In fluid dynamics experiments, flake-based flow visualization captures flow structures through light reflection from suspended flat tracers. However, the correspondence between visualization patterns and underlying flow physics remains unclear when the flows are unknown a priori. Inspired by the introduction of spin variable in quantum mechanics, we introduced the orientation variable into fluid dynamics and adapted the time-dependent Fokker-Planck framework to reveal the connection between experimental flake visualization patterns and virtual orientation probability fields from an Eulerian perspective. As a first example in which a dimensionless parameter distinguishes the dependency on the initial condition, we illustrated an analytical solution of the orientation probability in a rotating wave flow. With the inclusion of the diffusion term in the governing equation, the probability converges, beyond simple isotropization, to the flow-determined state with spatially varying anisotropy, eliminating dependency on initial conditions. As a second example, we solved the orientation probability field in the axisymmetric state in spherical Couette flow, demonstrating independence from initial conditions consistent with experimental reproducibility. An asymmetric pattern in experimental images, unexplained by the dynamics of the tracer orientation, was successfully recovered from the unique solution of the proposed equation.

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• Shear flows
• Taylor-Couette system
• Direct numerical simulations
• Probability theory