Stationary dynamics of a chain fountain

A rigorous mathematical treatment of the shape of a dissipative rope fountain

“…We plot y max versus x = v 0 /v ter for θ (0) = 1.15 in figure 4. This curve is well modelled by the parabola y max = −0.045732 + 0.26447(v 0 /v ter ) + 0.34639(v 0 /v ter ) 2 : the computed points, shown as small circles in figure 4, lie almost exactly on this curve. , so, knowing θ(σ ), it is straightforward to find τ (σ ).…”

“…Both [1] and [2] approximate the rope shape as a catenary, essentially a curve of the form y = a cosh(x/a) with a constant. From our mathematical description of this curve, we can show that this is a good approximation for ρ → 0, but not otherwise.…”

“…This explains the fact that the graph of τ (σ) for large σ has gradient zero in the free case, and negative gradient in the braked case. Both [3] and [9] approximate the rope shape as a catenary, essentially a curve of the form y = a cosh(x/a) with a constant. From our mathematical description of this curve, we can show that this is a good approximation for ρ → 0, but not otherwise.…”

“…The previous work on this problem in the context of chains includes (i) an experimental approach by Biggins & Warner [1], (ii) a discrete-to-continuum model due to Pfeiffer & Mayet [2], and (iii) various online media, a useful guide to which can be found in [1]. The rigorous paper of Virga [3] attacks chain paradoxes in configurations that do not correspond to chain fountains, but contains a thorough survey of the literature relating to falling chains.…”

On the non-integrability and dynamics of discrete models of threads

“…Beletsky with co-authors [21,22,23]. Recently, interest in studying thread dynamics has been revived by the so-called 'chain fountain' [24,25,26,27].…”