# The Chain Fountain

An interactive, entirely client-side laboratory for the chain fountain, also called the Mould
effect: a long bead chain poured from a beaker climbs into a standing arch above the rim instead
of simply pouring over it.

## What the page does

Two linked views, driven by one engine and one parameter.

1. **Fountain.** The steady solution, drawn to scale: the beaker, the pile, the arch and the
   falling leg, with beads advected along the solved shape. Sliders set the drop height, the
   pickup coefficient, the launch angle, the rim height above the pile and the beaker radius.
   Readouts give the chain speed, the fountain height, the anomalous reaction force, the
   effective tension at the pile, the power lost at the pickup and at the landing, the critical
   drop height below which no fountain forms, and the geometric ceiling above which none can.
2. **Falling chain.** The classic textbook problem, integrated live: a heap on a table falling
   link by link through a hole. A radio switches the pickup rule. A plot shows v squared against
   length for all three rules, and a bar chart shows the momentum audit, one bar per rule.

## The physics, in one line

Everything turns on c, the tension where the chain leaves the pile, in units of lambda v squared.
c = 1 is a fully inelastic pickup, c = 1/2 an energy-conserving one, and the second law forbids
less. The classic falling chain obeys a = g/(1+2c), so c = 1 gives the textbook g/3 and c = 1/2
gives g/2. In the steady fountain the landing condition forces the pickup tension to lambda g H
whatever the rule, so v squared = gH/c, and the two rules give steady speeds in the ratio
exactly root two. The fountain height is H(1-c)/c: a fully inelastic pickup gives no fountain at
all, which is Biggins and Warner's central claim, reached here independently.

## What is original here

The launch angle is a parameter of this model, not of any paper, and the geometric ceiling that
follows from it — an arch can never cross its own launch tangent — is this page's extension. So
is the critical-drop-height bisection, and the demonstration that a regularised pickup passes an
energy check while failing a momentum audit by a factor of 251,000.

## Sources

Biggins and Warner, Proc. R. Soc. A 470:20130689 (2014), arXiv:1310.4056. Pantaleone, Am. J.
Phys. 85, 414 (2017), arXiv:1910.03125. Flekkoy, Moura and Maloy, Front. Phys. 6:84 (2018).
Yokoyama, arXiv:1810.13008 (2018). Anghel, arXiv:1912.08682 (2019-20). Wong, Youn and Yasui,
arXiv:physics/0612250. Full provenance, including which papers were paywalled and unread, is in
CREDITS.txt.

## Terms

No account, no sign-in, no network calls, no telemetry, no cookies. A single localStorage key
remembers the slider positions on this browser. The code is MIT; see LICENSE.txt.