Marc Flandreau

Stefano Ugolini

The Crisis of 1866

Keywords: E42, E58, F33, N13 Financial crises, lending of last resort, financial stability, international currencies

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The so-called Overend-Gurney crisis of 1866 was a major turning point in British monetary and financial history. Following the emergence of a modern financial system in the 18 th century, the country had been plagued by recurrent panics. In every decade of the 19 th century until the crisis of 1866, there had been a point when, following a phase of credit expansion and "speculation", market conditions had deteriorated, and the money market seized. 1866 was just one more run in a series that included, among the most infamous, 1825, 1837-39, 1847, 1857. Yet the run that began in May 1866, was to be the last of that variety to occur in the United Kingdom until the outbreak of WWI. The only two relevant episodes of financial stress that occurred between 1866 and 1914 (1878 and the "Baring crisis" of 1890) did not have much in common with previous financial shocks neither in virulence nor in nature, since -unlike the previous crises -they were not accompanied by dislocations of the money market.

This surprising feat has fascinated economists and historians alike, and they have sought to understand its reasons. All eyes have been rightly turned towards the Bank of England, but what exactly were the actions that planted the seeds of financial stability remains disputed. [START_REF] Fetter | Development of British Monetary Orthodoxy 1797-1875[END_REF] characterized the 1870s as the era that saw the "victory of the Bagehot principle" during the 1870s [START_REF] Fetter | Development of British Monetary Orthodoxy 1797-1875[END_REF]. According to Giannini (1999) the newly acquired stability of the British financial system owes to the Bank of England's new willingness to undertake lifeboat operations (also see Mahate 1994). According to others, Britain's new financial resilience owed to consolidations in the Bank's macro-policy including its adherence to the Gold Standard (Schwartz 1987(Schwartz , 1995)): Restrictions on the Bank's discretionary power over the issue of high-powered money (the Gold Standard's "rule") would have stabilized the system by improving agents' ability to make forecasts. Still others emphasize microprudential aspects of crisis management, such as the emergence of automatic, anonymous lending on recognizably good collateral. For instance, according to Capie's frosted glass discount window metaphor (Capie 2002, pp. 310-1), the Bank's discount window was raised high enough to examine the quality of the collateral, without revealing the identity of the discounter: The central banker "does not know, nor does he care, who is on the other side of the window. He simply discounts good quality paper or lends on the basis of good collateral".

Other arguments emphasizing micro-prudential features include [START_REF] Calomiris | Banking Crises and the Rules of the Game[END_REF], who argues that the Overend-Gurney crisis served to establish the Bank of England's credibility and ability to act in a fully discretional way. The episode rebutted the principle of "too-big-to-fail" and signalled the Bank's decision to terminate "the put option inherent in the Bank's willingness to accommodate demand". 1 This chapter discusses what really happened in 1866. For one thing, [START_REF] Bagehot | Lombard Street: A Description of the Money Market[END_REF] never advised central bankers to engage into "lifeboat operations". He advised them to provide generous lending -essentially, to do away with credit rationing which had prevailed in previous crises. As recent research has shown, 1866 was indeed a watershed (something Bagehot himself had recognized). In 1866, the Bank had in practice -if not officially -acted as a lender of last resort, abstaining from credit rationing and as a result effectively becoming the place where the crisis was resolved [START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF]. Using new statistical evidence from previous joint research (Flandreau and Ugolini, 2013) this chapter goes one stage further and provides a fuller characterization of the events in 1866 and of the revolution that occurred then: In contrast with Capie, we argue that Britain's actual recipe for financial stability was the Bank of England's adoption of a principle of generous provision of non-anonymous lending. In other words, the Bank's window was fully raised so that the Bank could see the face of the discounter. The Bank was prepared to provide credit only to the extent that it liked what it saw. This meant that the counterparty had to abide by a number of behavioural norms: At the same time the Bank lent generously, it also performed strict monitoring over the banking system and thus protected itself against moral hazard. Tightened supervision and generous lending were the two sides of the new currency.

The remainder of the chapter is organised as follows. Section 1 reviews the structure of the English financial system on the eve of the Overend-Gurney Panic. Section 2 analyses in detail the Bank of England's actions during the crisis. In the light of this evidence, section 3 provides a characterization of Britain's newly formed approach to financial stability. Section 4 emphasizes the international aspects of the new policy adopted in 1866: We argue that the new policy helped entrench sterling as an unrivalled international currency. The conclusion sums up the findings and provides a comparison between the events in 1866 and the Baring Crisis of 1890.

1 Rejection of the too-big-to-fail principle is also a central theme of [START_REF] Capie | The Emergence of the Bank of England as a Mature Central Bank[END_REF]. One paper skeptical about the Bank of England's role in fostering financial stability is [START_REF] Batchelor | The Avoidance of Catastrophe: Two Nineteenth-Century Banking Crises[END_REF]. He argues that after 1866 there was an increase in the amount of information available to the public concerning the quality of collateral held by the banking system.

The Structure of the English Financial System on the Eve of 1866

Since the early modern era, European financial systems had been developing around a particular form of money market instruments: bills of exchange. Bills were negotiable promissory notes bearing multiple guarantees: bound to be paid at maturity by one person (the "acceptor" or insurer) who had agreed to certify the quality of the original debtor (the drawer), they were also secured by the signatures of all the people who had previously held and resold them (the endorsers). The bill market flourished in England during the Industrial Revolution, and specialized intermediaries (the bill brokers) started to emerge at the beginning of the 19 th century.

According to King (1936), the centrality of bill brokers within the system was established following the crisis of 1825 (initially known as "The Panic"). During this crisis, rampant credit rationing by the Bank of England made major London banks -which were heavily invested in bills -experience a serious maturity mismatch, which forced them to suspend payments. Scared by this episode, commercial banks vowed never to find themselves in the same situation again: Instead of keeping all of their assets in bills, the banks started to deposit large amounts of money "on call" with bill brokers, to whom the liquidity risk was shifted. This very episode transformed the English financial system into one which was unlike any other. Bill brokers (or "discount houses", as they came to be known later on) evolved from being brokers to being money market funds, taking deposits from banks and investing them directly (on their own account) in the bill market. In normal times risks were limited, and so were margins, encouraging substantial leverage: According to King (1935), the ratio of total assets to capital hovered around 10 in the mid-19 th century, and this was typically larger than what a "typical" bank would do. This is how Britain's variant of the modern "shadow banking system" was born. Initially, the Bank of England saw favourably an evolution, which was supposed to help it manage the money market at arm's length through the rediscounting system. However, relations gradually deteriorated. According to [START_REF] Wood | English Theories of Central Banking Control 1819-1858[END_REF] this was caused by the Act of 1844, which sanctioned the private company character of the Bank. Encouraged to compete directly on the discount market, the Bank started to see bill brokers as challengers and its own rediscount facilities as a free lunch provided to these challengers.

Figure 1 summarizes the basic features of the English financial system as it had been evolving after the turning points of 1825 and 1844. In this system, credit seekers obtained funding by drawing bills which, once accepted by specialized merchant banks (or "acceptance houses", as they came to be known later on), were sold (or "discounted") on the money market. This was the hunting ground for bill brokers, who sought a profitable re-employment of the funds deposited with them by commercial banks -which, in turn, collected deposits from the general public. As Figure 1 shows, the Bank and the bill brokers competed against one another and it is plausible that this competition was one aspect of the tensions that grew steadily between the Old Lady of Threadneedle Street and the bill brokers -in particular the biggest and most "prestigious" of them all, Overend, Gurney & Co. -setting the stage for the final denouement of May 1866 when the Bank refused to bail out "Overends".

The deteriorating relation between the Bank and the bill brokers (and principally Overends) cannot be solely explained in term of commercial rivalry. Because bill brokers were money market specialists, the Bank of England could, to some extent at least, "outsource" to them the screening of bills and just rely on the guarantee provided by them. Moreover, the generosity of bill brokers set the conditions of the money market: Attempts by the Bank of England to, say, tighten money market conditions could be frustrated by bill brokers' expansionary policies. In other words, the bill brokers had both a prudential role and a monetary policy role.

The Bank of England's need to make its rate "effective" in order to protect the gold reserve (a big theme of the 1850s) ran against this circumstance, and contained seeds of discord.

The conflict between the Bank of England and Overend, Gurney & Co. became overt in the aftermath of the crisis of 1857. During this panic, bill brokers resorted extensively to the Bank's rediscounting facilities, and Bank's directors started to think that the inherent inconsistency of the system had reached breaking point. By the end of November 1857, the Bank held £1.2m in bills rediscounted to Overends, which amounted to 3.37% of its total portfolio of commercial securities2 . This number might look unimpressive, but it reflected a broader patterns among all bill brokers, and this is what was perceived as unbearable. The argument was that, had the bill brokers refrained earlier from reckless lending, the crisis would have been avoided. Bill brokers should have reduced leverage by keeping higher cash reserves and it was their automatic access to the Bank's discount window that created moral hazard. To provide incentives in this direction, in March 1858 it issued a public statement which announced its willingness to shut down rediscounting facilities to bill brokers, except in exceptional circumstances. The press personalized the move as directed against Overend, Gurney & Co., and commented unfavourably (King 1936, pp. 202-3). Overends felt strong enough to indulge into retaliatory action: in April 1860, they orchestrated a mini-run on the Bank of England by building a position there and suddenly withdrawing £3m banknotes, thus creating disturbances in the money market. However, according to Bagehot (1873, p. 299) the attempt failed to achieve its probable objective of strengthening the bill brokers' case by exposing the vulnerability of the Bank of England: Instead, he says, the attack on the Bank's reserve made Overends unpopular with the City.

Bank of England Lending during the Overend-Gurney Crisis

The aftermath of the crisis of 1857 saw a considerable expansion of discount houses and joint-stock banks, which King (1936, p. 217) associates with the Companies Act. Another relevant aspect of the boom was the internationalization of the bill market (Hughes 1960).

