Yunbo Li
email: yunbo.li@mine-nantes.fr

Anne-Cécile Orgerie
email: anne-cecile.orgerie@irisa.fr

Jean-Marc Menaud
email: menaud@mine-nantes.fr

Balancing the use of batteries and opportunistic scheduling policies for maximizing renewable energy consumption in a Cloud data center

Keywords: Green data centers, Cloud computing, renewable energy, energy storage, opportunistic scheduling

The fast growth of cloud computing considerably increases the energy consumption of cloud infrastructures, especially, data centers. To reduce brown energy consumption and carbon footprint, renewable energy such as solar/wind energy is considered recently to supply new green data centers. As renewable energy is intermittent and fluctuates from time to time, this paper considers two fundamental approaches for improving the usage of renewable energy in a small/medium-sized data center. One approach is based on opportunistic scheduling: more jobs are performed when renewable energy is available. The other approach relies on Energy Storage Devices (ESDs), which store renewable energy surplus at first and then, provide energy to the data center when renewable energy becomes unavailable. In this paper, we explore these two means to maximize the utilization of on-site renewable energy for small data centers. By using realworld job workload and solar energy traces, our experimental results show the energy consumption with varying battery size and solar panel dimensions for opportunistic scheduling or ESDonly solution. The results also demonstrate that opportunistic scheduling can reduce the demand for ESD capacity. Finally, we find an intermediate solution mixing both approaches in order to achieve a balance in all aspects, implying minimizing the renewable energy losses. It also saves brown energy consumption by up to 33% compared to ESD-only solution.

I. INTRODUCTION

The fast growing demand for Cloud services, emphasized by the data deluge, has lead to a boom in the number of data centers worldwide. Consequently, the global electricity part dedicated to their consumption has reached unprecedented levels. In 2012, the number of data centers worldwide was estimated at 509,147 consuming roughly the output of 30 nuclear power plants [START_REF] Glanz | Power, Pollution and the Internet[END_REF]. In 2016, another study estimates that worldwide the data centers use 91 billion kilowatt-hours of electricity -enough to power New York City twice over -and their consumption is still growing rapidly [START_REF] Research | Coal Computing: How Companies Misunderstand Their Dirty Data Centers[END_REF]. This situation raises major environmental, economic and social concerns.

One way to save energy at a data center level consists in locating it close to where the electricity is generated, hence minimizing transmission losses. For example, Western North Carolina, USA, attracts data centers with its low electricity prices due to abundant capacity of coal and nuclear power following the departure of the region's textile and furniture manufacturing [START_REF]How dirty is your data?[END_REF]. Currently, this region has three super-size data centers from Google, Apple and Facebook with respective power demands of 60 to 100 MW, 100 MW and 40 MW [START_REF]How dirty is your data?[END_REF]. However, such huge facilities represent only a small fraction of the global consumption of data centers. Indeed, small-and medium-sized server rooms continue to account for nearly half the electricity consumption of the market [START_REF] Whitney | Data Center Efficiency Assessment scaling up energy efficiency across the Data Center Industry: evaluating Key Drivers and Barriers[END_REF].

For economical, environmental or marketing reasons, other companies opt for greener sources of energy. For example, Quincy (Washington, USA) supplies electricity to data facilities from Yahoo, Microsoft, Dell and Amazon with its lowcost hydro-electrics left behind following the shutting down of the region's aluminum industry [START_REF]How dirty is your data?[END_REF]. Several renewable energy sources like wind power, solar energy, hydro-power, bioenergy, geothermal power and marine power can be considered to power up super-sized facilities. The production variability of most renewable sources leads data center facilities to only partially rely on them and to depend also on the regular electrical grid as a backup or main supplier.

While using renewable sources may reduce energy costs, reduce peak power costs, or both [START_REF] Goiri | Parasol and GreenSwitch: Managing Datacenters Powered by Renewable Energy[END_REF], they are mostly intermittent and fluctuating over time (like sun and wind for instance). These variations may lead to electricity losses if the computing workload does not match the renewable production. Cloud infrastructures, on the other hand, can take advantage of multiple locations to increase their green consumption with approaches such as follow-the-sun and follow-the-wind [START_REF] Figuerola | Converged Optical Network Infrastructures in Support of Future Internet and Grid Services Using IaaS to Reduce GHG Emissions[END_REF]. As sun and wind provide renewable sources of energy whose capacity fluctuates over time, the rationale is to place computing jobs on resources using renewable energy, and migrate jobs as renewable energy becomes available for resources in other locations.

In the case of a single data center, such follow-the-sun approaches are not feasible. But instead, opportunistic scheduling algorithms can make advantage of renewable energy availability to perform jobs with low priorities [START_REF] Li | Opportunistic Scheduling in Clouds Partially Powered by Green Energy[END_REF]. Opportunistic policies distinguish two kind of computing jobs: jobs requiring to run continuously (like web servers) and jobs that can be delayed and interrupted, but with a deadline constraint (such batch jobs include monthly payroll computation for example).

The jobs of the second type wait for renewable energy surplus to be scheduled, thus reducing the overall consumption part of brown energy. However, such scheduling policies make use of virtual machine migrations and suspend/resume functions that have a cost in terms of energy consumption [START_REF] Orgerie | A survey on techniques for improving the energy efficiency of large-scale distributed systems[END_REF].

Another possible method for improving the effective utilization of intermittent and fluctuating renewable energy consists in using batteries to store green production surplus, and to use it during low production periods [START_REF] Goiri | Parasol and GreenSwitch: Managing Datacenters Powered by Renewable Energy[END_REF]. Typically for solar sources, energy can be stored during the day -if not fully consumed -and be utilized during nights when there is no production. However, batteries have an inherent energy efficiency (their yield) that leads to energy losses. So, is it greener to use opportunistic scheduling or batteries?

In this paper, we discuss these two approaches for maximizing the utilization of renewable energy in small and medium data centers. We compare these two solutions in terms of renewable energy utilization and total energy consumption in order to estimate whether the losses due to the battery efficiency balances or not the losses due to migration costs incurred by opportunistic scheduling policies. We also evaluate an intermediate solution mixing both approaches. This study investigates two types of batteries (lead-acid and lithium-ion, but can be easily generalized to other types of ESD), the optimal size of photovoltaic panels, several sunlight profiles and real-world workload traces from a medium-sized data center.

The remainder of the paper is organized as follows. Section II presents recent work on reducing energy consumption in data centers. Section III describes our data center and Energy Storage Device (ESD) models. Section IV describes the job scheduling algorithms: baseline algorithm with ESD, opportunistic scheduling without ESD and opportunistic scheduling combined with ESD. Section V presents our experimental setup including an analysis of real-world workload traces and the simulation-based methodology to find the optimal solar panel dimension and battery size for a given data center and a given workload. Section VI presents the results of our simulations which show the relationship between brown energy consumption and different solar panel dimension/battery size. Lastly, Section VII concludes this work and discusses the challenges for future research.

II. BACKGROUND AND RELATED WORK

Data center consumption reduction. Much of the early work focuses on the reduction of the brown energy consumption in data centers [START_REF] Goiri | Parasol and GreenSwitch: Managing Datacenters Powered by Renewable Energy[END_REF], [START_REF] Buyya | Energy-efficient management of data center resources for cloud computing: a vision, architectural elements, and open challenges[END_REF]- [START_REF] Srikantaiah | Energy aware consolidation for cloud computing[END_REF]. In order to save energy in mono-site data center, a classical goal consists in reducing the number of powered-on servers as idle servers can typically consume half of their peak power consumption [START_REF] Orgerie | A survey on techniques for improving the energy efficiency of large-scale distributed systems[END_REF]. Job scheduling, virtual machines (VMs) placement and consolidation directly affect the number of poweredon servers. In [START_REF] Ghribi | Energy efficient vm scheduling for cloud data centers: Exact allocation and migration algorithms[END_REF], Chaima et al. propose an energy-aware allocation and consolidation algorithm to switch off the underutilized servers in cloud data centers. For powered-on servers, dynamic voltage and frequency scaling (DVFS) allows CPU to work at multiple frequencies by increasing/decreasing voltage to meet the performance demand and to reduce the average power consumption [START_REF] Wang | Towards energy aware scheduling for precedence constrained parallel tasks in a cluster with dvfs[END_REF]- [START_REF] Etinski | Understanding the future of energy-performance trade-off via dvfs in hpc environments[END_REF]. Lastly, resource overcommitment increases server utilization and minimizes the number of powered-on servers. In [START_REF] Zhang | Virtual machine migration in an over-committed cloud[END_REF], Zhang et al. design a migration algorithm that reduces the risk of overload and minimizes the number of VM migrations in over-committed data centers.

Integrate renewable energy into data center. Unlike traditional infrastructures where energy sources are controllable, integrating renewable energy into a data center is difficult due to its intermittent and variable nature. Usually, solar energy is considered as an admissible renewable source as solar panels are easy to install, they present a reasonable efficiency and the variations in their electricity production are not too abrupt (as for wind) [START_REF] Goiri | Parasol and GreenSwitch: Managing Datacenters Powered by Renewable Energy[END_REF]. In [START_REF] Krioukov | Integrating renewable energy using data analytics systems: Challenges and opportunities[END_REF], Andrew et al. point out that the slack time is a key feature enabling the jobs to be delayed until the renewable energy becomes available. They propose an off-line scheduling algorithm to align the brown energy consumption with the renewable energy supply. However, it is arduous to make accurate long-range workload prediction. In the real-world workloads, a number of jobs cannot be delayed or interrupted, such as web services for instance. In our previous work [START_REF] Li | Opportunistic Scheduling in Clouds Partially Powered by Green Energy[END_REF], we propose an on-line solution making use of opportunistic scheduling for increasing solar energy utilization in a small-/medium-size data center without energy storage. This approach leverages two ideas: 1) delay part of the jobs (interruptible ondes) until solar energy becomes available; 2) when the renewable energy production cannot fully support the entire workload energy consumption, the system switches off the under-utilized servers with the help of consolidation techniques. However, it resorts to a consequent number of migrations caused by the consolidation algorithm and thus incurs a non negligible energy overhead. Furthermore, it may waste some renewable energy if there is not enough work to consume it all during production peaks. Batteries can solve this issue by partially or fully storing renewable energy surplus.

