Hydrodynamic Instability of Liquid Films on Moving Fibers

Transcription

1 Journa of Cooid and Interface Science 215, (1999) Artice ID jcis , avaiabe onine at on Hydrodynamic Instabiity of Liquid Fims on Moving Fibers Konstantin G. Kornev* and Aexander V. Neimark,1 *Institute for Probems in Mechanics, Russian Academy of Sciences, Prospect Vernadskogo 101, Moscow , Russia; and TRI/Princeton, 601 Prospect Avenue, Princeton, New Jersey Received December 14, 1998; accepted March 30, 1999 The stabiity of iquid fims on moving fibers is studied with a focus on effects caused by fim soid adhesion interactions and hydrodynamic interactions between the fim and the surrounding gas. We show that at high fiber veocities (arge Reynods numbers) the fim gas hydrodynamic interactions induce instabiity in fims, which woud, otherwise, be stabiized by the adhesion interactions at static conditions. Two types of unstabe modes caused by hydrodynamic factors are found: the first corresponds to the Kevin Hemhotz waves induced by inertia effects; the second, induced by viscosity effects, is observed at smaer (yet sti arge) Reynods numbers when the Kevin Hemhotz instabiity is suppressed. Linear stabiity anaysis of the system of couped hydrodynamic equations of fim and gas fow is performed by the method of norma modes. We derive the genera dispersion reation as an impicit equation with respect to the mode frequency. In the imit of vanishing fiber veocities, the obtained equation provides the dispersion equation for motioness fibers. The conditions of fim stabiity and unstabe modes are anayzed in terms of two dimensioness parameters, the adhesion factor, which quantifies the intensity of fim fiber interactions, and the Weber number, which quantifies the intensity of fim gas interactions. The Reynods number determines the regions of vaidity of the inviscid and viscous regimes of fim instabiity. The resuts are iustrated by estimates of the stabiity conditions for moving fibers in terms of the stabiity diagram, the critica fim thickness, the fastest unstabe mode, and the corresponding characteristic break-up time. Athough the methods deveoped are appicabe for any type of iquid soid interaction, the estimates were made for the ongrange van der Waas interactions. Practica appications incude various technoogies reated to fabrication and chemica modifications of fibers and fiber products, e.g., spin finishing and ubrication Academic Press Key Words: iquid fims; fibers; hydrodynamic instabiity; adhesion. I. INTRODUCTION The probem of stabiity of thin iquid fims on cyindrica soid surfaces has been drawing permanent attention of researchers on account of many important practica appications, 1 To whom correspondence shoud be addressed. Fax: incuding coating of fibers and wires, spin finishing, chemica modification, and drying of texties and other fibrous materias. It is we documented in the iterature that wetting properties of curved substrates may differ significanty from those of pane surfaces (1 15). Specific curvature-dependent effects come from the competition between the capiarity and ong-range intermoecuar forces. Much work has been done to expore the conditions of fim stabiity on motioness fibers (6 8, 10 12, 16 26). Convex surfaces of fibers 2 impy positive curvatures of coating iquid fims and, therefore, a positive Lapace pressure acting on the outer fim interface: P c p 0 p g 0 R f h. [1] Here, is the iquid gas surface tension, R f is the fiber radius, p i 0 are pressures in the gas, i g, and iquid, i, phases, and h is the fim thickness. Lapace pressure, when positive, tends to squeeze the iquid out of the fim. This effect causes a we-known phenomenon of capiarity-driven instabiity of fims coating cyindrica soid surfaces, simiar to the Rayeigh instabiity of jets (27). Goren (16), and then other researchers (Johnson et a. (23), Yarin et a. (25)), studied in detai the Rayeigh instabiity of iquid fims on cyindrica fibers without taking into consideration iquid soid interactions, or, in other words, adhesion. They concuded that the uniform iquid coating is aways unstabe and coapses into a periodic array of dropets, independent of fim thickness, fiber radius, and iquid properties, such as viscosity, surface tension, and density. The above-mentioned physica parameters determine the rate of fim destruction. Adhesion between iquid and soid may stabiize the fims. Thermodynamic equiibrium and stabiity of thin fims on curved soid surfaces are determined by competition between capiarity and adhesion forces. Whie the curvature-dependent Lapace pressure favors the iquid spreading over the soid surface, the thickness-dependent fim soid interactions oppose 2 Beow, we use the term fiber for any cyindrica soid substrate (fibers, wires, threads, etc.) /99 $30.00 Copyright 1999 by Academic Press A rights of reproduction in any form reserved.

2 382 KORNEV AND NEIMARK the fim thinning. An equiibrium iquid configuration is observed when these forces are in baance. Derjaguin (28, 29) deveoped a concept of disjoining pressure, which is currenty regarded as the most practica rationae to account for fim soid interactions (1 3). In the so-caed Derjaguin approximation (1 3), it is assumed that the disjoining pressure, (h), depends excusivey upon the fim thickness and is additive to the Lapace pressure. In particuar, for a uniform fim of thickness h 0 on a cyindrica fiber the pressure difference across the fim gas interface is represented as a sum of the Lapace pressure [1] and the disjoining pressures, (h 0 ): R f h h 0 p 0 g p 0 P c. [2] Starov and Churaev (6) discussed the curvature-dependent effects in terms of Eq. [2] for mode isotherms of disjoining pressure. In most practica systems, the behavior of thin fims is mainy determined by ong-range dispersion, or van der Waas forces. Brochard-Wyart (7, 8) has shown that iquid fims on fibers can be stabiized due to the action of dispersion forces. She anayzed conditions of thermodynamic equiibrium and derived a stabiity criterion for the van der Waas fims on fibers. The roe of dispersion interactions has been studied by Nowakowski and Ruckenstein (13), with respect to the spreading of sma dropets, and recenty by Chen and Hwang (26), who rendered a noninear anaysis of fim stabiity in the ubrication approximation. Effects of short-range interactions (acid base, hydrophobic, etc.) have been recenty studied by Neimark (30) with appications to equiibrium configurations of iquid dropets and contact anges on fibers. A new equation for contact anges on cyindrica fibers has been derived to account for the fiber curvature and the iquid soid interactions expressed in terms of the disjoining pressure isotherm. In this paper, we anayze the stabiity of iquid fims on moving fibers that requires a better understanding of specific interactions among fim, fiber, and fowing air. This probem has important practica appications to spin finishing and fiber coating technoogies, in which fiber veocities may exceed 100 km/h (31). We show that hydrodynamic interactions between the coating fim and the surrounding air cause an additiona source of instabiity, responsibe for fim destruction, dropet formation, and singing. 3 To the best of our knowedge, this effect, which we have named the wind effect, has not been considered in the iterature so far. We deveop a genera theory of the stabiity of fims on moving fibers for different types of fuid soid interactions, and formuate conditions of thermodynamic and hydrodynamic equiibrium. Stabiity diagrams and asymptotic estimates are derived for different adhesion and hydrodynamic regimes. 3 The term singing is used in the fiber industry to describe the dropet break-away from the moving fibers. FIG. 1. Disturbances of fim surface and schematic expanation of the Kevin Hemhotz mechanism of fim instabiity. II. THERMODYNAMIC STABILITY OF LIQUID FILMS ON FIBERS Let s first discuss conditions of thermodynamic stabiity of fims on motioness fibers. Lapace pressure across a curved interface depends on two principa radii of curvature, P L 1 R 1 1 R 2. [3] For a uniform fim on a cyindrica fiber, R 1, R 2 R f h, and Eq. [3] reduces to Eq. [1]. Consider the perturbation h h 0 ( z, t) of a gas iquid interface; here h 0 is the thickness of the undisturbed uniform fim (Fig. 1). A disturbance of the iquid gas surface causes oca changes in the capiary pressure. Using genera differentia forms for the principa radii of curvature, the Lapace pressure is expressed through the second and first derivatives of surface dispacement (32 34): P L 1 2 3/ 2 1 R f h / 2 R f h 0 R f h 0 2. [4] To suppress the interface disturbances, the upward (positive ) surface dispacements have to cause a oca increase, and correspondingy, the downward (negative ) dispacements have to cause a oca decrease in the iquid pressure. In other