The massive expansion in international trade during the 1850s conspired London's comparative advantage in trade finance to encourage the multiplication of originators (merchant or joint stock banks) and money market funds (bill brokers). Those with foreign connections and a London base could make large profits because they could take advantage of local information and London facilities. Figures provided by Roberts (1992) for Schroders, a merchant bank with contacts with the Continent and the US, suggest that in the early 1860s the profitability of acceptances was enormous -between 4 and 6% of the amounts accepted.

Figures quickly declined afterwards, possibly reflecting the effects of competition. 3 The result was internationalization of the London money market -sterling acceptances becoming a funding and investing vehicle of choice.

As was bound to be the case, not all those joining the craze were prudent: Many sought to enhance returns by investing short-term resources in long term or illiquid resources. Overends bucked the trend, and in the early 1860s they got increasingly invested in speculative-grade bills: as one investment after the other failed, they ended up with non-performing assets and 3 Figure computed using data provided by Roberts (1992,p p. 527 and 532). Acceptances were for a few months (typically 90 days), so that to compute the annual rate of return, one must compare annual revenues from acceptances and acceptances outstanding on the balance sheet at the end of the year (assuming they were essentially rolled over). Data for the early 1860s also include consignments made by Schroders: Controlling for these on the basis of later proportions gives the lower bound reported. Note that technically, acceptances appeared for the same amount on the asset and liabilities side of the balance sheet, and it was understood that some capital had to be set aside to meet contingencies. their liquidity declined. 4 In an attempt to attract fresh capital, the partnership was transformed into a limited liability company and floated on the stock exchange in 1865. Bagehot (1873, pp. 274-5) saw this move as the real cause of the firm's eventual fall, because henceforth losses became of public knowledge and its reputation got irremediably tarnished. Further shocks included a long period of high interest rates in London, which exacerbated the company's refinancing difficulties (figure 2), the stock market collapse of late 1865 and early 1866, and the failure of a number of customers. Calls to other bankers were unsuccessful and, in a last ditch effort, the Bank of England was approached in early May, but the "Governor took the view that the Bank could not assist one concern unless it was prepared to also assist the many others which were known to be in similar plight."5 This was decided after a confidential report was commissioned to investigate whether assistance by the Bank or a consortium of London commercial banks was an option. At 3:30 p.m., May 10, 1866, Overend Gurney & Co. suspended payments. The immediate reaction was described as the "wildest panic". Contemporaries compared the event to an "earthquake". From King (1936, p. 243): It is "impossible to describe the terror and anxiety which took possession of men's minds for the remainder of that and the whole of the succeeding day."

Figure 2 about here How did the Bank react to the panic? In the months preceding the crisis, market interest rates had almost constantly coincided with the Bank's discount rate. This was evidence of money market tension, which the Bank was nonetheless accommodating [START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF]. After the announcement of Overend's suspension, the official discount rate was raised from 7% to 9% and then 10%, but again, the market rate never exceeded this threshold (figure 2). In other words, the Bank continued to meet the considerably increased demand for cash. Both channels through which the Bank provided liquidity to the banking system -discounts of bills and advances on securities (including "parcels", i.e. bundles, of bills) -were heavily resorted to, and very few demands for cash were rejected (figure 3). In other words, the Bank was not losing its cool and continued to inject liquidity in the system.

Who was coming to the Bank's standing facilities? There comes an interesting finding: In spite of the official ban of March 1858, bill brokers dominated the stage. They were, by far, the biggest users of the discount window (figure 4a) and they resorted heavily to the advance facility -where, however, commercial banks held the lion's share (figure 4b). The notable feature of the episode is that bill brokers and commercial banks were not usual customers of the Bank, which in "normal" times used to have only merchant banks and trading houses coming to its discount window (Flandreau and Ugolini, 2013). 6 Figures 3,4a, and 4b about here What were applicants bringing to the Bank in exchange for cash? The composition of discounts, for which precise information is available, provides some elements of answer. The broad answer is: The same material as usual. If we sort out the paper discounted by the Bank in May 1866 according to acceptors (i.e. the firms which, after having been drawn upon, had underwritten the bills and thus bore the "initial" responsibility) and then compare it to the situation one year earlier, we get a very stable composition of the underlying material. To the extent that acceptors had to be rated for their paper to be recognized by the Bank of England, this implies that the underlying quality of the material they accepted was very stable. Tables 1a-b illustrate this. They give the Bank's twenty-five biggest exposures to acceptors (amounting to 39.5% of the total) in May 1866 as well as one year before the crisis. The shares and ranking of acceptors did not change considerably. Paper that was discounted by the Bank in normal times could be expected to be discounted during crises.

Tables 1a-b about here

To summarize, it appears that when the panic erupted, the Bank merely continued to do its "usual" business -although on a much grander scale. It kept lending, only it did much more and met demand for liquidity from all sides (despite the 1858 ban of bill brokers from the discount window). Next, underlying this "continuity" of operation, the nature (and thus quality) of the instruments considered eligible for refinancing operations did not change during the crisis. Guarantors (acceptors) deemed reliable before the crisis remained so during the crisis. In fact, as we also found, even in those cases in which the guarantee turned out to be dubious later (when the acceptor failed as a result of the crisis, as in the case of the Agra & 6 It is interesting to note that following its going public and diversification of its business into a financial conglomerate in the 1860s, Overend, Gurney & Co. had no longer been considered by the Bank as a "bill broker" -its account being transferred to the commercial banks' ledger since 1865 (Bank of England Archive, C24/1). This means that, already by this time, Overend was formally outside the scope of the 1858 rule, implying that the reason for refusing support had nothing to do with the rule itself.

Masterman Bank or the Consolidated Bank), the Bank of England adhered to the principle whereby it sustained its earlier policy. This supports the notion that a key aspect of crisis lending was indeed the question of the instrument upon which it lent, perhaps more than the identity of those to whom it lent. Indeed, it turns out that, among the discounters who received the biggest volumes of Bank's liquidity, were some of the main casualties of the crisis (such as Agra & Masterman or the Bank of London). At first sight, this is consistent with Capie's idea that the important question for lending of last resort is "What do you accept?" -not "From whom?". However, as the next section shows, the "what" and "from whom" issues are more intertwined than a superficial reading of the crisis of 1866 would suggest.

The Raised Eyebrow: From Lending of Last Resort to Banking Supervision

Although generally originated in the course of commercial transactions (such as for the finance of physical commodity shipping), bills of exchange were not backed by a physical security. In case of default of the acceptor, the holder of the bill had the right to turn to previous endorsers, but in no case could they seize (say) the bales of cotton collateral that might have been mentioned on the bill.7 Hence, the "value" of a bill of exchange consisted of the names written on it and of those names alone. Exposure was exposure to names, and this meant that -willy-nilly -the Bank actually had to know and to care about who was on the other side of the discount window: "what" and "whom" were the two sides of the same coin.

This helps explain the sophisticate system the Bank of England developed in order to monitor discounting risks. First, not anybody could be a discounter. This was a privilege and admittance in the discounters' list required being presented by other members of the club and provide material guarantees. 8 Second, risk management took the shape of a system of ledgers that permitted a real time control of exposure: Each bill that the Bank took in portfolio gave rise to two entries -one for the acceptor who had underwritten the bill, and one for the discounter who had presented it. 9 By examining its ledgers, the Bank could see at a glance its exposure to any given signature. The quality of acceptors was periodically reviewed and recorded in so-called "rating books". The evidence from the rating books also suggests that there were thresholds for exposure to any single risk [START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF].

The arrangement, which if we are to believe ledgers expanded after 1844, allowed the Bank to implement a close monitoring of the financial system. This monitoring was hardly anonymous. It enabled the Bank to keep a close eye on both the origination and the distribution of bills of exchange across the system (Flandreau and Ugolini, 2013). Critically, this allowed the Bank to observe when potentially speculative positions were in the course of being built by abnormal drawing or accepting of bills. Suppose for instance that some acceptor relaxed its standards and began accepting more paper than its capital permitted. The signature, thus indiscriminately thrown upon the market would have flowed back to the Bank of England, who would have immediately noticed the problem. For instance, in 1890 the Bank realized that Barings were getting into trouble long before the crisis, because the amount of Baring-accepted bills flowing into its portfolio through third-party discounts had become unusually large. The result were exchanges whereby the Bank of England pressured Barings to fall into line. 10 The strategy may not have worked, but such monitoring made sure that the Bank would not be caught sleeping at the wheel (more on this in the conclusion).

This sheds light on the Bank's attitude towards the financial system and in particular, towards Overend, Gurney & Co. during the crisis of 1866. By controlling its exposure it limited the amount of credit that could be granted to any single individual: It was not enough to present paper on an acceptable signature, one had also to make sure that not too much of this paper would be presented to the Bank. Thus, if for any reason one financial agent had excess exposure to another, it would find itself on the hook for the excess. If such rules were understood by all participants, then no single failure could be a serious threat for the system at large. On the other hand, all those who had behaved "well" (by the Bank of England's standards, which meant -among other things -being properly diversified) were eligible for assistance -assistance of the "ordinary" variety, through the discount window 11 . Under the new "regime" that was definitively established with the Crisis of 1866, the Bank was happy to 10 While in October 1889 the Bank had had in its portfolio no more than £80,000 in bills accepted by Baring Bros, in early October 1890 its exposure had climbed to £500,000. At this date, George J. Goschen, then Chancellor of the Exchequer, noted down in his diary: "Went to the Bank, things queer! Some of the first houses talked about" (Clapham 1944, pp. 327-8). At the peak of the crisis (18 th November 1890), the Bank's exposure to Barings would reach £715,000 (Bank of England Archive, C22/43). The "anomalous" supply of acceptances by Barings is also manifest from the figures reported in Chapman (1984, p. 121). 11 It is crucial to note that the Bank never refused "ordinary" assistance to Overend, Gurney & Co.: because they were probably short of eligible securities, Overends never showed up at the discount window in the period preceding the crash, and only went to Threadneedle Street -when things were already beyond repair -to ask for "extraordinary" assistance (Bank of England Archive, C24/1). discount proportionate amounts of bills guaranteed by all those signatures which were considered eligible in ordinary times (including those whose solvability was at risk of getting into trouble afterwards). Given the detailed knowledge that the Bank had of the money market and the redundant guarantees that it took from acceptors and discounters, given also the extreme division of risks in a system where the biggest exposure remained limited (as seen, the top acceptor was only 6% of the Bank's portfolio) the policy was a very narrowly calculated risk. And this does not even include the strong interest discounters had in doing whatever they could in order not to lose access to the discount window.

