- Power-law fluid
__NOTOC__A Power-law fluid is a type of
generalized Newtonian fluid for which theshear stress , "τ", is given by:au = K left( frac {partial u} {partial y} ight)^n
where:
*"K" is the "flow consistency index" (SI units Pa•s"n"),
*∂"u"/∂"y" is theshear rate or thevelocity gradient perpendicular to the plane of shear (SI unit s−1), and
*"n" is the "flow behaviour index" (dimensionless).The quantity
:mu_{operatorname{eff = K left( frac {partial u} {partial y} ight)^{n-1}
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 realnon-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 thephysical chemistry at themolecular 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 , or "shear-thinning" fluids have a lower apparent viscosity at higher shear rates, and are usuallysolution 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 orglycerine 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 fluid is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate::au = mu frac {partial u} {partial y}
These fluids have a constant viscosity, "μ", across all shear rates and include many of the most common fluids, such as
water , most aqueoussolution s,oil s,corn syrup ,glycerine ,air and othergas es.Dilatant fluids
Dilatant , or "shear-thickening" fluids increase in apparent viscosity at higher shear rates. They are rarely encountered, but one common example is an uncooked paste ofcornstarch andwater . Under high shear the water is squeezed out from between thestarch 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 aviscous 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 primary driven axle starts to spin due to loss of traction under one or both tires.ee also
*
Power law
*Rheology
*Navier-Stokes equations
*Fluid
*First-Order Fluid
*Cross fluid
*Carreau fluid
*Generalized Newtonian fluid
*Herschel-Bulkley fluid
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