Newtonian fluids transition from laminar to turbulent flow when inertial forces overwhelm viscous forces — the Reynolds number exceeds a critical value. Yield-stress fluids add a complication: the fluid doesn't flow at all below a threshold stress. A plug of unyielded material can persist even as the surrounding fluid turbulently flows around it.
The first complete direct numerical simulations of this transition for Herschel-Bulkley fluids reveal three regimes (arXiv:2603.11363). Below a generalized Reynolds number of 1,735: laminar. Between 1,735 and 2,920: transitional. Above 2,920: fully turbulent. These thresholds are higher than for Newtonian fluids (Re ≈ 2,300 for pipe flow), because the yield stress stabilizes the flow.
The transition occurs only when local Reynolds stresses exceed the yield stress. This is the key mechanism. In Newtonian transition, the critical condition is global — the bulk Reynolds number. In yield-stress transition, the condition is local. A flow can have a globally supercritical Reynolds number while remaining laminar in regions where the Reynolds stresses don't locally exceed the yield stress. The plug persists wherever turbulent stresses can't break through.
The transition is not an event but a territory. The fluid transitions region by region, each region yielding independently when local stresses cross the threshold. The global Reynolds number determines whether transition is possible; the local stress distribution determines where it occurs.