Thus, the sophisticate supervisory system put in place by the Bank of England around the mid-19 th century allowed it to extend lending-of-last-resort operations without provoking an increase in moral hazard. As a matter of fact, throughout the 19 th century the amounts-at-risk for the Bank experienced a secular decline, and by the late 1860s they had become basically negligible [START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF]. This, we argue, was Britain's 19 th -century recipe for financial stability. It consisted neither in reliance on abstract market discipline, nor on automatic and anonymous lending of last resort. It had little to do with the Gold Standard, and nothing at all with rescue operations. Rather, it rested on a strict monitoring system or, if one prefers, on a de facto central bank regulation. To this de facto regulation banks and money market participants had to submit, if they wanted to be in good terms with the central bank. Such good terms were valuable when crisis hit and the Bank was the last lender around -literally, the lender of last resort. This qualifies the oral tradition that in the late 19 th century, supervision was minimal and dealt with the "Bank of England Governor's eyebrow". 12 According to this view, it was enough for the Governor of the Bank to raise an eyebrow for bankers to put their house in order. There was, of course, much more to the raised eyebrow than the inconvenience of a stern look. There was detailed information by the Bank, and there was the Bank's power to act by denying discounting facilities. For those needing a reminder, the corpse of Overends could be shown. 13

12 For a printed variant, Withers (1910, p. 56) has an intriguing digression on the Bank of England being the "final arbiter" when the credit of a house came under suspicion. 13 Another interesting case discussed in Flandreau and Ugolini ( 2013) is that of the House of Vagliano, who after a dispute with the Bank of England was excluded from the discount window and had to leave banking (Chatziioannou and Harlaftis 2007, pp. 38-9)

Twin Success: International Aspects of the Crisis of 1866

But the panic of 1866 did not have solely domestic significance. As already indicated, the London market was relied upon by non-residents. London was a place where international supply and demand for short-term credit were cleared 14 . Foreign balances were held and funding was sought, notably -as indicated -trade finance. Figure 5 illustrates this, showing the breakdown of the Bank of England's portfolio of bills according to the geography of drawers: It shows totals drawn by domestic ("inland") vs. foreign drawers (i.e. foreign and colonial). We see the predominance of foreign originations. In May 1866, 65% of the bills discounted by the Bank in May had been drawn by agents residing outside the country. As a result, both the crisis and the Bank of England's actions to handle it were bound to have a significant impact on sterling as an international currency.

The crisis translated into increased risk aversion on behalf of international investors, and contemporaries pointed to what is today known as a "sudden stop" [START_REF] Calvo | Capital Flows and Capital-Market Crises: The Simple Economics of Sudden Stops[END_REF]). The mechanism they had in mind is today known as that of twin crises: In the instance, a credit crisis that was becoming a currency crisis. The sterling confidence crisis that developed was triggered by the surge in credit risk associated with the failure of Overends (Patterson 1870, pp. 227-8 is perhaps the most articulate illustration). 15 Unsure about the prospects of the London market, foreign investors liquidated positions and repatriated balances. The consequence was, in the immediate aftermath of the crisis, a weakening of sterling in spite of the exceptionally high interest rates the Bank maintained. In fact, if we use market rate differentials to compute the "forward" sterling-franc exchange rate and compare it to the gold points (Figure 6), we see that the credibility of sterling which was already under (mild) suspicion before the crisis came under serious doubt during the panic (a "forward" rate below the gold export point suggests that sterling suffered a credibility crisis and indeed, in April, the forward exchange rate raced away from the gold points). As for the spot rate it stayed discouragingly close to the gold export point following the Overend collapse, despite a baffling 6% interest rate differential between London and Paris. 16 Thus the 1866 crisis was also a currency crisis. It was arguably the worst experienced by sterling during the whole "classical" gold standard period , at least if we are to 14 For an account of the international foreign exchange market in the mid-19 th century, see Ugolini (2012). 15 See also Juglar (1889, p. 368); Wirth (1890, pp. 434-5);Macleod (1891, pp. 833-4). 16 Hawtrey (1919, pp. 149-50) argues that foreign investors were questioning the viability of the gold standard. This may be exaggerated: After losing £1.3m in the first two weeks after 11 th May, the gold reserve surpassed its pre-crisis level in June, and never sank below it in spite of the fact that an additional drain took place in July (figure 6). Thus, the case for a confidence crisis is strong (Ugolini 2010). evaluate this from the length and extent of the "high rates" period: Faced with reserve losses, the Bank of England had to keep the interest rate at a record level (10%) for more than three months, the longest period ever (Clapham 1944, pp. 429-32). Many observers saw the crisis as a blow to the reputation of London as an international money market. The shock was deterring foreign creditors from investing in English money market instruments despite the high rates. This opinion was taken seriously by the British Foreign Office, which felt it was necessary to send a circular to all diplomatic representations reassuring foreigners about the solidity of the English financial system 17 . Indeed, the policy followed by the Bank of England resulted from the same reading of the crisis -i.e., that the currency crisis would be resolved if the credit crisis was resolved. Such was also the logic of Bagehot's exhortations for generous lending against good collateral, while the "high rates" he also advised would take care of the currency crisis.

The consequences of this policy were equally important. At the end of the day, those investors who had not panicked and kept their money in London fared very well compared, say, to a counterfactual investment in Paris. Consider for instance an investor who would have converted his sterling into French francs bills with short maturities one month before the crisis (mid-March 1866) and then reinvested the bills as they matured, say week after week, during 6 months (until mid-September 1866) and then converted the proceeds in sterling again, and compare this investment with a similar one this time in sterling all the way through. At the end of the period the difference in yield between franc and sterling was 2.14% in favour of sterling, or an annualized differential of 4.28%. 18 This is one measure of the extent to which the more versatile -those who had fled to seek safety abroad --were encouraged to be faithful in the future. We conclude that the Bank of England's adoption of lending-of-last-resort policies in 1866 was one aspect of the process through which the role of sterling as a key reserve asset was established. 19 In successfully dealing with the financial shock, the Bank of England acquired enormous financial clout, and this cannot have been an irrelevant aspect of its subsequent triumphs.

Figure 5 about here 17 The text of the circular is found in Patterson (1870, pp. 234-5). 18 Authors computations from data in Ugolini (2010). Details available from authors. 19 The fact that a central bank should be expected to behave in such a way could not have been taken for granted at the time: for instance, the Bank of France's inability to do so after the events of 1870 seriously compromised the fate of the franc as an international currency. Bagehot himself noted this with satisfaction, but not without preoccupation: High-minded concern about the increased responsibility falling upon the Bank of England after the French debacle may have been a key motivation for Lombard Street (Bagehot 1873, pp. 31-2).

Conclusion

Relying on new statistical evidence (Flandreau and Ugolini 2013) this chapter has surveyed crucial elements of the management of the crisis of 1866. First, our assessment sheds new light on how Britain found her path towards financial stability in the second half of the 19 th century. By investigating the structure of the money market and the central bank's actions during the crisis, we have shown that Britain's recipe for financial stability consisted of a combination of generous liquidity provision and strict monitoring, made possible by the credible threat of exclusion from the central bank's standing facilities. The Bank's decisive adoption of this approach in 1866 reflected the end of credit rationing that had characterized crisis management until in 1857 [START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF] and paved the way for the establishment of a more resilient financial system in the following decades.

Second, we have emphasized the significance of the new policy beyond Britain and the British financial system. The successful handling of the situation by the Bank of England, we argued, had implications at the international level. The Bank had dealt with a twin crisis: The seizure experienced by the London money market in the few hours that followed the collapse of Overend, Gurney & Co. tested the resilience of British trade credit institutions and the stability of sterling. Therefore, the Bank's eventual success contributed to establishing the role of sterling as an international currency.

Last, in a volume devoted to British financial crises, it seems natural to end this chapter on the crisis of 1866 with a comparison. We suggest one with the Baring crisis of 1890. One major difference between the two crises is that, while the Bank let Overends fail, it did organize a rescue of Barings -although it made sure that the bankers paid dearly for it, so that the frequently used wording "bail out" is inappropriate. We can think of one main difference between the two crises that may account for the contrasted behaviour. The fall of a bill broker was bound to inflict losses to commercial banks but, since bill brokers did not play a firststage role in the origination of money market instruments (see figure 1), there was nothing in Overends' fall that would impair the operation of the money market: If cash was needed, the Bank could always provide it, and no large amount of information would be destroyed. By contrast, the fall of a first-order merchant bank such as Barings shattered the foundations of the London money market. Barings, unlike Overends, were large acceptors whose paper was "normally" received by the Bank of England: Refusing Barings' paper would undermine the London money market and send shockwaves throughout the system. 20 There was this, and there was also the large exposure that the Bank of England had to Barings -in contrast with the lack of exposure it had to Overends. 20 The systemic importance of merchant banks will again be proved by another crucial event in British financial history -i.e., the crisis of 1931 [START_REF] Accominotti | London Merchant Banks, the Central European Panic, and the Sterling Crisis of 1931[END_REF]).

