Understanding flake-based flow visualization through orientation probability dynamics

This study originated from an intriguing question about flakey tracer orientation in steady flows: whether orientations completely align or return to isotropy. To address this fundamental question, we introduced the concept of an "orientation variable" into fluid dynamics, inspired by the spin variable in quantum mechanics, and derived a time-dependent Fokker-Planck equation describing the orientation probability density field of flakes suspended in fluid from an Eulerian perspective. Our approach resolves contradictory intuitions about orientation behavior in quasi-steady flows through two key examples.

First, in rotating wave flows, we analytically demonstrated that without diffusion, although the system may appear to converge, the existence of conserved quantities mathematically guarantees the persistence of initial condition dependence. Conversely, introducing diffusion effects eliminates this dependence, leading to convergence toward unique equilibrium states with spatially varying anisotropy determined solely by the flow field.

Second, in spherical Couette flow, we performed numerical analysis in five-dimensional space (three physical coordinates plus two orientation angles) to obtain equilibrium solutions with anisotropic distributions. This successfully reproduced asymmetric patterns observed in experiments, which were previously unexplainable by deterministic tracer dynamics alone.

The proposed framework is applicable to flake-based visualization across a wide range of flow fields, from quasi-steady flows to turbulence, offering significant advances in understanding particle-laden flows and flow visualization techniques that bridge experiment and theory while uncovering novel physical principles within seemingly ordinary fluid visualization methods.