Sound effect on dynamics and stability of fluid sloshing in zero-gravity
Theoretical study of acoustic interaction affecting the dynamics and stability of limited fluid volume in zero-gravity is carried out. Two main acoustic effects on a fluid surface are analyzed. The first is the change of dynamic characteristics of fluid sloshing in zero-gravity due to acoustic loadi...
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Lukovsky, I.A. Timokha, A.N. 2008-07-21T17:29:36Z 2008-07-21T17:29:36Z 1999 Sound effect on dynamics and stability of fluid sloshing in zero-gravity / I.A. Lukovsky, A.N. Timokha // Акуст. вісн. — 1999. — Т. 2, N 3. — С. 69-83 — англ. 1028-7507 https://nasplib.isofts.kiev.ua/handle/123456789/1101 532.595 Theoretical study of acoustic interaction affecting the dynamics and stability of limited fluid volume in zero-gravity is carried out. Two main acoustic effects on a fluid surface are analyzed. The first is the change of dynamic characteristics of fluid sloshing in zero-gravity due to acoustic loading; the second is the movement of a "fluid cork" along the tube (acoustic pumping). Mathematical analysis is based on the averaging of original free interface problem. This allows to reduce a free interface problem to a free boundary problem on surface waves with additional nonlinear terms in the dynamic condition on an unknown surface. Nonlinear phenomena are described per structuring a series of analytical and numerical-analytical solutions. These examples concern the cylindrical vessel with gravity vector along the directrix and, hence, comparison of the results with solutions of capillary problem becomes available. The experimental conclusion that acoustic loads can give rise to equilibrium shapes contrasting to capillary surfaces is confirmed. Also the phenomena of acoustic stabilization and destabilization of "fluid-gas" interface are demonstrated including the case when such a destabilization causes the acoustic pumping. Проведено теоретичне дослідження акустичної взаємодії, яка визначає динаміку та стійкість обмеженого об'єму рідини у невагомості. Проаналізовано два основних типи впливу акустичного поля на вільну границю. Перший полягає у зміні динамічних характеристик плескань рідини у невагомості під впливом акустичного навантаження. Другий асоціюється у технічних застосуваннях з рухом "рідкої пробки" вздовж труби (акустичний насос). Математичний аналіз базується на осередненні вихідної задачі з вільною границею разділу двох середовищ, що дозволяє звести задачу до задачі з вільною границею про поверхневі хвилі з додатковими членами у динамічній умові на невідомій вільній поверхні. Нелінійні ефекти описуються через побудову ряду аналітичних і чисельно-аналітичних розв'язків цієї задачі. Приклади наведено для випадку циліндричної посудини, коли вектор гравітації направлений вздовж вісі циліндра, що дозволяє порівняти їх з розв'язками задачі про капіляр. Теоретичні дослідження підтверджують висновки, одержані у експериментах, про те, що акустичний вплив може призводити до положень рівноваги на границі розділу, які відрізняються від капілярних поверхонь. Окрім того, продемонстровані ефекти динамічної акустичної стабілізації та дестабілізації поверхні розділу, включаючи випадок коли дестабілізація обумовлює ефект акустичного насоса. Проведено теоретическое исследование акустического взаимодействия, определяющего динамику и устойчивость ограниченного объема жидкости в невесомости. Проанализированы два основных типа воздействия акустического поля на свободную границу. Первый состоит в изменении динамических характеристик плесканий жидкости в невесомости под воздействием акустического нагружения. Второй ассоциируется в технических приложениях с движением "жидкой пробки" вдоль трубы (акустический насос). Математический анализ базируется на усреднении исходной задачи со свободной границей раздела двух сред, что позволяет свести задачу к задаче со свободной границей о поверхностных волнах с дополнительными членами в динамическом условии на неизвестной свободной поверхности. Нелинейные эффекты описываются путем построения ряда аналитических и численно-аналитических решений этой задачи. Примеры относятся к случаю цилиндрического сосуда, когда вектор гравитации направлен вдоль оси цилиндра, что позволяет сравнить их с решениями задачи о капилляре. Теоретические исследования подтверждают выводы, полученные в экспериментах, о том, что акустическое воздействие может приводить к положениям равновесия на границе раздела, которые отличаются от капиллярных поверхностей. Кроме того, продемонстрованы эффекты динамической акустической стабилизации и дестабилизации поверхности раздела, включая случай, когда дестабилизация обуславливает эффект акустического насоса. en Інститут гідромеханіки НАН України Sound effect on dynamics and stability of fluid sloshing in zero-gravity Вплив звуку на динаміку та стійкість хлюпання рідини за відсутності тяжіння Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| spellingShingle |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity Lukovsky, I.A. Timokha, A.N. |
| title_short |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| title_full |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| title_fullStr |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| title_full_unstemmed |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| title_sort |
sound effect on dynamics and stability of fluid sloshing in zero-gravity |
| author |
Lukovsky, I.A. Timokha, A.N. |
| author_facet |
Lukovsky, I.A. Timokha, A.N. |
| publishDate |
1999 |
| language |
English |
| publisher |
Інститут гідромеханіки НАН України |
| format |
Article |
| title_alt |
Вплив звуку на динаміку та стійкість хлюпання рідини за відсутності тяжіння |
| description |
Theoretical study of acoustic interaction affecting the dynamics and stability of limited fluid volume in zero-gravity is carried out. Two main acoustic effects on a fluid surface are analyzed. The first is the change of dynamic characteristics of fluid sloshing in zero-gravity due to acoustic loading; the second is the movement of a "fluid cork" along the tube (acoustic pumping). Mathematical analysis is based on the averaging of original free interface problem. This allows to reduce a free interface problem to a free boundary problem on surface waves with additional nonlinear terms in the dynamic condition on an unknown surface. Nonlinear phenomena are described per structuring a series of analytical and numerical-analytical solutions. These examples concern the cylindrical vessel with gravity vector along the directrix and, hence, comparison of the results with solutions of capillary problem becomes available. The experimental conclusion that acoustic loads can give rise to equilibrium shapes contrasting to capillary surfaces is confirmed. Also the phenomena of acoustic stabilization and destabilization of "fluid-gas" interface are demonstrated including the case when such a destabilization causes the acoustic pumping.
Проведено теоретичне дослідження акустичної взаємодії, яка визначає динаміку та стійкість обмеженого об'єму рідини у невагомості. Проаналізовано два основних типи впливу акустичного поля на вільну границю. Перший полягає у зміні динамічних характеристик плескань рідини у невагомості під впливом акустичного навантаження. Другий асоціюється у технічних застосуваннях з рухом "рідкої пробки" вздовж труби (акустичний насос). Математичний аналіз базується на осередненні вихідної задачі з вільною границею разділу двох середовищ, що дозволяє звести задачу до задачі з вільною границею про поверхневі хвилі з додатковими членами у динамічній умові на невідомій вільній поверхні. Нелінійні ефекти описуються через побудову ряду аналітичних і чисельно-аналітичних розв'язків цієї задачі. Приклади наведено для випадку циліндричної посудини, коли вектор гравітації направлений вздовж вісі циліндра, що дозволяє порівняти їх з розв'язками задачі про капіляр. Теоретичні дослідження підтверджують висновки, одержані у експериментах, про те, що акустичний вплив може призводити до положень рівноваги на границі розділу, які відрізняються від капілярних поверхонь. Окрім того, продемонстровані ефекти динамічної акустичної стабілізації та дестабілізації поверхні розділу, включаючи випадок коли дестабілізація обумовлює ефект акустичного насоса.
Проведено теоретическое исследование акустического взаимодействия, определяющего динамику и устойчивость ограниченного объема жидкости в невесомости. Проанализированы два основных типа воздействия акустического поля на свободную границу. Первый состоит в изменении динамических характеристик плесканий жидкости в невесомости под воздействием акустического нагружения. Второй ассоциируется в технических приложениях с движением "жидкой пробки" вдоль трубы (акустический насос). Математический анализ базируется на усреднении исходной задачи со свободной границей раздела двух сред, что позволяет свести задачу к задаче со свободной границей о поверхностных волнах с дополнительными членами в динамическом условии на неизвестной свободной поверхности. Нелинейные эффекты описываются путем построения ряда аналитических и численно-аналитических решений этой задачи. Примеры относятся к случаю цилиндрического сосуда, когда вектор гравитации направлен вдоль оси цилиндра, что позволяет сравнить их с решениями задачи о капилляре. Теоретические исследования подтверждают выводы, полученные в экспериментах, о том, что акустическое воздействие может приводить к положениям равновесия на границе раздела, которые отличаются от капиллярных поверхностей. Кроме того, продемонстрованы эффекты динамической акустической стабилизации и дестабилизации поверхности раздела, включая случай, когда дестабилизация обуславливает эффект акустического насоса.