-Flandreau, Marc, and Stefano Ugolini ( 2013 -£ 1 000 000

Figure

  Figure 1 about here

Figure

  Figure 6 about here

  ), "Where It All Began: Lending of Last Resort and Bank of England Monitoring during the Overend-Gurney Panic of 1866", in Michael D. Bordo and William Roberds (eds.), The Origins, History, and Future of the Federal Reserve: A Return to Jekyll Island, Cambridge: Cambridge University Press, pp. 113-61. -Giannini, Curzio (1999), "Enemy of None but a Common Friend of All"? An International Perspective on the Lender-of-Last-Resort Function, Princeton: Princeton University. -Hawtrey, Ralph G. (1919), Currency and Credit, London: Longmans. -Juglar, Clément (1889), Des crises commerciales et de leur retour périodique en France, en Angleterre et aux États-Unis, 2 nd ed., Paris: Guillaumin. -King, Wilfred, T.C., (1935), "The Extent of the London Discount Market in the Middle of the Nineteenth Century", Economica (New Series), 2:7, pp. 321-6. -King, Wilfred T.C. (1936), History of the London Discount Market, London: Routledge. -Macleod, Henry D. (1891), The Theory of Credit, II.2, London: Longmans. -Mahate, Ashraf A. (1994), "Contagion Effects of Three Late Nineteenth Century British Bank Failures", Business and Economic History, 23:1, pp. 102-15. -Patterson, Robert H. (1870), "On Our Home Monetary Drains, and the Crisis of 1866", Journal of the Statistical Society of London, 33:2, pp. 216-42. -Roberts, Richard (1992), Schroders: Merchants and Bankers, Basingstoke: Macmillan. -Schwartz, Anna J. (1987), "Real and Pseudo-Financial Crises", in ead. (ed.), Money in Historical Perspective, Chicago: University of Chicago Press, pp. 271-88. -Schwartz, Anna J. (1995), "Why Financial Stability Depends on Price Stability," Economic Affairs, 15:4, pp. 21-5. -Seyd, Ernest (1868), Bullion and Foreign Exchanges, Theoretically and Practically Considered, London: Effingham Wilson. -The Economist, 1857, 1865-6. -Ugolini, Stefano (2010), "The International Monetary System 1844-1870: Arbitrage, Efficiency, Liquidity", Norges Bank Working Paper 2010/23. -Ugolini, Stefano (2012), "Foreign Exchange Reserve Management in the Nineteenth Century: The National Bank of Belgium in the 1850s", in Anders Ögren and Lars F.

Figure 1 :

 1 Figure 1: Stylized structure of the English financial system in the 19 th century.

Figure 5 :

 5 Figure 5: Geographical origin of the bills discounted by the Bank in May 1866 (per kind of discounter). Source: Flandreau and Ugolini (2013).

Figure 6 :

 6 Figure 6: International aspects of the 1866 crisis: variation of the Bank of England's gold reserve, and spot and "forward" franc-pound exchange rates. Sources: Ugolini (2010); The Economist (1865-6); Seyd (1868).

  Daily discounts and advances by the Bank of England in May 1866. Source: Flandreau and Ugolini (2013). Top 30 discounters from the Bank in May 1866. Source: Flandreau and Ugolini (2013).Figure 4b: Top 30 advances from the Bank in May 1866. Source: Flandreau and Ugolini (2013).

	Com-mercial Banks £ 1 600 000 £ 1 800 000 Savers deposit Bills Discounted deposit Figure 3: £ 0 £ 200 000 £ 400 000 £ 600 000 £ 800 000 £ 1 000 000 £ 1 200 000 £ 1 400 000 01/05/1866 02/05/1866 03/05/1866 04/05/1866	05/05/1866	Bill Brokers Bills Rejected invest 07/05/1866 08/05/1866 09/05/1866 10/05/1866 11/05/1866 12/05/1866	Bill Market Merchant Banks Borrowers finance guarantee Advances Accorded 14/05/1866 15/05/1866 16/05/1866 17/05/1866 18/05/1866 19/05/1866	Bank of England Figure 4a: £ 0 invest Advances Rejected 21/05/1866 22/05/1866 23/05/1866 24/05/1866 25/05/1866 26/05/1866 28/05/1866 29/05/1866 30/05/1866 31/05/1866 Alexanders Cunliffes & Co The National Discount Co Barclay & Co Harwood Knight & Allen Oriental Bank Corporation The City Bank London & County Bank Brightwen Gillett & Co Drake Kleinwort & Cohen Smith Fleming & Co The Colonial Co Ltd Oppenheim S & Sons Raphael & Sons Huth F & Co Frith Sands & Co Merchant Banks and Trading Houses Commercial Banks Bill Brokers Agra & Masterman Bank £ 100 000 £ 0 London & Westminster Bank Bank of London Harwood Knight & Allen Alliance Bank The Discount Corporation Ltd Lawes R & Co Frith Sands & Co Barclay & Co Smith Fleming & Co Barnett & Co Sheppards Pelly & Co Bank of Hindustan Ebbw-Vale Company Cunliffe R Son & Co Merchant Banks and Trading Houses Commercial Banks Bill Brokers	£ 200 000 £ 100 000	£ 300 000 £ 200 000	£ 400 000 £ 300 000	£ 500 000 £ 400 000	£ 600 000 £ 500 000	£ 700 000	£ 800 000 £ 600 000	£ 700 000 Top 30 Discounters £ 800 000 Top 30 Advances

Authors' computations on Bank of England Archive C25/3 and The Economist (21 st November 1857).

See Xenos (1869) for an informed -but not fully impartial -account of Overends' unhappy Greek investments.

King (1936, p. 242).

Seyd (1868, pp.

81-3). 8 "Membership" shrank dramatically throughout the 19 th century[START_REF] Bignon | Bagehot for Beginners: The Making of Lender-of-Last-Resort Operations in the Mid-Nineteenth Century[END_REF].However, it seems that this owed to the consolidation of the banking system, not to the Bank striking out previous customers. 9 It also recorded information on drawers. See Flandreau and Ugolini (2013) for details.


N M Ribe

M Habibi

Daniel Bonn

Stability of Liquid Rope Coiling

Keywords:

A thin 'rope' of viscous fluid falling from a sufficient height coils as it approaches a rigid surface.

Here we perform a linear stability analysis of steady coiling, with particular attention to the 'inertiogravitational' regime in which multiple states with different frequencies exist at a fixed fall height.

The basic states analyzed are numerical solutions of asymptotic 'thin-rope' equations that describe steady coiling. To analyze their stability, we first derive in detail a set of more general equations for the arbitrary time-dependent motion of a thin viscous rope. Linearization of these equations about the steady coiling solutions yields a boundary-eigenvalue problem of order twenty-one which we solve numerically to determine the complex growth rate. The multivalued portion of the curve of steady coiling frequency vs. height comprises alternating stable and unstable segments whose distribution agrees closely with high-resolution laboratory experiments. The dominant balance of (perturbation) forces in the instability is between gravity and the viscous resistance to bending of the rope; inertia is not essential, although it significantly influences the growth rate.

I. INTRODUCTION

The coiling of a thin 'rope' of viscous fluid falling from a sufficient height onto a surface is a common fluid dynamical instability that occurs in situations ranging from food processing to lava flows. The history of its investigation spans nearly fifty years, and includes laboratory experiments 1-10 , linear stability theory for incipient coiling [START_REF] Cruickshank | Low-Reynolds-number instabilities in stagnating jet flows[END_REF][START_REF] Tchavdarov | Buckling of thin liquid jets[END_REF] , finite-amplitude scaling analysis for high-frequency coiling [START_REF] Mahadevan | Correction: fluid 'rope trick' investigated[END_REF] , and direct numerical simulation [START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF][START_REF] Ribe | Coiling of viscous jets[END_REF] . Fig. 1 shows the configuration considered in most of these studies, in which fluid with constant density ρ, viscosity ν and surface tension coefficient γ is injected at a volumetric rate Q from a hole of diameter d ≡ 2a 0 and then falls a distance H onto a solid surface.

The present study is motivated by our recent numerical and experimental results [START_REF] Maleki | Liquid rope coiling on a solid surface[END_REF][START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF][START_REF] Ribe | Coiling of viscous jets[END_REF] , which are summarized in Figs. 2 and3. Fig. 2a shows a schematic view of the experimental apparatus, in which a thin rope of silicone oil is extruded downward from a syringe pump driven by a stepper motor. In a typical experiment, the fluid was injected continuously at a constant rate Q while the fall height H was varied over a range of discrete values, sufficient time being allowed at each height to measure the coiling frequency. Anticipating the possibility of hysteresis, we made measurements both with height increasing and decreasing, and in a few cases we varied the height randomly. For fall heights within a certain range, we observed two or three different steady coiling states with different frequencies, each of which persisted for a time before changing spontaneously into one of the others. Fig. 2b andc show the low-and high-frequency coiling states observed for ν = 5000 cm 2 s -1 , d = 0.15 cm, Q = 0.0066 cm 3 s -1 , and H = 20 cm. The coexistence of two states at the same fall height reflects the multivalued character of the curve of frequency vs. height, which is illustrated in more detail in Fig. 3. The symbols show coiling frequencies measured in an experiment performed using viscous silicone oil (ρ = 0.97 g cm -3 , ν = 1000 cm 2 s -1 , γ = 21.5 dyne cm -1 ) with d = 0.068 cm and Q = 0.00215 cm 3 s -1 , and the solid line shows the curve of frequency vs. height predicted numerically for the same parameters using the method of Ribe [START_REF] Ribe | Coiling of viscous jets[END_REF] . As the fall height H increases, the coiling traverses four distinct dynamical regimes.

For small heights H < 0.7 cm, both gravity and inertia are negligible in the rope, and coiling occurs in a viscous (V) regime with a frequency 14

Ω ∼ Q Ha 2 1 ≡ Ω V , (1)
where a 1 is the radius of the 'coil' portion of the rope (Fig. 1). The frequency decreases strongly with height, and is independent of viscosity because the fluid velocity is determined kinematically by the injection speed. For 1 cm≤ H ≤ 5 cm, coiling occurs in a gravitational (G) regime in which the viscous forces that resist bending in the coil are balanced by gravity.

