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

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 behaviour index (dimensionless).

The quantity

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 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)