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:

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

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

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

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

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

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

• Non-Newtonian fluids
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