THE CHAIN FOUNTAIN — credits, provenance and honest limits ========================================================== WHAT THIS IS ------------ An independent reimplementation of the PHYSICS of the chain fountain. There is no original software here to reimplement: the chain fountain is a physical demonstration, not a program. Every line of code in this bundle was written for it; no third-party library is vendored and nothing is loaded from a CDN. THE DEMONSTRATION ----------------- The effect is commonly called the CHAIN FOUNTAIN or the MOULD EFFECT, after Steve Mould, who brought it to wide attention in 2013. Biggins and Warner credit him in the opening of their paper as having "brought [it] to widespread attention". This page is not affiliated with Steve Mould, with the Cavendish Laboratory, or with any of the authors below. THE PHYSICS THIS PAGE IMPLEMENTS, AND WHERE IT COMES FROM --------------------------------------------------------- * J. S. Biggins and M. Warner, "Understanding the chain fountain", Proceedings of the Royal Society A 470(2163):20130689 (2014), doi 10.1098/rspa.2013.0689. Read as arXiv:1310.4056v2. The Royal Society page returned HTTP 403 to this project and the journal version was NOT read; nothing here is attributed to it. * J. Pantaleone, "A quantitative analysis of the chain fountain", American Journal of Physics 85(6), 414-421 (2017), doi 10.1119/1.4980071. Read as arXiv:1910.03125. The AIP page returned HTTP 403; the journal version was NOT read. * E. G. Flekkoy, M. Moura and K. J. Maloy, "Mechanisms of the Flying Chain Fountain", Frontiers in Physics 6:84 (2018), doi 10.3389/fphy.2018.00084. Open access, read in full. * H. Yokoyama, "Reexamining the Chain Fountain", arXiv:1810.13008 (2018). Preprint. * D.-V. Anghel, "The theory of the chain fountain revisited", arXiv:1912.08682 (2019-2020). Preprint. * C. W. Wong, S. H. Youn and K. Yasui, on the falling chain, arXiv:physics/0612250 — the source for "a = g/3 if the motion is nonconservative, but g/2 if the motion is conservative". * Calkin and March (1989), American Journal of Physics 57, 154, on the folded chain that falls faster than g. CITED SECOND-HAND ONLY; that paper was not opened by this project. THE MODEL --------- One parameter: c = T0 / (lambda v^2), the tension where the chain leaves the pile. c = 1 fully inelastic pickup a = g/3 in the falling-chain problem, no fountain c = 1/2 energy-conserving pickup a = g/2, and the tallest arch the second law allows c = 0 a control, physically wrong, shipped only so the audits have something to reject Biggins and Warner's anomalous reaction force is R = alpha * lambda * v^2 with alpha = 1 - c. REPRODUCED FAITHFULLY (derived here first, then found to agree to 1e-12) * their eqs. (0.2), (0.3), (0.4), (0.5), (0.6) and (0.7) * alpha = (1/2)(1 - I/((1/4) m b^2)), giving alpha = 1/6 for a three-bead link * the bound alpha + beta <= 1/2, and "alpha = 0 gives h2 = 0, that is no fountain" * v = sqrt(h1 g) in the classical case * the inverted-catenary shape of the moving chain, their eq. (0.13) * Flekkoy, Moura and Maloy's eq. (7), lambda g h = Delta F_z * Pantaleone's parameter-free eq. (7), V/V_freefall = sqrt((1 + h2/h1)/2) WHAT IS THIS PAGE'S OWN, AND NOT IN ANY PAPER * The launch angle theta0 is a parameter of THIS model. No paper sets it. * The GEOMETRIC CEILING that follows from it — an arch can never cross its own launch tangent, so if (beaker radius) x tan(theta0) does not exceed the rim height, NO drop height produces a fountain above the rim — is this page's extension, not a published result. Treat it as such. * The critical drop height H_crit and its bisection are this page's own. * The demonstration that a regularised pickup passes an energy check and fails a momentum audit is this page's own. WHAT DIFFERS FROM THE REAL DEMONSTRATION * One dimension and a steady state. The fountain view draws the solved steady shape with beads advected along it; it does not simulate a start-up transient. The falling-chain view IS a live transient, integrated with Runge-Kutta 4. * The chain is perfectly inextensible and perfectly flexible. A real ball chain has finite bending stiffness, which several of the papers above argue matters. * beta, the anomalous force at the floor, is fixed at zero. * No air drag. Pantaleone proposes drag to explain a measured speed deficit this model reproduces but cannot explain. * No entanglement, no chain-on-chain friction in the pile, and no model of the roughness of the underlying packing that Flekkoy, Moura and Maloy argue is essential. THE DISPUTE IS OPEN, AND THIS PAGE TAKES NO SIDE ------------------------------------------------ Biggins and Warner (2014) and Pantaleone (2017) argue the pile must push up on the chain. Flekkoy, Moura and Maloy (2018) accept the push but show the rigid-rod kick alone fails in simulation. Yokoyama (2018) and Anghel (2019-20) argue the push is unnecessary. Both dissents appear to remain preprints; that is evidence about peer review, not about correctness. In a one-dimensional steady theory "the beaker pushes back" and "the pickup is not fully inelastic" are THE SAME NUMBER, so this page ships the knob rather than a verdict. TRADEMARK SEARCH — WHAT WAS RUN AND WHAT IT RETURNED ---------------------------------------------------- Run 2026-09-20 against TMview (POST https://www.tmdn.org/tmview/api/search/results), criteria "C" (contains), all offices, all statuses, all Nice classes. HTTP 200 on every query. "chain fountain" 3 results, none of them the mark CHAIN FOUNTAIN: FOUNTAIN CHAIN CAPITAL — CN, Registered, class 36 FOUNTAIN HOTELS CHAIN — IL, Ended, class 42 "Chain of confidence with fountain logo" — BN, Registered, classes 35 and 41, Dart Industries Inc. "chainfountain" 0 results "mould effect" 1 result, a long descriptive New Zealand mark about condensation and moulds, Ended, class 11 — unrelated "self siphoning" 0 results "chain fountain toy" 0 results "bead chain" 6 results: BEAD CHAIN (US, Bead Industries / Bead Chain Manufacturing, classes 6 and 28 — the class-28 entries are Ended, one class-6 entry is Registered); BEAD AND CHAIN (CA, Ended, classes 14 and 35) FINDINGS: no mark reading CHAIN FOUNTAIN was returned by any office, and nothing in class 28 (games and toys) matched the phrase. THIS IS NOT A CLEARANCE OPINION and none is claimed. It is a record of what one contains-match search returned on one day. It does not cover unregistered or common-law rights, trade dress, or personality rights, and TMview's coverage is not complete. LICENCE ------- See LICENSE.txt. The code is MIT. The physics is not anyone's to licence.