Sofia Martinez
Plastic viscosity is a property of fluids that describes their resistance to flow freely. It is influenced by the viscosity of the liquid and the number, shape, and size of solids present in the fluid. This concept is particularly important in drilling engineering and slurry handling, where it is used to model the flow of materials such as mud and concrete. While there is no simple formula for viscosity, which depends on the interactions of molecules in a system, various models and equations have been developed to describe and predict it. These include the Buckingham-Reiner equation, the Swamee-Aggarwal equation, and models by Perzyna, Roscoe, Murata and Kikukawa, and others. The study of plastic viscosity and viscoplasticity is essential for understanding and engineering materials' behaviour under different conditions.
| Characteristics | Values |
|---|---|
| Definition of Plastic Viscosity | Plastic viscosity (PV) is the resistance offered by a fluid to flow freely. |
| Formula for Plastic Viscosity | There is no simple formula for plastic viscosity as it depends on the interaction of molecules in a system. |
| Factors Affecting Plastic Viscosity | Number, shape, and size of solids, viscosity of the liquid phase, temperature and pressure. |
| Applications of Plastic Viscosity | Drilling fluids, cementitious mix in concrete, and Newtonian fluids. |
| Related Concepts | Viscoplasticity, Bingham plastics, shear stress, and shear rate. |
| Key Contributors | Eugene C. Bingham, Prandtl, Reuss, Hohenemser and Prager, Perzyna, Hoff, Rabotnov, Hult, Lemaitre, Kratochvil, Malinini, Khadjinsky, Ponter, Leckie, Chaboche, Henri Tresca, Saint Venant, and Levy. |
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What You'll Learn
- Eugene C. Bingham proposed the Bingham plastic model in 1916
- Buckingham published the first description of friction loss for Bingham plastics
- The Buckingham-Reiner equation can be written in a dimensionless form
- Darby and Melson developed an expression in 1981
- The Swamee-Aggarwal equation is an approximation of the Buckingham-Reiner equation
Eugene C. Bingham proposed the Bingham plastic model in 1916
In 1916, Eugene C. Bingham proposed the Bingham plastic model, which is a linear model used to describe the flow behaviour of fluids that exhibit both solid-like and liquid-like behaviour. These fluids, known as Bingham plastic fluids, are characterised by a yield point and plastic viscosity, requiring a minimum shear stress to be exceeded for flow to initiate. Below the yield stress, a Bingham plastic fluid behaves as a solid, but once the yield stress is exceeded, it flows like a Newtonian fluid with constant viscosity.
The Bingham plastic model is particularly useful for treating drilling fluids, where it is commonly employed to estimate pressure loss in turbulent conditions and indicate the nature of contamination in the drilling fluid. It is defined by the equation τ=τ0+μ0dV/dy, where τ represents shear stress, τ0 is the yield stress, and dV/dy is the shear rate. This model is advantageous in drilling engineering as it enables the drilling fluid to suspend cuttings and solids when circulation stops, a property known as thixotropic behaviour.
The plastic viscosity of a Bingham plastic fluid is a critical parameter, representing the resistance offered by the fluid to flow freely. It is influenced by the interaction between the liquid and the solids or large molecules present, resulting in a weak solid structure that requires a certain amount of stress to break. Once the fluid starts flowing, there is a linear relationship between shear stress and shear rate, with the slope of this line representing the plastic viscosity.
The Bingham plastic model is a two-parameter model, incorporating both the yield stress and the plastic viscosity of the fluid. While it is widely used in the drilling industry, it has limitations in calculating pressure losses and matching viscosities over a large range of shear rates. However, it provides valuable insights into the behaviour of fluids with solid-like and liquid-like characteristics, making it a significant contribution to the field of materials science by Eugene C. Bingham.
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Buckingham published the first description of friction loss for Bingham plastics
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. A common example of a Bingham plastic 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.
Bingham plastic fluids are used as a common mathematical model of mud flow in drilling engineering and in the handling of slurries. In a pipe, if the pressure at one end 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, 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.
The first description of friction loss for Bingham plastics in fully developed laminar pipe flow was published by Buckingham. His expression, the Buckingham-Reiner equation, can be written in a dimensionless form as:
FL = {64 / Re} [1 + He / 6Re – 64 / 3 (He4 / f3Re7)]
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.
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The Buckingham-Reiner equation can be written in a dimensionless form
Plastic viscosity is a parameter of the Bingham plastic model, which is used to describe the behaviour of materials that behave as rigid bodies at low stresses but flow as viscous fluids at high stresses. The Bingham plastic model was proposed by Eugene C. Bingham in 1916, and it has found applications in various fields, including drilling engineering and the handling of slurries.
An important aspect of the Bingham plastic model is the concept of friction loss, which describes how the fluid behaves in a fully developed laminar pipe flow. This phenomenon was first described by Buckingham, who developed an expression known as the Buckingham-Reiner equation. This equation can be written in a dimensionless form as shown below:
${\displaystyle f_{\text{L}}={\frac {64} {\operatorname {Re} }} \left[1+{\frac {\operatorname {He}} {6\operatorname {Re} }}-{\frac {64} {3}} \left({\frac {\operatorname {He} ^{4}} {f^{3}\operatorname {Re} ^{7}}} \right) \right]}$
In this equation, ${f_{L}}$ represents the friction factor, Re is the Reynolds number, and He is the Hagen number. The Buckingham-Reiner equation provides an exact description of friction loss for Bingham plastics in fully developed laminar pipe flow.
However, due to the complexity of the solution, the Buckingham-Reiner equation is rarely used directly. Instead, researchers have developed approximations and alternative equations to simplify the calculation of the friction factor for Bingham plastic fluids. These include the Swamee-Aggarwal equation, the Darby-Melson equation, and the use of the Adomian decomposition method. These approximations offer similar accuracy to the Buckingham-Reiner equation while being more accessible and practical for calculations.
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Darby and Melson developed an expression in 1981
Plastic viscosity (PV) is defined as the resistance offered by a fluid to flow freely. It is a parameter of the Bingham plastic model, which is used to mathematically model mud flow in drilling engineering and the handling of slurries. In 1916, Eugene C. Bingham proposed the mathematical form for Bingham plastic.
In 1981, Darby and Melson developed an expression to get a single friction factor equation valid for all flow regimes. They used the approach of Churchill and of Churchill and Usagi. 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. The expression is given by:
> f = [f_L^m + f_T^m]^1/m
Darby and Melson's expression is an explicit approximation of the Buckingham-Reiner equation, which is an implicit equation that is rarely used due to its complexity. The Buckingham-Reiner equation was first published by Buckingham and can be written as:
> f_L = 64 / Re [1 + He / 6Re - 64 / 3(He^4 / f^3 Re^7)]
The Swamee-Aggarwal equation is another approximation of the Buckingham-Reiner equation and is given by:
> f_L = 64 / Re + 64 / Re (He / 6.2218 Re)^0.958
The 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.
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The Swamee-Aggarwal equation is an approximation of the Buckingham-Reiner equation
Plastic viscosity is a parameter of the Bingham plastic model. It is the slope of the shear stress/shear rate line above the yield point. In materials science, a Bingham plastic is a viscoplastic substance that behaves as a solid under low stress but flows as a viscous fluid under high stress. It was named after Eugene C. Bingham, who proposed its mathematical form in 1916.
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. 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 the complexity of the solution, it is rarely used.
Therefore, researchers have tried to develop explicit approximations for the Buckingham-Reiner equation. The Swamee-Aggarwal equation is one such approximation used to solve directly for the Darcy-Weisbach friction factor f for laminar flow of Bingham plastic fluids. It is given by:
FL = (64 / Re) + (64 / Re) * (He / 6.2218 / Re) ^ 0.958
The discrepancy between the Swamee-Aggarwal equation and experimental data is well within the accuracy of the data. This equation is a valuable tool for calculating the friction factor associated with the flow of non-Newtonian fluids, which can then be used to determine the pressure drop in a piping network for a given flow using the Darcy-Weisbach equation.
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Frequently asked questions
Plastic viscosity (PV) is the resistance offered by a fluid to flow freely. This resistance is a result of friction between the liquid undergoing deformation under shear stress and the solids and liquids present.
Plastic viscosity is calculated by subtracting the 300 r/min concentric cylinder viscometer reading from the 600 r/min concentric cylinder viscometer reading. However, there is no simple formula for plastic viscosity as it depends on the interaction of molecules in the system.
The concept of plastic viscosity is related to the Bingham plastic model, which was proposed by Eugene C. Bingham in 1916.
Plastic viscosity is a specific type of viscosity that pertains to the Bingham plastic model. It is a measure of the high-shear-rate viscosity, which is dependent on the number, shape, and size of solids, as well as the viscosity of the liquid phase.