3 INSTABILITY OF LIQUID FILMS ON FIBERS 383 words, to ensure stabiity, the iquid pressure disturbances, represented by the second term in the RHS of Eq. [4], must have the same sign as the surface disturbances, 2 0. [5] R f h 0 For harmonic perturbations, A cos kz (Fig. 1), the stabiity condition [5] reads as k 2 1 R f h [6] Inequaity [6] is the cassica Pateau criterion of jet stabiity (32). For thin fims, Eq. [3] is modified by the disjoining pressure, (h), which acts in a fim in addition to the capiary pressure (1 3), P P g 1 R 1 1 R 2 h. [7] For an undisturbed uniform fim, Eq. [7] reduces to Eq. [2]. For a disturbed fim, the iquid pressure disturbance depends on variations of the fim curvature and of the fim thickness, P 2 h. [8] R f h 0 h hh 0 Correspondingy, the stabiity criterion, inequaity [6], written for the wave numbers, is modified as k 2 R f h R f h 0 2 h hh 0 Here, the thermodynamic stabiity factor, k 2 R f h [9] 1 R f h 0 2 h 0, [10] is introduced. The modes are eveed out if condition [9] is fufied. Thus, the adhesion expands the range of admissibe perturbations. Moreover, when the thermodynamic factor is negative, any infinitesima disturbance is suppressed due to the action of the fuid soid interactions independent of the waveength (7). For thick fims, the dispersion forces are dominant in fim substrate couping. The dispersion component of the disjoining pressure is (h) a 2 HL /h 3 (7); the constant a HL is the modified Hamaker Lifshitz constant, a HL O (Å) (3, 8). From the criterion of thermodynamic stabiity, [9] and [10], it foows that the fims of thickness h 0 h c 3 1/4 a HL R f are stabe (7). An estimate gives h c 0.1 m for fibers of R f 100 m. Fims of thickness h 0 h c are unstabe with respect to disturbances of waveength, 2/k 2(R f h 0 )/ 2R f /1 (h c /h) 4 (7). The above anaysis concerns the motioness fiber. If the fiber is moving, hydrodynamic interactions between the fim and the surrounding gas cause interfacia disturbances, which may ead to destruction of thermodynamicay stabe fims. III. HYDRODYNAMIC STABILITY: THE PROBLEM FORMULATION Consider a fiber moving with veocity U in the direction. For fiber coating appications such as spin finishing the foowing data are typica: R f 10 m, U 10 4 cm/s (31). Then, the Reynods number for air Re g UR f g / g is arge, on the order of 10 2 ( g 10 3 g/cm 3 is the gas density and g 10 4 ps is the gas viscosity), so that the gas can be considered inviscid. The hypothesis of negigibe air friction is aso supported by the kinetic reasons. Since the free path ength in the air at the atmospheric pressure is on order of 0.1 m, the therma excitations shoud wash out the boundary ayer whose characteristic size is estimated as R f /UR f g / g 1 m (34). Under these fow conditions, the characteristic time 1/ of growth of a wave with a ength comparabe with the fiber radius is much smaer than the diffusion time of propagation of associated vortices in the fim. Therefore, the strong inequaity / /k 2 takes pace, where is the fuid density and is the fuid viscosity. For exampe, for water soutions/ suspensions on typica textie fibers, k 1/R f 10 3 cm 1, U/R f 10 4 /10 3 s 1, 10 2 ps, 10 3 g, and we have got /k / / 10 2 cm 2 /s. If the waveength is smaer, k 1/h 0, both times are comparabe. Thus, when the Reynods number for a iquid fim Re Uh 0 / is arge, the viscous terms in the Navier Stokes equations may be negected, so that both fuids, iquid and gas, can be considered inviscid. For the above-mentioned parameters, the range of fim thicknesses acceptabe for the purey inviscid mode is estimated as Re Uh 0 / (h 0 /R f ) (UR f / ) 10 3 (h 0 /R f ) 1. When the fim is so thin that this strong inequaity does not hod, the viscous effects in the fim must be considered. However, the gas fow can be aways treated in the inviscid approximation independent of the fim thickness. In case of inviscid fuids, we dea with a so-caed Kevin Hemhotz mechanism of interfacia instabiity (Drazin and Reid (33), Landau and Lifshitz (34)). The Kevin Hemhotz mechanism is caused by inertia and is not reated to interfacia tension and ratio of densities. The Kevin Hemhotz instabiity is schematicay iustrated in Fig. 1, where the reference

