Power-law fluid
Encyclopedia
A Power-law fluid, or the Ostwald
–de Waele
relationship, is a type of generalized Newtonian fluid
for which the shear stress
, τ, is given by
where:
The quantity
represents an apparent or effective viscosity
as a function of the shear rate (SI unit Pa•s).
Also known as the Ostwald
–de Waele
power law this mathematical relationship is useful because of its simplicity, but only approximately describes the behaviour of a real non-Newtonian fluid
. For example, if n were less than one, the power law predicts that the effective viscosity would decrease with increasing shear rate indefinitely, requiring a fluid with infinite viscosity at rest and zero viscosity as the shear rate approaches infinity, but a real fluid has both a minimum and a maximum effective viscosity that depend on the physical chemistry
at the molecular level. Therefore, the power law is only a good description of fluid behaviour across the range of shear rates to which the coefficients were fitted. There are a number of other models that better describe the entire flow behaviour of shear-dependent fluids, but they do so at the expense of simplicity, so the power law is still used to describe fluid behaviour, permit mathematical predictions, and correlate experimental data.
Power-law fluids can be subdivided into three different types of fluids based on the value of their flow behaviour index:
s of large, polymer
ic molecules in a solvent with smaller molecules. It is generally supposed that the large molecular chains tumble at random and affect large volumes of fluid under low shear, but that they gradually align themselves in the direction of increasing shear and produce less resistance.
A common household example of a strongly shear-thinning fluid is styling gel, which primarily composed of water and a fixative such as a vinyl acetate/vinylpyrrolidone copolymer (PVP/PA). If one were to hold a sample of hair gel in one hand and a sample of corn syrup
or glycerine in the other, they would find that the hair gel is much harder to pour off the fingers (a low shear application), but that it produces much less resistance when rubbed between the fingers (a high shear application).
is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate:
These fluids have a constant viscosity, μ, across all shear rates and include many of the most common fluids, such as water
, most aqueous solution
s, oil
s, corn syrup
, glycerine, air and other gas
es.
While this holds true for relatively low shear rates, at high rates most oils in reality also behave in a non-Newtonian fashion and thin. Typical examples include oil films in automotive engine shell bearings and to a lesser extent in geartooth contacts.
, or shear-thickening fluids increase in apparent viscosity at higher shear rates. They are rarely encountered, but one common example is an uncooked paste of cornstarch
and water
. Under high shear the water is squeezed out from between the starch
molecule
s, which are able to interact more strongly.
While not strictly a dilatant fluid, Silly Putty
is an example of a material that shares these viscosity characteristics. Another use is in a viscous coupling in which if both ends of the coupling are spinning at the same (rotational) speed, the fluid viscosity is minimal, but if the ends of the coupling differ greatly in speed, the coupling fluid becomes very viscous. Such couplings have applications as a lightweight,
passive mechanism for a passenger automobile to automatically switch from two-wheel drive to four-wheel drive such as when the vehicle is stuck in snow and the primary driven axle starts to spin due to loss of traction under one or both tires.
in a circular pipe gives a quadratic velocity profile (see Hagen–Poiseuille equation), a power-law fluid will result in a power-law velocity profile,
where
is the (radially) local axial velocity, 
is the pressure gradient along the pipe, and 
is the pipe radius.
Wilhelm Ostwald
Friedrich Wilhelm Ostwald was a Baltic German chemist. He received the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria and reaction velocities...
–de Waele
Armand de Waele
Armand de Waele was a British chemist, noted for his contributions to rheology, and after whom the Ostwald-de Waele relationship for non-Newtonian fluids is named....
relationship, is a type of generalized Newtonian fluid
Generalized Newtonian fluid
A generalized Newtonian fluid is an idealized fluid for which the shear stress, τ, is a function of shear rate at the particular time, but not dependent upon the history of deformation.\tau = F\left where:...
for which the shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
, τ, is given by
where:
- K is the flow consistency index (SISiSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units Pa•sn), - ∂u/∂y is the shear rate or the velocityVelocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
perpendicular to the plane of shear (SI unit s−1), and - n is the flow behaviour index (dimensionless).
The quantity
represents an apparent or effective viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
as a function of the shear rate (SI unit Pa•s).
Also known as the Ostwald
Wilhelm Ostwald
Friedrich Wilhelm Ostwald was a Baltic German chemist. He received the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria and reaction velocities...
–de Waele
Armand de Waele
Armand de Waele was a British chemist, noted for his contributions to rheology, and after whom the Ostwald-de Waele relationship for non-Newtonian fluids is named....
power law this mathematical relationship is useful because of its simplicity, but only approximately describes the behaviour of a real non-Newtonian fluid
Non-Newtonian fluid
A non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of non-Newtonian fluids is not independent of shear rate or shear rate history...
. For example, if n were less than one, the power law predicts that the effective viscosity would decrease with increasing shear rate indefinitely, requiring a fluid with infinite viscosity at rest and zero viscosity as the shear rate approaches infinity, but a real fluid has both a minimum and a maximum effective viscosity that depend on the physical chemistry
Physical chemistry
Physical chemistry is the study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems in terms of physical laws and concepts...
at the molecular level. Therefore, the power law is only a good description of fluid behaviour across the range of shear rates to which the coefficients were fitted. There are a number of other models that better describe the entire flow behaviour of shear-dependent fluids, but they do so at the expense of simplicity, so the power law is still used to describe fluid behaviour, permit mathematical predictions, and correlate experimental data.
Power-law fluids can be subdivided into three different types of fluids based on the value of their flow behaviour index:
| n | Type of fluid |
| <1 | Pseudoplastic |
| 1 | Newtonian fluid Newtonian fluid A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:... |
| >1 | Dilatant Dilatant A dilatant material is one in which viscosity increases with the rate of shear strain. Such a shear thickening fluid, also known by the acronym STF, is an example of a non-Newtonian fluid.... (less common) |
Pseudoplastic fluids
Pseudoplastic, or shear-thinning fluids have a lower apparent viscosity at higher shear rates, and are usually solutionSolution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
s of large, polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
ic molecules in a solvent with smaller molecules. It is generally supposed that the large molecular chains tumble at random and affect large volumes of fluid under low shear, but that they gradually align themselves in the direction of increasing shear and produce less resistance.
A common household example of a strongly shear-thinning fluid is styling gel, which primarily composed of water and a fixative such as a vinyl acetate/vinylpyrrolidone copolymer (PVP/PA). If one were to hold a sample of hair gel in one hand and a sample of corn syrup
Corn syrup
Corn syrup is a food syrup, which is made from the starch of maize and contains varying amounts of maltose and higher oligosaccharides, depending on the grade. Corn syrup is used in foods to soften texture, add volume, prevent crystallization of sugar, and enhance flavor...
or glycerine in the other, they would find that the hair gel is much harder to pour off the fingers (a low shear application), but that it produces much less resistance when rubbed between the fingers (a high shear application).
Newtonian fluids
A Newtonian fluidNewtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate:
These fluids have a constant viscosity, μ, across all shear rates and include many of the most common fluids, such as water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
, most aqueous solution
Solution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
s, oil
Oil
An oil is any substance that is liquid at ambient temperatures and does not mix with water but may mix with other oils and organic solvents. This general definition includes vegetable oils, volatile essential oils, petrochemical oils, and synthetic oils....
s, corn syrup
Corn syrup
Corn syrup is a food syrup, which is made from the starch of maize and contains varying amounts of maltose and higher oligosaccharides, depending on the grade. Corn syrup is used in foods to soften texture, add volume, prevent crystallization of sugar, and enhance flavor...
, glycerine, air and other gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
es.
While this holds true for relatively low shear rates, at high rates most oils in reality also behave in a non-Newtonian fashion and thin. Typical examples include oil films in automotive engine shell bearings and to a lesser extent in geartooth contacts.
Dilatant fluids
DilatantDilatant
A dilatant material is one in which viscosity increases with the rate of shear strain. Such a shear thickening fluid, also known by the acronym STF, is an example of a non-Newtonian fluid....
, or shear-thickening fluids increase in apparent viscosity at higher shear rates. They are rarely encountered, but one common example is an uncooked paste of cornstarch
Cornstarch
Corn starch, cornstarch, cornflour or maize starch is the starch of the corn grain obtained from the endosperm of the corn kernel.-History:...
and water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
. Under high shear the water is squeezed out from between the starch
Starch
Starch or amylum is a carbohydrate consisting of a large number of glucose units joined together by glycosidic bonds. This polysaccharide is produced by all green plants as an energy store...
molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s, which are able to interact more strongly.
While not strictly a dilatant fluid, Silly Putty
Silly Putty
Silly Putty , is the Crayola-owned trademark name for a class of silicone polymers. It is marketed today as a toy for children, but was originally created by accident during research into potential rubber substitutes for use by the United States in World War II...
is an example of a material that shares these viscosity characteristics. Another use is in a viscous coupling in which if both ends of the coupling are spinning at the same (rotational) speed, the fluid viscosity is minimal, but if the ends of the coupling differ greatly in speed, the coupling fluid becomes very viscous. Such couplings have applications as a lightweight,
passive mechanism for a passenger automobile to automatically switch from two-wheel drive to four-wheel drive such as when the vehicle is stuck in snow and the primary driven axle starts to spin due to loss of traction under one or both tires.
Velocity profile in a circular pipe
Just like a Newtonian fluidNewtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
in a circular pipe gives a quadratic velocity profile (see Hagen–Poiseuille equation), a power-law fluid will result in a power-law velocity profile,
where
is the (radially) local axial velocity, is the pressure gradient along the pipe, and is the pipe radius.See also
- Power lawPower lawA power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
- RheologyRheologyRheology is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force....
- Navier-Stokes equationsNavier-Stokes equationsIn physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
- FluidFluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
- First-Order Fluid
- Cross fluid
- Carreau fluid
- Generalized Newtonian fluidGeneralized Newtonian fluidA generalized Newtonian fluid is an idealized fluid for which the shear stress, τ, is a function of shear rate at the particular time, but not dependent upon the history of deformation.\tau = F\left where:...
- Herschel-Bulkley fluidHerschel-Bulkley fluidThe Herschel–Bulkley fluid is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency k, the flow index n, and the yield shear stress \tau_0...