Physics: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:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}(\dot{\gamma })={\mu }_{\mathrm{\infty }}+\frac{{\mu }_0-{\mu }_{\mathrm{\infty }}}{1+(m\dot{\gamma }{)}^n}$

where ${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}(\dot{\gamma })$ is viscosity as a function of shear rate, ${\mu }_{\mathrm{\infty }}$ is the infinite-shear-rate viscosity, ${\mu }_0$ is the zero-shear-rate viscosity, $m$ is the time constant, and $n$ is the shear-thinning index.

The zero-shear viscosity ${\mu }_0$ is approached at very low shear rates, while the infinite shear viscosity ${\mu }_{\mathrm{\infty }}$ is approached at very high shear rates.

When ${\mu }_0$ > ${\mu }_{\mathrm{\infty }}$, the fluid exhibits shear thinning (pseudoplastic) behavior where viscosity decreases with increasing shear rate; when ${\mu }_0$ < ${\mu }_{\mathrm{\infty }}$, 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.

See also

• Navier-Stokes equations
• Fluid
• Carreau fluid
• Power-law fluid
• Generalized Newtonian fluid

References

• Kennedy, P. K., Flow Analysis of Injection Molds. New York. Hanser. ISBN 1-56990-181-3