The corresponding coiling frequency is 14

Ω ∼ gQ 3 νa 8 1 1 4 ≡ Ω G . (2)
For 7 cm ≤ H ≤ 15 cm, a complex inertio-gravitational regime (IG) is observed in which viscous, gravitational, and inertial forces are all significant. The curve of frequency vs.

height is now multivalued, with up to seven different frequencies at a given height. The (rightward-and downward-facing) peaks in the curve correspond to resonant oscillations of the 'tail' portion of the rope with frequencies equal to the eigenfrequencies of a whirling viscous string [START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF] . The scaling law for all these frequencies is

Ω ∼ g H 1/2 ≡ Ω IG , (3)
with constants of proportionality that depend weakly on the dimensionless parameter

≡ Ω I . (4)
The existence of the four regimes just described has now been confirmed experimentally by Mahadevan et al. [START_REF] Mahadevan | Fluid 'rope trick' investigated[END_REF] for the inertial regime, by Maleki et al. [START_REF] Maleki | Liquid rope coiling on a solid surface[END_REF] for the viscous, gravitational, and inertial regimes, and by Ribe et al. [START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF] for the inertio-gravitational regime. The experimental observations in the IG regime are of particular interest. As shown in Fig. 3, the observed frequencies in this regime are concentrated along the roughly horizontal 'steps' of the Ω(H) curve, leaving the steeper portions with negative slope ( 'switchbacks') empty. The absence of observed steady coiling states along the switchbacks suggests that such states may be unstable to small perturbations. Here we investigate this question by means of a formal linear stability analysis.

II. GOVERNING EQUATIONS FOR AN UNSTEADY ROPE

The starting point of our analysis is a set of equations governing the unsteady motion of a thin viscous rope, i.e.., one whose 'slenderness' ≡ a 0 /L 1, where a 0 is a characteristic value of the rope radius and L is the characteristic length scale for the variations of the flow variables along the rope. Equations for a thin viscous rope have been derived by Entov and Yarin 15 , who described the geometry of the rope's axis using the standard triad of basis vectors from differential geometry (the unit tangent, the principal normal, and the binormal). However, such a description can lead to numerical instability when the total axial curvature is small, as it is over most of the length of a coiling liquid rope. Here we present an alternative formulation in which the basis vectors normal to the rope's axis are material vectors that are convected with the fluid. Because our goal is to perform a linear stability analysis of steady coiling, we write the equations in a reference frame that rotates with angular velocity Ωe 3 relative to a fixed laboratory frame. The Einstein summation convention over repeated indices or subscript/superscript pairs is assumed. Greek indices range over the values 1 and 2 only. Latin indices range over the values 1, 2, and 3 except for the Euler parameters q i , in which case i = 0, 1, 2, and 3. The quantity ijk is the usual alternating tensor. Finally, derivatives with respect to arclength along the rope axis are denoted by primes.

A. Geometry

Fig. 4 shows the geometry of an element of a thin viscous rope. Let x(s, t) be the Cartesian coordinates of a point on the rope's axis, where s is the arc length along it and t is time, and let d i (s, t) be a triad of orthogonal unit vectors defined at each point on the axis. The tangent vector to the axis is d 3 , and d 1 and d 2 ≡ d 3 × d 1 are material vectors normal to the axis that follow the rotation of the fluid. The rope's cross-section is assumed to be circular, with radius a(s, t), area A = πa 2 , and moment of inertia

I = πa 4 4 . (5)
.

The rate of change of the axial coordinates x as a function of arclength is

x = d 3 . (6)
The rates of change of the local basis vectors d i are in turn given by the generalized Frenet relations

d i = κ × d i , (7)
where κ ≡ κ i d i is the generalized curvature vector. The components κ i are related to the total curvature κ and the torsion τ of the axis by

κ = κ 2 1 + κ 2 2 1/2 , τ = κ 3 + κ -2 (κ 1 κ 2 -κ 2 κ 1 ) . (8)
To avoid the polar singularities associated with the traditional Eulerian angles, it is convenient to describe the orientation of the basis d i using four 'Euler parameters' q i (i = 0, 1, 2, 3), which are related to the direction cosines d ij (s) ≡ d i (s)•e j by 16

d ij =      q 2 1 -q 2 2 -q 2 3 + q 2 0
2(q 1 q 2 + q 0 q 3 ) 2(q 1 q 3 -q 0 q 2 ) 2(q 1 q 2 -q 0 q 3 ) q 2 0 + q 2 2 -q 2 1 -q 2 3 2(q 2 q 3 + q 0 q 1 ) 2(q 1 q 3 + q 0 q 2 ) 2(q 2 q 3 -q 0 q 1 ) q

2 3 + q 2 0 -q 2 1 -q 2 2      , (9)
The Euler parameters satisfy identically the relation q 2 0 + q 2 1 + q 2 2 + q 2 3 = 1, so only three of them are independent. The inverse Frenet relations for the curvatures κ i in terms of the Euler parameters and their derivatives are κ 1 = 2(q 0 q 1 + q 3 q 2 -q 2 q 3 -q 1 q 0 ) (10a) κ 2 = 2(-q 3 q 1 + q 0 q 2 + q 1 q 3 -q 2 q 0 ) (10b) κ 3 = 2(q 2 q 1 -q 1 q 2 + q 0 q 3 -q 3 q 0 ) (10c)

We now turn from the geometry of the axis to that of the rope as a whole. Given the basis vectors d i , the Cartesian coordinates X of an arbitrary point within the rope can be written

X(y 1 , y 2 , y 3 , t) = x(y 3 , t) + y 1 d 1 (y 3 , t) + y 2 d 2 (y 3 , t) ≡ x + y, (11)
where y 1 and y 2 are coordinates normal to the rope axis and y 3 ≡ s is an alternate notation for the arclength along the axis. In the following derivation, we shall make frequent use of the notation ∂ i = ∂/∂y i (i = 1, 2, 3).

The variable transformation (11) defines a set of (covariant) basis vectors g i = ∂ i X:

g 1 = d 1 , g 2 = d 2 , g 3 = hd 3 -κ 3 (y 2 d 1 -y 1 d 2 ) (12)
where

h ≡ (g 1 × g 2 )•g 3 = 1 -κ 2 y 1 + κ 1 y 2 . ( 13
)
Note that away from the rope axis, the vectors g i are not orthogonal if κ 3 = 0. It is therefore necessary also to use a set of contravariant (reciprocal) base vectors g i which satisfy g i •g j = δ i j , whence

g 1 = d 1 + h -1 κ 3 y 2 d 3 , g 2 = d 2 -h -1 κ 3 y 1 d 3 , g 3 = h -1 d 3 . (14)
The covariant and contravariant components of the metric tensor for the rope are then

g ij = g i •g j , g ij = g i •g j . (15)
Note that there is no distinction between covariant and contravariant basis vectors on the rope axis itself, because

g i = g i = d i = d i when y 1 = y 2 = 0.
In what follows it will sometimes be useful to replace the coordinates y 1 and y 2 by the polar coordinates

r ≡ y 2 1 + y 2 2 1/2 , θ = tan -1 y 2 y 1 . (16)
The covariant and contravariant base vectors for these coordinates are g r = g r = cos θd 1 + sin θd 2 ,

g θ = r(cos θd 2 -sin θd 1 ), g θ = r -2 g θ -κ 3 g 3 . (17)
A final geometric parameter of interest is the mean curvature H of the rope's surface, which determines the pressure associated with surface tension. In the limit a 1, aa 1,

2H ≈ - 1 a + κ 2 cos θ -κ 1 sin θ. (18)

B. Kinematics

Because the axis y 1 = y 2 = 0 of the rope is defined geometrically, it is not precisely a material line. Accordingly, two different velocity fields on the axis must be distinguished.

The first is simply the true fluid velocity u evaluated on the axis, viz. u(0, 0, y 3 , t) ≡ U (y 3 , t).

A second velocity field is

V (y 3 , t) = D t x, (19)
where

D t = ∂ t + W ∂ 3 , (20)
W (y 3 , t) = W (0, t) + y 3 0 ∆(s, t) ds, (21)
∆(y 3 , t) is the (yet to be determined) stretching rate of a material element that is aligned with the axis at time t, and W (0, t) is the velocity at which the rope is injected at y 3 = 0.

The convective derivative D t is the rate of change measured by an observer traveling at a speed (W ) equal to the sum of the injection speed and an additional velocity increment due to distributed stretching of the rope along its length [START_REF] Buckmaster | The buckling and stretching of a viscida[END_REF] . Unlike the usual convective derivative, D t applies only to field variables defined on the rope axis.

Although U = V in principle, V -U ∼ 2 U in the slender rope limit → 0 18 . Accordingly, we shall ignore the distinction between V and U from now on, and use the symbol U for both. The near-equality of U and V implies that the axis of the rope is very nearly a material line. The velocity W defined by (21) can therefore be regarded (with negligible error) as the rate of change of the arclength coordinate y 3 of a material point.

Expressions for the stretching rate ∆ and the convective rate of change of d 3 are obtained by differentiating (19) with respect to y 3 and using the generalized Frenet relations. We thereby obtain

∆ = U 3 -κ 2 U 1 + κ 1 U 2 , (22)
D t d 3 = ω 2 d 1 -ω 1 d 2 , (23)
where

ω 1 = -U 2 -κ 3 U 1 + κ 1 U 3 , (24a)
ω 2 = U 1 -κ 3 U 2 + κ 2 U 3 . (24b)
Now because the lateral unit vectors d α are material, their angular velocity about the rope axis is just the rate of rotation ω 3 of the fluid (note that ω 3 is a primitive variable, unlike ω 1 and ω 2 which are defined in terms of the velocity and geometry of the rope axis.) In certain situations, moreover, one must allow the unit vectors d 1 and d 2 at the end y 3 = 0 of the rope to rotate relative to the fluid with an additional spin ω 0 . For steady coiling with angular frequency Ω, for example, ω 0 ≡ -Ω is the additional spin required to make the base vectors d i (s) independent of time along the whole length of the rope (see Ribe [START_REF] Ribe | Coiling of viscous jets[END_REF] for further discussion.) The generalization of ( 23) is therefore

D t d i = (ω + ω 0 ) × d i , ω = ω i d i , ω 0 = ω 0 d 3 . (25)
As noted in § II A, the orientation of the basis vectors d i can be described by the Euler parameters q i . The evolution equations for these parameters that correspond to (25) are

D t q 0 = 1 2 [-ω 1 q 1 -ω 2 q 2 -(ω 3 + ω 0 )q 3 ] ( 26a
)
D t q 1 = 1 2 [ω 1 q 0 -ω 2 q 3 + (ω 3 + ω 0 )q 2 ] ( 26b
)
D t q 2 = 1 2 [ω 1 q 3 + ω 2 q 0 -(ω 3 + ω 0 )q 1 ] (26c)
D t q 3 = 1 2 [-ω 1 q 2 + ω 2 q 1 + (ω 3 + ω 0 )q 0 ] (26d)
The final kinematic equation needed describes the evolution of the rope's thickness. Consider a material element of the rope with (infinitesimal) length l(t), and let the arclength coordinate of its center be s(t). The incompressibility of the fluid requires that the volume V ≡ A(s(t), t)l(t) of this element be constant. Setting dV /dt = 0, we find

(∂ t A + ṡ∂ s A) l + A l = 0, (27)
where dots denote total time derivatives. Now ṡ = W and l/l = ∆, where W is the rate of change of the arclength (21) and ∆ is the stretching rate (22). Eqn. (27) then becomes

D t A = -A∆. ( 28
)
The convective rate of thinning of the rope is proportional to the rate of stretching of a material line that lies along the axis.

