  Non-Newtonian fluid flow might be driven by spatially nonlocal velocity, the
dynamics of which can be described by promising fractional derivative models.
This short communication proposes a space FrActional-order Constitutive
Equation (FACE) that links viscous shear stress with the velocity gradient, and
then interprets physical properties of non-Newtonian fluids for steady pipe
flow. Results show that the generalized FACE model contains previous
non-Newtonian fluid flow models as end-members by simply adjusting the order of
the fractional index, and a preliminary test shows that the FACE model
conveniently captures the observed growth of shear stress for various velocity
gradients. Further analysis of the velocity profile, frictional head loss, and
Reynolds number using the FACE model also leads to analytical tools and
criterion that can significantly extend standard models in quantifying the
complex dynamics of non-Newtonian fluid flow with a wide range of spatially
nonlocal velocities.