Energy Storage Devices. The variable and intermittent nature of renewable energy -like solar energy -makes it difficult to manage. In order to increase the usage of renewable energy, one way consists in carefully scheduling the workload to align with the time-varying available renewable energy. An alternate solution consists in using ESDs [START_REF] Ghiassi-Farrokhfal | Toward a realistic performance analysis of storage systems in smart grids[END_REF] to store the renewable energy and generate electricity for later usage.

The main parameters to be considered when dealing with ESDs are:

1) Efficiency: The energy used to charge a battery is higher than what can be used at a later time. 2) Battery charging and Discharging Rate Limit: This charging/discharging rate limit is determined by the type of battery. Typically the discharge/charge ratio is larger than 1 for most batteries. 3) Self-Discharge: There is an energy loss which is proportional to the storage time. 4) Depth-of-Discharge (DoD): Many factors may impact the battery lifetime such as the charging/discharging cycles [START_REF] Chen | Progress in electrical energy storage system: A critical review[END_REF], [START_REF] Divya | Battery energy storage technology for power systems?an overview[END_REF]. DoD can also impact the battery lifetime: in order to extend the battery lifetime to a reasonable time, we cannot use the full capacity of battery. We now present the ESD that includes re-chargeable batteries technologies (Electrochemical). In this paper, we consider two kinds of batteries: Lead-Acid Battery (LA) and Lithium-Ion (LI) which are prevalent in current data centers. In comparison with LA, LI battery has higher energy density, energy efficiency and lower discharge rate, but also higher cost. In the rest of the paper, battery and ESD terms are interchangeable.

III. CONTEXT AND ASSUMPTIONS

This paper is focusing on maximizing the use of renewable energy in a small/medium-scale data center with on-site solar panels. This section describes the context of this work.

A. Small and medium data centers

The considered data center comprises between 20 to 150 servers. Each server has limited resources in terms of CPU and RAM. We assume that there is no centralized storage system in the data center: each server has its own hard disk [START_REF] Beldiceanu | Towards energy-proportional Clouds partially powered by renewable energy[END_REF]. The data center is equipped with photovoltaic (PV) panels and an ESD. It has dual brown (from regular grid) and renewable energy supplies. If the renewable energy cannot be entirely consumed by the data center, the ESD stores the surplus of renewable energy for future use. We also assume that each server has a switch connected with renewable and brown energy supplies and the ESD. Specifically, the server can only opt for using one of the three sources at the same time.

A job can be submitted to the data center at anytime and it consists of an individual Virtual Machine (VM) to execute for a given duration. A VM is considered as the basic unit of resource allocation. We assume each VM has two constraints, namely CPU and RAM, and each job has its own duration and a predefined deadline. When a job finishes, the VM is destroyed and it releases its reserved resources back to the server. The job management system assumes that time is divided into slots. The VM resource allocation operations are performed periodically at the beginning of each time slot.

B. ESD model

As shown in Figure 1, PV panels turn solar energy into electrical energy which can be directly supplied for the data center or collected by the ESD. The ESD is composed of rechargeable batteries which first collect and store energy (generated from solar energy only), and then power the data center when scheduled.

Solar photovoltaic

ESD

Data center

Grid

Fig. 1: Energy sources of the data center

The capacity of the ESD is finite. Herein, we use parameter C to express the maximum capacity of an ESD. At a given time t, C available (t) represents the energy that has been collected and is stored by the ESD. In order to keep a longer battery lifetime, we take into account the DoD constraint [START_REF] Divya | Battery energy storage technology for power systems?an overview[END_REF], which stipulates that the remaining energy stored in an ESD has to be larger than the DoD threshold. So, in other terms, the available stored energy is smaller than a higher bound ηC (0 < η < 1, e.g, η = 0.8). By considering the DoD constraint, one can see that C available (t) never reaches C. Formally, we have 0 ≤ C available (t) ≤ ηC.

An ESD has two significant functionalities: charging (collects energy from solar panels) and discharging (powers the data center). In our system, we consider that charging and discharging are two independent procedures. It implies ESD is never under charging and discharging states simultaneously. The charging rate has an upper bound λ depending on the ESD type and capacity. During a given time period [t i , t j ] (t j > t i ), if we suppose the available green energy (supplied by PV cells) is E(t i , t j ), we use formula 1 to compute the amount of energy E in (t i , t j ) that can be collected by an ESD.

E in (t i , t j ) = min(E(t i , t j ), λ(t j -t i ), C available (t i )) × σ (1)
Parameter σ is constant and expresses the energy efficiency of the battery's charging procedure. The discharging rate also has an upper bound denoted µ. During a consecutive time period [t i , t j ], we use formula ?? to compute the amount of energy E out (t i , t j ) that can be provided by the ESD. Parameter E self-discharge (t j -t i ) represents the energy loss due to the self-discharging of batteries.

Eout(t i , t j ) = min(µ(t j -t i ), ηC -C available ) -E self-discharge (t j -t i )
In this paper, we consider only solar energy as renewable energy source. Due to the variable and intermittent nature of solar energy, an energy production prediction is performed when a job scheduling decision has to be taken. It only predicts the solar energy for the following time slot (1 hour), so that such short-time prediction may have a high accuracy [START_REF] Goiri | Parasol and GreenSwitch: Managing Datacenters Powered by Renewable Energy[END_REF]. To simplify the problem, we assume that there is no prediction error in our validation methodology.

IV. VM SCHEDULING

A. Baseline algorithm

Here we describe the baseline algorithm which will be used as a comparison reference. Whether the renewable energy is sufficient for the workload energy or not, we expect to minimize the total energy consumption of the servers. The minimization can be done at different levels such as infrastructure or application for instance. As mentioned previously, a server in idle state consumes roughly half of its peak power. Therefore, an effective approach consists in reducing the number of powered-on physical servers.

Given a set of VMs with different resources requirements and a set of servers with fixed capacities, we want to find the minimum number of servers needed to contain all VMs, such that the amount of VMs' resource requirements assigned to each server does not exceed its capacity. The VM placement problem can be modeled as a n-dimensional binpacking with finite number of bins, where the different VM resource requirements can be modeled as different sizes of items and the various server's resources correspond to different bin sizes. In this paper, we consider CPU and RAM as the constraints for both servers and VMs. The VM placement problem then become a 2D bin-packing problem which is an NP-Hard problem. To solve this problem, we adopt the First Fit Decreasing (FFD) heuristic algorithm. A proof shows that for every list L, F F D(L) ≤ 11/9 OP T (L) + 1 where OP T (L) denotes the minimum number of bins that L can be all packed in and F F D(L) denotes the number of bins used by FFD [START_REF] Yue | A simple proof of the inequality ffd (l)? 11/9 opt (l)+ 1,? l for the ffd bin-packing algorithm[END_REF].

The regular FFD scheduling algorithm usually considers VM resource requirements as the resources' constraints for its placement. However, Cloud jobs typically have resource utilization levels well below their resource requirements on average over time [START_REF] Li | Opportunistic Scheduling in Clouds Partially Powered by Green Energy[END_REF]. The resource over-commitment technique increases the server utilization by considering lower bounds than the actual user requirements for allocating resources to VMs and thus, putting more VMs on a single server. As a consequence, when this lower bound is reached by each VM running on the server, some VMs have to be suspended or migrated in order to free resources. As resource over-commitment is widespread in Cloud infrastructures [START_REF] Zhang | Virtual machine migration in an over-committed cloud[END_REF], [START_REF] Dabbagh | Efficient datacenter resource utilization through cloud resource overcommitment[END_REF], we combine FFD and resource over-commit in order to increase server resource usage and reduce the number of powered-on servers. However, as the over-commitment configuration can greatly increase both CPU and RAM utilization, it can lead to overload the server. Consequently, we will need to migrate the VMs from the over-loaded servers to others thus incurring an extra energy consumption and performance degradation. Hence, for these reasons, the jobs need to be provisioned for their peak draw by analyzing the history of jobs behaviors and seeking a safety over-commitment configuration. We study a real-world job trace and, in a similar way as our previous work [START_REF] Li | Opportunistic Scheduling in Clouds Partially Powered by Green Energy[END_REF], we set a reasonable over-commitment configuration avoiding RAM overloads.

We take into account of the VM creation and VM live migration energy overhead. Unlike the VM creation, the energy consumption of VM migration depends on the VM disk size and the number of dirty pages in RAM that impacts the migration time.

The considered baseline algorithm implements both FFD and over-commit resources techniques, as our opportunistic algorithm does. At any time, the jobs are submitted and the broker directly places them on the servers. The baseline consumes the solar energy when it is available. The battery is charged when a surplus solar energy appears. Otherwise, the workload first discharges the battery and then uses the brown energy. Note that, there is no opportunistic job scheduling mechanism in the baseline algorithm.

