Lattice Boltzmann simulation reveals supercritical bifurcation in flow mode transitions of power-law fluids in the four-roll mill

The four-roll mill has been traditionally viewed as a device generating simple extensional flow with a central stagnation point. However, our systematic investigation using a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model reveals two previously unreported supercritical bifurcation modes: a quadrifoliate vortex mode featuring four symmetrical counter-rotating vortices, and a dumbbell-shaped quad-vortex mode where vortices detach from but remain symmetric about the stagnation point. The numerical framework, representing the first successful extension of TRT-RLB method to power-law fluid dynamics, enables comprehensive mapping of flow characteristics across Reynolds numbers (1≤Re≤50), power-law indices (0.7≤n≤1.3), and geometric configurations. We found that the transition from quadrifoliate vortex mode exhibits distinct pathways depending on the power-law index: at relatively small n, the flow undergoes a direct supercritical bifurcation to simple extensional flow, as the higher viscosity in the central region suppresses curvature formation at the edge of vortices; while at relatively large n, it evolves through an intermediate dumbbell-shaped state, because the lower viscosity in the central region allows smaller vortices to be flattened and separated by incoming flows. Further analysis reveals that the seemingly anomalous phenomenon of vortex generation at high Re, where vortices diminish in size or disappear as Re increases, is fundamentally attributed to the symmetrical geometric structure of the viscous-force-dominated region. Among geometric configurations, the roller radius r emerges as the dominant factor controlling bifurcation points and vortex dimensions, whereas the roller-container gap δ exerts minimal influence on flow regimes.