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1028-7507 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/1101 |
| citation_txt |
Sound effect on dynamics and stability of fluid sloshing in zero-gravity / I.A. Lukovsky, A.N. Timokha // Акуст. вісн. — 1999. — Т. 2, N 3. — С. 69-83 — англ. |
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ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83UDC 532.595SOUND EFFECT ON DYNAMICS AND STABILITY OFFLUID SLOSHING IN ZERO-GRAVITYI. A. LUKOVSKY, A. N. T IMOKHAInstitute of Mathematics of NAS of Ukraine, KyivReceived 12.05.99 � Revised 17.08.99Theoretical study of acoustic interaction a�ecting the dynamics and stability of limited
uid volume in zero-gravity iscarried out. Two main acoustic e�ects on a
uid surface are analyzed. The �rst is the change of dynamic characteristicsof
uid sloshing in zero-gravity due to acoustic loading; the second is the movement of a \
uid cork" along the tube(acoustic pumping). Mathematical analysis is based on the averaging of original free interface problem. This allowsto reduce a free interface problem to a free boundary problem on surface waves with additional nonlinear terms in thedynamic condition on an unknown surface. Nonlinear phenomena are described per structuring a series of analytical andnumerical { analytical solutions. These examples concern the cylindrical vessel with gravity vector along the directrix and,hence, comparison of the results with solutions of capillary problem becomes available. The experimental conclusion thatacoustic loads can give rise to equilibrium shapes contrasting to capillary surfaces is con�rmed. Also the phenomenaof acoustic stabilization and destabilization of \
uid { gas" interface are demonstrated including the case when such adestabilization causes the acoustic pumping.�஢¥¤¥® ⥮à¥â¨ç¥áª®¥ ¨áá«¥¤®¢ ¨¥ ªãáâ¨ç¥áª®£® ¢§ ¨¬®¤¥©á⢨ï, ®¯à¥¤¥«ïî饣® ¤¨ ¬¨ªã ¨ ãá⮩稢®áâì®£à ¨ç¥®£® ®¡ê¥¬ ¦¨¤ª®á⨠¢ ¥¢¥á®¬®áâ¨. �à® «¨§¨à®¢ ë ¤¢ ®á®¢ëå ⨯ ¢®§¤¥©áâ¢¨ï ªãáâ¨ç¥áª®-£® ¯®«ï ᢮¡®¤ãî £à ¨æã. �¥à¢ë© á®á⮨⠢ ¨§¬¥¥¨¨ ¤¨ ¬¨ç¥áª¨å å à ªâ¥à¨á⨪ ¯«¥áª ¨© ¦¨¤ª®á⨢ ¥¢¥á®¬®á⨠¯®¤ ¢®§¤¥©á⢨¥¬ ªãáâ¨ç¥áª®£® £à㦥¨ï. �â®à®© áá®æ¨¨àã¥âáï ¢ â¥å¨ç¥áª¨å ¯à¨«®¦¥¨ïå ᤢ¨¦¥¨¥¬ \¦¨¤ª®© ¯à®¡ª¨" ¢¤®«ì âàã¡ë ( ªãáâ¨ç¥áª¨© á®á). � ⥬ â¨ç¥áª¨© «¨§ ¡ §¨àã¥âáï ãá।¥-¨¨ ¨á室®© § ¤ ç¨ á® á¢®¡®¤®© £à ¨æ¥© à §¤¥« ¤¢ãå á।, çâ® ¯®§¢®«ï¥â ᢥá⨠§ ¤ çã ª § ¤ ç¥ á® á¢®¡®¤®©£à ¨æ¥© ® ¯®¢¥àå®áâëå ¢®« å á ¤®¯®«¨â¥«ì묨 ç«¥ ¬¨ ¢ ¤¨ ¬¨ç¥áª®¬ ãá«®¢¨¨ ¥¨§¢¥á⮩ ᢮¡®¤®©¯®¢¥àå®áâ¨. �¥«¨¥©ë¥ íä䥪âë ®¯¨áë¢ îâáï ¯ã⥬ ¯®áâ஥¨ï àï¤ «¨â¨ç¥áª¨å ¨ ç¨á«¥®- «¨â¨ç¥áª¨åà¥è¥¨© í⮩ § ¤ ç¨. �ਬ¥àë ®â®áïâáï ª á«ãç î 樫¨¤à¨ç¥áª®£® á®á㤠, ª®£¤ ¢¥ªâ®à £à ¢¨â 樨 ¯à ¢«¥¢¤®«ì ®á¨ 樫¨¤à , çâ® ¯®§¢®«ï¥â áà ¢¨âì ¨å á à¥è¥¨ï¬¨ § ¤ ç¨ ® ª ¯¨««ïà¥. �¥®à¥â¨ç¥áª¨¥ ¨áá«¥¤®¢ ¨ï¯®¤â¢¥à¦¤ î⠢뢮¤ë, ¯®«ãç¥ë¥ ¢ íªá¯¥à¨¬¥â å, ® ⮬, çâ® ªãáâ¨ç¥áª®¥ ¢®§¤¥©á⢨¥ ¬®¦¥â ¯à¨¢®¤¨âì ª¯®«®¦¥¨ï¬ à ¢®¢¥á¨ï £à ¨æ¥ à §¤¥« , ª®â®àë¥ ®â«¨ç îâáï ®â ª ¯¨««ïàëå ¯®¢¥àå®á⥩. �஬¥ ⮣®, ¯à®-¤¥¬®áâ஢ ë íä䥪âë ¤¨ ¬¨ç¥áª®© ªãáâ¨ç¥áª®© áâ ¡¨«¨§ 樨 ¨ ¤¥áâ ¡¨«¨§ 樨 ¯®¢¥àå®áâ¨ à §¤¥« , ¢ª«îç ïá«ãç ©, ª®£¤ ¤¥áâ ¡¨«¨§ æ¨ï ®¡ãá« ¢«¨¢ ¥â íä䥪⠪ãáâ¨ç¥áª®£® á®á .�஢¥¤¥® ⥮à¥â¨ç¥ ¤®á«÷¤¦¥ï ªãáâ¨ç®ù ¢§ õ¬®¤÷ù, ïª ¢¨§ ç õ ¤¨ ¬÷ªã â áâ÷©ª÷áâì ®¡¬¥¦¥®£® ®¡'õ¬ãà÷¤¨¨ ã ¥¢ £®¬®áâ÷. �à® «÷§®¢ ® ¤¢ ®á®¢¨å ⨯¨ ¢¯«¨¢ã ªãáâ¨ç®£® ¯®«ï ¢÷«ìã £à ¨æî. �¥à訩¯®«ï£ õ ã §¬÷÷ ¤¨ ¬÷ç¨å å à ªâ¥à¨á⨪ ¯«¥áª ì à÷¤¨¨ ã ¥¢ £®¬®áâ÷ ¯÷¤ ¢¯«¨¢®¬ ªãáâ¨ç®£® ¢ â ¦¥ï.�à㣨© á®æ÷îõâìáï ã â¥å÷ç¨å § áâ®á㢠ïå § àã宬 \à÷¤ª®ù ¯à®¡ª¨" ¢§¤®¢¦ âà㡨 ( ªãáâ¨ç¨© á®á). � â¥-¬ â¨ç¨© «÷§ ¡ §ãõâìáï ®á¥à¥¤¥÷ ¢¨å÷¤®ù § ¤ ç÷ § ¢÷«ì®î £à ¨æ¥î à §¤÷«ã ¤¢®å á¥à¥¤®¢¨é, é® ¤®§¢®«ïõ§¢¥á⨠§ ¤ çã ¤® § ¤ ç÷ § ¢÷«ì®î £à ¨æ¥î ¯à® ¯®¢¥à奢÷ 墨«÷ § ¤®¤ ⪮¢¨¬¨ ç«¥ ¬¨ ã ¤¨ ¬÷ç÷© 㬮¢÷ ¥¢÷¤®¬÷© ¢÷«ì÷© ¯®¢¥àå÷. �¥«÷÷©÷ ¥ä¥ªâ¨ ®¯¨áãîâìáï ç¥à¥§ ¯®¡ã¤®¢ã àï¤ã «÷â¨ç¨å ÷ ç¨á¥«ì®- «÷â¨ç¨å஧¢'離÷¢ æ÷õù § ¤ ç÷. �ਪ« ¤¨ ¢¥¤¥® ¤«ï ¢¨¯ ¤ªã 樫÷¤à¨ç®ù ¯®á㤨¨, ª®«¨ ¢¥ªâ®à £à ¢÷â æ÷ù ¯à ¢«¥¨©¢§¤®¢¦ ¢÷á÷ 樫÷¤à , é® ¤®§¢®«ïõ ¯®à÷¢ï⨠ùå § ஧¢'離 ¬¨ § ¤ ç÷ ¯à® ª ¯÷«ïà. �¥®à¥â¨ç÷ ¤®á«÷¤¦¥ï ¯÷¤â¢¥à-¤¦ãîâì ¢¨á®¢ª¨, ®¤¥à¦ ÷ ã ¥ªá¯¥à¨¬¥â å, ¯à® â¥, é® ªãáâ¨ç¨© ¢¯«¨¢ ¬®¦¥ ¯à¨§¢®¤¨â¨ ¤® ¯®«®¦¥ì à÷¢®¢ £¨ £à ¨æ÷ ஧¤÷«ã, ïª÷ ¢÷¤à÷§ïîâìáï ¢÷¤ ª ¯÷«ïà¨å ¯®¢¥àå®ì. �ªà÷¬ ⮣®, ¯à®¤¥¬®áâ஢ ÷ ¥ä¥ªâ¨ ¤¨ ¬÷ç®ù ªãáâ¨ç®ù áâ ¡÷«÷§ æ÷ù â ¤¥áâ ¡÷«÷§ æ÷ù ¯®¢¥àå÷ ஧¤÷«ã, ¢ª«îç îç¨ ¢¨¯ ¤®ª ª®«¨ ¤¥áâ ¡÷«÷§ æ÷ï ®¡ã¬®¢«îõ ¥ä¥ªâ ªãáâ¨ç®£® á®á .NOMENCLATURE� Q(W (x; y; z)�0) is the interior of a tank;� Q2(t) is
uid subdomain in Q;� Q1(t) and Q3(t) are gas subdomains;� �(t) : �(x; y; z; t)=0 (or �1(t), �2(t) for thethird problem) are \
uid { gas" interfaces;� h�i(� )(�(x; y; z; � )=0); �i(� )(x=Hi(y; z; � )),i=1; 2, are averaged interfaces describing theslow-time interface
uctuations;� �0(�0(x; y; z)=0; x=H0(y; z)) (or �0i(x == H0i(y; z)), i=1; 2) is capillary{ acoustic equi-
librium shape (averaged in time pro�le of inter-face);� S1(S3) is tank's wall touching the gas;� S2 is tank's wall touching the
uid;� S0�S1 is a sound vibrator;� 'i(x; y; z; t), i=1; 2, are the velocity potentialsof gas and
uid;� l is size of the tank;� �i(x; y; z; t), i=1; 2, are densities of gas and
u-id;� pi(x; y; z; t), i=1; 2, is pressure;c
I. A. Lukovsky, A. N. Timokha, 1999 69
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83� � is frequency of acoustic excitation;� n is the outer normal to Q (or outer normal toQ2 on the interface �);� � is the coe�cient of surface tension;� g is virtual gravity acceleration;�
i, i=1; 2, are Theta's constants;� �0i, i=1; 2, are averaged densities;� @� is \
uid{ gas { tank" contact line;� � is \
uid {gas { tank" contact angle;� K1+K2 is mean curvature;� p01 is \atmospheric" pressure;� c is the sound speed in gas;� Bo=gl2�02=� is Bond number;� k=�l=c is wave number of acoustic �eld;� The expression F;x means a derivative of func-tion F by variable x.INTRODUCTIONA limited
uid volume occupying hal
y a tank ofsome vehicle (missile, marine tanker, petroleum cis-tern etc.) or orbiting satellite (e. g., spacelab or shut-tle) performs a complete wave motion associated withmobility of its free surface (sloshing). In ground con-ditions the sloshing is caused by dominating gravityand inertial forces. In zero-gravity these factors aresu�ciently small to e�ect the sloshing. Then the sur-face tension is the only primary force returning the
uid volume in its unperturbed stationary capillaryequilibrium position. The another short-life excita-tions having determinated or random nature neverlead to stabilization of capillary shape. Typically,they form the accident waves and destroy continuityof the media. There exist two engineering problemsinvolving
uid sloshing in zero-gravity that motivatethe study of behavior of a
uid volume in an orbitingvehicle. The �rst is the problem of keeping of
u-ids in prescribed subdomain of a vessel (positioning).The second is creation of driving forces to achievepumping between the tanks (pumping). The usualway to overcome these technical problems is basedon use of either active methods (acceleration of vehi-cle, electromagnetic �elds, ejecting membranes etc.)or passive devices suppressing the
uid (partitions,obstacles, membranes, etc.). Both methods require
the additional constructions and devices with rathermassive components.A number of experimental studies, in which the ef-fect of high-frequency periodic loading on
