Power-law fluid

In continuum mechanics, a power-law fluid, or the Ostwald–de Waele relationship, is a type of generalized Newtonian fluid (time-independent non-Newtonian fluid) for which the shear stress, τ, is given by

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

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

${\displaystyle \mu _{\mathrm {eff} }=K\left({\frac {\partial u}{\partial y}}\right)^{n-1}}$

represents an apparent or effective viscosity as a function of the shear rate (SI unit Pa s). The value of K and n can be obtained from the graph of ${\textstyle \log(\mu _{\mathrm {eff} })}$ and ${\textstyle \log \left({\frac {\partial u}{\partial y}}\right)}$. The slope line gives the value of n – 1, from which n can be calculated. The intercept at ${\textstyle \log \left({\frac {\partial u}{\partial y}}\right)=0}$ gives the value of ${\textstyle \log(K)}$.

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: