Carreau fluid

Carreau fluid 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)

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

${\displaystyle n}$ = power index

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 ${\displaystyle \mu _{\operatorname {\inf } }}$.

The model was first proposed by Pierre Carreau.