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. Our systematic investigation using a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model reveals unexpected richness in the flow physics, identifying 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 ($1 \leq Re \leq 50$), power-law indices ($0.7 \leq n \leq 1.3$), and geometric configurations. 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, while at relatively large $n$, it evolves through an intermediate dumbbell-shaped state. Among geometric parameters, the roller radius $r$ emerges as the dominant factor controlling bifurcation points and vortex dimensions, whereas the roller-container gap $\delta$ exerts minimal influence on flow regimes. The transitions between flow modes can be precisely characterized through the evolution of vortex dimensions and velocity gradients at the stagnation point, providing quantitative criteria for flow regime identification. These findings enrich our fundamental understanding of bifurcation phenomena in extensional devices and provide quantitative guidelines for achieving desired flow patterns in four-roll mill applications.