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,

.

For

, this reduces to a Newtonian fluid with

. For

, we obtain a power-law fluid with

.

The infinite-shear viscosity

and the zero-shear viscosity

of the Carreau fluid are taken equal to

and

, respectively. The relaxation parameter

is set equal to

.

For the flow of a Carreau fluid in a pipe, only numerical solutions are available.

[1] H. Binous, "Introducing Non-Newtonian Fluid Mechanics Computations with

*Mathematica* in the Undergraduate Curriculum,"

* Chemical Engineering Education*,

**41**(1), 2007 pp. 59–64.

[2] J. O. Wilkes,

*Fluid Mechanics for Chemical Engineers*, Upper Saddle River, NJ: Prentice Hall, 1999.