B. Opportunistic job scheduling

As time in our system is discrete, the optimization operations are performed periodically at each slot. According to the job characteristics, opportunistic scheduling approaches classify the jobs into two types called here web jobs and batch jobs. The web job is defined as non-interruptible job. It has the highest priority on scheduling. When both types of jobs arrive, the broker pushes them respectively to the web job pool and then to the web queue, the batch job to the batch pool. We adopt the FFD algorithm to place the web jobs, all the jobs in web queue are immediately placed on the servers which have sufficient resources. Unlike the web jobs, batch jobs can be suspended with a slack time that may increase the potential chance to exploit the renewable energy. When the slack time reaches 0, the batch job is promoted as a web job. After all the web jobs have been placed, the broker seeks among the running servers which meet the batch jobs resource requirement. The web job placement and batch job placement are independent algorithms.

Recall that, opportunistic scheduling targets two problems: 1) when workload energy consumption is higher than the solar energy supply, it runs partially the workload: it suspends the batch jobs which have a non-null slack time and performs VM consolidation in order to switch-off more servers. This is to reduce the brown energy consumption. 2) Otherwise, it runs the entire workload and the batch jobs which were delayed before. This is to maximize the solar energy usage.

Due to the ESD efficiency, there is an energy consumption during battery charging. In contrary, opportunistic scheduling can reduce the effect due to battery efficiency by delaying the batch jobs to consume the solar energy directly instead of storing it in the battery. However, the more batch jobs are delayed, the more numerous VM migrations will be due to consolidation. As shown in Figure 2, the purple curve w(t) denotes the workload energy consumption and the green curve g(t)

C. Battery charge/discharge model

denotes the solar power. We observe in Figure . 2 that for areas a 1 , a 2 and b 1 , b 2 , the workload energy consumption is higher than the solar energy supply. If there is no ESD, the total energy consumption from the grid can be expressed as:

E brown = t 1 t=0 (w(t) -g(t)), ∀(w(t) > g(t)), t ∈ T (2)
When the workload energy demand is less than the solar energy, the amount of surplus solar energy is defined as:

E surplus = t 2 t=t 1 (g(t) -w(t)), ∀(w(t) < g(t)), t ∈ T (3)
For day 1 on Figure 2, E brown = a 1 + a 2 and E surplus = c 1 .

The battery has to be charged when w(t) < g(t). When the solar energy is not sufficient to supply for the current workload energy, we first discharge the battery. Once the battery runs out, the servers then consume the brown energy from the grid.

In the rest of paper, we seek for the relationship between c 1 and (a 2 + b 1 ) in different cases. Ideally, if c 1 is much larger than (a 2 + b 1 ), the amount of energy produced by solar panels is sufficient to offset the whole workload energy consumption that takes into account an ideal ESD (it can store all the surplus solar energy). In a real case, the battery often has limited size and solar energy is not sufficient to compensate the workload energy consumption (c 1 < (a 2 + b 1 )).

If we assume the ESD is ideal for both solutions, then in baseline, the energy loss is mainly caused by the battery efficiency. The solar energy needed can be formulated as:

c 1 × σ > (a 2 + b 1 ) (4)
where σ denotes the battery energy efficiency.

V. EXPERIMENTATION CONDITIONS

We developed a trace-driven simulator to compare different resource allocation and scheduling policies and estimate their energy consumption using a power model based on real measurements. The simulator integrates the data center and ESD models described in Section III. We use it to evaluate the impact on energy consumption with different configuration and ESD technologies. This section presents the experimentation conditions used in the simulator.

A. Workload trace

For all the simulations, we use the real-world trace from a medium-scale private Cloud data center provided by Easyvirt, a French SME which has a research partnership with this Cloud provider. The original trace was collected from 26th of March 2014 to 5th of July 2014. We extracted a non-holiday week: the data consists of 787 web jobs and 3148 batch jobs. It precises each job's initial VM resource requirement, the instantaneous CPU and RAM utilization. In our scenario, the CPU and RAM utilization is averaged over 1 hour in order to eliminate the noise. Each web job takes roughly 12 hours and each batch job takes about 6 hours with a deadline equal to 12 hours (it has to be executed within the 12 hours following its submission).

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Fig. 3: Solar energy production with solar panels of 5.52 m 2

B. Server energy consumption model

A server power consumption is related to its different components. Most previous studies [START_REF] Orgerie | A survey on techniques for improving the energy efficiency of large-scale distributed systems[END_REF] agree on the fact that the dynamic server power consumption mainly depends on the CPU frequency. We performed power consumption measurements on the Grid'5000 testbed [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF]. The used servers are Dell PowerEdge R720 with two 6-cores Intel Xeon E5-2630 processors (2.3GHz), 32 GB of RAM and 600 GB of disk. The power consumption was taken for different CPU load profiles as described in [START_REF] Li | Opportunistic Scheduling in Clouds Partially Powered by Green Energy[END_REF]. Our experimental results shows in particular that a server on idle state can roughly consume half of its maximal power consumption.

C. Solar energy trace

For solar energy production, we use a mini-scale solar power farm which was set up in the campus of University Nantes 1 . It is composed by 8 identical panels Sanyo HIP-240-HDE4 and SMA Sunny Boy 1200 inverter. The theoretical max power is 240 Watt per panel. Then the instantaneous peak power can be expressed as a function of m: e.g. for a given solar power farm with 8 panels, max E spv = 240 × m watt where m = 8. As shown in Figure 3, we choose the trace of a random week (22-28 June 2015) which is mostly sunny.

VI. RESULTS

In this section, we compare the opportunistic scheduling approach and the battery approach for maximizing solar energy utilization in a data center. First, we determine the optimal solar panel and the optimal battery size depending on the approach. Then we compare both approaches under various conditions, and we combine them.

A. Find the optimal solar panel dimension

We assume that the battery size and the charging/discharging rate both approach infinite in this early experiment to determine the optimal solar panel dimension for the given workload. We seek for a ideal solar panel that can supply the entire workload energy consumption. We first find a solution for this problem in the baseline case. Since the workload energy consumption can be estimated statistically, the area of solar panels can be trivially determined via calculation.

In this scenario, we assume an infinite battery size. So, it enables to store all the surplus solar energy in the region c 1 and c 2 as shown in Figure 2. Due to the battery efficiency 1 Traces available online: http://photovolta2.univ-nantes.fr Fig. 4: Workload energy/solar energy supply ratio and maximum solar energy that can be generated per unit time and limited charge/discharge rates, the amount of surplus solar energy has to be strictly greater than the required workload energy in order to compensate for these intrinsic ESD losses.

Recall that, we are looking for c 1 which satisfies c 1 × σ ≥ a 2 + b 1 where σ indicates the battery efficiency. This formula describes the energy stored during the day which furnishes the workload energy until the solar energy becomes available again. Figure 4 illustrates the workload energy/solar energy supply ratio. As the size of solar panels increase, the brown energy consumption of the entire workload decreases. When the solar panel area increases to 226.32 m 2 (about 15 × 15 meters), the workload energy consumption reaches 0, under the assumption of a battery with infinite size and infinite charging/discharging rate. We later determine the optimal battery size.

B. Find optimal battery size in ideal case

In this scenario, we assume that the size of solar panels are ideal, so equal to 226.32 m 2 . Thus, it can provide sufficient energy for the workload needs. Figure 5 shows the brown energy consumption of baseline algorithm and of opportunistic scheduling with the same type of battery (LI battery) depending on the battery size (purple and green curves). We also indicate the corresponding volume taken by the batteries depending on their capacity for both LI and LA types (blue and yellow curves). The brown energy of both solutions decreases when we increase the battery size. We can see that opportunistic scheduling always requires a smaller battery size need than baseline, as expected. We can also see that the workload does not consume anymore brown energy when battery size is larger than 90 kWh (600 L for a LI ESD) for opportunistic and 140 kWh (950 L for a LI ESD) for baseline. So, using opportunistic scheduling decreases the ideal battery capacity and its corresponding volume by 36%.

With opportunistic scheduling, the batch jobs are delayed in order to be executed when solar energy is available. This has two effects: 1) a part of solar energy is directly consumed instead of storing it the battery; 2) despite the batch jobs delayed run, the surplus solar energy is reduced. This is why it can use a smaller battery size than baseline to reach the same brown energy consumption. Figure 5 also shows that LA volume is larger than LI. Using the price values provided in Table I, for a 90 kWh battery, for LI, it represents 600 L and 47,250 $, while for LA, it represents 1,150 L and 18,000 $. Both price and volume can impact the adoption of such an approach for logistics and financial reasons. While LI and LA batteries exhibit different characteristics, their loading and discharging schemes behave similarly. In the remaining of the paper, we use LI batteries without loss of generality.

C. Opportunistic vs. baseline when solar energy is not sufficient for the workload In this scenario and for all the experiments presented below, we assume that there is not enough solar energy to fulfill the entire workload needs. We consider the solar panel dimension which is not able to provide sufficient solar energy to compensate the workload energy consumption. We compare the brown energy consumption with variable battery sizes for the both solutions, opportunistic and baseline. The different configurations of the opportunistic scheduling are defined as delaying 30%, 50%, 70%, 90%, 100% batch jobs (instead of directly executing them when solar energy is available, and thus storing it). This represents possible tradeoff between the opportunistic scheduling approach and the ESD-based approach. We can see that the both opportunistic and baseline reduce brown energy when the battery size becomes large. For a given battery size which is inferior to 73 kWh, opportunistic always consumes less brown energy in comparison with baseline. After this point, the brown energy consumption of opportunistic approach does not decrease any more when battery size becomes lager. In contrast, the baseline brown consumption continues to decrease while the battery size increases up to 110 kWh. Although all the other configurations of opportunistic approach can get lower brown energy with a larger battery compared with pure opportunistic scheduling, they are still higher than baseline in this particular case.

