Understanding the dispersion of active particles is crucial for predicting their macroscopic transport. While classical Taylor dispersion explains solute spreading in shear flows, active suspensions introduce additional complexity due to the coupling between motility and external flow. In this talk, I will first compare several continuum models—the Smoluchowski, Pedley–Kessler (PK), and Generalised Taylor Dispersion (GTD) models—using a gyrotactic swimmer suspension with buoyancy–flow coupling effect in a vertical pipe as a benchmark. I will then present a new Smoluchowski-based dispersion framework for oscillatory flows, in which the PK and GTD models become either inefficient at low frequencies or invalid at high frequencies. To address this, we introduce two time variables to capture both effective and oscillatory dynamics without assuming time-scale separation, thereby greatly simplifying the application of Aris' method of moments. The framework shows excellent agreement with Brownian dynamics simulations across the full frequency spectrum and reveals two distinct dispersion regimes of active particles depending on the oscillation frequency. Moreover, the two time-variable framework can be extended to other oscillatory dispersion processes, such as periodic variations in swimming speed and shape due to flagellar beating, as well as systems involving multiple oscillatory frequencies.