uid slosh-ing is examined, allows one to suggest the use of vi-brations as an e�cient approach for actuation of the
uid and solving the management problems. The ef-fect of sound (or vibration) on a limited
uid volumein microgravity can change the dynamic features offree surface [1 { 8]. Devices producing the acoustic orvibrational �elds are not massive and do not requirethe considerable energy expenditure. In addition,if the acoustic �eld produces an e�ective pressure(acoustic radiation) positioning the
uid in containerthen the dynamics of container (and, of course, thevehicle motions) does not undergo a change [1, 8, 9],therefore no orbit correction is required. However,implementation of acoustic methods is questionablewithout detailed preliminary theoretical investigationof the problem. The reason for this state of art isthe rise of paradoxical phenomena in behavior of thesloshing in tank under vibroloading. The papers be-low report the distortion of capillary forms visualizedas craters or fountains on a free surface [2,3,5,6], sta-bilization and destabilization of th efree surface at-tended by active evaporation from \
uid{ gas" inter-face (for cryogenic case) [1, 4{ 6, 9, 10], deformation,levitation, rotation and the destroying of
uid dropsin standing acoustic �eld [7,8,10] etc.In this paper we develop an analytical approachbased on the methods of theory of sloshing in micro-gravity. It allows us to examine some of the men-tioned phenomena. Originating from the original hy-drodynamic problem on coupled
ows of two com-pressible media we apply an asymptotic method andaveraging technique to derive a new free boundaryproblem (asymptotic model) describing the slow-timeinterface vibration. The analysis of analytical andnumerical { analytical solutions of this free boundaryproblem gives the explanation of nature of a numberof nonlinear vibroacoustic phenomena.1. ACOUSTIC RADIATION PRESSURE AS AGOVERNING FORCEIn �g. 1 the three typical situations involving thesound loading for control of a
uid volume in mi-crogravity are presented. They are associated withlevitating, destroying or assembling drops, stabiliza-tion of shape of
uid domain and pumping of bound-ed
uid volume (\sliding
uid cork") along a tube.On contrary to vibrations of the vehicle, this \vi-bromethod" does not involve the vessel in a coupledpulsation. Local pulsations of vibrating subarea on70 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83
a b cFig. 1. Three implementations of sound control method in zero-gravity conditions:a { assembling
uid drops; b { positioning a
uid domain; c { pumping a \
uid cork"S1 are transmitted by standing acoustic wave in Q1upon the \
uid {gas" interface. This standing wave
uctuates slowly in time-varied gas domain Q1(t)driven by the slow-time sloshing of re
ecting surface(interface) �(t). In addition, standing acoustic �eldyields in Q1 a \radiant" energy implying itself a sta-tionary (in sense of averaged pulsations) acoustic ra-diation pressure as well as acoustic vibration. Thismeans, that the pressure distribution upon �(t) hasboth quick-time (pulsation) and slow-time (averagedacoustic radiation pressure
uctuations) components.The pressure distribution determines the sloshing ofthe re
ecting interface. The above speculations ex-plain the reasons of studying of mutual \interface {acoustic radiation pressure" in
uence.In
uence of the acoustic radiation pressure on thesurface waves is low-investigated in experiments. Themost of works study a special class of ultrasonic phe-nomena associated with either ultrasonic capillary ef-fect (
uid
ow in thin narrow
exible tube causedby progressive elastic waves along it) or acoustical-ly driven jet emergency from the free surface whenlocal ultrasonic vibrator is situated near the bot-tom. Moreover, only few papers examine the e�ectof standing acoustic �eld onto \
uid { gas" interface.Just these investigations concern the subject of ourresearch.We should distinguish two di�erent situations of\standing acoustic wave { interface" interaction. Theboth should be referred to pure microgravity hy-dromechanics. The �rst one appears when the acous-tic wavelength is longer then characteristic spatialsize of interface. For the another situation thewavelength and the interface length are comparable.Acoustically levitated large drops and acousticallyforced interface sloshing are two typical examples ofthis last situation. One is evident, that the key prob-lem in the analysis of these microgravity phenome-na is the question of stability of the interface due toacoustic load.Preliminary analysis in the experimental
works [1,3,7,11,12] uses either phenomenological orenergetic approaches to predict the interface insta-bility. Estimation of the acoustic radiation energya�ecting the evaporation allows to explain qualita-tively the
uid pumping when the exciting frequencyis situated in the neighborhood of the �rst acousticnatural tone in gas. However, these approaches arenot able to describe the interface pro�le and calculatethe frequency range where instability of the interfaceoccurs.The another approach is to consider a lumped ener-gy amount (surface tension, gravity plus the energy ofacoustic radiation pressure) to derive a minima prin-ciple for \potential energy". Mentioned approach wasproposed for some classes of the surface wave prob-lems in [13]. Such phenomenological method in [11]was used to analyze the stability of levitated
uiddrops in crossed standing acoustic waves due to a cho-sen
uctuation of drop's shape. In discussed paperthe one-parametric family of
uctuations was chosenand the problem was reduced to minima problem fora function of one variable.)We consider the problem on \gas {
uid" interfacesloshing exposed to acoustical excitation by vibratorsituated in gas domain. When launching from mod-el disposition of continuum media in accordance toscheme shown in �g. 1,b we implement an averag-ing technique coupled with methods developed forclassical capillary sloshing problem [14{16]. Thisapproach allows us to overcome the principal theo-retical di�culties and apply well-known spectral ap-proach used to analyze the stability of capillary equi-libria [16]. We get the problem on capillary{ acousticequilibria (pro�le of the interface is determined by abalance of surface tension, gravitation, and acous-tic radiation pressure). This equilibrium interfacecan be treated as an averaged in time interface'sshape if steady-state motion of \
uid{ gas" systemoccurs. The capillary{ acoustic equilibrium shapedi�ers from capillary one. In this paper we showthat the relative slow-time sloshing with respect toI. A. Lukovsky, A. N. Timokha 71
