Cross fluid

In fluid dynamics, a Cross fluid is a type of generalized Newtonian fluid whose viscosity depends upon shear rate according to the Cross power law equation:

$${\displaystyle \mu _{\mathrm {eff} }({\dot {\gamma }})=\mu _{\infty }+{\frac {\mu _{0}-\mu _{\infty }}{1+(m{\dot {\gamma }})^{n}}}}$$

where ${\displaystyle \mu _{\mathrm {eff} }({\dot {\gamma }})}$ is viscosity as a function of shear rate, ${\displaystyle \mu _{\infty }}$ is the infinite-shear-rate viscosity, ${\displaystyle \mu _{0}}$ is the zero-shear-rate viscosity, ${\displaystyle m}$ is the time constant, and ${\displaystyle n}$ is the shear-thinning index.

The zero-shear viscosity ${\displaystyle \mu _{0}}$ is approached at very low shear rates, while the infinite shear viscosity ${\displaystyle \mu _{\infty }}$ is approached at very high shear rates.¹

When ${\displaystyle \mu _{0}}$ > ${\displaystyle \mu _{\infty }}$, the fluid exhibits shear thinning (pseudoplastic) behavior where viscosity decreases with increasing shear rate; when ${\displaystyle \mu _{0}}$ < ${\displaystyle \mu _{\infty }}$, the fluid displays shear thickening (dilatant) behavior where viscosity increases with shear rate.

It is named after Malcolm M. Cross who proposed this model in 1965.²³