C. Local dynamical equations

We turn now to the equations of conservation of mass and momentum satisfied at each point in the rope. The strain rate tensor relative to general nonorthogonal coordinates is

e ij = 1 2 g i •∂ j u + g j •∂ i u . ( 29
)
Incompressibility of the fluid requires g ij e ij = 0, or

∂ α (hu α ) + ∂ 3 u 3 + κ 3 (y 2 ∂ 1 u 3 -y 1 ∂ 2 u 3 ) = 0. ( 30
)
Turning now to the momentum equations, we note first that the stress tensor relative to the local basis vectors g i and per unit local surface area is

τ ij = -pg ij + 2µg ik g il e kl , ( 31
)
where p is the pressure. However, it is more convenient to work with the modified (nonsymmetric) stress tensor

σ ij = σ ij = hτ ik g k •d j , (32)
which represents the stresses relative to the axial basis vectors d i and per unit area of a reference surface at the axis. Note that

σ ij = σ ij because d i = d i . Unlike τ ij , σ ij can
meaningfully be integrated over cross-sections, because the basis vectors and the surface to which it is referred do not vary across the rope. The equations of equilibrium in terms of σ ij are (Green and Zerna [START_REF] Green | Theoretical Elasticity[END_REF] , p. 150)

ρh Ẍ = ∂ i (σ ij d j ) + ρhg, ( 33
)
where Ẍ is the acceleration of a fluid particle and g i d i ≡ g is the gravitational acceleration. Although the standard notation for gravity is similar to that for the covariant and contravariant basis vectors in § II A, the different numbers of subscripts and superscripts used in the two cases prevents confusion. For later use, we define the stress vector

σ i = σ ij d j . ( 34
)
that acts on a surface whose normal is parallel to d i .

We now calculate explicitly the acceleration Ẍ. To first order in the lateral coordinates y 1 and y 2 , the velocity field within the rope as measured in the rotating reference frame is

u = U - 1 2 y∆ + ω × y, (35)
where y = y 1 d 1 + y 2 d 2 . The total velocity u is the sum of the axial velocity U , a radial inflow associated with stretching at a rate ∆, and a velocity ω × y associated with bending in two mutually perpendicular planes (at rates ω 1 and ω 2 ) and twisting (at a rate ω 3 ). The acceleration corresponding to (35), measured now relative to the fixed laboratory frame, is

Ẍ = Ω × [Ω × (x + y)] + 2Ω × u + D t U + (y 2 D t ω 1 -y 1 D t ω 2 ) d 3 + D t ω 3 (d 3 × y) + (ω•y)(ω 3 d 3 -ω 0 ) +[(ω × y)•d 3 ](ω × d 3 ) -∆(ω × y) - 1 2 D t ∆ - ∆ 2 4 + ω 2 3 y. ( 36
)
The first two terms on the right side of (36) are the additional centrifugal and Coriolis accelerations associated with the angular velocity Ω of the rotating frame relative to the fixed laboratory frame. Note that the vectors u, U and ω that appear in (36) are measured relative to the rotating frame.

D. Global force and torque balance

The equations of global force balance are obtained by integrating the momentum equations (33) together with (36) over a cross-section S of the rope with area A, yielding

ρAJ = N + P (37)
where

N = S σ 3 dS, (38)
is the stress resultant vector,

J = Ω × (Ω × x) + 2Ω × U + D t U , (39)
is the acceleration averaged over the cross-section,

P = ρAg + C [(g r •d α )σ α -a σ 3 ] dl, ( 40
)
is the applied load vector, and C is the (circular) contour around the cross-section.

The equations of global torque balance are obtained by applying the operator y× to (33) and then integrating over the cross-section. This yields

ρIK = M + d 3 × N + M, ( 41
)
where I is the moment of inertia (5) of the cross-section,

M = S y × σ 3 dS, (42)
is the bending/twisting moment vector,

K α = D t ω α + κ α (D t U 3 + βγ3 ω β U γ ) +Ω 2 (-κ α d 3β x β + αβ3 d β3 d 33 ) +Ω [2d β3 ( αβ3 ω 3 + βγ3 κ α U γ ) -d α3 ∆] -ω α ∆ + αβ3 ω β (ω 3 -ω 0 ), (43a)
and

K 3 = 2D t ω 3 -κ α D t U α +Ω 2 d αβ κ α x β +2Ω { αβ3 [d α3 (ω β + κ β U 3 ) + d 33 κ α U β ] -d 33 ∆} -2ω 3 ∆ + αβ3 κ α [(ω 3 + ω 0 )U β -ω β U 3 ] , (43b)
are the components of the average moment of the acceleration, and

M = ρI [(g × d 3 )κ -(κ × g)d 3 ] + C y × [(g r •d α )σ α -a σ 3 ] dl ( 44
)
is the applied moment vector.

E. Applied loads and moments

We now determine simplified expressions for the applied load and moment vectors (40) and (44), assuming that the outer surface r = a of the rope is acted upon by surface tension but is otherwise stress-free. Accordingly, the stress vector there is

τ ij n i g j | r=a = 2γHn ( 45
)
where H is the mean curvature (18),

n ≡ n i g i = h 2 c + a 2 -1/2 (h c g r -a d 3 ) (46)
is the unit vector normal to the surface and

h c ≡ h| r=a = 1 + a(κ 1 sin θ -κ 2 cos θ). (47)
By projecting (45) onto the base vector d k and rewriting the result in terms of σ ij using (32), we obtain

[(g r •d α )σ α -a σ 3 ] r=a = 2γH(h c g r •d i -δ i3 a )d i . (48)
Eqn. (48) together with ( 17), (18), and (47) permits evaluation of the line integrals in ( 40) and ( 44), which yields

P = ρAg + 2πγ(aκ × d 3 + a d 3 ), (49)
M α = ρIκ α g 3 + γAκ α a , M 3 = -ρIκ α g α . (50)
F. Constitutive relations

The dynamical equations are completed by constitutive relations for the stress resultant N 3 , the bending moments M 1 and M 2 , and the twisting moment M 3 . These can be derived by asymptotic expansion of the governing equations in powers of the slenderness = a 0 /L 1, following a procedure similar to that of Ribe [START_REF] Ribe | A general theory for the dynamics of thin viscous sheets[END_REF] . To facilitate the derivation, define dimensionless variables ŷα = y α /a 0 , â = a/a 0 , ŷ3 = y 3 /L, and κi = Lκ i , and the dimensionless derivative ∂3 = ∂/∂ ŷ3 .

To determine the constitutive relation for the axial stress resultant N 3 , we consider slow (inertia-free) deformations dominated by stretching. Suppose for definiteness that gravity is the primary force responsible for the deformation of the rope, as is the case e.g. for a vertical liquid rope stretching under its own weight. The scales for the velocity and pressure within the rope are then

u ∼ ρgL 2 µ , p ∼ ρgL. (51)
We also suppose that the magnitude of the surface tension coefficient γ is such that

γ ρgL 2 ≡ B-1 = O(1), ( 52
)
where B is a modified inverse Bond number.

The scales (51) suggest that the velocity and pressure fields can be represented by asymptotic expansions of the form

u i = ρgL 2 µ j=0 m=0 n=0 j ŷm 1 ŷn 2 u (jmn) i (ŷ 3 ), ( 53a
) p = ρgL j=0 m=0 n=0 j ŷm 1 ŷn 2 p (jmn) (ŷ 3 ), (53b)
where the coefficients u (jmn) i and p (jmn) are dimensionless functions of the arclength ŷ3 . To simplify the notation, let

∆ (ijk) = ∂3 u (ijk) 3 + κ1 u (ijk) 2 -κ2 u (ijk) 1 , (54)
We now substitute the expansions (53) into the continuity equation (30), the momentum equations ( 33) with the inertial term on the left-hand side neglected, and the boundary condition (45), and then require terms proportional to the same powers of , ŷ1 , and ŷ2 in each equation to vanish separately. This yields a set of coupled linear algebraic equations for the coefficients u (jmn) i and p (jmn) that can be solved sequentially. A Mathematica 20 script that implements this solution procedure is available upon request from the first author. The leading-order expression for the (dimensional) stress resultant N 3 is

N 3 ρgLA = -p (000) + 2∆ (000) , (55)
and the sequential solution procedure described above gives

p (000) = -∆ (000) + B-1 . (56)
Substituting (56) into (55) and redimensionalizing using ∆ ≈ ρgL∆ (000) /µ, we find