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83this equilibrium surface is also di�erent from capil-lary waves in tank and have drastically di�erent hy-drodynamic characteristics of these \acoustic" slosh-ing (natural frequencies, natural modes, stability re-sponse due to diverse excitations etc.).The last experimental investigations of levitationof the drops in standing acoustic waves conductedin spacelab USML-2 [9, 11] con�rmed the strong de-pendence of stability of the drop on acoustic radia-tion. The analysis of this phenomenon being donein [7, 8, 10, 11] is based on theoretical estimation ofdistribution of the acoustic radiation pressure acrossthe drop's surface. Such approach proposed earli-er in [13] introduces the additional nonlinear termscorresponding to acoustic radiation pressure in gov-erning equations. Mentioned terms can be treated asthe averaging vibroforces �rst proposed by P. Kapit-sa [17] for analysis of vibrostabilization of verticalpendulum (see also [18]). The experimental resultsobtained for rigid body dynamics are in good agree-ment with vibrophenomena occurring in hydrody-namic systems [4 { 6]. This means that we can ex-tent some results of rigid body vibromechanics ontothe examined case accounting only the basic balanc-ing forces, namely, the surface tension, gravitationand the acoustic radiation pressure. Also, it shouldbe noted that the model of perfect compressible
uidand gas is applicable to describe the basic nonlinearphenomena known from experiments. Hereinafter weconsider the potential
ows.Note, that similar assumption (inviscid potential
ows, gravity �eld and surface tension) forms thebase of classical free boundary problem on sloshingof incompressible
uid in vessels (capillary{ gravitywaves): �' = 0 in Q(t);@'@n = 0 on S(t);ZQ(t) dQ = const;@'@n = � �;tjr�j on �(t);@'@t + 12(r')2 +A� = 0 on �(t); (1)where the interface is �(t) : �(x; y; z; t)=0. The ve-locity potential'(x; y; z; t) in general should be deter-mined; A is the operator of potential forces (gravity,surface tension etc.). This problem (1) di�ers fromthe problem below so long as it neglects the com-pressibility.Some aspects of theory of sloshing were developed
in [15, 16, 19]. This theory used to be implement-ed into industry problems associated with coupled\body {
uid" motions and calculation of the dynam-ics of capillary
uid in spacecraft vehicles. The prob-lem on capillary equilibrium shape follows immedi-ately from (1) if ' and � are not dependent on t. Werefer the readers who interest in the results of corre-sponding theory to the transactions [16,19{ 23].In order to investigate the sloshing exposed to high-frequency excitation we should take into account thecompressibility. Preprint [24] presents the elements ofsuch theory for mathematical problems modeling the\vibro-sloshing". Krylov{Bogolyubov{Mitropolskiiaveraging technique is used in developed theory. Inthe present paper we follow the same way to ana-lyze a coupled \gas {
uid" sloshing noted in �g. 1 asproblems b and c. The main governing free bound-ary problem derived from the original hydrodynamicproblem contains the nonlinear terms coinciding withthe expression for Langevin acoustic radiation pres-sure. These terms appear in dynamic condition on\
uid {gas" interface.2. PROBLEM ON SLOSHING OF COMPRESS-IBLE MEDIA DUE TO ACOUSTIC EXCITA-TION. ASYMPTOTIC (AVERAGED) PROB-LEMWe examine wave motions of \
uid Q2(t) {gas Q1(t)" interface �(t). The gas and the
uid aresuggested to be compressible. The sound vibrator issituated on part of the wall S0 in way to be alwaystouching only gas domainQ1. It produces an acoustic�eld in the gas. Note, that for all numerical exam-ples we consider the following parameters: frequencyrange of acoustic �eld 1�3 kHz, Bond numbers 0�30for the characteristic size of the tank 0:05�1 m.2.1. Statement of the problemGoverning equations in the both continuum media(i=1; 2) are the following:@�i@t + div(�ir'i) = 0;�ir�@'i@t + 12 (r'i)2 + gx� = �rpi;�i = �0i� pip0i�1=
i in Qi(t): (2)Zero Neumann (no-slip) condition is ful�lled on thewall of the tank Q:@'i@n = 0 on Si; (3)72 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83kinematic and dynamic boundary conditions shouldbe satis�ed on interface:@'i@n = � �;tjr�j on �(t); i = 1; 2;�p1 = �p2 + �(K1 + K2) on �(t) (4)along with the condition on contact line:� (rW;r�)jrW j = cos� jr�j on @�(t): (5)The distribution of normal velocity V0(x; y; z) sin(�t)on the vibrator�1 @'1@n = �01V0(x; y; z) sin(�t) on S0 (6)completes this interface value problem.The solution of (2) { (6) is the set of functions'i(x; y; z; t), pi(x; y; z; t), �i(x; y; z; t) and �(x; y; z; t).Let l be a characteristic size of Q and t�=1=� be acharacteristic time. Then the dimensionless problemtakes the following form:@�i@t + div(�ir'i) = 0;�ir�@'i@t + 12 (r'i)2 +Box��2� � = �rpi;�i = � pip0i�1=
i in Qi(t);@'i@n = 0 on Si;@'i@n = � �;tjr�j ; on �(t);�p1 �01�02 = �p2 + ��2� (K1 +K2); on �(t);�1 @'1@n = sup jV0(x; y; z)jc�0 �� �0k V (x; y; z) sin t on S0;� (rW;r�)jrW j jr�j = cos� on @�(t);
(7)
where V (x; y; z) = V0(x; y; z)= sup jV0j;�2� = �2l3�02/�:2.2. Asymptotic analysisThe dimensionless problem (7) has several smallparameters. One of them "=sup jV0j=(c�0)�1 ex-presses the smallness of amplitude of vibrations in gas
domain with respect to sound speed M=sup jV0j=cwhat is typically correct in acoustic approximationof compressible
ow. Value of �0 is actually the rela-tion between M and ":�0 = M" ; j�0j � 1:The Bond number Bo implies the relation betweengravitation and surface tension.We assume, that other small parameters depend on" according to the following relations:�01�02 = �1"; j�1j � 1;��2� = ��1"3; j�j � 1: (8)The �rst relation implies the smallness of gas densitywith respect to
uid density. This means, that thesecond order component of pressure in Q1 yields thethird order terms on interface when interaction be-tween the subdomains occurs. The dimensionless pa-rameter ��2� is su�ciently small for large �. If V0=0the original problem is reduced to the problem on freecoupled vibrations of compressible
uid and gas in atank. Natural (eigen) vibration of such system wereinvestigated in [25]. The analysis of correspondingspectral problem has shown that each natural modeof the interface has an in�nite set of eigenvalues. Thelowest elements of these sets complete a subsequenceof eigenvalues (sub-spectrum). This sub-spectrumcorresponds to natural motions of the system dueto mobility of the interface; it is close to spectrumof the problem on capillary natural vibration of in-compressible
uid. Other eigenvalues correspond tovibration of the system due to compressibility. Theyare su�ciently large-scaled ones.If t�=��1� 1 is su�ciently small and wave num-ber k�1 (� is situated in the vicinity of dominatingnatural tone of acoustic pulsation in gas), then valueof the lowest (\sloshing" spectral component men-tioned above) is ordered as p��2� (both the surfacetension and gravitation have order ��2� in the dy-namic conditions). The acoustic radiation pressurehas asymptotic order "2�01=�02=�1"3 in the dynamiccondition. To provide the similar order between thesethree interacting forces we should suggest ��2� ��1"3or suppose the second relation from (8) to be ful�lled.That is why � implies the relation between potentialforces (surface tension, gravitation and acoustic radi-ation pressure):� = �l�3�2"2�01 = 1�2�"2�01��102 :We can use asymptotic technique and the methodof separation of quick-time (pulsation) and slow-timeI. A. Lukovsky, A. N. Timokha 73
