Santra, SrimantaSrimantaSantra2026-10-202026-10-202026-10-06Physics of Fluids 38 (10): 103606 (2026)https://hdl.handle.net/11420/65344Mixing of a passive scalar in a time-dependent flow is usually judged from the variance decay of a nominal flow. In many applications, though, stirring protocols and flow parameters are uncertain, and the more relevant question is whether the required level of homogenization is reached within a finite time. We study this reliability question for incompressible advection–diffusion with uncertain forcing parameters, using a time-periodic double-gyre flow as the main benchmark and a periodically stirred cavity as a secondary test. Direct numerical simulations show that delayed mixing is not a fluctuation around the mean variance decay curve. Instead, slow realizations retain coherent scalar remnants near weakly exchanging regions of the flow, which produces a persistent upper tail in the mixing time distribution. To identify these slow events without running a full simulation for every parameter sample, we build a parameter- and phase-dependent reduced transport operator by combining a proper orthogonal decomposition-based Koopman representation with a polynomial-chaos expansion in the uncertain parameters. From this operator, we compute a finite-time scalar-retention score based on the leading singular direction of the learned propagator in a scalar-energy metric. The diagnostic ranks parameter realizations by their tendency to retain scalar energy, identifies the realizations contributing to the upper tail of the mixing time distribution, and recovers the scalar structures responsible for delayed homogenization. Reliable mixing under uncertainty is therefore controlled not only by the mean variance decay, but also by the least-damped scalar directions of the uncertain transport operator, which suggests that designs targeting reliable mixing should act on these directions directly.en1089-7666202610American Institute of Physics (AIP)https://creativecommons.org/licenses/by/4.0/Natural Sciences and Mathematics::530: Physics::530.4: States of Matter::530.42: Fluid PhysicsNatural Sciences and Mathematics::519: Applied Mathematics, ProbabilitiesJournal Article10.1063/5.034559810.15480/882.18911