D. Solar energy losses with variable battery size

Fig. 7: Solar energy lost due to the limited battery size Figure 7 shows the energy losses with varying battery sizes. This energy losses stem from the battery limited charging rate and size. Since the opportunistic approach delays the batch jobs until the solar energy becomes available, the workload here consumes solar energy directly and stores the rest of solar energy to the battery. Thus, opportunistic can use a smaller battery while achieving the equivalent effect as baseline. Finally, when battery size is 80 kWh for opportunistic and 110 kWh for baseline, they both reach zero solar energy loss caused by the battery's limited charging rate and size.

E. Opportunistic scheduling migration vs. baseline battery loss

Fig. 8: Migration cost vs. battery efficiency loss Figure 8 shows the amount of energy lost with variable battery size. Here, we focus on 2 types of energy loss: 1) battery energy efficiency; 2) VM migration energy cost. In baseline, as the size of battery increases, the energy loss increases; this happens because of battery energy efficiency and of very few VM migrations due to overloaded servers (over-commit policy). Hence, the choice of battery type highly impact the waste of energy.

For the opportunistic approach, the energy loss mainly depends on: 1) migrations caused by consolidation; 2) the rest is same as baseline that is the battery efficiency. Since the solar energy was not sufficient enough for the workload needs, the opportunistic algorithm has to suspend some batch jobs and to perform consolidation in order to keep a low number of powered-on servers. And the delayed batch jobs then are executed when solar energy become available again. The delayed workload directly consumes the solar energy and the remaining solar energy is stored in the battery. Thus the opportunistic approach stores less energy than baseline in the ESD, and thus the loss due to battery efficiency are lower with the opportunistic approach. However, the total solar energy is not sufficient for the entire workload in this case, the opportunistic approach periodically performs VM consolidations that may lead to a number of VM migrations. This migration energy cost compensates that gain. For this reason, it can be better to partially delay the batch jobs (respectively 10, 30, 50 and 70% are delayed). In fact, when we delay less batch jobs, it leads to less migrations by consolidation, but more energy will be stored in the ESD. There is a balance for the opportunistic approach between the energy loss caused by migrations and by battery efficiency. We observe that when there are 30% batch jobs delayed, the energy loss which contains the migration energy overhead and battery efficiency, the opportunistic approach gets a lower energy loss than baseline when battery size is greater than 40 kWh.

F. FFD scheduling impact

For a given workload such as ours, the opportunistic approach has a lower energy loss in comparison with baseline when it delays 30% batch jobs. Unfortunately, in Figure 6, the opportunistic approach consumes more brown energy than baseline. It seems that the result is in conflict with the result in Figure 8. After our analysis, there is also an impact from the two consecutive FFD algorithms (for web and batch jobs) that leads the opportunistic approach to need more servers to place the same amount of jobs compared with baseline.

There is a performance degradation when we change the input list size; e.g. for any list L with a length l, F F D(L) ≤ 11/9OP T (L) + 1 as mentioned in Section IV-A. If we divide the list L into 2 sub-lists L 1 with a length 0 < l 1 ≤ n -t and L 2 with length 0 < l 2 ≤ t, F F D(L 1 ) ≤ 11/9OP T (L 1 ) + 1 and F F D(L 2 ) ≤ OP T (L 2 ) + 1. The performance may be different between FFD(L) and (FFD(L 1 ) + FFD(L 2 )). In case of our simulation results, we observe that there is a performance degradation with the number of bins needed: FFD(L) (FFD(L 1 ) + FFD(L 2 )). This explains why the opportunistic scheduling has a lower energy loss than baseline in Figure 8, but it consumes more brown energy than baseline in Figure 6 when the battery size is large enough.

VII. CONCLUSION

Integrating renewable energy into data centers significantly reduces the traditional energy consumption and carbon footprint of these energy-hungry infrastructures. As renewable energy is intermittent and fluctuates with time, it is usually under-utilized. In this paper, we address the problem of improving the utilization of renewable energy for a single data center by using 2 approaches: opportunistic scheduling and energy storage. Our first result deals with analyzing the workload to find ideal solar panel dimension and battery size, this is used to power the entire workload without any brown energy consumption. However, in reality, either the solar panel dimension or the battery size are limited, and we still have to address the problem of matching the workload consumption and renewable energy production. The second result shows that opportunistic scheduling can reduce the demand for battery size while the renewable energy is sufficient. The last results demonstrate that for different battery sizes and solar panel dimensions, we can find an optimal solution combining both approaches that balances the energy losses due to different causes such as battery efficiency and VM migrations due to consolidation algorithms. Our future work includes studying the pertinence of both approaches with other renewable energy sources, like wind for instance. As wind energy presents a completely different production profile compared to solar energy, we would like to investigate whether the trade-off between the opportunistic scheduling and energy storage approaches which is proposed in this paper remains the same or would be different.

Fig. 2 :

 2 Fig. 2: Workload energy consumption and solar energy

Fig. 5 :

 5 Fig. 5: First two curves (purple and green): brown energy consumption with an LI ESD. Last two curves (blue and yellow): ESD volume.

Fig. 6 :

 6 Fig. 6: Brown energy consumption with varying battery size

  Table I shows the different constraints per battery kind.

		LA	LI
	DoD	0.8	0.8
	Charge rate / ESD size (%)	12.5	25
	Efficiency	0.75	0.85
	Self-discharge (per day)	0.3%	0.1%
	Discharge rate / charge rate ratio	10	5
	Price ($/kWh)	200	525

TABLE I :

 I The battery characteristics (data from[START_REF] Chen | Progress in electrical energy storage system: A critical review[END_REF]-[START_REF] Wang | Energy storage in datacenters: what, where, and how much?[END_REF])

ACKNOWLEDGMENT

This work has received a French state support granted to the CominLabs excellence laboratory and managed by the National Research Agency in the "Investing for the Future" program under reference Nb. ANR-10-LABX-07-01.


Pierre-Thomas Brun

Basile Audoly

Alain Goriely

Dominic Vella

THE SURPRISING DYNAMICS OF A CHAIN ON A PULLEY: LIFT-OFF AND SNAPPING

Keywords: strings and chains, dynamic contact mechanics, curvature-reversal, whip, Atwood machine

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. Here, we consider the dynamics of the chain when one of the weights is removed and, thus, one end is pulled with constant acceleration. This simple change has dramatic consequences for the ensuing motion: at a finite time, the chain 'lifts off' from the pulley and the free end subsequently accelerates faster than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack, or snap, of a whip. We combine experiments, numerical simulations, and theoretical arguments to explain key aspects of this dynamical problem.

Introduction

When publishing the design of his machine in 1784, little could Atwood have known that, more than two centuries later, students would be asked to predict the outcome of his experiment. The problem, traditionally offered as an illustration of the principles of Newtonian mechanics, consists in deriving the acceleration a of two masses M > m subject to the action of gravity g while attached to a massless and inextensible chain passing over a frictionless pulley. The well-known result is that [START_REF] Goriely | Shape of a cracking whip[END_REF] a = Mm M + m g for the heavier of the two, with the second having the opposite acceleration.

Little attention has been paid to the seemingly trivial case in which both m = 0 and the chain has a finite linear density. If the mass of the chain nevertheless remains small compared to M , then equation [START_REF] Goriely | Shape of a cracking whip[END_REF] immediately gives that the remaining mass falls with acceleration a = g. However, a simple experiment (see, for example, figure 1b) reveals that this apparent simplification actually has a dramatic effect on the resulting motion: the chain 'lifts off' from the pulley in a complex motion. This lift-off has some surprising features that we explore in detail in this paper. In particular, we show that the free end accelerates faster than the end that is being pulled by the mass: in this sense, the free end 'beats' the free fall of the mass. We also show that the chain eventually 'snaps' in a manner that is reminiscent of the crack of a whip [START_REF] Goriely | Shape of a cracking whip[END_REF][START_REF] Mcmillen | Whip waves[END_REF].

Thin filamentary structures are as important in applications as they are ubiquitous in nature and industry; examples range from macromolecules such as DNA [START_REF] Neukirch | Extracting DNA Twist Rigidity from Experimental Supercoiling Data[END_REF] to the kilometric transoceanic cables laid on the ocean bed from telecommunication vessels [START_REF] Carter | Submarine cables and the oceans: connecting the world[END_REF]. On a human scale, textiles, hair and ropes are other examples of thin elongated structures in which one dimension greatly exceeds the two others (so that the objects may be modeled as one-dimensional). The thinness of such rod-like structures also makes them flexible so that they are frequently subject to large deformations in the three dimensional environment in which they evolve. This flexibility in shape in turn leads to rich bifurcation landscapes [START_REF] Brun | An introduction to the mechanics of the lasso[END_REF], striking pattern formation [START_REF] Habibi | Coiling of elastic ropes[END_REF], and intricate dynamical behavior.