4 384 KORNEV AND NEIMARK frame is chosen as moving with veocity U/ 2 aong the fiber, so that the phases are moving with equa veocities, U/2, in opposite directions. In case of harmonic perturbations of the interface, the fuid is acceerated at the convex regions of the deformed interface, points A, A, and A, and deceerated at the concave regions, points B, B, and B. In accordance with the Bernoui equation, the smaer the speed, the greater the pressure. Thus, perturbations of the interface cause changes in veocity that in turn induces changes in pressure. The resuting pressure differences across the interface are directed so that they tend to ampify the initia disturbances. Consider the instabiity mechanism in fims on fibers in more detai without using the inviscid approximation for the iquid fim fow in the initia equations. In the reference frame attached to the moving fiber the basic undisturbed fow is as foows. In the region of fim R f r R f h 0 (h 0 is the fim thickness in the unperturbed state), the viscous fuid of density g remains at rest. In the region r R f h 0, the inviscid gas moves with a uniform veocity U in the direction. Gravity is negected. Then the system of equations for the fow perturbations takes the foowing form. Equations for the Fim Fuid The fim fuid is considered as incompressibe and satisfies the equation of continuity 0. [11] We restrict ourseves to the initia stage of instabiity evoution, and therefore the inearized Navier Stokes equations may be used for description of the fow pattern within the fim, t p. [12] Using the formua from vector anaysis, ( ) cur(cur) (34), and accounting for the incompressibiity condition [11], we can represent the veocity fied as a sum of an irrotationa veocity fied and a rotationa one, u V. The irrotationa veocity fied admits treatment in terms of a potentia with u. The rotationa components of the veocity fied obey the incompressibiity condition [11] and the foowing equations (34): V r t V r V 2 r, [13] r V z t V z. [14] The suggested subdivision of the veocity fied aows us to treat the fow pattern as caused separatey by inertia and viscous forces. No-sip at the soid boundary of the iquid annuus, r R f, gives z V z 0, [15] r V r 0. [16] Equations for the Surrounding Gas Because of a high veocity regime of gas fow, the gas compressibiity may be appreciabe. So, we write the continuity equation for air in a genera form (34), g t g g 0. [17] Subdividing the pressure and air density into the background vaues and perturbations as p g g U 2 /2 p, g g, we can use for perturbations the equation of state (34) p c 2, [18] where c is the sound veocity. Substituting Eq. [18] into Eq. [17], we obtain after inearization p t c 2 g g 0. [19] Substituting in Eq. [19] the characteristic scae for pressure perturbations, g U 2 / 2, the characteristic time, R f /U, and the characteristic scae for veocity gradients, U/R f, we concude that the compressibiity effect is important when the asymptotic equation U 2 /c 2 O (1) hods. Using for an estimate U 10 4 cm/s and c cm/s, we find the vaue U 2 /c Thus, we sti may treat the gas as incompressibe, but the safety factor is sma. Beow, we use the continuity equation in the form g 0. [20] Because of arge Reynods numbers, we may mode the fow of air as the fow of an inviscid fuid. The veocity fied g U w g obeys Euer s equations, g t g g 1 p g, [21] g and the boundary condition at infinity; namey, the perturbation of gas veocity must vanish at infinity, w g 3 0, as r 3. [22]

5 INSTABILITY OF LIQUID FILMS ON FIBERS 385 We aso assume that the initiay disturbed fow is irrotationa and suppose the veocity potentia g with g U g to exist. Then, the incompressibiity condition [20] is reduced to the Lapace equation for the veocity potentia. Thus, both gas and iquid veocity potentias i, i, g obey the Lapace equation (we suppose that the motion is axiay symmetric, i.e., that a the derivatives with respect to ange theta, in cyindrica coordinates, are zeros): 1 r r r i r 1 r 2 2 i 2 2 i 2 0. [23] Couping at the Fim Gas Interface Since the viscosity ratio g / is a sma parameter, we can ignore the shearing action of air, r z z r g g r g z z r 1. [24] Despite the seemingy kindred statement for ordinary fows of inviscid fuids, both parts of the veocity fied are connected by the boundary conditions at the interface and fiber surface. We ascribe a variations of the pressure fied to those caused by the irrotationa component of the veocity fied. Hence, the pressure can be obtained by using Bernoui s theorem as t p 0. [25] Simiary, inearization of Bernoui s equation by negecting products of the increments gives the gas pressure at the interface as g g U g t p g 0. [26] The kinematic condition that each surface partice remains on the surface gives U t g r, [27] t r V r. [28] Since the tangentia stress at the free surface of iquid is zero, Eq. [24], we have 2 2 zr V r z V z 0. [29] r We aso have to take into account that the norma component of the stress tensor jumps by the Lapace and disjoining pressures at the interface, r r 0 R f h 0, p g r z 2 V r p h 2 hh 0 r. [30] Thus, the hydrodynamic mode for stabiity anaysis consists of the cosed system of inear equations [11], [13], [14], [23] subjected to the boundary conditions [22], [24] [30]. IV. SOLUTION In stabiity anaysis we use the method of norma modes (33, 34), whereby sma disturbances are resoved into harmonic modes, which may be treated separatey. Consider a typica wave component with, t Ae itk, i r, z, t R i re itkz, i g, ; V j r, z, t V j re itkz, j r, z; [31] p i r, z, t P i re itkz, i g,. [32] Here k is a rea positive wave number. A mode with Im 0 wi be ampified, growing exponentiay with time unti it becomes so arge that the noninear effects come to the forefront. If Im 0, the mode is said to be neutray stabe, and if Im 0, asymptoticay stabe or stabe (33). Equations [13], [14], and [22] with assumed t and z variations, Eqs. [31] and [32], are reduced to the system of Besse equations d 2 V r dr 2 d 2 R i dr 2 1 r d 2 V z dr 2 1 r dr i dr k 2 R i 0, [33] dv r dr 2 V r V r r 2 0, [34] 1 r dv z dr 2 V z 0, [35] where 2 (i / ) k 2. The soution to the Besse equations [33] [35] has the foowing form: R i r c i I 0 kr d i K 0 kr, V z r a I 0 r b K 0 r, V r r e I 1 r f K 1 r. [36] Here, I n ( z), K n ( z) are the modified Besse functions of order