N 3 = 3µA∆ -πγa. (57)
To determine the constitutive relations for M 1 , M 2 and M 3 , we consider slow deformations dominated by bending and twisting. The velocity and pressure then scale as

u ∼ ρgL 2 2 µ , p ∼ ρgL , (58)
and the appropriate asymptotic expansions are

u i = ρgL 2 2 µ j=0 m=0 n=0 j ŷm 1 ŷn 2 u (jmn) i (ŷ 3 ), ( 59a
) p = ρgL j=0 m=0 n=0 j ŷm 1 ŷn 2 p (jmn) (ŷ 3 ), (59b)
Let

ω (ijk) 1 = -∂3 u (ijk) 2 -κ3 u (ijk) 1 + κ1 u (ijk) 3 , (60a)

III. LINEARIZED EQUATIONS FOR STABILITY OF STEADY COILING A. Basic state

The basic states whose stability we shall analyze are numerical solutions for the steady coiling of a viscous rope, obtained using the continuation method described in Ribe [START_REF] Ribe | Coiling of viscous jets[END_REF] . The equations governing steady coiling are obtained from the full unsteady equations in § II by

setting ∂ t = U 1 = U 2 = 0, U 3 = W ≡ U ,

B. Perturbation expansion

The next step is to write each of the twenty-one unsteady dependent variables as the sum of a steady (barred) value and an exponentially growing perturbation. Denoting the spatially varying parts of the perturbation variables by hats, we have

A = Q/ Ū + ÂE, x i = xi + xi E, q j = qj + qj E, U α = Ûα E, U 3 = Ū + Û3 E, W = Ū + Ŵ E, ω α = κα Ū + ωα E, ω 3 = ω3 + ω3 E, N i = Ni + Ni E, M i = Mi + Mi E, (65)
where E = exp(σt) and σ is the growth rate. By substituting (65) into the equations derived in § II and linearizing in the usual way, we obtain a set of twenty-one coupled linear ODEs for the perturbation variables  ≡ 2πāâ, x1 , x2 , x3 , q0 , q1 , q2 , q3 , Û1 , Û2 , Û3 , Ŵ , ω1 , ω2 , ω3 , N1 , N2 , N3 , M1 , M2 , and M3 . These equations are given explicitly in the Appendix.

C. Boundary conditions

The boundary conditions satisfied by the dependent variables are most conveniently obtained by working in a fixed laboratory reference frame. The velocity and angular velocity vectors in the laboratory frame are related to those in the corotating frame by

V lab = V + Ωe 3 × x, ω lab = ω + Ωe 3 . ( 66
)
Consider first the boundary conditions at the injection point s = 0. The cross-sectional area of the rope, the Cartesian coordinates of its axis, and the advection velocity W are all fixed there, requiring

A(0) -A 0 = x 1 (0) = x 2 (0) = x 3 (0) = W (0) -U 0 = 0, ( 67
)
where A 0 = πa 2 0 and U 0 = Q/A 0 . With no loss of generality, we stipulate that the local basis vectors d i (0) at the injection point are constant in the rotating frame, which implies

d ij (0) =      q1 (0) 2 -q2 (0) 2 2q 1 (0)q 2 (0) 0 2q 1 (0)q 2 (0) q2 (0) 2 -q1 (0) 2 0 0 0 -1      , (68)
where q1 (0) and q2 (0) are the Euler parameters for the steady coiling solution. The fluid velocity at s = 0 is equal to the imposed injection velocity, or

V (0) = -U 0 e 3 . ( 69
)
The vanishing of the angular velocity in the laboratory frame requires

ω(0) + Ωe 3 = 0. ( 70
)
Turning now to the contact point s = , we note that the vertical coordinate of the rope's axis there is just (minus) the total fall height less a small correction for the finite radius of the rope, or

x 3 ( ) = -H + a( ). ( 71
)
Because fluid typically piles up beneath the coiling rope in laboratory experiments (Fig. 2), the fall height H appearing in (71) should be interpreted as an effective value, i.e., the total fall height less the height of the fluid pile. The mobility of the contact point requires that the rope's axis there must be horizontal and have zero curvature about a horizontal axis normal to the rope, or

d 33 ( ) = d 33 ( ) = 0. ( 72
)
The no-slip condition requires the velocity of the rope axis at the contact point to be zero in the laboratory frame, except for a small vertical velocity proportional to the rate of change of the rope's radius. Thus

V ( ) + Ωe 3 × x( ) = e 3 d dt a( ), (73)
Finally, the vanishing of the angular velocity vector in the laboratory frame requires

ω( ) + Ωe 3 = 0. ( 74
)
All the above boundary conditions are valid for an arbitrary time-dependent motion of the rope. To determine the boundary conditions satisfied by the perturbation (hatted) variables, we begin by recalling the boundary conditions for steady coiling, which are 14

0 = Ā(0) -A 0 = x1 (0) = x2 (0) = x3 (0) = q0 (0) = q3 (0) = V3 (0) -U 0 = W (0) -U 0 = ω1 (0) = ω2 (0) = ω3 (0) -Ω (75)
at the injection point s = 0 and

= x1 ( ¯ ) -Ω -1 Ū ( ¯ ) = x2 ( ¯ ) = x3 ( ¯ ) + H -ā( ¯ ) = q0 ( ¯ ) = q1 ( ¯ ) = q2 ( ¯ ) -2 -1/2 = q3 ( ¯ ) + 2 -1/2 = ω1 ( ¯ ) = ω2 ( ¯ ) -Ω = ω3 ( ¯ ). ( 76
= Â(0) = x1 (0) = x2 (0) = x3 (0) = q0 (0) = q1 (0) = q3 (0) = V1 (0) = V2 (0) = V3 (0) = Ŵ (0) = ω1 (0) = ω2 (0) = ω3 (0). ( 77
)
The derivation of the boundary conditions at the contact point s = is somewhat more complicated, because both the rope length ≡ ¯ + ˆ and the base vectors d i ( ) ≡ di ( )+ di ( ) change with time. To first order in the perturbation quantities, therefore, the expansion of a generic scalar or vector variable φ( ) at the contact point is

φ( ) ≈ φ( ¯ ) + ˆ φ ( ¯ ) + φ( ¯ ). ( 78
)
Linearizing the boundary conditions ( 71) through (74) about the steady conditions (76) with the help of (78), we obtain the following boundary conditions for the perturbation variables at s = ¯ :

0 = x3 ( ¯ ) -ˆ ā ( ¯ ) -â( ¯ ) = q2 ( ¯ ) + q3 ( ¯ ) = V1 ( ¯ ) -Ω x1 ( ¯ ) d31 ( ¯ ) -x2 ( ¯ ) = V2 ( ¯ ) + σ ˆ ā ( ¯ ) + â( ¯ ) = V3 ( ¯ ) -Ω x1 ( ¯ ) d32 ( ¯ ) + x1 ( ¯ ) + ˆ ∆( ¯ ) = ω1 ( ¯ ) + ˆ M1 ( ¯ ) 3µ Ī( ¯ ) -Ω d21 ( ¯ ) = ω2 ( ¯ ) + ˆ M2 ( ¯ ) 3µ Ī( ¯ ) -Ω d23 ( ¯ ) = ω3 ( ¯ ) + ˆ M3 ( ¯ ) 2µ Ī( ¯ ) -Ω d22 ( ¯ ). (79)
In (79) and henceforth, dij (s) are the perturbations of the direction cosines ( 9), e.g. d12 = 2(q 1 q2 + q2 q1 + q0 q3 + q3 q0 ). The constitutive relations (64) have been used to eliminate ω i ( ¯ ) from the boundary conditions on ωi ( ¯ ) in (79).

In summary, (77) and ( 79) are the twenty-two boundary conditions required to constrain the twenty-one perturbation variables and the unknown perturbation ˆ of the rope length.

IV. NUMERICAL STABILITY ANALYSIS

The equations (A.1) together with the boundary conditions (77) and (79) constitute a linear two-point boundary-value problem of order 21 that has nontrivial solutions only for particular values of the growth rate σ. We solve this eigenvalue problem numerically using a continuation method implemented by the software package AUTO 97 [START_REF] Doedel | Numerical analysis and control of bifurcation problems, Part I[END_REF][START_REF] Doedel | Numerical analysis and control of bifurcation problems, Part II[END_REF] (freely available at http://indy.cs.concordia.ca/auto/). The basic idea (e.g., Keller [START_REF] Keller | Numerical Methods for Two-Point Boundary-Value Problems[END_REF] , p. 235) is to introduce into the boundary conditions three new adjustable real parameters β i (i = 1, 2, 3) which are then varied gradually to refine an initial guess for the (possibly complex) eigenvalue σ. In particular, we introduce a new boundary condition

M1 ( ¯ ) = β 1 + iβ 2 , (80)
and modify the boundary condition on ω1 ( ¯ ) from (79) to

ω1 ( ¯ ) = -ˆ M1 ( ¯ ) 3µ Ī( ¯ ) + Ω d21 ( ¯ ) + β 3 . ( 81
)
The problem is initialized by setting β 1 = β 2 = β 3 = 0 and making an initial guess for the growth rate σ. The solution procedure then comprises two steps. First, we 'pull' β 3 away from 0 to some finite value (e.g., 1) with σ fixed, letting β 1 and β 2 float freely. Then β 3 is 'pushed' gradually back to 0 with β 1 and β 2 fixed, leaving the real and imaginary parts of σ free to float. At the end of this process, one has both an eigenvalue σ and the full set of associated complex eigenfunctions for the twenty-one perturbation variables. High accuracy is ensured by solving the equations for the steady basic state simultaneously in the same program, on the same numerical grid as the perturbation equations. The resulting system is of order 59 (17 steady variables plus the real and imaginary parts of 21 perturbation variables).

