View source for Power-law fluid

__NOTOC__

A '''power-law fluid''', or the '''[[Wilhelm Ostwald|Ostwald]]–[[Armand de Waele|de Waele]] relationship''', is a type of [[generalized Newtonian fluid]] (time independent Non-Newtonian fluid) for which the [[shear stress]], ''τ'', is given by

:<math>\tau = K \left( \frac {\partial u} {\partial y} \right)^n </math>

where:
*''K'' is the ''flow consistency index'' ([[SI]] units Pa&nbsp;s<sup>''n''</sup>), 
*{{sfrac|∂''u''|∂''y''}} is the [[shear rate]] or the [[velocity]] [[gradient]] perpendicular to the plane of shear (SI unit s<sup>−1</sup>), and
*''n'' is the ''flow behavior index'' (dimensionless).

The quantity

:<math>\mu_\mathrm{eff} = K \left( \frac {\partial u} {\partial y} \right)^{n-1} </math>

represents an ''apparent'' or ''effective [[viscosity]]'' as a function of the shear rate (SI unit Pa&nbsp;s). The value of K and n can be obtained from graph off log(µ <sub>eff</sub>) and <math>log\left( \frac {\partial u} {\partial y} \right) </math>. The slope line gives the value of n-1 from which n can be calculated. The intercept at <math>log\left( \frac {\partial u} {\partial y} \right) =0 </math> gives the value of K.

Also known as the [[Wilhelm Ostwald|Ostwald]]–[[Armand de Waele|de Waele]] power law<ref name=OdW>e.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]] ''et al.'', ''Kolloid Zeitschrift'' (1933) 65 (1) 44-62</ref><ref name=Ostwald1929>Ostwald called it the de Waele-Ostwald equation: ''Kolloid Zeitschrift'' (1929) 47 (2) 176-187</ref> 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:
{|class="wikitable
!''n''
!Type of fluid
|-
|<1
|[[Pseudoplastic]]
|-
|1
|[[Newtonian fluid]]
|-
|>1
|[[Dilatant]] (less common)
|}

{{clear|left}}

== Pseudoplastic fluids ==
"[[Pseudoplastic]], 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 [[solution]]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)." <ref>{{cite book |last1=Saramito |first1=Pierre |title=Complex fluids: Modeling and Algorithms |date=2016 |publisher=Springer International Publishing Switzerland |location=Cham, Switzerland |isbn=978-3-319-44362-1 |page=65 |url=https://www.springer.com/cda/content/document/cda_downloaddocument/9783319443614-c2.pdf?SGWID=0-0-45-1593584-p180195027}}</ref>

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 increase or decrease in shear rate. Pseudo plastic fluids show this behavior over a wide range of shear rate; however often approach a limiting Newtonian behavior at very low and very high rates of shear. These Newtonian regions are characterized by the viscosities <math>\mu 0 </math> and <math>\mu \infty </math> respectively. 

== Newtonian fluids ==
A [[Newtonian fluid]] is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate:

:<math>\tau = \mu \frac {\partial u} {\partial y} </math>

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.

== Dilatant fluids ==
[[Dilatant]], or ''shear-thickening'' fluids increase in apparent viscosity at higher shear rates. 

They are in common use in [[viscous coupling]]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 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 insures 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 axel and the rear axel spinning at the same rate in four-wheel drive passenger automobiles. 

Dilatant fluids rarely encountered in everyday situations. One common example is an uncooked paste of [[cornstarch]] and [[water]], sometimes known as [[Non-Newtonian fluid#Oobleck|oobleck]]. Under high shear rates the water is squeezed out from between the [[starch]] [[molecule]]s, which are able to interact more strongly, enormously increasing the viscousity.

While not strictly a dilatant fluid, [[Silly Putty]] is an example of a material that shares these viscosity characteristics.

== Velocity profile in a circular pipe ==

Just like a [[Newtonian fluid]] 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,

:<math>u(r) = \frac{n}{n+1}\left(\frac{dp}{dz}\frac{1}{2K}\right)^\frac{1}{n}\left(R^\frac{n+1}{n}-r^\frac{n+1}{n}\right) </math>

where ''u''(''r'') is the (radially) local axial velocity, ''{{sfrac|dp|dz}}'' is the pressure gradient along the pipe, and ''R'' is the pipe radius.

==See also==
*[[Power law]]
*[[Rheology]]
*[[Navier–Stokes equations]]
*[[Fluid]]
*[[First-order fluid]]
*[[Cross fluid]]
*[[Carreau fluid]]
*[[Generalized Newtonian fluid]]
*[[Herschel–Bulkley fluid]]

==References==
{{reflist}}

[[Category:Non-Newtonian fluids]]