Alexei Volkov
In fluid dynamics, Newtonian fluids are those that have a linear relationship between shear stress and shear rate, meaning that they flow smoothly and their viscosity remains constant. On the other hand, non-Newtonian fluids exhibit either shear-thinning or shear-thickening behaviour, and some exhibit a yield stress, meaning they behave like a solid until a certain stress level is reached. One of the most common viscosity models used to describe non-Newtonian fluids is the Bingham Plastic model, which is used in hydraulic analysis and has applications in drilling engineering and the handling of slurries. So, while not a Newtonian fluid itself, the Bingham Plastic model is an important tool for understanding and working with non-Newtonian fluids.
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What You'll Learn
Bingham Plastic Model
In materials science, a Bingham plastic is a viscoplastic material that acts like a rigid body under low stress but flows as a viscous fluid when the stress is high. It is named after Eugene C. Bingham, who first 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.
The Bingham Plastic model is a commonly used model for non-Newtonian fluids in hydraulic analysis. It describes the relationship between the shear rate and the applied shear stress on a fluid. The model is defined by the following equation:
Du/dy = τ − τ0
Where du/dy is the shear rate (or velocity gradient), τ is the applied shear stress, and τ0 is the yield stress. The yield stress represents the minimum stress required for the fluid to start flowing.
The Bingham Plastic model also introduces the concept of plastic viscosity or the coefficient of rigidity, denoted as μ∞. This parameter represents the apparent viscosity of the fluid and is assumed to be constant for Bingham plastics. If the value of μ∞ is 0, the fluid behaves like a Newtonian fluid.
Some fluids may exhibit Bingham pseudoplastic behaviour, where the relationship between shear stress and shear rate is non-linear. In such cases, the Herschel-Bulkley viscosity model can be used for modelling. To apply the Bingham Plastic model or the Herschel-Bulkley model, values for τ0 and μ∞ are required, or rheological test data can be used.
The Bingham Plastic model is useful for modelling fluids with complex behaviours, such as mud flow in drilling engineering and the handling of slurries. It provides a more accurate description of non-Newtonian fluids, which have variable viscosities and require more than just shear stress to describe their flow behaviour.
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Non-Newtonian fluids
The Bingham Plastic model is one of the most common viscosity models used in hydraulic analysis. It is used to describe non-Newtonian fluids. The model states that the shear rate resulting from a given shear stress applied to the fluid can be described using an equation involving the shear rate (or velocity gradient), the applied shear stress, the yield stress, and the plastic viscosity or coefficient of rigidity.
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. 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 Bingham Plastic model uses a constant value for plastic viscosity, meaning a value of zero would result in a Newtonian fluid. Some fluids may exhibit Bingham pseudoplastic behaviour, where shear stress and shear rate do not have a linear relationship. These fluids can be modelled using the Herschel-Bulkley viscosity model. The Herschel-Bulkley model is one of several models used to characterise non-Newtonian fluids, including the Power Law, Casson, and Bingham Plastic models.
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Rheology models
Herschel-Bulkley Model
This model is used to describe fluids that exhibit a non-linear relationship between shear stress and shear rate, such as in Bingham pseudoplastic behaviour. It requires values for yield stress and plastic viscosity or rheological test data.
Power Law Model
This model describes a power-law relationship between shear stress and shear rate, where the viscosity decreases as the shear rate increases.
Casson Model
The Casson model is one of the rheology models available in FluidFlow software. It is used to characterise non-Newtonian fluids by defining data points of shear rate versus shear stress.
