Generalized-Newtonian fluid transport by an instability-driven filament

Cilia are micro-scale hair-like organelles. They can exhibit self-sustained oscillations which play crucial roles in flow transport or locomotion. Recent studies have shown that these oscillations can spontaneously emerge from dynamic instability triggered by internal stresses via a Hopf bifurcation. However, the flow transport induced by an instability-driven cilium still remains unclear, especially when the fluid is non-Newtonian. This study aims at bridging these gaps. Specifically, the cilium is modelled as an elastic filament, and its internal actuation is represented by a constant follower force imposed at its tip. Three generalized Newtonian behaviours are considered, i.e. the shear-thinning, Newtonian and shear-thickening behaviours. Effects of four key factors, including the filament zero-stress shape, Reynolds number (Re), follower-force magnitude and fluid rheology, on the filament dynamics, fluid dynamics and flow transport are explored through direct numerical simulation at Re of 0.04 to 5 and through a scaling analysis at Re approximate to 0. The results reveal that even though it is expected that inertia vanishes at Re << 1, inertial forces do alter the filament dynamics and deteriorate the flow transport at Re >= 0.04. Regardless of Re, the flow transport can be improved when the flow is shear thinning or when the follower force increases. Furthermore, a linear stability analysis is performed, and the variation of the filament beating frequency, which is closely correlated with the filament dynamics and flow transport, can be predicted.