Many filamentary structures, despite the range of length scales and materials encountered, are well modelled by the Kirchhoff equations for elastic rods [START_REF] Audoly | Elasticity and geometry: from hair curls to the non-linear response of shells[END_REF]. A particularly interesting limit of these equations is the case of inextensible strings in which inextensible rods have negligible resistance to bending and twisting: their behaviour is dictated to a large extent by the geometric constraint of inextensibility. In fact, this constraint by itself is enough to give rise to complicated dynamics: while the simplest case of a straight string accelerating along it length may be understood by a simple application of Newton's second law, any closed shape is a solution of the governing equations with any constant tangential velocity [START_REF] Healey | Steady axial motion of strings[END_REF]. It is therefore the combination of acceleration and 'turning a corner' that gives rise to the most interesting dynamics. In particular, moving inextensible strings form surprisingly complicated shapes including arches [START_REF] Hanna | Slack dynamics on an unfurling string[END_REF] and the mesmerising 'chain fountain' [START_REF] Biggins | Understanding the chain fountain[END_REF][START_REF] Biggins | Growth and shape of a chain fountain[END_REF].

In this paper we consider a chain moving around a pulley subject to a constant acceleration at one end and free at the other end. We model the chain as an inextensible string in partial contact with a disk (the pulley) and investigate key features of the motion. This setup is perhaps the simplest geometry in which one can study how a string or chain 'turns a corner' since the curvature of this corner is simply that of the pulley. We first describe in some detail the different phenomena that are observed experimentally and then analyze the lift-off in detail by combining theoretical arguments with numerical simulations. Finally, we describe the snapping that is associated with the reversal of curvature that ultimately occurs close to the free end.

Experimental observations

In the idealized setup of our problem, a mass M is attached to the end of a chain of linear density (mass per unit length) ρ placed around a circular pulley of radius R. The mass M is released at time t = 0 and subject to a constant acceleration a. The chain is initially held on the pulley with a known length of chain, L, hanging free from the first point of contact C (see Fig 1a).

The realization of this thought-experiment was carried out by using conventional ball and link chains purchased from the local do-it-yourself shop and hung over horizontal Pyrex beakers (Fisher Scientific) that were used as the frictionless pulley (the surface of the beaker being smooth, the chain slides over the surface with very little friction). The mass M ρ L is released at time t = 0 resulting in an acceleration a ≈ g.

Typically, the time evolution of the chain passes through three, qualitatively different, phases shown in figure 1b: • For 0 ≤ t < t LO the chain follows the dynamics that might naively be expected: it moves around the pulley at the speed that is imposed by the accelerating mass. • At a time t = t LO the chain starts to partially lose contact with the pulley, lifting off so that for t > t LO the hanging part of the chain goes faster than the rest of the chain. Due to the excess length of the chain, an arch forms between points C and C * (t) defining an angle θ * (t) (see figure 1b). • After the free end goes past the last contact point C (first image of the last row in figure 1b) the arch flares up, its curvature increasing rapidly until the free end eventually snaps at time t = t * (the curvature there changes sign). To complement our experiments, numerical simulations were carried out using the Discrete Elastic Rod method, which provides a discretization of the equations of motion for thin elastic rods using a Lagrangian formulation [START_REF] Bergou | Discrete elastic rods[END_REF]. This method, and its counterpart for thin viscous threads, have been used and validated previously [START_REF] Jawed | Coiling of elastic rods on rigid substrates[END_REF][START_REF] Audoly | A discrete geometric approach for simulating the dynamics of thin viscous threads[END_REF]. In the simulations, we use a string model (i.e. the bending and twisting stiffnesses are both set to zero), we prescribe the acceleration of the chain's end that is attached to the mass in our experiments. The frictionless contact with the pulley is implemented by a geometrical methodby, i.e. by alternating dynamic steps which ignore contact forces, with projection steps where the configuration is projected onto the manifold of admissible configurations. Figure 2 shows a direct comparison between experiments and numerical simulations without any adjustable parameter. The favorable agreement between the two validates the approximations made in modelling the chain as a string that is driven at constant acceleration (our experiments are in fact conducted with a constant force M g but the two methods are approximately equivalent since ρ L M ). Of particular interest is the possibility offered by these simulations to access the successive space derivatives of the basic physical variables (such as curvature) with controllable time resolution covering long periods of time. Additionally, physical quantities that are otherwise challenging to measure experimentally, such as the tension within the chain, are readily available from simulations.

At any given time before the chain snaps (0 < t < t * ), the chain may be divided into four regions: (I) and (IV) denote the two straight, vertical parts, respectively, (II) denotes the part of the chain not in contact with the pulley and (III) the contact region. For example, in figure 2b-d, only a small portion of the chain (III) remains in contact with the pulley. In this region the chain's speed is prescribed by the free fall of the mass, as in region (IV). In region (I), the chain moves significantly faster than the imposed acceleration, while remaining perfectly straight and tangent to the pulley. In addition, the tension in the chain is larger than if the lift-off were artificially suppressed (shown as a dashed line in figure 2d). The area (II) bridges the areas (I) and (III)-(IV) and their corresponding physical quantities, resulting in a ballooning shape (see figure 2).

The length of the chain that is initially hanging freely, L, turns out to play an important role in the problem and, therefore, we use L as the length scale of the problem. It is then natural to use the quantity (L/a) 1/2 as the time scale of the problem. If the acceleration of the freely hanging part of the chain were to remain at a throughout the motion, the length of the chain hanging beneath the point C then shrinks to zero at a time t = 2L/a.

From the 6 dimensional quantities of interest, we identify three dimensionless parameters:

(2)

π 1 = ρ L/M, π 2 = R/L, and π 3 = g/a.
In our experiments M is chosen such that π 1 1; ρ therefore enters in the problem as a multiplying factor for forces only. The value of π 2 may easily be varied experimentally by varying the radius of the pulley or, more simply, by varying the length of chain that initially hangs freely. By contrast, varying π 3 is more difficult. Our experiments are always performed with a = g and hence π 3 = 1. However, the inclusion of a body force acting on the chain complicates some of the analysis without changing the qualitative behaviour, as shown in figure 2. Therefore, in much of the analysis we shall neglect the acceleration due to gravity, i.e. we assume π 3 = 0. Experimentally, this case could be obtained by placing the chain, pulley and the mass on a smooth horizontal surface: at t = 0, the mass is then pushed off the table into the air [START_REF] Hanna | Jump conditions for strings and sheets from an action principle[END_REF] and allowed to fall under gravity so that the acceleration of the chain is a = g but the effective gravity acting directly on the chain vanishes. In the simulations, the value of π 3 is set to 1 or 0 depending on wether the results are to be compared with experiments or with our analysis.

Prior to lift-off

We consider a pulley of radius R around which a chain with linear density ρ is hung. We begin by including gravity as a body force, though later we will neglect it to simplify the ensuing calculations. Initially, the length of chain between the free end (at arc-length position s = 0) and the chain's first point of contact with the pulley is L (i.e. the point C in figure 1 has arc-length coordinate s = L at t = 0 -).

At t = 0 + , the far end at arc length s = ∞ is pulled with constant acceleration a. The end at s = 0 remains free, so that the tension there vanishes, T (0, t) = 0. To achieve the constant acceleration a, the tension within this straight portion of the chain must increase along the length as a simple consequence of Newton's second law. The conservation of angular momentum for the chain reads

∂ ∂s (T t) + ρ g + P n = ρ ∂ ∂t (vt), (3)
where t(s) and n(s) are the unit tangent and normal vectors to the chain axis, respectively. Here, P represents the (frictionless) reaction from the pulley on the chain, such that P = 0 whenever there is no contact. Projecting (3) onto the tangential and normal directions, and using the Frenet formula ∂t/∂s = -κn with κ the curvature, we find that

∂T ∂s -ρ g cos φ = ρ ∂v ∂t (4) -T κ -ρ g sin φ + P = -ρ v 2 κ, ( 5
)
where φ is the angle between the local tangent to the chain and the upward pointing vertical and κ = ∂φ/∂s is the curvature. In particular, while the chain remains in contact with the pulley, t < t LO , we know that

φ(s, t) =      0, 0 ≤ s < s C s-s C R , s C ≤ s < s C + πR π, s C + πR ≤ s (6)
where

(7) s C (t) = L -at 2 /2
is the arc-length position of the first contact point between the pulley and the chain (the point labelled C in figure 1). In the hanging part φ(s, t) = 0, and so, integrating (4) subject to T (0, t) = 0, we find that the tension T may be written:

(8) T (s, t) = ρ (a + g)s, s ∈ [0, s C ]
In the region where the chain is in contact with the pulley, (4) leads to

(9) T (s, t) = ρ as + ρ gR sin φ + C 1 , s ∈ [s C , s C + πR]
where the constant of integration C 1 = ρ g s C is found by requiring the tension to be continuous at s = s C . In the freely hanging portion of the chain, ( 5) is identically satisfied by P = 0. Elsewhere, (5) can be viewed as an equation for the reaction of the pulley on the chain required for this motion to occur. From a simple rearrangement, we find that [START_REF] Biggins | Understanding the chain fountain[END_REF]

) P = -ρ v 2 R + 2ρ g sin φ + 1 R ρ (as + gs C ), s ∈ [s C , s C + πR].
Note that P is a function of arc length s and time t through the angle φ and arc length of the contact point, s C (t), in addition to the explicit dependence on s. The minimum value of P at any particular time t < t LO is attained at s = s C :

(11) P min (t) = ρ s C (a + g) -v 2 /R = ρ (a + g)L -1 2 at 2 (3a + g) /R.
The key to understanding why lift-off occurs is held by the expression for the smallest reaction force P min (t) [START_REF] Biggins | Growth and shape of a chain fountain[END_REF]. At sufficiently early times P min > 0, and the contact pressure is everywhere positive (it prevents the chain from penetrating the pulley). However, P min vanishes at a time t = t LO given by [START_REF] Bergou | Discrete elastic rods[END_REF] t LO = 2L(a + g) a(3a

+ g) = 2 L a 1 + π 3 3 + π 3 ,
and then becomes negative. For time t > t LO , the assumption (6) that the string is in contact with the pulley over half a circle breaks down, as it predicts a negative pressure (an adhesive force would be required to maintain contact). The negative values of P are first attained in the neighbourhood of C, which suggests that lift-off takes place there. At time t = t LO , the length of the hanging chain is

(13) s LO C = s C (t LO ) = 2a 3a + g L = 2 3 + π 3 L.
In [START_REF] Bergou | Discrete elastic rods[END_REF], t LO is less than the time 2L/a when the hanging part (I) shrinks to a point the mathematical solution (6) ignoring the lift-off. This confirms that hanging part (I) has a finite length when the lift-off takes place. This is also consistent with

s LO C = s C (t LO ) > 0.
Apart from factors obviously required by dimensional analysis, both ( 12) and ( 13) appear to depend only on the ratio π 3 = g/a of the acceleration due to gravity to the imposed acceleration. They do not depend on the dimensionless group π 2 = R/L involving the size of the pulley. Furthermore, we note that in the limit of zero gravity, π 3 = 0, one third of the original hanging chain has passed around the pulley with two-thirds still hanging; when pulling the chain using a heavy mass in free fall, i.e. imposing π 3 = 1, then one-half of the original hanging chain has passed around the pulley with one-half still hanging. Here, incorporating gravity makes a quantitative, but not qualitative, difference; either way, an appreciable fraction of the initially hanging chain remains below the first contact point C.