6 386 KORNEV AND NEIMARK n, and a, b, c i, d i, e, f are constants to be determined by the boundary conditions. Appication of the boundary conditions [22], [24] [30] eads to a system of four inear agebraic equations, which has a nontrivia soution provided that the determinant of the coefficients vanishes. The determinant represents the desired dispersion reation I 1kR f I 1 R f K 1 kr f K 1 R f ki 0 kr f I 1 R f kk 0 kr f K 0 R f D, k 2k 2 I 1 kr 0 2 k 2 I 1 r 0 2k 2 K 1 kr 0 2 k 2 K 1 r 0 d 41 d 42 d 43 d [37] Here, and d 41 2 I 0 kr 0 k, ki 1 kr 0 i ki 0 kr 0 I 2 kr 0, d 42, ki 1 r 0 i I 0 r 0 I 2 r 0, d 43 2 K 0 kr 0 k, k K 1 kr 0 i kk 0 kr 0 K 2 kr 0, d 44, k K 1 r 0 i K 0 r 0 K 2 r 0,, k g ku 2 K 0kr 0 kk 1 kr 0 h hh 0 r kr 0 2. If we set g 0, 0, U 0, then Eq. [37] wi coincide with Goren s dispersion reation for motioness fibers (16). Equation [37] is an impicit equation with respect to frequency,, that cannot be soved expicity except for two imiting cases, the case of arge Reynods numbers and the case of highy viscous iquids. In this paper, the main effort focuses on understanding the arge Reynods number regime of fow. V. NEGLIGIBLE VISCOSITY: THE KELVIN HELMHOLTZ INSTABILITY In this section we consider the inviscid approximation that impies the strong inequaity Uh 0 / 1. The dispersion reation between the frequency and the wave number for an inviscid fuid can be derived from Eq. [37] considering the imit of vanishing. However, it is easier to dea with the inviscid equations of motion directy. The two methods must yied the same resuts. We address this question beow. Considering the inviscid mode, we have to designate a 0, b 0, e 0, f 0 in Eqs. [36]. Yieding a the resuting equations, we obtain the dispersion reation as D 0, k ku 2 g K 0 kr 0 K 1 kr 0 2 I 0 kr 0 K 1 kr f I 1 kr f K 0 kr 0 I 1 kr 0 K 1 kr f I 1 kr f K 1 kr 0 k r 0 2 kr r 2 0 h hh 0 0. [38] The two first terms in the LHS of Eq. [38] are positive for any r 0, R f. Indeed, the properties of the Besse functions impy that because r 0 R f, for any wave number k, we have I 1 (kr 0 ) I 1 (kr f ) and K 1 (kr 0 ) K 1 (kr f ), and accordingy, I 1 (kr 0 )/ I 1 (kr f ) K 1 (kr 0 )/K 1 (kr f ). The expression in the parentheses in the ast term in the LHS of Eq. [38] is positive for stabe fims (see the stabiity condition [9]), which impies that the third term in the LHS of Eq. [38] is negative for some disjoining pressure isotherms and wave numbers corresponding to the adhesion hindered perturbations. Thus, Eq. [38] has at east one root, even in the case of a motioness fiber. Motioness Fibers For motioness fibers, U 0, the soution of Eq. [38] corresponds to aowed capiary waves with the frequency c f c kr 0, kr f, a k 2 r 0 kr r 2 1/ 2 0 h hh 0. [39] The frequency depends on the thermodynamic dispersion reation, Eq. [9], and the hydrodynamic factor f c kr 0, kr f, a a K 0kr 0 K 1 kr 0 I 1/ 2 0kr 0 K 1 kr 0 I 1 kr f K 0 kr 0, [40] I 1 kr 0 K 1 kr f I 1 kr f K 1 kr 0 where a g / (1).

7 INSTABILITY OF LIQUID FILMS ON FIBERS 387 In the imiting case of utrathin fims, h 0 R f, Eq. [40] can be simpified. Taking into account the formua (35) I x K x I x K x 1/x, [41] with accuracy O(a) O(kh 0 ) 2 one obtains f c kr 0, kr f, a kh 0 [42] Thus, Eq. [42] has the same wave number dependence as that obtained in the ubrication approximation (26), yet the thickness-dependent factors differ. Beow we address this point. Genera Case In a genera case, U 0, Eq. [38] is a quadratic equation; hence the number of roots depends on the sign of the discriminant Di k r 0 2 where kr r 2 0 h hh 0 C D ku 2 DC, C g K 0 kr 0 K 1 kr 0, D I 0 kr 0 K 1 kr f I 1 kr f K 0 kr 0 I 1 kr 0 K 1 kr f I 1 kr f K 1 kr 0. [43] For stabe modes, the discriminant Di must be positive. In the opposite case of the negative Di, Eq. [38] has two imaginary roots, 1,2, one of which corresponds to the ampifying mode. The stabiity condition can be written in the dimensioness form where kr 0 2 Wf d kr 0, R f /r 0, a 0, [44] interactions the third term in the LHS of the inequaity [44] is aways positive and pays a destabiizing roe. This term is proportiona to the Weber number. The dimensioness function f d (kr 0, R f /r 0 ) depends excusivey upon geometry (fiber radius and fim thickness) and inertia properties of fuids. When the ratio of densities is arge, a g / 1, the function f d (kr 0, R f /r 0, a) becomes density independent. Thus, the stabiity condition for moving fibers depends on two dimensioness factors, the thermodynamic factor and the hydrodynamic factor W. When the Weber number, W, vanishes, the stabiity condition [44] is reduced to the thermodynamic stabiity condition, Eq. [9]. Asymptotes Because of the compexity of the function f d (kr 0, R f /r 0, a), we cannot derive expicity the stabiity condition. However, there are two imiting cases for which Eq. [38] can be soved expicity. 1. Very short wave ength, h 0 r 0,R f ;kr 0 1. In this case, the waves represent rippes on the fim interface with waveengths much smaer than the fim thickness; parameters kr f and kr 0 are arge compared to unity. Using the asymptotic expressions (35) for the Besse functions for z 1, I () (e /2) and K () (/2)e, the dispersion reation is reduced to the foowing form: g ku 2 2 coth kh 0 k r 0 2 kr 0 2. [46] Equation [46] is simiar to the known dispersion reation for shaow waves (36), where the parameter /r 2 0 stands instead of gravity. For very short waves, h 0, the dispersion reation [46] is reduced to the ordinary dispersion reation for capiary waves, or so-caed Kevin Hemhotz waves (33, 34). A detaied anaysis of the Kevin Hemhotz waves can be found in (36), see aso (33), (34), (36). The stabiity condition is represented as the foowing inequaity: g U 2 r 0 2 g. [47] The critica wave number must exceed 1/h 0, and f d kr 0, R f /r 0, a kr 0 DC/ g C D [45] k cr r 0 1 h 0. [48] W U 2 g r 0 / is the Weber number; it measures the intensity of inertia forces reative to capiary ones. Comparing with the stabiity condition for motioness fibers, inequaity [9], we see that due to hydrodynamic In other words, the Kevin Hemhotz waves form ony for quite stabe fims, actuay, if the condition (r 0 /h 0 )is fufied. For van der Waas fims the ast condition is modified to a HL 3 h 0. This is an upper estimate for the fim thickness aowing the propagation of short waves. For organic fuids on dieectric fibers the modified Hamaker Lifshitz constant is rather sma, a HL O (Å) (3, 8), so that the imit is not