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83components to derive an averaged asymptotic prob-lem describing the slow-time interface sloshing.2.3. Asymptotic averaged problemWhen introducing the quick-time t and the slow-time � ="3=2t we assume the functions 'i, pi and �to be depending on x; y; z; t and � . Then the solutionof (7) can be extended into series'i =Xk "k=2'(k=2)i (x; y; z; t; � );pi =Xk "k=2p(k=2)i (x; y; z; t; � );� =Xk "k=2�(k=2)(x; y; z; t; � ): (9)When setting (9) into (7) (with taking into ac-count (8)) we reduce the solving of the original prob-lem to a sequence of the problems with respect to'(k=2)i , p(k=2)i , �(k=2); i=1; 2, k=0; 1; 2; : : : The low-est order asymptotic solution depends on zero-, �rst-,second- and third-order approximations. The anoth-er approximations can be found recursively from theabove ones. The averaging procedure selects out twoindependent functions determining the slow-time in-terface sloshing:�2(x; y; z; � )= h'2i;t == "3=2'(3=2)2 (x; y; z; "3=2t) + o("3=2);and �(x; y; z; � )= h�i;t=�0(x; y; z; � )+o("2):Finally, we introduce�1(x; y; z; � ) = '(1)1 (x; y; z; �; t)= sin t(here �1 is wave function of the acoustic �eld in gas)describing the slow-time
uctuation of the acoustic�eld in Q1.The asymptotic procedure transforms the origi-nal free interface problem to nonlinear approximate
(asymptotic) boundary value problem:��2 = 0 in hQ2i(� );@�2@n = 0 on hSii;@�2@n = 0 on hS2i;@�2@n = � �;�jr�j on h�i(� );�2� + 12(r�2)2 + ��1(Bo x� (K1 +K2))++ 14�1(k2(�1)2 � (r�1)2) == const(� ) on h�i(� );� (rW;r�)jrW j jr�j = cos� on @h�i(� );Z<Q2> dQ = const; (10)
��1 + k2�1 = 0 in hQ1i(� );@�1@n = 0 on hS1i [ h�i(� );@�1@n = �0V (x; y; z)k on S0: (11)2.4. Asymptotic approximate problem as a slosh-ing problemThe problem (10), (11) is an analogy of the prob-lem on sloshing (1). The dynamic condition in theboundary problem (10) includes the nonlinear oper-ator A corresponding to a \potential force"A� = ��1(Box� (K1 +K2))++ 14�1(k2(�1)2 � (r�1)2) on h�i(� );where �1(x; y; z; � ) is the solution of system (11).Above �2 depends on �1 and �1 depends paramet-rically on the pro�le of sub-boundary h�i(� ) so longas it satis�es Neumann boundary problem (11) withvaried h�i(� ).To compare this problem with well-investigatedproblem on capillary{ gravity waves we should setV (x; y; z)�0. Then �1(x; y; z)=const and the de-rived problem is immediately transformed to problemon capillary waves in a tank Q [16,24].The theory of capillary waves in tank consists of thetheory of capillary equilibria and the theory of rela-tive capillary waves [16, 19{ 23]. If relative capillary74 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83waves are su�ciently small, the problem is rewrittento the spectral problem with spectral parameter onunperturbed free surface [16,19]. The solutions of thisproblem describe the natural vibration. When ana-lyzing the signs of the eigenvalues for various physicalparameters we make the conclusions on the stabilityof capillary equilibrium shape.Below we place our emphasis on case of non-zerofunction V (x; y; z).3. CAPILLARY{ACOUSTIC EQUILIBRIAThe problem on equilibria for given case can beobtained under the assumption that we have no slow-time waves on the interface (�=�(x; y; z)).If the free surface h�i(� ) does not evolute in time� then �2=const, �1=�01(x; y; z), �=�0(x; y; z).The problem (10), (11) is reduced to a stationaryfree boundary problem in averaged domainQ0 df=hQ1iwith unknown free boundary �0 df=h�i:�(Box� (K1 + K2))++14(k2(�01)2 � (r�01)2) = const on �0;� (rW;r�0)jrW j = cos� jr�0j on @�0;ZhQ2i dQ = const; (12)��01 + k2�01 = 0 inQ0;@�01@n = 0 on hS1i [ �0;@�01@n = �0V (x; y; z)k on S0: (13)Here �0(�0(x; y; z)=0) is the averaged shape of�(t). It was named as the capillary{ acoustic equi-librium (CAE) shape. The problem (12) impliesthe dependence between the shape of averaged sur-face (shape of
uid) and geometry of the acoustic�eld in gas. The �rst equation from (12) expressesthe relation between surface tension, gravitation andthe acoustic radiation pressure when interacting witheach other.The solution of the problem (12), (13) consists oftwo functions: �0 and �01. The �rst function de-scribes the averaged interface �0, while the secondone should be found from the inhomogeneous Neu-mann boundary value problem (for this last prob-lem a part of boundary coincides with �0). More-over, governing equation on �0 includes the nonlin-
ear terms depending on �01. Thus, wave function�01 e�ects the capillary{ acoustic equilibrium shape�0 and the pro�le of this equilibrium mode e�ectsthis wave function.We can introduce the nonlinear operator A1(�0)and formally reduce the boundary value problem oncapillary{ acoustic equilibria to a class of capillaryequilibrium problem with a special nonlinear opera-tor corresponding to the acoustic pressure distribu-tion on �0:A1(�0) = 12�k2�201 � (r�01)2����0 ; (14)where �01(x; y; z) is the solution of (13).The boundary problem (12) can be interpreted asa capillary problem on equilibria augmented by theacoustic radiation pressure. This problem takes theform�(K1 +K2) + Box+ 1�A1(�0) = const;� (rW;r�0)jrW jjr�0j = cos� on @�0;ZhQ2i dQ = const: (15)Solvability of the nonlinear boundary problem (15)so far remains an open question even for a tankof simple shape. It is basically caused by di�cul-ties arising in the analysis of main capillary term�(K1+K2) present in governing equation. The solv-ability theorems on capillary equilibria were estab-lished only for cylindrical and conical tanks [16, 20,21, 26]. The case of exotic tanks requires detailedanalysis as well as the additional experimental inves-tigations on orbiting station [20{23].In this investigation we suppose that Q has thecylindrical shape. This allows us to structure the ana-lytical and numerical{ analytical solutions, and com-pare the obtained data with known results of capillarytheory.4. STABILITY OF CAE SHAPEWe assume the interface h�i(� ) to be initially per-turbed with respect to CAE. If this displacement issu�ciently small and the CAE is stable, the mag-nitude of the interface sloshing is also small. Thismeans that we can consider the problem on linearvibration.Let Q has the cylindrical shape with vertical walldetermined by equation W (y; z)=0 and bottomsx=�hf ; x= hg. Here hf is the height of
uidI. A. Lukovsky, A. N. Timokha 75
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83column; hg is the height of a gas (in unperturbedstate). CAE surface can be described by equationx=H0(y; z) when the perturbed surface h�i(� ) al-lows the explicit form x=H(y; z; � ). Linearized evo-lutional problem takes the form��2 = 0 in Q0;@�2@n = 0 on hS2i;8><>: @�2@n = H�q1 + (rH0)2 ;�2� + ��1AH = 0 on �0; (16)whereAH=�div� rHq1+(rH0)2�� (rH;rH0)rH0�1+(rH0)2�3=2�++ 2� �k2�01�01;xH�(r�01;r�01;x)H++ k2�01��(r�01;r�)��0 ++ BoH on �0;(W;yH;y+W;zH;z)(W;yH0;y +W;zH0;z) == (rH0;rH)q1+(rH0)2 on @�0;Z�0 Hdydz=0; (17)��+ k2� = 0 in Q0;@�@n = 0 on hS1i [ S0;@�@n = h�01;xxH ��01;zH;z � �01;yH;y�� [�01;xyH0;y + �01;xzH0;z]Hi�� 1.q1 + (rH0)2 on �0: (18)Here �2=�2(x; y; z; � ) determines the small motionof
