  In this study, we numerically investigate hysteretic behaviors of secondary
bifurcations in the Taylor-Couette flow of a semidilute, neutrally buoyant, and
noncolloidal suspension. We consider a suspension with a bulk particle volume
fraction of 0.1, a radius ratio of 0.877, and a particle size of 60. The
suspension balance model (SBM) is introduced for numerical simulations to model
the dynamics of particles undergoing shear-induced particle migration with
rheological constitutive laws. The suspension flow transitions from circular
Couette flow (CCF) via ribbons (RIB), spiral vortex flow (SVF), and wavy spiral
vortex flow (WSVF) to wavy vortex flow (WVF) with the increase of suspension
Reynolds number. The primary transition arises very slowly with an oscillatory
critical mode and appears through a supercritical (or non-hysteretic)
bifurcation. However, we find hysteretic behaviors in subsequent bifurcations
(spiral vortex flow (SVF) to wavy spiral vortex flow (WSVF) and WSVF to wavy
vortex flow (WVF)) during increasing-Re and decreasing-Re procedures with a
rapid-step change near the transition boundaries. The WSVF and WVF states are
more sustained below the transition boundaries when the Reynolds number is
rapidly decreased in stages. However, the SVF and WSVF become WSVF and WVF more
sharply with increasing Re, respectively. To conclude this study, we also
examine in detail, a standing wave of weak counter-rotating vortices that
occurs as the primary instability and analyze the wave that oscillates in time
but is stationary in space in the RIB state.