8 388 KORNEV AND NEIMARK reaized. However, this imiting case is justified for heium fims on quartz fibers, a HL 10 7 cm (38). 2. Long waves, h 0 r 0,R f ; kr 0 1. Consider now the possibiity of forming the waves with a ength arger than the fiber radius. Using the asymptotic expansions (35) for the Besse functions for z 1, K 0 (z) n z, K 1 (z) z 1, I 0 (z) 1, I 1 (z) z/2, we can empoy the foowing approximations: K 0 kr 0 K 1 kr 0 kr 0n kr 0, I 0 kr 0 K 1 kr f 1, kr f I 1 kr f K 0 kr 0 kr f 2 n kr 0, I 1 kr f K 1 kr 0 R f 2r 0, I 1 kr 0 K 1 kr f r 0 2 R f. Then, the second fraction in the LHS of the dispersion reation [38] is transformed into I 0 kr 0 K 1 kr f I 1 kr f K 0 kr 0 I 1 kr 0 K 1 kr f I 1 kr f K 1 kr 0 2 kr f 2 n kr 0 2kh 0, and the dispersion reation [38] takes on the form ku 2 g kr 0 n kr kr 0 2 c 2 n kr 0 2kh 0 k r 0 2 kr 0 2, [49] where c R f /r 0. The discriminant of this quadratic equation has the form Di Wb c 2 n n a 2 2 n b c 2 n. [50] Here kr 0, a g /, and b r 0 /2h 0. Assuming that the ratio of densities is arge, a 1, we rewrite the discriminant as Di W 2 n c 2 n b. [51] The second cofactor in Eq. [51] is positive for any 1, i.e., within the range of vaidity of our approximations. Hence we have to determine a positivity condition for the first cofactor ony. The critica condition Di 0 is expressed as n 1 W 2 1. [52] The root of Eq. [52] first appears when the ogarithmic branch in the LHS touches the hyperboic branch in the RHS. The atter eads to the foowing system of equations: n 1 W 1, [53] 2 W W. [54] Inequaity [53] means that the smaer the stabiity factor, the smaer the admissibe Weber number. An asymptotic anaysis of Eq. [54] can be done in terms of the variabe (2/W). Introducing the atter, Eq. [54] is rewritten in the form n, [55] where 1 n() is a arge parameter. It can be proved that the soution to Eq. [55] has the asymptotic form (37) 1 k0 c km n m km, m1 in which c Substituting this expression in the definition of, we obtain the asymptotic behavior of the imiting vaue of the Weber number, W, as a function of the thermodynamic factor, ( 1): W 2 n 1 n1 n n1 n o n 1. [56] It is noteworthy that even for very sma absoute vaues of the thermodynamic factor the critica Weber numbers are appreciabe. That is, the critica Weber number has a perceptibe vaue of W 0.26 when the thermodynamic factor is as sma as Consider for exampe organic iquid fims on 10 m fibers. The critica fim thickness is estimated as h c 0.1 m at 30 dyne/cm (7, 8), and the thermodynamic factor corresponds to the fim thickness h h c h c /4. These fims become unstabe when the fiber veocity exceeds the critica vaue U 0.26/R f g 10 3 cm/s. Thus, the adhesion properties of fibers pay a dominant roe in the prevention of ong waveength instabiity in the high veocity spinning processes. VI. THE EFFECTS OF VISCOSITY In this section we consider a criterion of vaidity of the inviscid fuid approximation and discuss physica mechanisms of the fim instabiity of thin fims assuming that h 0 R f. In the subsection

9 INSTABILITY OF LIQUID FILMS ON FIBERS 389 on stabiity conditions, we derive viscosity-induced corrections to the stabiity criterion obtained in the inviscid fuid approximation (Eqs. [44], [45]). The subsection on characteristic times contains resuts on characteristic times of growth of unstabe waves. Consider the cases for which the moduus of the compex attenuation constant is smaer than the growth rate, i.e., when the diffusion time of propagation of vortices, diffusion /k 2, is greater than the characteristic time of wave ampification. The atter can be roughy estimated from Eq. [39] as wave 1/k/ r 0 2. For arge Reynods numbers and ong waves the asymptotic equaity wave r 0 /U diffusion hods, thus indicating that the inertia forces dominate the viscous ones. Hence, the attenuation constant can be cacuated assuming that the veocity profie is simiar to that for an inviscid fuid (see the Appendix). When the fiber veocity is sma, U r 0 / diffusion, the viscous forces dominate the inertia ones and the inviscid mode becomes ineigibe. Stabiity Conditions Consider expansion of the dispersion equation [37] in powers of. The inviscid approximation, Eq. [38], is the principa, viscosity-independent term in this expansion. We expect that for sufficienty arge Reynods numbers, Re Uh 0 /, a viscous correction to the critica frequency is sma. Indeed, the first correction to Eq. [38] is D 1, k I 1kR f K 1 kr f d 41 d 43 0, which can be represented as D 1, k D 0, k 2i k 2 I 1kR f K 1 K 1 kr f I 1 0. [57] K 1 kr f I 1 I 1 kr f K 1 Let 0 be a root of D 0 (, k) 0. For arge Reynods numbers we may take 0, where is a sma attenuation constant. Then, is determined from D 0, k 0 or 2i 0 k 2 I 1kR f K 1 K 1 kr f I 1 K 1 kr f I 1 I 1 kr f K 1, 2i 2 0 k I 1kR 2 f K 1 K 1 kr f I 1 K 1 kr f I 1 I 1 kr f K D 0, k 0. [58] The numerator of Eq. [58] is positive due to the properties of the modified Besse functions: functions I i ( z) monotonicay increase and their derivatives are positive; functions K i ( z) monotonicay decrease and their derivatives are negative. When the denominator is negative, the imaginary part of is negative for rea 0, which eads to instabiity. This fact provides another method for cacuations of dangerous fow regimes. Soution of the dispersion reation [43] gives 0 kuc C D Di C D, [59] where the discriminant, Di, and constants, C and D, are defined by Eq. [43]. Substituting Eq. [59] into denominator of Eq. [58], we obtain 0 2 D 0, k 0 0 k r 0 2 kr r 2 0 h hh 0 C D ku 2 DC. [60] The sign in Eq. [60] corresponds to that taken in Eq. [59]. Thus, the denominator in Eq. [58] is negative when the frequency of the mode corresponding to the negative sign in Eq. [59] is positive. The reation for the critica wave number (or critica Weber number) is 0 0 (negative branch); i.e., k*uc 2 Di*. Using the dimensioness variabes, the atter takes the form 2 Wa K 2 0 Di, a, W 0, [61] K 1 where we introduced the dimensioness discriminant as Di, a, W r 3 0 Di. After simpe agebra, Eq. [61] is rewritten as W 1 2 K 1 K 0. [62] The dangerous mode first appears just after the ower branch of

10 390 KORNEV AND NEIMARK the soution [59] tangents the k-axis. This condition gives the foowing constraint: Wa d d K 2 0 ddi, a, W K 1 d d d W K 0 K [63] Note, neither the thermodynamic factor nor the Weber number depends upon the ratio of densities. Equations [62], [63] provide an expicit parameterized form of the W stabiity diagram (the parameter is the critica wave number). Generay speaking, the critica conditions cacuated by using Eqs. [62], [63] differ from those obtained by anayzing the roots of the discriminant (cassica Kevin Hemhotz anaysis). Actuay, the presence of the first terms in the LHS of Eqs. [61] and [63] makes it possibe to distinguish the Kevin Hemhotz instabiity from the discussed one. The difference is significant ony if the compex Wa becomes of the order of unity or greater. For a reasonabe veocity range, no more than U 10 4 cm/s, this may be appicabe for the thick fibers ony. Indeed, the ratio of densities cannot vary in a broad range and, as a rue, fuctuates around Due to their sma surface energy, the apoar iquids have more chances to undergo the Kevin Hemhotz instabiity than the poar ones. Therefore, underestimating the surface tension by the vaue 10 dyne/cm one gets that the fiber radius shoud be 1 cm or greater. But the associated Reynods number fas into the range of turbuence, so that the mode becomes inappicabe for such thick fibers. At the same time, the difference between the criteria for the Kevin Hemhotz instabiity and those for the viscosity induced instabiity coud be significant near the phase transition where both phases have amost the same density and the surface tension is very sma. Thus, the instabiity may arise for the gas veocities smaer than the critica veocity of the Kevin Hemhotz instabiity discussed in Section V. Taking into account formua [41] and eiminating the Weber number in Eq. [63], the critica thermodynamic factor can be represented in eading order as 2 2 K 1 2 K K K 2 0 K 1 1. [64] K 0 Equations [62] [64] aow us to eiminate the wave number and pot a stabiity diagram in terms of the thermodynamic stabiity factor,, and Weber number, W (Fig. 2). For thin van der Waas fims, h 0 R f, we arrive at the reation between the critica fim thickness and the critica wave number h c W h c W 3 1/4 a HL R f W, [65] FIG. 2. Stabiity diagram. where (1 ) 1/4, and the thermodynamic factor,, is defined by Eq. [64]. For motioness fibers, Eq. [65] reduces to the critica fim thickness of thermodynamicay stabe fims (7); see Section II. To resume, there are two types of unstabe modes: the first corresponds to the Kevin Hemhotz waves with the negative imaginary part of 0, whie the second type constitutes the waves with the zeroth imaginary part of 0 and with the negative imaginary part of the viscous correction. The second type of waves are transformed into the Kevin Hemhotz waves of the first type when the two branches of Eq. [59] intersect. The Kevin Hemhotz instabiity is affected by the ratio of fuid densities, whie the viscosity-induced instabiity is not. Characteristic Times Consider the characteristic break-up time of unstabe modes. The characteristic growth rate (or break-up time T) can be represented as 1 T 1 1, for the Kevin-Hemhotz waves, [66] 1 T 1, for the viscosity-induced waves. [67] Here, is the moduus of the imaginary part of Eq. [59], and is the moduus of Eq. [58] in which ony the imaginary part of 0 is retained. One can see that the fuid viscosity is a stabiizing factor for the Kevin Hemhotz waves, and a destabiizing one for waves of the second type. As shown in the

11 INSTABILITY OF LIQUID FILMS ON FIBERS 391 Appendix, the denominator of Eq. [58] is the wave or excess energy, consisting of the potentia energy caused by the capiary and surface forces, as we as of the kinetic energy of moving perturbations. The excess energy becomes negative when the wave excitation owers the tota energy of the unperturbed moving system (Pierce (40), Bendjamin (41), Cairns (42), Stepanyants and Fabrikant (37)). In the case under consideration, the energy becomes negative whenever the fim thickness osciations are in phase with the fow veocity osciations. An interpretation of the physica mechanism of viscous destabiization is given in the Appendix. Under practica conditions of fim coating, the fim thickness may be inconsistent with its vaue in thermodynamic equiibrium. That is, the fim is basicay unstabe even for motioness fibers. Thus, the Kevin Hemhotz instabiity is more important for technoogica appications, and we turn our attention to this cass. Consider some properties of the Kevin Hemhotz waves. First, et s find a criterion of dynamic stabiity of thin fims. Comparing the moduus of 0 and, we see that our theory is appicabe if and ony if the inequaity / 1 hods. Otherwise, the viscous forces dominate the inertia ones and the fim stabiity is expected. The wave number k * of the fastest growing mode corresponds to the maximum of the imaginary part of 0 with respect to the wave number k. This maximum aways exists because the interva of wave numbers admissibe for instabiity has an upper boundary equaed to the critica wave number. For arger wave numbers (smaer waveengths), the capiary forces suppress the excitations. Using that the parameter (1 c)/c h 0 /R f is sma and accounting for Eq. [41], the characteristic break-up time can be written from Eq. [59] as where and f W, R f r 0 3 /h 0, [68] f W, f 2 f 2 f 2 1 K 0 f, u K 1 f [69] u 2 K 0 f K 1 f f 2 f 2 K 2 0 f. [70] K 1 f In Eqs. [68] [70], f is the critica dimensioness wave number of the fastest growing mode. The f is a positive soution of the equation W 2 f 2 /u f. [71] Invoking asymptotic properties of the modified Besse functions and taking into account that the function K 0 ()/K 1 () is monotone, one obtains that u, Eq. [70], varies monotonicay from zero to 3/2. Therefore, a positive soution to Eq. [71] exists in the interva f [/2, ). Hence, if the fiber is motioness, the fastest growing mode corresponds to the wave number /2 (this concusion aso foows from Eqs. [39], [42]). If the fim is so thick that the disjoining pressure can be negected ( 1) yet the inequaity h 0 R f hods, the critica wave number is f 1/ (see aso Eqs. [39], [42]). It is a itte bit greater than Rayeigh s capiary mode for jets, Rayeigh (Rayeigh (27)). The greatest vaue of the dimensioness break-up time is f (0, 1) 2. Thus, for thin fims h 0 R f, the time of rippe growth is inversey proportiona to the square root of the fim thickness, yet it is impicity affected by the fim thickness via the thermodynamic factor. Assuming for a moment f O (1) (motioness fibers), 30 dyne/cm, R f 10, and 1 g/cm 3, one obtains an estimate for the characteristic break-up time as R f /h s. In the same case, h 0 R f, the characteristic time of wave damping caused by viscous forces is estimated as W, 2 r 0 R 2 f r 0 3. [72] h 0 f Thus, in the imiting case h 0 R f, the time of viscous damping depends on the fim thickness as R f /h 0 and via the thermodynamic factor as we. This dependence differs significanty from the expected reverse cubic aw of Boussinesq and Reynods due to a different physica situation. In the ordinary ubrication theory one suggests that the fim fuid works out the Poiseuieian profie to satisfy the no-sip and no-stress boundary conditions at the fiber surface and the iquid gas interface, respectivey. But for arge Reynods numbers the physica scheme of fuid fow is akin to that observed at the sudden stretching of a nyon stocking. Amost a the partices have the same veocity and ony a very sma transition ayer forms a Poiseuieian profie. In our case the stretching force is the compressing hydrauic head, F extension 2R f U 2, driving the fim to extend. Because of viscosity, the iquid fim resists stretching with the force F friction 2R f h 0 U/R f. Accounting for both forces and assuming U R f /, R f, one gets the dimensiona factor in Eq. [72]. For f O (1), R f 10, 1 g/cm 3, and 10 2 ps, we obtain the estimate R f /h s. The estimate shows that the inequaity / 1 hods for any water fims with reasonabe thickness; i.e., they are unstabe. It shoud be noted that Eq. [72] hods true for the viscosityinduced waves as we, provided that f ies in the aowed range specified by the inequaities 0 0 ( f, W, ) 0 ( f, W, ), where the pus and minus subscripts denote the corresponding branches of Eq. [59].