uid; �(x; y; z; � ) describes the small parametricevolution of wave function in Q0 occurring due tothe interface
uctuation (x=H(y; z; � )).We introduce harmonic dependences of �2, Hon � : H(y; z; � )=exp(i�� )h(y; z) and �2(x; y; z; � ) =
= i� exp(i�t)�(x; y; z) [16]. Natural modes h, � andnatural frequencies � can be found from the followingspectral problem:�� = 0 in Q0;@�@n = 0 on hS2i;@�@n = hq1 + (rH0)2 on �0;��2�+ ��1Ah = 0 on �0: (19)The problem (19) includes the squares of �. This isthe reason, why the sign of �2 subjects the stability ofCAE. If the spectrum is strongly positive then CAEis stable. Else it is unstable in view of small initialperturbation. This allows to formulate the following\dynamic" spectral criterion of stability:� CAE is stable if and only if �2l >0 for all l.The another way to analyze the stability of CAE isassociated with studying the pressure balance
uctu-ation on interface �0 due to its displacement. If thepressure balance is always distributed in such a man-ner that returning forces are directed to keep CAEshape, then �0 is stable. Else �0 is unstable. Sincethe pressure balance on CAE surface is given by gov-erning equation (12) and operator A corresponds to aperturbation of this balance for a small displacementH the following \static" spectral criterion holds true:� � > 0 for spectral problem Ah = �h.Implementation of the spectral criteria is the e�-cient method to establish the stability properties fordiverse capillary problems. In the next section we ap-ply them to investigate CAE shapes for cases whenthe solution can be found in analytical form. In par-ticular, spectral criteria of stability allow to establishthat1) planar CAE shape can demonstrate stability un-der negative over-critical gravitation (stabiliza-tion);2) \
uid {gas" interface can demonstrate instabili-ty (resonance distortion) for positive Bond num-ber when capillary surface is stable.5. STABILITY OF PLANAR CAE SHAPEThe spectral problem on capillary sloshing in ver-tical circular cylinder allows the analytical solutionfor right contact angle (�=�=2). It has been ana-lyzed by many authors (see, for example, the mono-graph [16]). In this case capillary surface is planarand perpendicular to directrix of the cylinder. The76 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83spectral problem on natural sloshing can be solvedby method of separation of spatial variables. Thismeans, that �0(x; r; �)=x�H0(r; �) = 0, H0(r; �)=0.Planar capillary surface is stable if and only ifBo > ��11; (20)where J 0p(�pq)=0; Jp(r) is Bessel function.Natural frequencies e�pq of capillary waves canbe found from spectral problem (19) (setting�01�0(V � 0)), i. e.e�2pq = ��1�pqth (�pqhf )(Bo + �pq);hpq(r; �) = Jp(�pqr) sincos (p�) : (21)Here we chose the values of characteristic dimensionand characteristic time similar to that for CAE prob-lem to make the comparison of capillary and capil-lary{ acoustic problems easier.Below we consider the problem on CAE for�=�=2. Depending on the shape of acoustic �eldmodulated on S0 this equilibrium interface state canbe planar or non-planar one. The �rst (planar) casewill be an object of our detailed analysis.Under the above assumptions we suggest for a ver-tical circular cylindrical tank (in numerical examplesit has radius l=0:1 m) to be partially �lled by waterand air at temperature 20�C and atmospheric pres-sure. The gravity is 10�5�10�7g0, the acoustic fre-quency is 700�2000 Hz.5.1. Non-planar capillary{ acoustic equilibriaThe pro�le �0 depends on not only contact angle,but also the distribution of normal velocity compo-nent on S0. We show this via numerical examplescalculated under the above assumptions.If the acoustic �eld in gas over
uid is pla-nar (V0(r; �)=v0=const, "=�v0=c sin(khg), �0 == � sin(khg), V (r; �)=1) then CAE shape is alsoplanar and problem (12) has the analytical solutionH0(r; �)�0 (\trivial" solution) with wave function�01(x; y; z)=k�2 cos(kx). If the sound vibrator onS0 modulates a non-planar acoustic �eld (and, there-fore, the right hand side of boundary condition onS0 is not constant), then CAE shape is not pla-nar. We present some shapes of �0 calculated whenV (r; �)=0:2+0:2J0(�01r) in �g. 2.5.2. Planar capillary{ acoustic equilibrium shape(V0=const, �=�=2)The case of planar CAE shape is convenient objectto study the dependence of interface stability from
acoustic loading. The case V0=const corresponds toa vibrating piston situated on the ceiling of the cir-cular cylinder. For this case capillary and capillary{acoustic equilibrium shapes coincide with each oth-er. We consider the natural vibration with respect tothese \trivial" equilibrium pro�les. Method of sepa-ration of spatial variables x; r; � in cylindrical coordi-nate system gives the natural frequencies and naturalmodes in analytical form. One can show that naturalmodes hpq (21) are the same for the both cases. How-ever, the natural frequencies di�er from each other.The frequencies �pq of capillary{ acoustic waves arecalculated as follows:�2pq = �1�pqth (�pqhf )��(Bo +�2pq)��12 ( 1=(� th (�hg)); k2 < �2pq ;�1=(� tg (�hg)); k2 � �2pq �;� =qjk2 � �2pqj: (22)The stability depends on the sign of �2pq and e�2pq .For capillary equilibria the squares of natural fre-quencies are positive if the condition (20) is satis�ed.CAE is stable if the inequality �2pq>0 for all pq holdstrue.We depict the stability response of planar CAEshape on wave number in �g. 3 (Bo=10, "=0:007,hg=2). Because of the positiveness of Bond num-ber the planar capillary equilibrium shape is alwaysstable. Obviously, planar CAE shape must be stablefor k lower then critical value. This value dependson type of perturbation. We interchange the pertur-bations in accordance with the sequence of naturalmodes hpq. The ranges of stability for such pertur-bations are �gured on axes pq. Note, that each axispq includes the range (Opq; O0pq). Here the point Opqcoincides with the origin of axis pq (it correspondsto k=0). This range is caused by positiveness of Boand the point that CAE tends to capillary equilib-ria when k!0, �(k)!1. We assume that k�1.This one requires a speculative choice of k1 on axisOk to exclude small k from consideration. Minimalrange (O11; O011) appears for h11. It de�nes the crit-ical value k2 on Ok and the �rst range of stabilityI=(k1; k2). When k increases the ranges of stabilityand instability alternate each other.Conclusion 1. If planar capillary interface is sta-ble then planar CAE shape is stable for su�cient-ly small k2 (k1; k2). The stability loss occurs for ksituated in right hand to O011. The �rst instabilityrange (k2;�11) is caused by the �rst acoustic reso-nance (�11 is the �rst natural frequency). Hence,the instability phenomenon has resonance character.I. A. Lukovsky, A. N. Timokha 77
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83
a bFig. 2. Some non-planar CAE shapes (radial cross-section of circular cylinder):a { "=0:0025, k=1:1108; a { "=0:0025, k=1:3305;
Fig. 3. Stability diagram for positive Bond number78 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83
a bFig. 4. Stabilization phenomenon: (AB) corresponds to stable planar CAE shapeAnalogous phenomenon was found for pendulum on
