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Edited by: Miguel Rubi, University of Barcelona, Spain

Reviewed by: Alexandre De Castro, Brazilian Agricultural Research Corporation (EMBRAPA), Brazil; Rodolfo Morales, National Polytechnic Institute, Mexico

This article was submitted to Computational Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A chain which is made to flow from a container, forms a striking arch that rises well above the container top. This phenomenon is caused by the well known Mould effect and is explained by a supply of momentum from the container, causing an upwards kick. Here we introduce a theory that allows for dynamic fluctuations of the chain and compare with corresponding simulations and experiments. The predictions for the chain velocity and fountain height agree well with experiments. We also explore the underlying mechanism for this momentum transfer for different chain models and find that it depends subtly on the nature of the chain as well as on the container.

The dynamics of ropes and chains have attracted scientific attention for centuries. They are found in biological systems, many different technologies, as well as in everyday life. Examples include our tenants, the DNA molecule, the tail of a cat, the line of a fly-caster, a whip, or the chain of a falling anchor. Hanging chains were studied by Galileo in the 1600's [

When the end of a chain is dropped from a pile contained at some height above the floor, gravity will set it in motion, and eventually the whole chain will have flowed over the edge of the container. As this happens the chain rises far above the rim of the container. The first to communicate this striking effect was Mould [

The flying chain resulting from a 5 m drop of the chain, photographed outside the physics building at the University of Oslo.

An analysis of the process was first carried out by Biggins [

In a recent paper Flekkøy et al. [

In this follow-up article, we proceed with a closer quantitative comparison between theory, simulations and experiments where the parameter space of the simulations is explored in some detail. Part of this exploration is a study of an experiment that may serve to distinguish between the different mechanisms that are responsible for the container force. This experiment measures the fountain height as a function of container width, and it is shown that different chain structures cause different dependencies between these quantities.

Even though strong spatial fluctuations are clearly visible in experiments, theoretical treatments of the chain generally assume a time-independent trajectory, that is, a steady state. In the present paper we develop a theory that goes beyond that simplification by allowing fluctuations, assuming only a statistical steady state where quantities such as momentum, are assumed steady only when averaged over sufficiently long times, or when ensemble averages are taken. We compare the predictions of the theory with measurements of the chain velocities. This is an interesting quantity in this context because it is sensitive to fluctuations around the steady state that is often assumed when analysing the chain dynamics. We find that the inclusion of dynamical fluctuations indeed improves the agreement with such measurements.

Descriptions of the momentum balance of the chain exists in several text books, such as that on chain dynamics and shape [

Our Eulerian formulation of the chain dynamics uses the lab-frame of reference, since then it is easy to express the mechanical steady states.

We start with the hydrodynamic style of expressing mass conservation

where the mass density ρ = λ/

where the brackets denote an ensemble average. Equation (1) then gives

Integrating this equation over a suitable volumes containing piece of the chain we may apply Gauss' theorem.

_{t}, integration gives the effective replacement _{t}|. These are the areas of intersections with the chain. Carrying out such an integration of Equation (3) yields

where the unit vectors are defined in

Just like in the case of the mass we may assume that the momentum inside a given volume remains constant in the sense that the average ∂_{t}〈ρ

where

The steady state assumption leaves us with the averaged force-balance condition

Carrying out a volume integration of Equation (7) yields

where we have used the fact that _{t}/_{t} = _{F} = 0 there.

When the chain shape is stationary, on the other hand, _{t}_{t} everywhere, Equation (8) reduces to

with

Taking the tangential and normal components of this equation allows the integration of both _{t} along the chain. Once such a solution is found with a tension _{0}(

In the following we will work out the fountain height _{C} and _{B} to be small enough to neglect gravity and only in _{B} will _{B} gives a force

Taking the _{C} we get^{1}

where the subscript _{C}, and the + sign corresponds to the intersection of the upwards moving chain where _{tz} > 0. We have used that the unit normal _{z}.

In Equation (11) both the _{tz}-factors will contribute to increase 〈_{C}〉 since they are smaller than one. The first factor exists because the tension acts along a variable direction and the second because momentum is advected over an intersection surface that depends on α as is illustrated in

Now, we may integrate over _{1}, which gives

where 〈_{1}〉 is the average chain length contained in _{1}. We will take the tension _{F} at the bottom of the volume to vanish.