From the above argument, we also gain some intuition into why the chain lifts off: the tension at the contact point s C (t) is decreasing with time (because the length of hanging chain is decreasing) while the tension required to turn the chain around the pulley increases (because the chain is moving faster and faster). Lift-off occurs when the excess tension in the chain due to the hanging length is 'used up'.

Lift-off

The prediction for the lift-off time, [START_REF] Bergou | Discrete elastic rods[END_REF], presents a natural test of the model assumptions, in particular with regard to our neglect of the bending stiffness of the chain and the assumption of a frictionless contact between chain and pulley. Experiments were performed with a = g (i.e. π 3 = 1) and a number of different pulley sizes and initial hanging lengths (i.e. varying π 2 ). As shown in figure 3, we find reasonable agreement between the theoretical prediction for t LO and that measured experimentally. In particular, experiments confirm the result that the size ratio π 2 = R/L does not affect the onset of lift-off. We also found that the value of t LO is insensitive to the type of chain used (ball chains and link chains give the same results).

However, we note that the onset of lift-off is difficult to assess accurately in experiments owing to its slow initial dynamics (to be discussed in detail in § 5); it is therefore natural that experimental values of t LO are slightly larger that theoretically predicted (though they remain within 10% of the theoretical value). Furthermore, the favorable agreement between experiments and numerical solution of the full model, which neglects any source of dissipation, shown in figure 2a suggests friction cannot be the cause of the discrepancy.

Behaviour just after lift-off

We now turn our attention to the early phase of the motion, just after liftoff. In this phase, our experiments and numerical simulations show that the chain 'balloons' off the pulley (see figure 1b). For simplicity, we shall henceforth neglect 12)) is verified within 10% (the solid curve corresponds to t LO 1.1 L/g) . In these experiments, a = g, i.e. π 3 = 1.

the role of gravity, i.e. we set π 3 = 0. This analysis is conducted hand in hand with numerical simulations for π 3 = 0, but these results should not be compared with experiments (for which π 3 = 1). In Appendix A we consider how the results of this section change when π 3 = 0; this analysis shows that the presence of gravity makes quantitative, rather than qualitative, changes to the motion around lift-off.

5.1. Velocity after lift-off. The numerical simulations and experiments both suggest that the portion of the chain that is freely hanging at the instant of first lift-off (i.e. 0 ≤ s < s C (t LO )) remains straight (within numerical accuracy) in the immediate aftermath of lift-off. We therefore attempt to repeat the calculation of §3 but this time we treat the acceleration in this straight portion, vhang (t), as an unknown. Motivated by numerical observations, we assume that the length of the straight portion of the chain is precisely the hanging length of the chain at t LO , namely s C (t LO ), which we denote by s LO C as in [START_REF] Jawed | Coiling of elastic rods on rigid substrates[END_REF]. Setting g = 0 in (4), together with φ = 0 (in the straight, hanging part), and integrating we find that ( 14)

T = ρ vhang s, 0 ≤ s ≤ s LO C
where the length of the straight portion s LO C = L-1 2 at 2 LO . To close the problem, we note that at the end of this straight portion, the radius of curvature must change to some finite value (by construction), and further that there is no reaction force (since The speed of the hanging part of the chain for g = 0 from numerical simulations (dotted blue curve) compared to the imposed velocity at the pulling end, v = at, (dashed line) and the prediction of eq. 16 (solid orange curve). Inset: Error δ between the estimate (16) for v hang and the numerical data, shows that the expansion for tt LO t LO in ( 17) is correct at O(tt LO ) 2 . (b) ∆s denotes the length of chain that has lost contact with the pulley for t > t LO . The corresponding angle is θ * representing a region on the pulley of length Rθ * . The excess length ∆L is defined as the difference between these two quantities ∆L = ∆ -Rθ * (c) The value of the relative excess length ∆L/L (solid) as predicted by equation ( 19) compares favourably with numerical data (dots); in particular the scaling prediction ∆L/L ∼ (tt LO ) 3 is borne out by the data. at this point the chain is no longer in contact with the pulley). Substituting these results into (5), we then have that T (s LO C , t) = ρ v 2 hang and hence, after combining with [START_REF] Audoly | A discrete geometric approach for simulating the dynamics of thin viscous threads[END_REF], that [START_REF] Hanna | Jump conditions for strings and sheets from an action principle[END_REF] vhang

(t) = v 2 hang (t) s LO C .
Integrating [START_REF] Hanna | Jump conditions for strings and sheets from an action principle[END_REF] subject to the initial condition v hang (t LO ) = at LO , we find that

(16) v hang (t) = 2 3 L 2t LO -t = a t LO 2 - t t LO -1
. This simple approximation predicts that the velocity of the hanging portion of the chain diverges as t → 2t LO . However, the argument above only holds in the immediate aftermath of lift-off, tt LO t LO . In the simulations, the reversal of curvature actually occurs at a time t * ≈ 1.365 L/a ≈ 1.67t LO (see figure 7b-c), which is earlier than that predicted by [START_REF] Schagerl | On the paradox of the free falling folded chain[END_REF].

The Taylor expansion of ( 16) for t -

t LO t LO reads (17) v hang (t) = a t LO 1 + t -t LO t LO + t -t LO t LO 2 + O t -t LO t LO 3 .
The first two terms in in the square bracket combine to give at, which is the imposed pulling velocity: the correction to the imposed constant acceleration is given by the following term and occurs at order (tt LO ) 2 . In figure 4a, we observe that our numerical results for the velocity are in good agreement with the prediction of eq. ( 16) for times close to t LO but then deviate with an error δ that is cubic in tt LO . This cubic error confirms that the first correction to the constant acceleration motion, namely the term of order (tt LO ) 2 in ( 17), is correct. By differentiating ( 16) with respect to time, we obtain an expression for the excess acceleration of the free part of the chain

(18) ∆a = vhang -a = a t 2 LO (2t LO -t) 2 -1 ,
which is positive for all t > t LO . Thus, even though the free end is not subject to external forces, it accelerates faster than the end that is actually subject to the imposed (constant) acceleration.

A quantity of interest is the arclength of the region over which the chain has lost contact with the pulley for t > t LO , which we denote by ∆s (see figure 4b). The above calculation does not give us enough information to determine this quantity. However, the result for v hang , [START_REF] Schagerl | On the paradox of the free falling folded chain[END_REF], can be used to determine the excess length of chain that has lost contact with the pulley, i.e. the difference between the length of chain that has lifted off the pulley, ∆s(t), and the arc length of the pulley from which it has lifted off, Rθ * (t), with θ * (t) the angle subtended between the two contact points C and C * (see figure 4b). Denoting this quantity by ∆L(t) = ∆s(t) -Rθ * (t) we find that

(19) ∆L(t) = t tLO [v hang (t ) -at ] dt = 2L 3 log t LO 2t LO -t - a 2 (t 2 -t 2 LO ).
We emphasize that this calculation does not allow us to determine ∆s and Rθ * separately, but only their difference. The Taylor series of ∆L(t) for t -

t LO t LO reveals that (20) ∆L(t) ≈ a (t -t LO ) 3 3t LO ,
i.e. the excess length grows only relatively slowly after the start of lift-off. Note that the appearance of a leading-order behaviour at order (tt LO ) 3 here is consistent with our earlier finding that the correction to the speed of motion is at order (tt LO ) 2 in ( 17), since we have integrated this quantity in time. Comparison of the predictions of this analysis, ( 16) and [START_REF] Moulton | The elastic secrets of the chameleon tongue[END_REF], with the numerical solution of the fully nonlinear problem (see figure 4) shows that these expressions are asymptotically correct immediately after lift-off, tt LO t LO , and validates the underlying assumption (that the hanging portion of the chain simply accelerates vertically upwards). 5.2. Characteristics. We now proceed to explain the size of the region of liftoff as a function of time. We show that the point of contact C * (t), as defined in fig 4b, follows a characteristic of the wave equation describing the evolution of the small perturbations to the motion of the string. In other words, the motion of C * (t) can be viewed as a traveling front produced by the initial lift-off at t LO . To do so, we now consider the length of chain ∆s that makes up the 'ballooning' arch forming after lift-off. This length is defined to be ∆s(t) = s * cs c , where s c and s * c correspond to the arc-length coordinate of the contact points C and C * , respectively (see figure 4b).