exible vibrating unit [18].We can chose k, " and hg , for which planar CAEshape is stable under negative over-critical Bondnumber (Bo+�211<0). This e�ect appears to be pos-sible in view to existence of post-resonance ranges ofstability (Apq ; A0pq), (Bpq ; B0pq) etc.First such a situation is shown in �g. 4,a (forhg=1; "=0:0035; Bo=�7:5; �5:0). Here negativeBond numbers Bo=�7:5; �5:0 lead to instability ofplanar capillary surface due to perturbation by h11(Bo+�211<0, Bo+�2pq>0, pq 6=11). The stabilityranges (O11; O011) are absent for the both examples.The next appropriate range k2 (AB) can be foundin the vicinity of the �rst resonance k>�11. Thisrange of stable planar CAE shape grows only whenincreasing Bo (see two lower diagrams in �g. 4, a).The second case is shown in �g. 4,b (hg=1:25;"=0:0025; Bo=�10:5; �7:5; �3:7; 0:0; Bo+�211<0,Bo+�221<0, Bo+�2pq>0, pq 6=11; 21). Here fork=�10:0 planar CAE shape is unstable when per-
turbing the modes h11; h21. It is necessary to use thesecond post-resonance range (AB) : k>�21 to reachits stabilization. This range (AB) also grows whenincreasing Bo.Conclusion 2. A high-frequency acoustic �eld ingas can stabilize or destabilize \
uid{ gas" interfacefor various Bond numbers. The dynamic stabiliza-tion (destabilization) e�ect is forced by acoustic res-onance.Let us compare the eigenvalues �pq and e�pq for dif-ferent indexes pq. Table gives these values for ac-tual parameters of \water { air" system (l=0:1 m,Bo=0:0, "=0:0025, hg=1:25l, hf =2:0l). One cansee, that the sound e�ect on natural frequencies withnumbers more than 31 is not principal and the hier-archy �211 < �221 < �201 < : : : (23)holds true for capillary waves. That is why pertur-bation of natural mode with index 11 (main asym-metric perturbation of interface) is most dangerous.I. A. Lukovsky, A. N. Timokha 79
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83Table. Hierarchy of eigenvalues in \water { air"system vs acoustic wavenumberModes k=0:0 k=1:48 k=1:85 k=3:7h11 0.4540 0.251 21.60 6.707h21 2.075 1.963 1.882 1.588h01 4.097 3.992 3.924 1.204h12 5.400 5.296 5.231 4.087h31 10.95 10.85 10.79 10.11This hierarchy can hold true for planar stable CAEshape (see k2 (k1; k2) in �g. 3 or the second columnof table). Moreover, for mentioned k the addition-al inequality �2pq<e�2pq holds true. The last means,that the acoustic �eld can lead to decrease of naturalfrequencies.However, the acoustic loading can change the hi-erarchy (23) (see the last two columns in the table).For k throughout (k3; k4) (see, �g. 3) the symmetricperturbation by mode 01 is most dangerous (because0<�201<�2pq , pq 6=01). Analogous example is present-ed in the last column of the table.Conclusion 3. The e�ect of acoustic loading of in-terface can be expressed in essential changes of natu-ral frequencies. It can lead to decrease of the properfrequencies or to increase of frequencies for the se-lected surface modes. The selected modes can losestability, although, sometimes, the acoustic �eld canstabilize a position of balance.6. ACOUSTIC PUMPINGIn this section we extend the previous results on-to case depicted in �g. 1, c. Vibration of one fromthe end-walls of the tube creates a standing acousticwave in gas domain. For the �rst time the prob-lem was experimentally investigated in [1] where theterm \problem on acoustic pumping in microgravi-ty" has been introduced in accordance with possi-ble implementation of acoustic methods for pump-ing the cryogenic
uid along the tube. This pump-ing (driven by evaporation phenomenon) is forced byresonance interaction of acoustic �eld with the inter-face, so that the resulting force upon the interfaceexceeds the acoustic radiation pressure [1]. Estima-tion of pressure shows that phenomenon of acousticpumping can be explained by intensi�cation of evap-oration. The increase of evaporation can be a�ectedonly by hydrodynamic instability of sloshing of theinterface touching the pulsating gas.If free surface �1 is stable, the pumping of cryo-genic
uid does not occur. In order to describe theacoustic pumping phenomenon we should use a reso-nance acoustic loading when the sound in Q1 desta-
bilizes and destroys the interface �1 keeping at thesame time the interface �2 stable.Under the above assumptions we suppose, that Qhas the form of circular tube and that the di�erenceof averaged pressures between gas domainsQ1(t) andQ3(t) on the interfaces �1(t) and �2(t) suppresses themobility of the mass center of the
uid Q2(t), name-ly, the small gravity and acoustic radiation pressuredo not force it's change. This condition can be easysupplied via relation between the averaged pressurep02 in domain Q3(t) and the averaged pressure p01 inQ1(t): p02 = p01 � (�02ghf )=jS0j: (24)Here jS0j is the area of cross-section of cylindricaltube.We suppose that the sound vibrator is situated onthe left hand end-wall S0 of Q and pose the origin ofcoordinate system Oxyz on S0 tracing Ox along thedirectrix of the cylinder. The vector g (small gravityacceleration) is parallel to Ox. The coe�cient of thesurface tension � and contact angle are assumed tobe constant.Let the equation of the interfaces �i take the formx=Hi(y; z; t). When repeating the averaging proce-dure described above in detail we arrive at the prob-lem on slow-time sloshing:��2 = 0 in Q2(� );@�2@n = 0 on S2;@�2@n = (�1)i Hi�(1 + (rHi)2)1=2 on �i(� );�2� + 12(r�2)2 + ��1���BoH1 + div rH1(1 + (rH1)2)1=2�++14�1 �k2(�1)2 � (r�1)2 + P � = 0 on �1(� );'2� + 12(r'2)2 + ��1���BoH1 � div rH1(1 + (rH1)2)1=2� = 0 on �2(� );(�1)iH;yW;y +H;zW;zjrW j == cos�p1 + (rHi)2 on @�i(� );80 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83��1 + k2�1 = 0 in Q1(� );@�1@n = 0 on S1 [ �1(� );@�1@x = V (y; z)k on S0;hZS0 �k2(�1)2 � (r�1)2 + P �dsi = 0;where P =const is chosen from the last integralcondition (averaged pressure on S0 is constant);"=supjV0j=c. The case V �0 corresponds to capil-lary
uid sloshing of a \
uid cork" in micro-gravity.The problem on CAE for this case is divided intotwo independent problems for�01= h�1i(x=H01(y; z);�01=�01(x; y; z))and �02= h�2i(x=H02(y; x));namely,� BoH01 + div rH01p1 + (rH01)2!++14(k2(�01)2 � (r�01)2 + P ) = 0on �01; (25)��01 + k2�01 = 0 in Q01;@�01@n = 0 on hS1i [ �01;@�01@n = V (y; z)k on S0 (26)and � BoH02 + div rH02p1 + (rH02)2! == ��P � �Bo hfjS0j � on �02; (27)where(�1)iH;yW;y +H;zW;zjrW j = cos�p1 + (rH0i)2on �0i; i = 1; 2;ZhQ2i dQ = const: (28)
In order to investigate the stability of �01 and �02we use \the static spectral criteria":� CAE shape (surface �01(x=H01(y; z)) plus surface�02(x=H02(y; z))) is stable if and only if all eigenval-ues �(1)l and �(2)l determined from the spectral prob-lems��(1)h+ A1h = 0 on �01;A1h = Boh+ div� rhq1 + (rH01)2�� (rh;rH01)rH01�1 + (rH01)2�3=2�++ 2��k2�01�01;xh� (r�01;r�01;x)h++k2�01�� (r�01;r�)��01 on �01;� (h;yW;y + h;zW;z)(H0;yW;y +H0;zW;z) == (rH0;rh)p1 + (rH012) on @�01;Z�01 hdydz = 0;
(29)
��+ k2� = 0 in Q01;@�@n = 0 on hS1i [ S0;@�@n = ���01;xxh ��01;zh;z � �01;yh;y�� [�01;xyH0;y +�01;xzH0;z]h �� 1.q1 + (rH0)2 on �01 (30)(here �01 is the solution of the problem (26);�(x; y; z) describes changing of wave functionin Q01 due to perturbation of free interfaceI. A. Lukovsky, A. N. Timokha 81
