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:

:$\backslash mu\_\backslash mathrm\{eff\}(\backslash dot\; \backslash gamma)\; =\; \backslash mu\_\backslash infty\; +\; \backslash frac\; \{\backslash mu\_0-\backslash mu\_\backslash infty\}\{1\; +\; (m\backslash dot\{\backslash gamma\})^n\}$

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

The zero-shear viscosity $\backslash mu\_0$ is approached at very low shear rates, while the infinite shear viscosity $\backslash mu\_\backslash infty$ is approached at very high shear rates.{{Cite web|url=http://www.rheologyschool.com/advice/rheology-tips/25-making-use-of-models-the-cross-model|title=Making Use Of Models: The Cross Model|last=Cunningham|first=Neil|website=www.rheologyschool.com|language=en|access-date=2018-02-28}}

When $\backslash mu\_0$ > $\backslash mu\_\backslash infty$, the fluid exhibits shear thinning (pseudoplastic) behavior where viscosity decreases with increasing shear rate; when $\backslash mu\_0$ < $\backslash mu\_\backslash infty$, 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.{{Cite journal |last=Cross |first=Malcolm M. |date=1965-06-01 |title=Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems |url=https://www.sciencedirect.com/science/article/abs/pii/009585226590022X |journal=Journal of Colloid Science |volume=20 |issue=5 |pages=417–437 |doi=10.1016/0095-8522(65)90022-X |issn=0095-8522|url-access=subscription }}{{Cite journal |last=Galindo-Rosales |first=F. J. |last2=Rubio-Hernández |first2=F. J. |last3=Sevilla |first3=A. |last4=Ewoldt |first4=R. H. |date=2011-12-01 |title=How Dr. Malcom M. Cross may have tackled the development of “An apparent viscosity function for shear thickening fluids” |url=https://www.sciencedirect.com/science/article/abs/pii/S0377025711002011 |journal=Journal of Non-Newtonian Fluid Mechanics |volume=166 |issue=23 |pages=1421–1424 |doi=10.1016/j.jnnfm.2011.08.008 |issn=0377-0257|url-access=subscription }}