# Carreau fluid

Carreau fluid in physics is a type of generalized Newtonian fluid where viscosity, ${\displaystyle \mu _{\operatorname {eff} }}$, depends upon the shear rate, ${\displaystyle {\dot {\gamma }}}$, by the following equation:

${\displaystyle \mu _{\operatorname {eff} }({\dot {\gamma }})=\mu _{\operatorname {\inf } }+(\mu _{0}-\mu _{\operatorname {\inf } })\left(1+\left(\lambda {\dot {\gamma }}\right)^{2}\right)^{\frac {n-1}{2}}}$

Where: ${\displaystyle \mu _{0}}$, ${\displaystyle \mu _{\operatorname {\inf } }}$, ${\displaystyle \lambda }$ and ${\displaystyle n}$ are material coefficients.

${\displaystyle \mu _{0}}$ = viscosity at zero shear rate (Pa.s)

${\displaystyle \mu _{\operatorname {\inf } }}$ = viscosity at infinite shear rate (Pa.s)

${\displaystyle \lambda }$ = characteristic time (s)

${\displaystyle n}$ = power index

The dynamics of fluid motions is an important area of physics, with many important and commercially significant applications.

Computers are often used to calculate the motions of fluids, especially when the applications are of a safety critical nature.

## Carreau Fluid Shear Rates

• At low shear rate (${\displaystyle {\dot {\gamma }}\ll 1/\lambda }$) a Carreau fluid behaves as a Newtonian fluid with viscosity ${\displaystyle \mu _{0}}$.
• At intermediate shear rates (${\displaystyle {\dot {\gamma }}\gtrsim 1/\lambda }$), a Carreau fluid behaves as a Power-law fluid.
• At high shear rate, which depends on the power index ${\displaystyle n}$ and the infinite shear-rate viscosity ${\displaystyle \mu _{\operatorname {\inf } }}$, a Carreau fluid behaves as a Newtonian fluid again with viscosity ${\displaystyle \mu _{\operatorname {\inf } }}$.

## Origin of Carreau Fluid Model

The model was first proposed by Pierre Carreau.