham plastic. The surface has ridges and peaks because Bingham plastics mimic solids under low shear stresses.]]

In materials science, a Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form in 1916.

It is used as a common mathematical model of mud flow in drilling engineering, and in the handling of slurries. A common example is toothpaste, which will not be extruded until a certain pressure is applied to the tube. It is then pushed out as a relatively coherent plug.

Explanation

thumb|left|302px|Figure 1. Bingham Plastic flow as described by Bingham

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. These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a Newtonian fluid.

thumb|right|302px|Figure 2. Bingham Plastic flow as described currently

Figure 2 shows the way in which it is normally presented currently. 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:

:<math>f = {2h_\text{f} gD \over LV^2}</math>

where:

  • <math>f</math> is the Darcy friction factor (SI units: dimensionless)
  • <math>h_\text{f}</math> is the frictional head loss (SI units: m)
  • <math>g</math> is the gravitational acceleration (SI units: m/s²)
  • <math>D</math> is the pipe diameter (SI units: m)
  • <math>L</math> is the pipe length (SI units: m)
  • <math>V</math> 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:

:<math>f_\text{L} = {64 \over \operatorname{Re\left[1 + {\operatorname{He} \over 6\operatorname{Re - {64 \over 3} \left({\operatorname{He}^4 \over f^3\operatorname{Re}^7}\right)\right]</math>

where:

  • <math>f_\text{L}</math> is the laminar flow Darcy friction factor (SI units: dimensionless)
  • <math>\operatorname{Re}</math> is the Reynolds number (SI units: dimensionless)
  • <math>\operatorname{He}</math> is the Hedstrom number (SI units: dimensionless)

The Reynolds number and the Hedstrom number are respectively defined as:

:<math>\operatorname{Re} = {\rho VD \over \mu},</math> and

:<math>\operatorname{He} = {\rho D^2\tau_o \over \mu^2}</math>

where:

  • <math>\rho</math> is the mass density of fluid (SI units: kg/m<sup>3</sup>)
  • <math>\mu</math> is the dynamic viscosity of fluid (SI units: kg/m s)
  • <math>\tau_o</math> is the yield point (yield strength) of fluid (SI units: Pa)

Turbulent flow

Darby and Melson developed an empirical expression

that was then refined, and is given by:

:<math>f_\text{T} = 4 \times 10^a \operatorname{Re}^{-0.193}</math>

where:

  • <math>f_\text{T}</math> is the turbulent flow friction factor (SI units: dimensionless)
  • <math>a = -1.47\left[1 + 0.146 e^{-2.9\times {10^{-5\operatorname{He\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:

:<math> f_L = {64 \over \mathrm{Re + {64 \over \mathrm{Re \left( {\mathrm{He}\over 6.2218 \mathrm{Re\right)^{0.958}</math>

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:

:<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>

where

:<math> K_1 = {16 \over \mathrm{Re + {16 \mathrm{He} \over 6\mathrm{Re}^2},</math>

and

:<math> K_2 = - {16 \mathrm{He}^4 \over 3\mathrm{Re}^8}.</math>

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: