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__NOTOC__ In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, a power-law fluid, or the Ostwald–de Waele relationship, is a type of
generalized Newtonian fluid A generalized Newtonian fluid is an idealized fluid for which the shear stress is a function of shear rate at the particular time, but not dependent upon the history of deformation. Although this type of fluid is non-Newtonian (i.e. non-linear) in ...
(time-independent
non-Newtonian fluid A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for ex ...
) for which the shear stress, , is given by :\tau = K \left( \frac \right)^n where: * is the ''flow consistency index'' ( SI units Pa s''n''), * is 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 ...
or the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
perpendicular to the plane of shear (SI unit s−1), and * is the ''flow behavior index'' (
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
). The quantity :\mu_\mathrm = K \left( \frac \right)^ represents an ''apparent'' or ''effective
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 in ...
'' as a function of the shear rate (SI unit Pa s). The value of and can be obtained from the graph of \log(\mu_\mathrm) and \log\left( \frac \right) . The slope line gives the value of , from which can be calculated. The intercept at \log\left( \frac \right) = 0 gives the value of . Also known as the Ostwaldde Waele power lawe.g. G. W. Scott Blair ''et al.'', ''J. Phys. Chem''., (1939) 43 (7) 853–864. Also the ''de Waele-Ostwald'' law, e.g
Markus Reiner Markus Reiner ( he, מרכוס ריינר, born 5 January 1886, died 25 April 1976) was an Israeli scientist and a major figure in rheology. Biography Reiner was born in 1886 in Czernowitz, Bukovina, then part of Austria-Hungary, and obtain ...
''et al.'', ''Kolloid Zeitschrift'' (1933) 65 (1) 44-62
Ostwald called it the de Waele-Ostwald equation: ''Kolloid Zeitschrift'' (1929) 47 (2) 176-187 this
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
relationship is useful because of its simplicity, but only approximately describes the behaviour of a real non-Newtonian fluid. For example, if 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 is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical me ...
at the
molecular A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
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:


Pseudoplastic fluids

Pseudoplastic 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, ...
, or ''shear-thinning'' are those fluids whose behaviour is time independent and which have a lower apparent viscosity at higher shear rates, and are usually solutions of large,
polymer A polymer (; Greek '' poly-'', "many" + '' -mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
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 is a food syrup which is made from the starch of corn (called maize in many countries) and contains varying amounts of sugars: glucose, maltose and higher oligosaccharides, depending on the grade. Corn syrup is used in foods to soften ...
or
glycerine Glycerol (), also called glycerine in British English and glycerin in American English, is a simple triol compound. It is a colorless, odorless, viscous liquid that is sweet-tasting and non-toxic. The glycerol backbone is found in lipids known ...
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). This type of behavior is widely encountered in solutions or suspensions. In these cases, large molecules or fine particles form loosely bounded aggregates or alignment groupings that are stable and reproducible at any given shear rate. But these fluids rapidly and reversibly break down or reform with an increase or decrease in shear rate. Pseudo plastic fluids show this behavior over a wide range of shear rates; however often approach a limiting Newtonian behavior at very low and very high rates of shear. These Newtonian regions are characterized by the viscosities \mu_0 and \mu_\infty respectively.


Newtonian fluids

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 ...
is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate: :\tau = \mu \frac These fluids have a constant viscosity, ''μ'', across all shear rates and include many of the most common fluids, such as
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
, most
aqueous solution An aqueous solution is a solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), in water would be ...
s, oils,
corn syrup Corn syrup is a food syrup which is made from the starch of corn (called maize in many countries) and contains varying amounts of sugars: glucose, maltose and higher oligosaccharides, depending on the grade. Corn syrup is used in foods to soften ...
,
glycerine Glycerol (), also called glycerine in British English and glycerin in American English, is a simple triol compound. It is a colorless, odorless, viscous liquid that is sweet-tasting and non-toxic. The glycerol backbone is found in lipids known ...
, air and other gases. 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

Dilatant A dilatant (, ) (also termed shear thickening) material is one in which viscosity increases with the rate of shear strain. Such a ''shear thickening fluid'', also known by the initialism ''STF'', is an example of a non-Newtonian fluid. This beha ...
, or ''shear-thickening'' fluids increase in apparent viscosity at higher shear rates. They are in common use in
viscous coupling A viscous coupling is a mechanical device which transfers torque and rotation by the medium of a viscous fluid. Design Rotary viscous couplings with interleaved, perforated plates and filled with viscous fluids are used in automotive system ...
s in automobiles. When both ends of the coupling are spinning at the same rotational speed, the viscosity of the dilatant fluid is minimal, but if the ends of the coupling differ in speed, the coupling fluid becomes very viscous. They are used to prevent all of the torque from going to one wheel when the traction on that wheel drops, e.g. when one wheel is on ice. The viscous coupling between the two driven wheels ensures that both wheels turn at the same rate, providing torque to the wheel that is not slipping. Viscous couplings are also used to keep the front axle and the rear axle spinning at the same rate in four-wheel drive passenger automobiles. Dilatant fluids are rarely encountered in everyday situations. One common example is an uncooked paste of
cornstarch Corn starch, maize starch, or cornflour (British English) is the starch derived from corn ( maize) grain. The starch is obtained from the endosperm of the kernel. Corn starch is a common food ingredient, often used to thicken sauces or ...
and
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
, sometimes known as oobleck. Under high shear rates, the water is squeezed out from between the
starch Starch or amylum is a polymeric carbohydrate consisting of numerous glucose units joined by glycosidic bonds. This polysaccharide is produced by most green plants for energy storage. Worldwide, it is the most common carbohydrate in human d ...
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s, which are able to interact more strongly, enormously increasing the viscosity. While not strictly a dilatant fluid,
Silly Putty Silly Putty is a toy based on silicone polymers that have unusual physical properties. It bounces, but it breaks when given a sharp blow, and it can also flow like a liquid. It contains a viscoelastic liquid silicone, a type of non-Newtonian fl ...
(
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
fluid) is an example of a material that shares these viscosity characteristics.


Velocity profile in a circular pipe

Just 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 ...
in a circular pipe gives a quadratic velocity profile (see
Hagen–Poiseuille equation In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow ...
), a power-law fluid will result in a power-law velocity profile, :u(r) = \frac\left(\frac\frac\right)^\frac\left(R^\frac-r^\frac\right) where ''u''(''r'') is the (radially) local axial velocity, ' is the pressure gradient along the pipe, and ''R'' is the pipe radius.


See also

*
Power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
*
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 plastic flow rather than deforming elastically in response to an appli ...
*
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
*
Fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
* First-order fluid *
Cross fluid A Cross fluid is a type of generalized Newtonian fluid whose viscosity depends upon shear rate according to the following equation: :\mu_\mathrm(\dot \gamma) = \mu_\infty + \frac where \mu_\mathrm(\dot \gamma) is viscosity as a function of sh ...
*
Carreau fluid Carreau fluid in physics is a type of generalized Newtonian fluid where viscosity, \mu_, depends upon the shear rate, \dot \gamma, by the following equation: : \mu_(\dot \gamma) = \mu_ + (\mu_0 - \mu_) \left(1+\left(\lambda \dot \gamma\right) ^2 ...
*
Generalized Newtonian fluid A generalized Newtonian fluid is an idealized fluid for which the shear stress is a function of shear rate at the particular time, but not dependent upon the history of deformation. Although this type of fluid is non-Newtonian (i.e. non-linear) in ...
*
Herschel–Bulkley fluid The 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 consistenc ...


References

{{reflist Non-Newtonian fluids