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83�01 : x=h1(y; z)) and from the spectral problem��(2)h+ A2h = 0 on �02;A2h = Boh � div� rhq1 + (rH02)2�� (rh;rH02)rH02�1 + (rH02)2�3=2 � on �02;h;yW;y + h;zW;zH0;yW;y +H0;zW;z == (rH0;rh)q1 + (rH02)2 on @�02;Z�02 hdydz = 0 (31)
are positive ones (�(i)l >0, i=1; 2). Negative �(1)lcorresponds to unstable �01 and negative �(2)l cor-responds to unstable �02.7. PLANAR CAE SHAPES AND INSTABILITYOF \FLUID CORK"The simplest way to show the e�ect of acoustic �eldis to consider the system when �=�=2, V =1. In thiscase the surfaces �01 and �02 (both in the presenceof planar acoustic �eld V =1 and sound free case) areplanar ones, i. e.,H01(y; z) = hg; H02(y; z) = hg + hf ;�01(x; y; z) = cos(k(x � hg))k2�0 :Here �0=sin(khg).The problems (29) and (31) are solved by methodof separation of spatial variables in cylindrical coordi-nate system. The natural modes on the free surfaces�01 and �02 areh(pq)i = �Jp(�pqr) sincos (p�)� ; J 0p(�pq) = 0:The eigenvalues are�(1)pq = ��1(�2pq � Bo)��12 ( 1=(� th (�h1)); k2 < �2pq;�1=(� tg (�h1)); k2 � �2pq;�(2)pq = ��1(�2pq +Bo); (32)
where � =qjk2 � �2pqj. Note, that for capillary sur-face natural modes are the same ones, bute�(1)pq = ��1(�2pq � Bo);e�(2)pq = ��1(�2pq + Bo):Capillary planar surface is stable if and only if��11<Bo<�11. However, in accordance with (32),there exists such a wave number k, that �(1)pq <0 forany Bond number, even for Bo=0. This means thatthe surface �01 can become unstable due to acousticloading. When analyzing the expression for �(1)11 indetails we �nd out that for any Bond number pla-nar �01 is not stable as k!�11� (the frequency ofsound is close to the �rst natural tone of acoustic�eld in gas), because �(1)11 tends to �1 as k!�11.Just such a resonance phenomenon was observed inthe experimental work [1].Conclusion 4. The acoustic pumping of a cryo-genic
uid in zero-gravity utilizes the hydrodynamicinstability phenomenon forced by acoustic resonance.CONCLUSIONSA number of analytical and numerical solutions ofaveraged problems on CAE and the slow-time slosh-ing of \
uid{ gas" interface under zero-gravity con-ditions exposed to acoustic vibrations of gas domaincon�rms the good agreement of the theory with ex-perimental data. The examples concern the case ofcylindrical vessel with low-gravity acceleration vectoralong the directrix. This limitation makes availablethe comparison with trivial (planar) solutions of cap-illary problem.We show that if the sound vibrator modulates anon-planar acoustic �eld then CAE shape is not pla-nar one even for positive Bond number. This meansthat CAE shape does not coincide with capillaryshape. However, even if the acoustic vibrator pro-duces planar acoustic �eld in gas and CAE shape isalso planar one, no conclusion about the stability ofthis interface can be made. Acoustic loads changedrastically the dynamic properties of the sloshing in-terface, so the analysis of new stability is necessary.We propose a simple analytical way how to �nd therange of exciting frequencies, for which the planarequilibrium shape is unstable in contrast to the pla-nar capillary surface. It is transparent treatment ofthe destabilization phenomenon. On the other hand,we have found out the stabilization phenomenon,namely, we calculated the ranges of exciting frequen-cies, for which the planar CAE shape is stable whencapillary planar surface is unstable. The both ranges82 I. A. Lukovsky, A. N. Timokha
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 3. �. 69 { 83are situated near a natural frequency of acoustic vi-bration of gas domain. In addition, the acoustic loadsalways lead to the drift of spectrum of the problemon natural vibration with respect to CAE. It is ex-pressed through the decrease of natural tones or totheir increase for selected surface modes.We implement the same technique to analyze theacoustic pumping phenomenon (\
uid cork" move-ment in a tube supplied by acoustic action upon onefrom interfaces). When utilizing the resonant excita-tion we have found the range of frequencies, at whichthe interface interacting with the acoustic �eld is un-stable for arbitrary Bond number.ACKNOWLEDGEMENTAuthors are grateful for support in part made byDeutsche Forschungsgemeinschaft.1. Wesseln Ph. S. Acoustic pumping in cryogenic liq-uid // Design. News.{ 1967.{ 22, N 12.{ P. 96{102.2. �®àä¥«ì¤ �., �®«®å®¢ �. �á¯ã稢 ¨¥ ¯®¢¥àå-®á⨠¦¨¤ª®á⨠¯®¤ ¢®§¤¥©á⢨¥¬ ã«ìâà §¢ãª //�®ª«. �� ����.{ 1955.{ 105, N 3.{ �. 476{477.3. Ingard U., Ross J. A. Jr. Some aspects of the interac-tion of sound with a liquid surface // Proc. 7-th Int.Congr. Acoust. Vol. 2.{ Budapest, 1971.{ P. 213{216.4. Wolf G. H. Dynamic stabilization of the interchangeinstability of a liquid { gas Interface // Phys. Rev.Let.{ 1970.{ 24, N 9.{ P. 444{446.5. � ¨¥¢ �. �., � ª¨§ �. �., � ¯¥ª® �. �. � ¤¨ -¬¨ç¥áª®¬ ¯®¢¥¤¥¨¨ ᢮¡®¤®© ¯®¢¥àå®á⨠¦¨¤-ª®á⨠¢ ãá«®¢¨ïå, ¡«¨§ª¨å ª ¥¢¥á®¬®áâ¨, ¯à¨ ¢¨-¡à 樮®¬ ¢®§¤¥©á⢨¨ // �ਪ«. ¬¥å.{ 1977.{13, N 5.{ �. 102{107.6. �¬®¢ �. �., �¥à¥¯ ®¢ �. �. � ¢®§¨ª®¢¥¨¨áâ 樮 ண® ५ì¥ä ¯®¢¥àå®áâ¨ à §¤¥« ¦¨¤ª®á⥩ ¢ ¢¨¡à 樮®¬ ¯®«¥ // �§¢. ������.�¥å ¨ª ¦¨¤ª®á⨠¨ £ § .{ 1986.{ N 6.{ �. 8{13.7. Tian Y., Holt R. G., Apfel R. E. Deformation andlocation of an acoustically levitated liquid drop //J. Acoust. Soc. Amer.{ 1995.{ 93.{ P. 3096{3104.8. Lee C. P., Anilkumar A. V., Wang T. G. Static shapeof an acoustically levitated drop with wave {,drop in-teraction // Phys. Fluids.{ 1994.{ 6, N 11.{ P. 3554{3566.9. Principal investigator: T. Wang / Drop physics mod-ule / Drop dynamics experiment // The second Unit-ed States Microgravity Laboratory (USML-2). 90-dayScience Report.{ March, 1996.{ P. 69{71.10. Apfel R. E., Tian Y., Jankovsky J., Shi T., Chen X.,Holt R. G., Trinh E., Croonguist A., Thorn-ton K. C., Sacco A. Jr., Colemen C., Leslie F. W.,Matthiesen D. H. Free oscillations and surfactant stu-
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uids in an acoustic �eld // ASA 127-th MeetingM.I.T.{ 1994, June, 6{10.{ P. 1{2.13. Lukovsky I. A., Timokha A. N.One class of boundaryvalue problems of the theory of surface waves // Ukr.Math. J.{ 1991.{ 43, N 3.{ P. 359{364.14. Faltinsen O. M. A nonlinear theory of sloshing inrectangular tanks // J. Ship Res.{ 1974.{ 18, N 4.{P. 224{241.15. �㪮¢áª¨© �. �. �¢¥¤¥¨¥ ¢ ¥«¨¥©ãî ¤¨ ¬¨ªã⢥à¤ëå ⥫ á ¯®«®áâﬨ, ç áâ¨ç® § ¯®«¥ë¬¨¦¨¤ª®áâìî.{ �.: � ãª. ¤ã¬ª , 1990.{ 296 á.16. Myshkis A. D., Babsky V. G., Kopachevskii N. D.,Slobozhanin L. A., Typsov A. D. Low-gravity
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ows with freeboundaries. Part I: Vibrocapillary equilibria // Uni-versitaet Leipzig. Preprint NTZ.{ N 1, 1999.{ P. 1{18.25. �㪮¢áª¨© �. �., �¨¬®å �. �. � ᢮¡®¤ëå ª®-«¥¡ ¨ïå á¨á⥬ë \¦¨¤ª®áâì { £ §" ¢ 樫¨¤à¨ç¥-᪮¬ á®á㤥 ¢ á« ¡®¬ £à ¢¨â 樮®¬ ¯®«¥ // �àï-¬ë¥ ¬¥â®¤ë ¢ § ¤ ç å ¤¨ ¬¨ª¨ ¨ ãá⮩稢®á⨬®£®¬¥àëå á¨á⥬.{ �., �áâ¨âãâ ¬ ⥬ ⨪¨�� ����, 1986.{ �. 5{12.26. �à «ì楢 �. �. � §à¥è¨¬®áâì § ¤ ç¨ ® ª ¯¨««ï-à å // �¥áâ. �¥¨£à. ã-â .{ 1975.{ 1, ¢ë¯. 1.{�. 143{149.I. A. Lukovsky, A. N. Timokha 83
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