For times close to t LO , lift-off takes place but it is mild; as a result, this lift-off can be described by (i) extending beyond t LO the solution maintaining contact with the upper half of the pulley (base solution) and (ii) perturbing it to remove any negative contact pressure. The equations governing the perturbation are obtained by a standard linearization, and are linear wave equations. In these wave equations, the wave speed c = T /ρ is both time and space-dependent (recall that T = T (s, t)). Assuming that the tension remains close to its value prior to lift-off, i.e. T ≈ ρ as as derived in eq.( 8), we have c = √ as, and the characteristics of the wave equation satisfy ( 21) ds dt = ± √ as.

yielding the expansion waves depicted in figure 5.

When lift-off occurs, the chain is locally freed from the pulley at the point s LO C and this information propagates within the envelope formed by the two characteristics that depart from this point in the (t, s) plot. Within this region, the tension in the chain is altered, i.e. the tension of the true solution does not match exactly that predicted by the base solution that ignores lift-off; beyond this region, information can only propagate at the wave speed at the edges, which is, by construction, given by the tension just before lift off. The limited propagation speed of information about lift-off results in the formation of an arch from the point C to a second contact point C * (see figure 4b). We hypothesize that the length of the arch ∆s(t) = s * cs c is simply the width of the envelope formed by the two characteristics that originate from C at t = t LO (see figure 5). Deriving ∆s from eq.( 21) yields [START_REF] Sm Deban | Extremely high-power tongue projection in plethodontid salamanders[END_REF] ∆s = 2 2 3 √ aL(tt LO ).

We use numerical simulations to test the estimate [START_REF] Sm Deban | Extremely high-power tongue projection in plethodontid salamanders[END_REF]. As shown in figure 6a the agreement between the two is asymptotically correct immediately after lift-off (with the correct prefactor). This indicates that the propagation of lift-off front is indeed limited by the rate at which the perturbations can travel through the string. This argument correctly accounts for the short times dynamics, ∆s ∼ (tt LO ). We also note that θ * , the angle from C to C * defined in figure 5b, follows the same scaling law and agrees well with equation [START_REF] Sm Deban | Extremely high-power tongue projection in plethodontid salamanders[END_REF]. Indeed, recall that ∆L = ∆s -Rθ * ; for t-t LO 1, we have shown ∆L ∼ (t-t LO ) 3 in equation ( 20) and ∆s ∼ (t-t LO ) above, it follows that ∆s and R θ * must be equal to leading order in (tt LO ), i.e. θ * ∼ ∆s/R as checked numerically in figure 6b.

Snapping

We now describe the kinematics of the free end during snapping. Let φ(s, t) be the angle between the tangent t and the upward pointing vertical axis. We define the curvature κ(s, t) = ∂φ(s, t)/∂s (recall that the free end corresponds to s = 0). We define snapping as a sudden change in maximal curvature of the chain as depicted in figure 7a-b, which shows the time evolution of the largest curvature of the chain, κ M . We observe that the arc-length position s M at which this maximum curvature is observed travels towards the free end of the chain as snapping approaches, as shown in figure 7b. During snapping, κ M has a singular behavior as well: the curvature strongly increases and changes sign, as shown in the last three configurations in figure 7a and in figure 7b. Note that for real ropes and the mesh size is ∆s = 6.3 × 10 -3 . Imposed acceleration at endpoint is a = 0.5, g = 0, R = 1 and L = 4π) this singularity will be regularized by their finite bending stiffness 1 . Therefore, the singular behavior predicted by our string model may be seen as the 'outer solution' of a regularized model. Note that snapping is associated with a rapid increase of the free end velocity as illustrated in figure 7c. These features are similar to what occurs in a cracking whip [START_REF] Goriely | Shape of a cracking whip[END_REF] and a falling chain [START_REF] Schagerl | On the paradox of the free falling folded chain[END_REF][START_REF] Tomaszewski | Dynamics of ropes and chains: I. the fall of the folded chain[END_REF][START_REF] Tomaszewski | The motion of a freely falling chain tip[END_REF]. Let us denote by t * the time of snapping, defined as the time where the curvature diverges near the free end. For optical accuracy, we analyze the snapping dynamics based on simulations having a relatively large value of π 2 = R/L = 1 2π . This allows us to use a fine and uniform spatial discretization of the string. To ease the comparison between this specific simulation and the rest of our results, we introduce a non-dimensionalisation with respect to the space and time scales of the problem, namely L and L/a. Dimensionless quantites are denoted by a bar, such as t * = t * / √ a L. The dimensionless snapping time t * is a function of π 2 and π 3 ; in the forthcoming analysis, we focus on the particular case π 2 = 1/(2π) and π 3 = 0.

To measure the divergence of curvature from the simulations, we use a curvature norm that is intended to capture the largest value of curvature, ( 23)

K(t) = 1 0 κ(s, t ) p ds 1 p-1
, where p ≥ 2 is an integer. The value of the integer p that is used results from a trade-off: one should use values of p as large as possible, and K will then accurately capture the maximum curvature; on the other hand, if p is too large then K fluctuates significantly as a result of discretization errors. In the following, we take p = 6; we have verified that the results are unchanged for p = 8.

We begin by assuming that as the singularity at t = t * is approached, the curvature diverges according to a power law κ ∼ (t *t) -α , for some exponent α to be determined, on a region of size ∼ 1/κ. The region of divergence contributes an amount ∼ (t *t) -(p-1) α to the integral in equation ( 23): as soon as α(p -1) > 0, this contribution makes the integral diverge at the time of snapping t * and t * can be identified from a numerical plot of t as a function of time-the condition α(p-1) > 0 is indeed satisfied as p = 6 and α > 0, see below. One can check that the quantity K(t) diverges as K ∼ (t *t) -α : a numerical plot of K versus t yields the exponent α as well, independently of the particular value of p chosen in the definition of K. The points collapse onto a straight line, aside from a thin band of width ≈ 15∆t where the large curvature is not well represented by the discretization. A linear fit in figure 8a yields both the snapping time t * (where the fitting line crosses the taxis) and the exponent α (the reciprocal of the slope). This fit yields a value of the exponent α = 0.659 which is numerically close to 2/3 -we do not know whether this is a coincidence -and t * = 1.415. The latter is close to (but less than) the singular time 2t LO a/L = 2 2/3 ≈ 1.63 found by extrapolating the asymptotic behavior in the immediate aftermath after lift-off, see equation [START_REF] Schagerl | On the paradox of the free falling folded chain[END_REF]. These numerical values of t * and α have been obtained in the particular case π 2 = 1/(2π) and π 3 = 0. We suspect, however, that the exponent α can be explained by a boundary-layer theory and that it is actually independent on π 2 and π 3 . Unfortunately, we can offer no proof for this statement. As observed in the experiments (see §2), the string remains straight in the simulations along an interval of length (t) comprising the free end, at all times until t * . This straight portion of the string is apparent in figures 7a and 8c. It defines the region denoted by (I), which is adjacent to the region (II) of large curvature (see Fig 2c). The dimensionless length = /L is found numerically to scale like ∼ (t *t) β , with β = 1.08 ± 0.2 (fit not shown). The numerical accuracy on this exponent β is not as good as that of α as the determination of the straight region is sensitive to numerical noise. It is clear from our numerical data, however, that the exponent β is larger than α, meaning that the length of the straight region near the free end vanishes more quickly than the minimum radius of curvature as t → t * , t < t * .

Next, we proceed to use our analysis to uncover a self-similar behavior in the curvature profile. Anticipating that the point of maximal curvature plays a central role we introduce a weighing to capture its position: (25) s (t) = In figure 8d, we plot the rescaled curvature κ(s, t)/K(t) as a function of the rescaled arc-length s and obtain a good collapse. Note that the unscaled maximum curvature K varies by a factor ∼ 5, from approximately 65 to 308, between the first and last snapshot in this series. Therefore, the collapse shows that the curvature distribution is self-similar close to the snapping time t * : consistently with our initial scaling assumption, the curvature scales like κ ∼ (t *t) -α in a region of size ss ∼ (t *t) α . Note that this analysis on the behavior of the chain prior to snapping, so that t < t * .

In summary, our analysis shows that the 'whipping' singularity may characterized by a self-similar solution, whose scaling behaviours have been identified numerically.

Conclusions

In this paper, we have considered a degenerate version of Atwood's machine, in which a single mass pulls a chain around a pulley. In stark contrast with the apparent simplicity of the setup we have found that the dynamics is extremely rich, successively displaying a ballooning instability of the chain and a snapping motion of its free end, reminiscent of what is seen in a cracking whip. We have shown that the chain dynamics is well captured by a frictionless string model and that some of its features may be captured by simple arguments. In particular we have shown that the geometry of the problem, through the imposed rotation of the chain around the pulley, is key to understanding how the end of the chain is able to 'beat' the free-fall that drives its motion.

Our observations can be used to speculate on the peculiar hunting techniques of a variety of amphibians. Indeed, instead of throwing their tongue in a straight motion (as observed in chameleons [START_REF] Cv Anderson | Ballistic tongue projection in chameleons maintains high performance at low temperature[END_REF][START_REF] Moulton | The elastic secrets of the chameleon tongue[END_REF]), certain species of toads [START_REF] Sm Deban | Thermal effects on the dynamics and motor control of ballistic prey capture in toads: maintaining high performance at low temperature[END_REF] and salamanders [START_REF] Sm Deban | Extremely high-power tongue projection in plethodontid salamanders[END_REF] adopt an unfurling tongue strategy. Of course, the reasons for such a mechanism are many and varied but we believe that the increase of tip velocity observed in the case of a chain is likely to reappear in problems involving a finite bending stiffness. It is then natural to wonder whether this geometrical amplification of acceleration may be used by these amphibians to allow them to maximize their chances of capturing a prey?