12 392 KORNEV AND NEIMARK FIG. 3. The correction factor as a function of the Weber number W. VII. NUMERICAL RESULTS AND DISCUSSION Anayses of the stabiity diagram can be done numericay. Fortunatey, when the ratio of densities is arge, a 1, the soutions of Eqs. [62] and [52] are practicay indistinguishabe for ong waves. For short waves, asymptotic anaysis of Eqs. [62], [63] for any ratio of densities can be found esewhere (36), (37). However, when a 1, the soution coincides with Eq. [47] with accuracy O(a). Therefore, numerica anayses may be performed in a finite range of wave numbers and then anayticay continued by asymptotes. A numerica soution of Eqs. [62], [63] is potted in Figs. 2 and 3. On moving fibers, the thermodynamicay unstabe fims characterized by the positive factor (inequaity [9]) remain unstabe. The iquid air interactions enhance interfacia fuctuations and can drive the thermodynamicay stabe fims, characterized by the negative factor, to the edge of stabiity. The ine W c () divides the pane of parameters into two regions, the stabe region beow and the unstabe region above. W c () gives the critica vaue of the Weber number that corresponds to the appearance of hydrodynamic instabiity in the primariy stabe fims. As approaches zero, W c () diminishes. For the van der Waas fims, the thermodynamic factor varies from 1, for sufficienty thick fims whose behavior is governed predominanty by the destabiizing action of capiarity, to negative vaues corresponding to the thermodynamicay stabe fims. We anayzed how the hydrodynamic interactions affected the stabiity conditions for the van der Waas fims. In Fig. 3, the correction factor, Eq. [65], is potted as a function of the Weber number. The critica thickness of thermodynamicay stabe fims on motioness fibers was estimated in Section II, and it corresponds to W 0. As W increases the critica thickness decreases, which refects the enhancement of interface disturbances due to the fim air interaction. The reduction of the critica fim thickness may be significant. For exampe, assuming typica vaues for parameters, a HL 1Å,R f 10, 30 dyne/cm, we obtain that the critica thickness for motioness fibers is h c 42 nm, whie the critica thickness reduces to h c 27 nm when the fiber speed becomes U 10 4 cm/s. The condition of thermodynamic stabiity shows that the east critica wave number is the zeroth one; i.e., the fim as a whoe oses its uniformity as the thermodynamic factor approaches zero, Eq. [9]. In contrast, the critica conditions of hydrodynamic stabiity impy that the critica waveength and, correspondingy, the critica wave number are finite (Fig. 4). At the same time, the break-up time of the critica mode is infinite. In other words, there is a barrier in waveengths of stabe periodic unduations that separates dynamicay stabe unduations from unstabe ones. The competition among the imposed pressure head, capiary pressure, and adhesion properties of fibers seects aowed modes. The break-up time of unstabe fims is aso affected by the fuid soid interactions expressed in terms of a disjoining pressure isotherm. However, since the growth of instabiity begins just after the fim is deposited, we can put for estimate 1. The atter impies that the disjoining pressure can be ignored for initia instants of time. In Figs. 5 and 6 we pot the characteristic time of wave ampification and the characteristic time of wave damping in dimensioness form. Using these diagrams, we can assess the critica fim thickness as h 0 /R f U cap (W, 1)/U vis f (W, 1), where U cap / R f is the characteristic veocity of capiary waves and U vis / R f is the characteristic veocity of propagation of disturbances in viscous fuids. If the fim thickness is arger than this vaue, the apparent veocity of capiary waves is greater, and the fim breaks down. And vice versa, for thin fims the viscous forces FIG. 4. The correction factor and the critica Weber number W c as functions of the dimensioness wave number.

