| chain speed v | – |
|---|---|
| v / free fall | – |
| fountain height | – |
| height / drop | – |
| alpha = R / λv² | – |
| pickup tension T₀ | – |
| effective tension at the pile | – |
| beaker reaction R | – |
| power lost at the pickup | – |
| power lost at the landing | – |
| critical drop height | – |
| geometric ceiling | – |
| rim clearance | – |
| c re-inferred from h/H | – |
| alpha from the rod impulse | – |
| margin monotone in H | – |
| arch: ODE vs closed form | – |
The pile is at the bottom, the rim is the dashed line, and the thin diagonal is the launch tangent the arch can never cross. The faint curve inside the magnified inset is the far-field projectile limit: the same arch under an effective gravity of g/(1−c).
| rule | – |
|---|---|
| length moving x | – |
| x from the closed form | – |
| speed v | – |
| acceleration, measured | – |
| acceleration, closed form | – |
| psi, measured | – |
| psi, predicted | – |
| dissipation rate | – |
psi is (dp/dt minus the weight) divided by the weight: what external force the motion demands beyond gravity. Only the fully inelastic rule closes at zero. The dotted points on the plot are the discrete oracle — a chain of rigid links stepped by an impulsive rule, sharing no code with the integrator.
What this is
A chain fountain is what happens when you tip a long bead chain out of a beaker and let one end fall to the floor: the chain does not simply pour over the rim, it climbs into a standing arch several centimetres above it and stays there for as long as the chain lasts. Steve Mould brought it to wide attention in 2013, which is why it is often called the Mould effect.
This page is an independent reimplementation of the physics, not of any program. There is no original software here to copy: the chain fountain is a physical demonstration, and the model below was derived from scratch and then checked against the published literature.
The one number
Everything turns on c, the tension in the chain where it leaves the pile, measured in units of λv². A fully inelastic pickup — every link jerked from rest to the chain speed, half the work lost — has c = 1. An idealised loss-free pickup has c = 1/2. The second law forbids anything below that. Biggins and Warner’s anomalous reaction force is R = αλv² with α = 1 − c: in a one-dimensional steady theory, “the beaker pushes back” and “the pickup is not fully inelastic” are the same single degree of freedom. This page ships the knob, not a verdict.
What is faithful, and what is this model’s own
- Faithful. The steady force balance, the closed forms h₂/h₁ = α/(1 − α − β) and v² = h₁g/(1 − α − β), the bound α + β ≤ 1/2, the inverted-catenary shape, and Pantaleone’s parameter-free relation V/Vfree fall = √(½(1 + h₂/h₁)). All were derived here independently and then found to agree with the papers to 1e-12.
- This model’s own. The launch angle θ₀ is a parameter of this page, not of any paper. The geometric ceiling that follows from it — an arch can never cross its own launch tangent, so a wide enough rim cannot be cleared at any drop height — is this page’s extension, not a published result.
- Simplifications. One dimension, a steady state, an inextensible chain, no air drag, no chain-on-chain friction in the pile, no transient start-up in the fountain view, and β (the anomalous force at the floor) fixed at zero. The falling-chain view is a genuine transient, integrated live.
- Not modelled. Entanglement, the finite bending stiffness of a real ball chain, the roughness of the packing that Flekkøy, Moura and Måløy argue is essential, and the drag Pantaleone proposes to explain his speed deficit.
The dispute is open
Biggins and Warner (2014) argue the pile must push up on the chain, and that without that push there is no fountain at all. Pantaleone (2017) confirms the push quantitatively with no adjustable parameters. Flekkøy, Moura and Måløy (2018) accept the push but show in simulation that the rigid-rod kick alone does not produce a fountain — the roughness of the underlying packing is needed, and a completely flexible bead chain fountains anyway. Yokoyama (2018) and Anghel (2019–20) argue the push is unnecessary and the fountain follows from energy-conserving chain dynamics. Both dissents appear to remain preprints. The most recent paper found here (January 2026) sides with Biggins and Warner. No paper resolving it was found.
Sources
- J. S. Biggins and M. Warner, “Understanding the chain fountain”, Proc. R. Soc. A 470(2163):20130689 (2014). Read as arXiv:1310.4056v2; the Royal Society page returned HTTP 403 and was not read.
- J. Pantaleone, “A quantitative analysis of the chain fountain”, Am. J. Phys. 85(6), 414 (2017). Read as arXiv:1910.03125; the AIP page returned HTTP 403.
- E. G. Flekkøy, M. Moura and K. J. Måløy, “Mechanisms of the flying chain fountain”, Front. Phys. 6:84 (2018). Open access, read in full.
- H. Yokoyama, “Reexamining the chain fountain”, arXiv:1810.13008 (2018).
- D.-V. Anghel, “The theory of the chain fountain revisited”, arXiv:1912.08682 (2019–2020).
- C. W. Wong, S. H. Youn and K. Yasui, on the falling chain, arXiv:physics/0612250 — the source for a = g/3 when the motion is non-conservative and a = g/2 when it is.
Trademark search
A TMview search was run on 2026-09-20 — POST to the TMview results API, contains-match, all offices, all statuses, all Nice classes. “chain fountain” returned 3 marks: 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”, “self siphoning” and “chain fountain toy” returned nothing. “mould effect” returned one unrelated New Zealand mark about condensation (Ended, class 11). “bead chain” returned 6, including BEAD CHAIN in class 28 (US, Ended). No mark reading CHAIN FOUNTAIN was returned by any office, and nothing in class 28 matched the phrase. This is what the search returned; it is not a clearance opinion. It does not cover unregistered or common-law rights, and TMview’s coverage is not complete.
Keyboard
1–4 switch sections. Space starts or stops the falling chain. R resets it.