The flow field is modeled using a Heaviside step function H(r − r*) acting as a gate: below the critical radius r*, the flow is in the laminar inflow regime; above r*, it is in the helical vortex regime.
When this gated field is differentiated in the distributional (weak) sense, a Dirac delta term δ(r − r*) appears at the transition radius. This is not an artifact of the model — it is a fundamental property of distributional derivatives of discontinuous functions.
The delta term encodes a concentration of energy exactly at the transition point. Once the flow crosses r*, energy is deposited there as a boundary term that has no smooth path back. The transition is thermodynamically one-way.
This proves irreversibility without invoking entropy or thermodynamics directly. It falls out of the distributional calculus of the gate function alone — a purer and more general result.
| Heaviside gate | H(r − r*)·v(r) encodes transition at r = r* |
| Weak derivative | d/dr[H(r − r*)] = δ(r − r*) |
| Boundary term | ∫ φ·δ(r−r*)dr = φ(r*) for test fn φ |
| Physical meaning | energy deposited at r* — no smooth return |