A space fractional constitutive equation model for non-Newtonian fluid flow

Non-Newtonian fluid flow can be driven by spatially nonlocal velocity, the dynamics of which can be described by promising fractional derivative models. This study reports a left-side, Caputo type, space fractional-order constitutive equation (FCE) using a nonlocal, fractional velocity gradient and then interprets physical properties of non-Newtonian fluids for steady pipe flow. Results show that the generalized FCE 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 FCE 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 FCE model also leads to analytical tools and criterion that can extend standard models in quantifying the complex dynamics of non-Newtonian fluid flow with a wide range of spatially nonlocal velocities. Model extension using the general two-side fractional derivative is also briefly discussed.