A Continuum Theory of Channeling in Espresso and the Grind Paradox

Finer coffee can brew weaker espresso, and the only mechanism proposed so far uses two lumped flow paths. We treat channeling as an instability of a continuous dissolving bed at fixed pressure: Darcy flow through a Kozeny–Carman bed that opens as its solubles dissolve, solute transport and first-order extraction, in a two-dimensional section of the puck with one calibrated constant. The extraction front moves at $\gamma_f q$, and a thin front grows at a closed-form rate, confirmed against a direct simulation within $0.11$ percent and a moving-frame eigenvalue within $5.0$ percent. The exact linear problem along the moving base puts the fastest channel spacing at $3.6$ mm at every grind and shows unevenness decaying over a shot coarser than $1.58$ times the reference grind. In 1264 simulated shots, millimeter-scale porosity noise costs up to $5.35$ percent of the extraction at the finest grind and flattens the extraction-grind curve without reversing it; streaks running the full height of the puck do not reverse it while they are $1$ mm wide. The curve turns over only when the unevenness runs the full height and is at least about one channel spacing wide, and then shallowly, by $0.39$ points or $1.7$ percent, far less than the 2020 experiment’s drop. Only better puck preparation reduces channeling, recovering $54$ percent of the loss.