It has been observed- at least for some types of chains- that the interaction between the falling chain and the floor may produce an added downwards force, causing freely falling chains to accelerate slightly faster than gravity [_{F} = 0. Then the _{F}-term above vanishes, and the first and third terms cancel due to Equation (11). This leaves the velocity equation

This equation shows that the velocity is governed by the weight of the downwards moving part of the chain, λ〈_{1}〉_{1}〉, will increase the velocity with which it falls.

In order to simplify (Equation 13) we note that |_{tz}| = cos Θ where Θ is the local chain-angle to the vertical. As the tension goes to zero toward the floor, it may be reasonable to assume a de-correlation between Θ and _{z}. Also, for small angles we may Taylor expand to get

with an error that enters only at 4. order in Θ. We may then write

This is the prediction that will be compared with the simulations and experiments.

In order to get a prediction for _{T} and _{B}. For beads at rest in the container gravity is balanced by the force from below that keep them from falling. We therefore introduce

the net force on the beads in _{B}. This force is non-zero only for moving beads. Furthermore, the momentum advection is non-zero only at the top of the _{B}-surface, so the

Doing the same for volume _{T} gives

where _{T} is the length of chain contained in _{T}. Here the first two terms on the left hand side cancels due to Equation (11) and the last two terms may be replaced by −〈Δ_{z}〉 due to Equation (19). This leaves the fountain equation

This shows that _{T} = 0 and thus _{T} and

In order to get a theoretical relationship between _{1} we need a model for the force Δ_{z}. The simplest possible model for the container force relies on the notion that it is caused by collisions between the beads that are accelerated along the bottom, and the beads that are still stationary. Each impact will happen at a rate ∝

The last equality follows since, if _{1}, then _{1}. By postulating, or measuring, constants of proportionality between Δ_{1}〉 and _{1}, and between 〈_{T}〉, it is straightforward- though not very enlightening- to produce the constant of proportionality between _{1}.

However, for the purpose quantifying the effect of fluctuations it is useful to compare the above theory with one that ignores them. It is therefore instructive to point to an elegant analytical formulation of the chain evolution, which is given by Biggins in Equations (11) and (12) in Biggins [

We have performed experiments using a 50 m long chain of metallic beads having diameters of 4.5 mm. We believe this to be the same kind of chain used by Biggins [

Images were acquired with three cameras operating on different modes. A high resolution Nikon D7200 DSLR camera is used to obtain images of the whole chain and is placed on a high tripod about 3 m away from the system. A Photron SA5 high speed camera was used to image the descending segment of the chain. The images were captured close to the ground at a framerate of 4000 fps. An additional set of high speed images was captured by a Nikon J4 camera at 1000 fps to capture the details of how the beads take off inside the container.

The Photron high speed images were used to measure the vertical speed component _{z}. This was done by tracking the motion of individual beads in the chain. _{t} are not necessarily parallel).

Example of bead tracking using a series of high speed images. The filled green circle corresponds to the current position of one particular bead being tracked and the red open circles correspond to later positions of the same bead. The tracking lasts for 50 frames here corresponding to a time span of 12.5 ms.

In order to determine how the momentum transfer takes place, we have performed an additional experiment where high speed footage inside the container was acquired, to image the chain take-off process. Typical images are shown in

A time sequence of 7.5 ms using a 50 m chain of 4.5 mm beads. Individual chain beads are traced by different colors. Adapted from Flekkøy et al. [

The simulations are based on a particle representation of the individual beads of the chain and integrate the equations of motion that derive from Newtons second law in 3 dimensions. The algorithm resembles that used by Vrbik [

The beads in the chain are taken to interact through a harmonic potential with an equilibrium separation _{0}, i.e., if the separation between two neighboring particles is Δ_{1} − _{2} then

Also, an interparticle dissipative force −βΔ_{c}_{∥}, where _{∥} is the horizontal velocity component, is included. This force dampens the motion inside the container that would remain for a long while if there were only the inter-particle dissipation.

Since the beads have only nearest neighbor interactions, the interaction between a bead and the underlying chain packing cannot be done by bead interactions. In stead we introduce a rough container bottom. Attempting to make this roughness correspond to the packing we use the bottom height function

where _{0} comes close to mimicking the geometry of the chain itself.

_{max}.