13 INSTABILITY OF LIQUID FILMS ON FIBERS 393 FIG. 5. number. Inertia time and Weber number as functions of the critica wave dominate the others and stabiize the interface. Accounting for typica physica parameters, we see that the viscous damping is important ony for short waves, whie the arge drops grow practicay nondamped. Due to the difference in physica causes of instabiities, the ubrication approximation predicts that the characteristic time is inversey proportiona to fim thickness cubed, 1/h 3 0 (Yarin et a. (25), Chen and Hwang (26)), whie in the discussed regime the time is inversey proportiona to fim thickness 1/h 0. Thus, the characteristic fow regimes can be distinguished by measuring the break-up time and fim thickness. fibers, the hydrodynamic effects enhance the interfacia instabiities and impose more severe conditions of fim stabiity. When the initia fim is thermodynamicay unstabe (0), hydrodynamic fow affects the growth rate and critica waveength. When the initia fim is thermodynamicay stabe ( 0), two types of unstabe modes are shown to be caused by hydrodynamic factors: the first corresponds to the Kevin Hemhotz waves induced by inertia effects; the second, induced by viscosity effects, is observed at smaer (yet sti arge) Reynods numbers when the Kevin Hemhotz instabiity is suppressed. The magnitude of fim gas hydrodynamic interactions is quantified by the Weber number, W, that measures the intensity of inertia forces reative to capiary forces. As the Weber number increases the critica thickness of stabe fims decreases and the characteristic break-up time for unstabe fims reduces substantiay. We have demonstrated that the aowed Weber numbers may be arge if and ony if the negative adhesion factor is arge. In this case, the instabiity ooks ike a rippe formation. If the negative adhesion factor is sma, the resuting instabiity eads to the formation of drops of characteristic size on the order of the fiber radius. We have derived conditions of fim stabiity and anayzed the unstabe modes in terms of two dimensioness parameters, the adhesion factor,, and the Weber number, W. The Reynods number, Re, determines the regions of vaidity of different approximations. Linear stabiity anaysis of the system of couped hydrodynamic equations of fim and gas fow has been performed by the method of norma modes. We derived the genera dispersion reation as an impicit equation with respect to the mode frequency. In the imit of vanishing fiber veocities, the obtained equation provides the dispersion equation for motioness fibers. VIII. CONCLUSIONS The stabiity of iquid fims on moving fibers has been anayzed with a focus on effects caused by fim soid adhesion interactions and hydrodynamic interactions between the fim and surrounding gas. We have shown that at high fiber veocities the fim gas hydrodynamic interactions induce instabiity in the fims, which woud be stabiized by the adhesion interactions at static conditions. This effect, specific for moving fibers, has been named the wind effect. The wind effect may ead to fim destruction and dropet formation, which are negative factors in various technoogies reated to fabrication and chemica modification of fibers and fiber products, e.g., spin finishing and ubrication. The fim fiber adhesion interactions are expressed in terms of the thickness-dependent disjoining pressure, which contros conditions of thermodynamic equiibrium on motioness fibers and determines the adhesion factor of thermodynamic stabiity,. The adhesion factor,, impies the upper imit of stabe fim thicknesses and wave engths of dangerous perturbations for a given iquid surface tension and fiber diameter. In the case of moving FIG. 6. number. Inertia and viscous times as functions of the critica wave

14 394 KORNEV AND NEIMARK This equation represents a generaization of the dispersion reation derived by Goren (16), in which the disjoining pressure effects were not considered. In the arge Reynods number regime, the obtained genera dispersion reation is expanded in the power series of inverse Reynods numbers, and is soved in the first order. The principa term of the series corresponds to the inviscid approximation. The expicit forms of the dispersion reation have been derived in two imiting cases of very short (much smaer than the fim thickness) and very ong (much arger than the fiber radius) waveengths. In the inviscid approximation, the fim instabiity is simiar to the Kevin Hemhotz mechanism of formation of capiary waves. The first correction refects viscosity effects in the growth rate of unstabe modes. Viscosity effects are essentia for veocities smaer than the critica veocity of the Kevin Hemhotz instabiity obtained in the inviscid approximation. Increasing the fiber veocity, we first encounter the fim instabiity caused by coherence in osciations of fim thickness and fow veocity. Connected via the no-sip boundary condition, viscous and potentia components of veocity thus transfer the kinetic energy of moving fiber into the potentia and kinetic energy of the interface. Thus, the fuid viscosity pays here a critica roe. For thin fims, with thickness much ess than the fiber radius, we have obtained the expicit formuas for the borders of stabiity expressed via the adhesion factor, Weber number, and critica waveength. For the van der Waas fims, the critica thickness is represented as a product of its thermodynamic vaue, characteristic for motioness fibers, and a correction factor, which depends upon the Weber number. The growth rates of the fastest growing modes are estimated for both types of unstabe modes. In contrast to the ubrication approximation, in which the break-up time is inversey proportiona to the fim thickness cubed (26), we have found that the break-up time is inversey proportiona to the square root of thickness with a mutipier, which impicity depends on the fim thickness through the adhesion factor. The damping time is found to be inversey proportiona to the fim thickness with a mutipier, which aso impicity depends on the fim thickness through the adhesion factor. When disjoining pressure is negigiby sma and the fiber is motioness, the critica wave number is sighty greater than the Rayeigh criterion for jet stabiity. The resuts are iustrated by the estimates of the stabiity conditions for moving fibers in terms of the stabiity diagram, the critica fim thickness, the fastest unstabe mode, and the corresponding characteristic break-up time. The estimates were done for practica conditions of fiber processing. Athough the methods deveoped are appicabe to any type of iquid soid interaction, practica estimates were made for the ong-range van der Waas interactions, which ikey dominate the fim fiber interactions in many practica situations. ACKNOWLEDGMENT This work was supported by a group of TRI corporate participants. APPENDIX To eucidate formua [58], it is convenient to consider the procedure of direct cacuation of the attenuation constant used by Landau and Lifshitz (34). Reca that the attenuation constant is defined as Ṡ 2E, [A1] where Ṡ is the rate of energy dissipation due to interna friction in the fuid and E is the wave energy. The ange brackets denote an average over the osciations. Beow, the time-fourier images have the same denotations as corresponding functions in the main text. Formua [A1] reads that the wave energy decreases with time exponentiay, and the cofactor 2 accounts for the fact that the energy is proportiona to the squared ampitude. The rate of energy dissipation for a viscous fuid can be written as Ṡ 4 2 i x k rdrdz, [A2] and the wave energy is represented as a sum of kinetic and potentia energies of the system. The kinetic energy is T g 2 dz r 0 U 2 g z U 2 2 r 2rdr r 2 0 dz R f 2 z 2 r 2rdr. [A3] The kinetic energy to second order in the veocity potentias and surface dispacement is rewritten as T g 2 dz r 0 2 g z 2 r 2rdr g 2 dz2u g z r 0 2r 0 r 2 0 dz R f 2 z 2 r 2rdr. [A4] Note that in the first and third terms of Eq. [A4], integrating by parts, the integras can eiminate z-derivatives. Integrating the remaining parts over the r-coordinate and using the boundary condition at the fiber, we can reduce the first and third integras to the integras over the unperturbed gas iquid interface. Re-