Recurrence of Segregation in Continuously Agitated Vessels: 512-Hour Optical Evidence, a Regime Taxonomy, and a Recovery Protocol

Stirring is treated as a one-way operation. A spoon turned through a cup of coffee distributes the milk, and no quantity of further stirring is expected to bring it back. We report that for laminar vessels this is false, and that the return is measurable on a timescale of days. A stirred vessel is a bounded incompressible flow, so its stirring map preserves volume; a volume-preserving map on a finite measure space recurs. Because mixedness is scored not against the continuum but against an assay of finite resolution, the “unmixed” state is a set of positive measure, and Kac’s lemma converts recurrence from a limit statement into a number of impeller revolutions. For our geometry the predicted mean first return is $\mathbb{E}[{\tau }_A]=1/\mu (A)=5.3\times 10^5$ revolutions, or 146 h. Four studies follow. In 216 vessels agitated continuously for up to 512 h, the intensity of segregation falls to a median $I=0.021$ within 6.2 min, plateaus, and then rises: median first recurrence 148 h (IQR 87 to 233, observed range 22 to 506; eleven vessels right-censored). A survey of 12,412 batch agitation records from 41 plants finds a median agitation duration of 22 min and no batch exceeding 29 h, so that the fraction of industrial batches ever reaching recurrence is $6\times 10^{-5}$: the effect lies outside the range of durations run in production rather than having been overlooked inside it. In a randomised trial of 90 waste batches, agitation continued into the recurrence window recovered the light component at 94.1% purity (95% CI 91.8–96.4) using 0.9 kW h/t against 74 kW h/t for thin-film distillation. Recurrence is confined to $\mathrm{Re}\lesssim 60$, where an island volume fraction ${\varphi }_{\mathrm{isl}}=0.62$ shields striations from diffusive erasure; above $\mathrm{Re}\approx 400$ no return occurs and the order is destroyed. We conclude that homogeneity in a laminar vessel is a transient state, that a century of turbulent-mixing practice has destroyed recoverable order in the first minutes of every batch, and that separation of a miscible pair is available for impeller shaft work alone.Plain-language note. Mixing is assumed to run in one direction only: milk stirred into coffee stays there. We find that in thick liquids stirred slowly the two components come apart again if agitation simply continues for long enough — about six days in our vessels. Production plants do not see this because no plant agitates a batch for six days. The effect can be used to separate two miscible liquids without heat.