The floor, which is located at _{F}_{z}. Newtons second law is then integrated using a Velocity Verlet scheme of 4'th order accuracy in time.

The distance _{0}/2, but may also be taken to be smaller, thus simulating smaller beads. The interaction between the particles and the boundaries (side walls and rough bottom) are derived from a harmonic potential so that the interaction force is implemented as a repulsive spring force which is linear in the overlap length.

The internal stiffness force is introduced to keep the chain from bending. It acts on the particle that is in the middle of a stiff 3-particle segment and in order to keep it an internal force, an equal and opposite counterforce is distributed on the two nearest neighbors. It is implemented by the following force

where _{2}, _{1} the unit vectors pointing between the neighbors, as is illustrated in _{0}, the _{0}(_{2}−_{1}) ≈ _{2} − _{1}. To obtain a piecewise rigid chain, _{s} is applied to every third bead. By choosing

It is possible to generalize this force in a way so that it only kicks in when the angle Θ between _{2} and _{1} exceeds a maximum value Θ_{max}.

For this purpose we introduce the transverse displacement vector Δ_{T} which is the displacement of a particle from the line that connects its two nearest neighbors. This vector is illustrated in _{max}. We require that the restoring stiffness force be linear in the increase in Δ_{T} above the Θ_{max} value, i.e.,

Now, this force acts along with the interaction forces that produce a particle separation near _{0}, so we may assume that these other forces have already caused such a separation, and we approximate Δ_{i} = _{0}_{i}. Assuming also that the new displacement vector

when Θ ≥ Θ_{max}. Using the fact that the _{i}'s are unit vectors with internal angles Θ and Θ_{max}, the force then takes the form

This force is applied to _{max} is zero this would result in a long rigid chain, so we choose it in stead to the measured value Θ_{max} = 63 degrees. When Θ_{max} is zero, and the force is applied to every third particle, the model of Equation (26) is reproduced.

As experiments with the piecewise rigid chain may be carried out using pieces of pasta, the corresponding model used in the simulations will henceforth be termed the

Initially, the chain is packed in the container in straight segments that extend between points on the container wall. As the beads are added and meet the wall, a random new direction pointing into the container, is chosen.

In the following we explore the mechanisms producing the force Δ

_{0} and Θ_{max} = 63°. The fountain must acquire its asymptotic height before 〈_{T} and

The length _{T} of the upwards moving part of the chain as a function of fountain height

In order to check the effect of variations of the container bottom, we measured _{0}, there are only weak variations in

Fountain height _{1} = 4 m. The container height is the vertical extent of the container and thus a minimum value for

Observing that a bumpy packing, or container bottom, is crucial to produce a fountain both for the realistic and fully flexible chain, we may still inquire if there are any other chains that do not rely on the structure of the bottom. Indeed, Biggins [

However, doing the same in our simulations, using (i) a fully flexible separated bead chain having rigid segments of length 3 cm, and the bottom roughness _{max} = π) separated bead model with a rough bottom.

A simulated time sequence of the fountain evolution using the separated beads model.

A simulated time sequence of the fountain evolution using the pasta model. The times are 0.5, 2.0, 2.5, and 3.0 s and _{1} = 1.8 m.

In _{1}. They are carried out by simulating the different chain and container models. We have also included the original measurements of Biggins and Warner [_{0}/4 agree well with these. We have truncated the measurements at _{1} = 4 m as the chain lengths which have been applied, do not allow the system to reach a steady state above that container elevation. Andrew et al. [_{1} using a chain of length 41.5 m and elevation heights up to 18 m. For such elevations one would need a chain more than 200 m long to reach the asymptotic _{1} and some transient value of

The fountain height as a function of elevation _{1} for the different models. Reproduced from Flekkøy et al. [

It is seen that the combination of a realistic chain and a smooth bottom produces no fountain, thus ruling out the kick-off mechanism as a complete explanation for the phenomenon. The kick-off mechanism by itself only works to explain the chain fountain of the pasta model. However, it is seen that while both the realistic and flexible chains rely on a rough bottom to produce a fountain, the existence of a realistic rigidity

Is there a crucial experiment that may serve to distinguish between the different mechanisms? The fountain produced by the pasta-model on a smooth container bottom (_{max} = π) model can only be explained by the bumpy take-off mechanism.

Fountain height as a function of container width _{1} = 2.5 m.

