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

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

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 and . The slope line gives the value of n – 1, from which n can be calculated. The intercept at gives the value of .

Also known as the Ostwaldde 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)
  1. ^ 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
  2. ^ Ostwald called it the de Waele-Ostwald equation: Kolloid Zeitschrift (1929) 47 (2) 176-187