Bingham Plastic Model
Named after Eugene C. Bingham, this model is used to describe the behaviour of viscoplastic materials that act as rigid bodies at low stresses but flow as viscous fluids at high stresses. The Bingham Plastic model is commonly used in hydraulic analysis and can be applied to situations like mud flow in drilling engineering and the handling of slurries. It uses a constant value for plastic viscosity, and when this value is set to zero, the model describes a Newtonian fluid. The model can be expressed with the following equation:
> η = du/dy = τ − τo/μ∞
Where:
- Η = Apparent viscosity of the fluid
- Du/dy = Shear rate (or velocity gradient)
- Τ = Applied shear stress
- Τo = Yield stress
- Μ∞ = Plastic viscosity or coefficient of rigidity
Swamee-Aggarwal and Darby-Melson Equations
These equations can be used to determine the friction factor of Bingham plastic fluids in any regime. The friction factor is a critical parameter in calculating pressure drop and flow rate in pipe flow systems.
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Yield stress
Mathematically, the yield stress of a Bingham plastic is denoted as τo (tau-zero) in the Bingham Plastic model, which describes the relationship between shear rate and applied shear stress. In this model, the shear rate (du/dy) represents the velocity gradient of the fluid, while τ (tau) represents the applied shear stress. The Bingham Plastic model is commonly used in hydraulic analysis and is particularly useful for modelling non-Newtonian fluids, such as drilling fluids and slurries.
The concept of yield stress is essential in understanding the unique behaviour of Bingham plastics. Unlike Newtonian fluids, which exhibit a linear relationship between shear stress and shear rate, Bingham plastics require a finite amount of stress to initiate flow. This is because Bingham plastics contain particles or large molecules that interact to form a weak solid structure. This structure must be overcome for flow to occur, resulting in a yield stress that must be exceeded.
The yield stress of a Bingham plastic can provide valuable information about the composition and quality of certain materials. For instance, in drilling engineering, an increase in the yield point of a drilling fluid can indicate chemical contamination or degradation of chemicals used to maintain the yield point. Similarly, an increase in plastic viscosity can suggest the presence of solid contamination in the fluid.
In summary, yield stress is a critical parameter in characterising Bingham plastics and understanding their behaviour. It represents the minimum stress required to initiate flow in these materials, and its value can provide insights into the underlying structure and composition of the fluid or solid. By considering yield stress, scientists and engineers can better model and work with non-Newtonian fluids, such as those encountered in drilling operations and the handling of slurries.
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Volumetric flow rate
The Bingham Plastic model is commonly used to describe non-Newtonian fluids in hydraulic analysis. This model, first proposed by Eugene C. Bingham in 1916, characterises fluids that behave as rigid bodies at low stresses but flow as viscous fluids when subjected to high stress.
However, Bingham plastics exhibit distinct behaviour regarding volumetric flow rate. Unlike Newtonian fluids, Bingham plastics require the application of a specific amount of stress, known as the yield stress, before they begin to flow. Below this critical stress, Bingham plastics behave as rigid bodies with no observable shear rate or velocity. Once the yield stress is exceeded, the volumetric flow rate of Bingham plastics increases steadily with increasing shear stress.
The volumetric flow rate of Bingham plastics is influenced by their unique rheological properties. These fluids require two parameters for description: the yield stress and the slope of the line representing plastic viscosity. The presence of particles or large molecules within the fluid creates a weak solid structure that must be broken down before flow occurs, resulting in the characteristic threshold stress behaviour.
Various equations have been developed to describe the behaviour of Bingham plastics, including the Bingham Plastic model, the Buckingham-Reiner equation, and the Swamee-Aggarwal and Darby-Melson equations. These equations aid in calculating friction factors and understanding the volumetric flow rate characteristics of these non-Newtonian fluids.
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Frequently asked questions
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.
A Newtonian fluid flows and gives a shear rate for any finite value of shear stress.
A Newtonian fluid flows and gives a shear rate for any finite value of shear stress. A Bingham plastic, on the other hand, does not exhibit any shear rate until a certain stress, called the yield stress, is achieved.
A common example of a Bingham plastic is toothpaste, which will not be extruded until a certain pressure is applied to the tube. Some fluids may also exhibit Bingham pseudoplastic behavior, where shear stress and shear rate do not have a linear relationship.