The predictions for the velocity of the chain given by Equation (17) involves the mean square velocity

This is indeed what is measured in the simulations and experiments. A comparison between measurements and the prediction of Equation (17) is shown in

Velocities measured 20 cm above the floor. The red line shows the experimental average _{z} is the vertical component of the velocity. The blue circles show simulation measurements taken at the same location but as a function of time. The indigo line shows a semi-analytic prediction derived from Equations (11) and (12) in Biggins [

This is most likely due to a combination of two factors: (1) lack of hydrodynamic drag from the air, and (2) lack of bending friction in the simulations. The hydrodynamic drag may be added to the simulations by including a force per bead _{air} is the mass density of the air, _{D} is the drag coefficient of a bead, in the force calculations. For single sphere _{D} ≈ 0.5, but our spheres are equipped with pins connecting to their neighbors and also move together in a direction that is often not parallel to the chain itself. These factors may cause _{D} to increase, and we have tentatively set _{D} = 1. The result, which is shown in _{1}-value than in the experiments. This is consistent with visual observations of the real and simulated chains, see

Mass conservation then implies that

The analytic, fluctuation free prediction of Biggins is seen to lie below the present simulations, at least asymptotically. Biggins asymptotic _{1}〉 〈cos Θ〉 → _{1} was done. This means that the disagreement is most likely due to the added mass in the downwards moving chain due to the fluctuations.

It should be noted that the prediction of Equation (17) for the average velocity 〈_{z}〉 itself reflects both the geometric effect of the fluctuating chain and the effect of velocity fluctuations δ_{z} around the average as

There is a fine point linked to the measurement of _{z} and the fact that our continuum theory of Equation (8) is compared to a discrete particle model. Indeed, the velocity varies discontinuously from particle to particle, and only when their velocities are averaged into a group does it make sense to represent them by a differentiable field as we do. The velocities, and by implication the cos Θ-values, shown in

_{0}, above which there can be no direct interaction between the bead and the underlying packing. The fact that the beads start slowing down above the distance _{0} means that there must be an upwards acting force that is transmitted through the chain to the particle when it is located between the two lines. This can be more clearly seen by visually comparing a group of three consecutive beads close to the packing (green square in

Velocities measured at different heights above the floor. The graph shows how the upwards forces, coming from the interaction with the floor, slow down the beads. The rectangular frames illustrate how a group of three beads is compressed as it hits the packing. The rightmost vertical line shows where forces from the floor are observed to slow the chain down. The leftmost line denotes the distance _{0}, above which there can be no direct interaction with the underlying packing.

This observation runs contrary to the observation by Grewal et al. [

In summary, we have developed a theory of the flying chain motion that includes fluctuations, and we have proceeded to demonstrate, mainly via simulations, that these fluctuations play a quantitative role that is necessary for the theory to agree with measurements. The simplest observation of this is the fact that chain buckling creates an increased effective mass in the chain by about 25%.

The existence of a fountain relies entirely on the upwards acting force from the container on the chain. But the mechanisms acting to produce this force varies depending on the construction of the chain. While a chain made of rigid segments may create a fountain even when it takes off from a smooth container bottom, this is not the case for a chain with flexible links between spherical beads. In this case the required momentum must be picked up from the container via collisions with beads in it or a rough container bottom. Yet, the introduction of a maximum bending angle between links in the chain, will enhance the effect of these collisions.

We have studied an experiment that may serve to distinguish the mechanisms at play, that is, the measurement of fountain height vs. container width. While the rigid segment tend to reduce its fountain height with increasing width, the opposite is true for the flexible chain without a maximum bending angle between links in the chain.

The datasets generated for this study are available on request to the corresponding author.

EF did the theory and simulations as well as the main part of the writing. MM did the experiments as well as the written description of those, while KM having developed the labs that made the experiments possible supervised these. All authors contributed to the discussions defining the work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We wish to thank Ellen Karoline Henriksen and Carl Anghell who first introduced us to the flying chain, Dragos-Victor Anghel for his valuable theoretical input and Frédéric Lindboe for his practical tips on how to pack the beads. We also thank the Research Council of Norway through its Centres of Excellence funding scheme, project number 262644.

^{1}This result may also be proven by integrating over only half the _{C} volume and assuming 〈_{z}_{x}〉 = 0 on the vertical surface of this volume.