Finally, we note that while we have been able to rationalize some of the observations from experiment and simulation, others remain elusive. For example, predicting the shape of the ballooning region either with simple arguments, or preferably analytically, remains beyond our reach. Difficulties in doing so arise from the fact that the 'base solution' of the problem is unsteady (since a > 0). An interesting avenue of research would be to explore the case where the chain is pulled at constant speed thus without any acceleration, potentially allowing for analytical developments. Similarly, there is hope that the self-similar exponent α and β identified in the previous section can be explained by some boundary layer theory in future work.

The form of (28) appears to be substantially different to that in the absence of gravity, [START_REF] Schagerl | On the paradox of the free falling folded chain[END_REF]. However, the qualitative behaviour is, in fact very similar: v hang (t) is an increasing function of t > t LO and has a singularity at This cubic growth of the excess length with time following lift-off echoes that found in the absence of gravity, given by [START_REF] Moulton | The elastic secrets of the chameleon tongue[END_REF]. As already seen, the only significant difference is in the prefactor.

Figure 1 .

 1 Figure 1. Experiments using ball chains. (a) Experimental setup: the free fall of a mass M forces a chain with initial hanging length L = 54 cm to slide around a glass cylinder of radius R = 4 cm. The initial length L of the hanging part denotes the section of chain from the free end s = 0 to the first contact point with the cylinder C before the mass is dropped (which occurs at t = 0). The vectors (t, n) denote the tangent and the normal of the chain, respectively. (b) Time sequence (∆t = 2.5 ms) of the chain dynamics successively lifting up from pulley (from t = t LO onwards) showing that a ballooning arch develops before the chain eventually 'snaps' at a time t = t * (reverses its curvature dramatically).

Figure 2 .

 2 Figure 2. Numerical simulations: (a) Comparison between the chain shape observed experimentally (images in the background) and the prediction of numerical simulations (overlaid curve -green online). Here there are no adjustable parameters (L = 54 cm, R = 4 cm, time between two frames ∆t = 0.9 ms). (b) At each instant before snapping, the chain may be divided into four zones. (c) Plot of the curvature κ (dashed curve) and the chain speed v (solid curve) as functions of the arc length. (d) A numerical reconstruction of the tension T within the chain shows that it varies linearly with arc length in regions (I), (III) & (IV). Region (II) bridges regions (I) and (III), each of which is characterized by a different acceleration. For comparison, the dashed line shows the tension as a function of arc length in the mathematical solution with uniform and constant acceleration a where lift-off is prevented (this artificial solution would give rise to a negative contact pressure).

Figure 3 .

 3 Figure 3. Experimental values of the time at which lift off occurs, t LO , plotted as a function of the chain length L. This data, gathered with pulleys of radius ranging from R = 2.5 cm to R = 7.5 cm and two types of chain, collapses on a single master curve. The prediction derived in §3. (see eq. (12)) is verified within 10% (the solid curve corresponds to t LO 1.1 L/g) . In these experiments, a = g, i.e. π 3 = 1.

  -Rθ *

Figure 4 .

 4 Figure 4. Behavior just after lift-off π 3 = 0 (the simulation timestep is ∆t = 15 × 10 -4 and the mesh size is ∆s = 6.3 × 10 -3 .Imposed acceleration at endpoint is a = 0.5, g = 0, R = 1 and L = 4π). (a) The speed of the hanging part of the chain for g = 0 from numerical simulations (dotted blue curve) compared to the imposed velocity at the pulling end, v = at, (dashed line) and the prediction of eq. 16 (solid orange curve). Inset: Error δ between the estimate (16) for v hang and the numerical data, shows that the expansion for tt LO t LO in (17) is correct at O(tt LO ) 2 . (b) ∆s denotes the length of chain that has lost contact with the pulley for t > t LO . The corresponding angle is θ * representing a region on the pulley of length Rθ * . The excess length ∆L is defined as the difference between these two quantities ∆L = ∆ -Rθ * (c) The value of the relative excess length ∆L/L (solid) as predicted by equation (19) compares favourably with numerical data (dots); in particular the scaling prediction ∆L/L ∼ (tt LO ) 3 is borne out by the data.

C a r aFigure 5 .

 5 Figure 5. The parameter space (t a/L, s/L) is mapped with the characteristics of the system, color coded as functions of the direction of propagation (given by the sign of ds/dt in this plot). The time evolution of the position of the contact point C, obtained analytically before t LO and numerically for t > t LO , is shown by solid blue curve. The envelope of characteristics emitted from C at t = t LO is highlighted by the shaded area. (The simulation timestep is ∆t = 15 × 10 -4 and the mesh size is ∆s = 6.3 × 10 -3 . Imposed acceleration at endpoint is a = 0.5, g = 0, R = 1 and L = 4π).

Figure 6 .Figure 7 .

 67 Figure 6. Arch structure for π 3 = 0. (a) The evolution of ∆s/L with time following lift-off determined numerically (blue dots) and compared to the theoretical expression in (22) (orange solid line). (b) Test of the predition θ * ∼ ∆s/R derived at the end of §5. In both cases the numerical data corresponds to a simulation with timestep ∆t = 15 × 10 -4 and the mesh size ∆s = 6.3 × 10 -3 . Imposed acceleration at endpoint is a = 0.5, g = 0, R = 1 and L = 4π.

Figure 8 .

 8 Figure 8. Searching for self-similar solutions in the snapping (for this particular simulation, the linear mass is ρ = 5 and the radius of pulley is R = 1 and L = 2π such that π 2 = R/L = 1/(2π). There is no gravity and no bending rigidity. The simulation timestep is ∆t = 2 × 10 -4 and the mesh size is ∆s = 3.14 × 10 -3 . Imposed acceleration at endpoint is a = 0.5). (a) Identifying the singularity exponent α for the curvature and the critical time t * using the quantity K(t) defined in (23). From our numerical simulations we find that (α, t * ) = (0.659, 1.41531) (red dashed line). (b) These values may be checked in a standard log-log plot. The wells are numerical artifacts: they correspond to values of |t * -t| as small as a few simulation steps, implying that the curvature is very peaked and prone to discretization errors. (c) Configurations right before and after the snapping time, 1.405 ≤ t ≤ 1.42 (t * ≈ 1.415): darker configurations correspond to times closer to snapping and reference configurations are denoted by the symbols ( ), ( ) and ( ) (d) Collapse of the curvature in rescaled variables (s, κ/K) using the same shading convention for 1.405 ≤ t ≤ 1.414 (before snapping occurs).

(

  The snapping time t * and the exponent α entering in the power law K(t) ∼ |tt * | -α are best determined from the simulations by plotting the quantity

	(24)	d(ln K) dt	-1	∼	t	* -t α
	as a function of t, see figure 8a.

  By design, s = 0 lies in the centre of the region with high curvature. In this definition, the offset s (t) allows us to ignore the straight region entirely (which shrinks according to a different exponent). We consider a set of simulation snapshots such that 1.405 ≤ t ≤ 1.414, i.e. such that 0.001 ≤ t

			1 0 s κ p (s, t) ds 1 0 κ p (s, t) ds	.
	By design, the weighting is concentrated in the region of large curvature: s (t)
	yields the typical value of s at time t in the region (II) where the curvature diverges.
	Finally we rescale the arc-length parameter to
	(26)	s =	s -s (t) 1/K(t)	,

*t ≤ 0.01. This corresponds to the configurations bounded by the symbols ( ) and ( ) in figure

8b-c

, with darker plots corresponding to times closer to the snapping time t * .

  LO ) 2 + O(tt LO )3 , so that the change in the pulling velocity occurs at O(tt LO ) 2 , as in the case without gravity (see figure4a).A.2. The excess length ∆L. From the expression (28) for the acceleration of the hanging portion of the chain, we find that the excess length absorbed by the lifted off portion is

	30)	t = t LO +	s LO C g	α.
	Furthermore, a Taylor expansion of (28) reveals that
	(31)	v hang = at + (t -t ∆L = -s LO a 2 t LO s LO C C log sinh α -g s LO C sinh α (t -t LO )	+	a 2	(t 2 LO -t 2 )
		= a + g

3 (tt LO ) 3 3t LO + O[(tt LO ) 4 ].

(32)

For chains made up of rigid links, this singularity is regularized too, by the contact between successive chain links.

Acknowledgment

This publication is based in part upon work partly funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 637334, GADGET to DV) and the grant KUK-C1-013-04, made by the King Abdullah University of Science and Technology (KAUST).

Appendix A. The effect of gravity immediately after lift-off For simplicity, the main portion of the paper focusses on the problem in the absence of gravity. In this Appendix, we show how the results derived in §55.1 for the behaviour immediately following lift-off are altered once gravity is included. These results confirm that, in this case at least, gravity affects the results quantitatively, rather than qualitatively.

A.1. The unknown chain acceleration. Immediately after lift-off, the free end of the chain accelerates at an unknown rate vhang . To determine this acceleration, we repeat the argument of § 55.1 incorporating gravity: the normal force arising from the tension at the material position that was at C when lift-off started (ρ ( vhang + g)s LO C κ) is equated with the force arising from the centripetal acceleration (ρ v 2 hang κ) for some undetermined curvature κ. We then have that ( 15