Power-law fluid

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

\tau = K \left( \frac {\partial u} {\partial y} \right)^n

where:

K is the flow consistency index (SI units Pa•sⁿ),
∂u/∂y is the shear rate or the velocity gradient perpendicular to the plane of shear (SI unit s⁻¹), and
n is the flow behavior index (dimensionless).

The quantity

\mu_{\operatorname{eff}} = K \left( \frac {\partial u} {\partial y} \right)^{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^[1]^[2] 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:

n	Type of fluid
<1	Pseudoplastic
1	Newtonian fluid
>1	Dilatant (less common)

Pseudoplastic fluids

Pseudoplastic, or shear-thinning fluids have a lower apparent viscosity at higher shear rates, and are usually solutions of large, polymeric 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).

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:

\tau = \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 aqueous solutions, oils, corn syrup, glycerine, 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, 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 rates the water is squeezed out from between the starch molecules, 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.

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,

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]

where $u(r)$ is the (radially) local axial velocity, $dp/dz$ is the pressure gradient along the pipe, and $R$ is the pipe radius.

References