Here we present the results of stability analyses for three of the laboratory experiments reported by Ribe et al. [START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF] , in each of which the coiling frequency Ω is measured as the fall height H is varied for fixed values of the hole diameter d, the flow rate Q, and the fluid properties ρ, ν, and γ. Each experiment is therefore defined by particular values of the dimensionless groups

Π 1 = ν 5 gQ 3 1/5 , Π 2 = νQ gd 4 1/4 , Π 3 = γd 2 ρνQ . (82)
To carry out the stability analysis for a given experiment, we first calculate numerically the dimensionless frequency Ω(ν/g 2 ) 1/3 ≡ Ω of steady coiling as a function of the dimensionless height H(g/ν 2 ) 1/3 ≡ H. This yields a curve similar to that shown (in dimensional form) in Fig. 3. Next, we choose a trial value of H, and use the 'pull/push' procedure described above to search for unstable modes having (σ) > 0. We then continue any such modes in both directions along the curve Ω( H), monitoring σ to identify the fall heights at which with the highest frequencies in Fig. 5, which lie close to an unstable segment of the calculated curve. However, the growth rate of the instability along this portion of the curve is very small (σ ≈ 0.02Ω), implying that the coiling rope executes Ω/2πσ ≈ 8 revolutions during the time required for a perturbation to grow by a factor e. This may explain why apparently steady states such as those in Fig. 5 are observed despite their instability sensu stricto.

V. DISCUSSION

A more detailed examination of our numerical solutions helps to understand the mechanism by which steady coiling becomes unstable. As an illustration, we consider the case Π 1 = 3690, Π 2 = 2.19, Π 3 = 0, and H = 0.894, for which the steady coiling frequency is Ω = 1.401 (solid black circle in Fig. 6). Setting Π 3 = 0 eliminates the uninteresting effect of surface tension, which increases the steady coiling frequency by only 5%.

where J is given by (39). Because γ = 0 for our illustrative example, (37) implies that fV + fG + fI = 0.

The steady coiling solution comprises an interior region in which bending is negligible ( M1 ≈ 0), and two boundary layers near the injection and contact points where significant bending is concentrated (Fig. 8b). In the interior, the rope behaves essentially as a 'whirling viscous string' 10 : the lateral deflection increases smoothly downward (Fig. 8a), and the gravitational force is balanced about equally by the viscous force associated with axial stretching and by (centrifugal) inertia (Fig. 8c). In the lower (and more dynamically significant) boundary layer, the gravitational force is balanced almost entirely by the viscous force associated with bending, with inertia playing a subsidiary role.

The most unstable eigenmode of the steady solution shown in Fig. 8 has a real eigenvalue σ = 0.625Ω, where Ω is the steady coiling frequency. The structure of this eigenmode is shown in Fig. 9, using the same variables (lateral deflection, bending moment, and forces per unit length) as for the steady solution. The perturbation forces fV , fG , and fI are given respectively by the viscous, gravitational, and inertial terms of (A.1l) with γ = 0 and α = 2, and satisfy fV + fG + fI = 0. The structural features of the eigenmode are concentrated in the lower boundary layer, where the gravitational force is balanced primarily by viscous forces (Fig. 9c). The mechanism of the instability therefore involves a balance between gravity and the viscous resistance of the rope to bending, with inertia playing a secondary role. This conclusion can be verified by 'turning off' all the inertial terms in the perturbation equations (A.1) while holding constant all the other parameters in the numerical code. The instability still occurs; but the growth rate σ = 1.42Ω is now more than double the 'true' growth rate σ = 0.625Ω predicted by the full numerical model with all inertial terms retained.

This demonstrates that inertia is not essential to the instability, but that it nevertheless significantly influences the growth rate.

A comparison of Figs. 5-7 raises a further question: how does the number N s of stable segments of the curve Ω(H) depend on the experimental parameters? The stable segments are confined for the most part to the roughly horizontal portions ('steps') of the Ω(H) curve.

Ribe et al. [START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF] showed that the total number N of (stable and unstable) steps in the curve scales as N ∼ Π 5/32 1 in the limit when Π 1 → ∞ and gravitational stretching of the rope is strong (a 1 a 0 ). Figs. 5-7 suggest that N s also increases with Π 1 : N s = 2 for Π 1 = 1220, and N s = 3 for Π 1 = 3690 and 10050. Moreover, the fourth step in Fig. 7 is only slightly unstable (σ ≈ 0.004Ω), suggesting that Π 1 = 10050 may be just below the value above which N s = 4. Unfortunately, numerical convergence becomes difficult to achieve when Π 1 and/or Ω is too large, and we were therefore not able to determine a scaling law for N s . For now, we can only speculate that it scales in the same way as the total number of steps, viz.,

N s ∼ Π 5/32 1 .
In conclusion, our linear stability analysis shows that steady coiling in the multivalued 'inertio-gravitational' (IG) regime is stable only along discrete segments of the frequency vs. height curve, the distribution of which agrees very well with high-resolution laboratory measurements. The stability analysis further shows that coiling is stable at all heights in the three remaining regimes (viscous, gravitational, and inertial), in agreement with the experiments of Maleki et al. 9 . Analytical theory, numerical analysis, and laboratory experiments thus come together to offer a consistent portrait of steady coiling over the whole range of fall heights and frequencies at which the phenomenon occurs.

gd 2 H 2 /

 22 νQ[START_REF] Ribe | Multiple coexisting states of liquid rope coiling[END_REF] . As the height increases further, the amplitude of the oscillations in Ω(H) gradually decreases until the curve becomes smooth again at H ≈ 18 cm. Viscous forces in the coil are now balanced almost entirely by inertia, giving rise to inertial (I) coiling with a

  and ω α = κ α U , where U (s) is the velocity parallel to the axis of the rope. The result is a seventeenth order two-point boundary-value problem for the variables x1 , x2 , x3 , q0 , q1 , q2 , q3 , Ū , κ1 , κ2 , ω3 , N1 , N2 , N3 , M1 , M2 , and M3 , where the overbars have been added to distinguish the variables of the basic state from the perturbation variables to be introduced in a moment. Because the flow is steady, each barred variable is a function of the arclength s only. Examples of the geometry of steady coiling are shown in Figs. 1 and 2, and the mathematical structure of the solutions is discussed in more detail in Ribe 14 .

  ) at the contact point s = ¯ , where Ū (s) = Q/ Ā(s). The boundary conditions satisfied by the perturbation (hatted) variables are obtained by linearizing the boundary conditions (67) through (74) about the steady boundary conditions (75) and (76). At the injection point, the resulting conditions are 0

  (σ) becomes zero, i.e. at which the mode in question becomes stable. By repeating this procedure for different trial values of H along the curve Ω( H), we determine the portions of the curve that represent unstable steady states.The results of this procedure are shown in Figs.5 -7for the parameters (Π 1 , Π 2 , Π 3 ) corresponding to the three laboratory experiments referred to above. In each figure, the symbols indicate experimental measurements obtained in series with H increasing (squares), decreasing (circles), and varied randomly (triangles.) The continuous curve in each figure shows the numerically calculated curve Ω( Ĥ) for steady coiling, and its solid and dashed portions indicate stable and unstable steady states, respectively. Overall, the agreement between the numerical calculations and the experiments is very close: the observed steady states are concentrated along the stable portions of the calculated curves, leaving the unstable portions almost entirely 'unpopulated'. The only significant exceptions are the three measurements

Fig. 8

 8 Fig.8shows the lateral displacement x1 (s) of the rope's axis (in the plane containing the injection point and the contact point with the plate), the bending moment M1 (s), and the viscous, gravitational, and inertial forces per unit rope length in the d 2 -direction. These forces are defined (using an obvious notation) as

FIG. 1 :FIG. 4 :FIG. 3 :

 143 FIG. 1: Steady coiling of viscous corn syrup (photograph by N. Ribe.) Fluid with density ρ, viscosity ν and surface tension coefficient γ is injected at volumetric rate Q through a hole of diameter d ≡ 2a 0 and falls a distance H onto a solid surface. The angular coiling frequency is Ω, the radius of the 'coil' portion of the rope is R, and the radius of the rope at the base of the coil is a 1 .

FIG. 5 :FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 : 8 .

 567898 FIG.5: Stability of steady coiling with Π 1 = 1220, Π 2 = 2.09, and Π 3 = 0.019. The continuous curve shows the numerically calculated frequency of steady coiling as a function of height. The solid and dashed portions of the curve indicate stable and unstable steady states, respectively, as predicted using the numerical stability analysis described in the text. Symbols indicate experimental measurements 10 obtained in series with H increasing (squares), decreasing (circles), and varied randomly (triangles.)

Acknowledgments

We are grateful to A. Boudaoud, A. Davaille, and J. Lister for helpful discussions, and to J. Lister for suggesting the method used to determine the eigenvalues of the linear stability problem. We also thank two anonymous referees for comments that led to significant improvements in the presentation. Symbolic mathematical manipulations were performed with the aid of Mathematica 20 . This is IPGP contribution number 2150.

.

(60b)

The leading-order expressions for the (dimensional) moments M i can then be written

M 2 ρgI = p (010) -2∆ (110) + 2κ 3 u

The sequential solution procedure described previously yields

,

= -ω

= ω

= κ2 ω

Substituting (62) into (61) and redimensionalizing using the relations

,

we obtain

G. Summary

The unsteady flow of a liquid rope is described by the twenty-one variables A, x 1 , x 2 ,

, and M 3 . The twenty-one differential equations they satisfy are ( 6), (21) in the form W = ∆, (24), ( 26), (28), (37), (41), (57), and (64). The auxiliary definitions required to close the system are ( 9), ( 22), (39), ( 43), (49), and (50).

APPENDIX: PERTURBATION EQUATIONS

The twenty-one first-order ODEs satisfied by the perturbation variables are given below.

Here P ≡ P is an alternate notation for the perturbation of the enclosed quantity, and

The perturbation curvatures κi are eliminated from the above equations using the auxiliary relations Ū κ1 = ω1 -κ1 Ŵ + 2σ ( q 1 , q 0 + q 2 , q 3 ) , (A.2a) Ū κ2 = ω2 -κ2 Ŵ + 2σ ( q 2 , q 0 + q 3 , q 1 ) , (A.2b) Ū κ3 = ω3 -κ3 Ŵ + 2σ ( q 3 , q 0 + q 1 , q 2 ) , (A.2c) which are themselves obtained by combining the perturbation forms of ( 10) and (26).

1 G. Barnes and R. Woodcock, "Liquid rope-coil effect," Am. J. Phys. 26, 205 (1958).