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. ^{[2] [3]}

See also

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

References

1. Cunningham, Neil. "Making Use Of Models: The Cross Model". www.rheologyschool.com. Retrieved 2018-02-28.
2. Cross, Malcolm M. (1965-06-01). "Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems" . Journal of Colloid Science. 20 (5): 417–437. doi:10.1016/0095-8522(65)90022-X. ISSN   0095-8522.
3. Galindo-Rosales, F. J.; Rubio-Hernández, F. J.; Sevilla, A.; Ewoldt, R. H. (2011-12-01). "How Dr. Malcom M. Cross may have tackled the development of "An apparent viscosity function for shear thickening fluids"" . Journal of Non-Newtonian Fluid Mechanics. 166 (23): 1421–1424. doi:10.1016/j.jnnfm.2011.08.008. ISSN   0377-0257.

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