A non-Newtonian fluid has a viscosity that changes with the applied shear force. For a Newtonian fluid (such as water), the viscosity is independent of how fast it is stirred, but for a non-Newtonian fluid the viscosity is dependent on the stirring rate. It gets either easier or harder to stir faster for different types of non-Newtonian fluids. Different constitutive equations, giving rise to various models of non-Newtonian fluids, have been proposed in order to express the viscosity as a function of the strain rate.
In power-law fluids, the relation
is assumed, where
is the power-law exponent and
is the power-law consistency index. Dilatant or shear-thickening fluids correspond to the case where the exponent in this equation is positive, while pseudo-plastic or shear-thinning fluids are obtained when
. The viscosity decreases with strain rate for
, which is the case for pseudo-plastic fluids (also called shear-thinning fluids). On the other hand, dilatant fluids are shear thickening. If
, the Newtonian fluid behavior can be recovered. The power-law consistency index is chosen to be
. For the pseudo-plastic fluid, the velocity profile is flatter near the center, where it resembles plug flow, and is steeper near the wall, where it has a higher velocity than the Newtonian fluid or the dilatant fluid. Thus, convective energy transport is higher for shear-thinning fluids when compared to shear-thickening or Newtonian fluids. For flow in a pipe of a power-law fluid, an analytical expression is available [1, 2].
According to the Carreau model for non-Newtonian fluids, first proposed by Pierre Carreau,
, this reduces to a Newtonian fluid with
, we obtain a power-law fluid with
The infinite-shear viscosity
and the zero-shear viscosity
of the Carreau fluid are taken equal to
, respectively. The relaxation parameter
is set equal to
For the flow of a Carreau fluid in a pipe, only numerical solutions are available.
 H. Binous, "Introducing Non-Newtonian Fluid Mechanics Computations with Mathematica
in the Undergraduate Curriculum," Chemical Engineering Education
(1), 2007 pp. 59–64.
 J. O. Wilkes, Fluid Mechanics for Chemical Engineers
, Upper Saddle River, NJ: Prentice Hall, 1999.