Differential and variational formalism for an acoustically levitating drop

Starting with the most general problem on interface waves between two ideal fluids, treated here as an ullage gas and a liquid, respectively, and separating fast and slow time scales, differential and variational formalism for an acoustically levitating drop and its time-averaged shape (the drop vib...

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Дата:	2015
Автори:	Chernova, M.O., Lukovsky, I.O., Tymokha, O.M.
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Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Нелінійні коливання
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Цитувати:	Differential and variational formalism for an acoustically levitating drop / M.O. Chernova, I.O. Lukovsky, O.M. Tymokha // Нелінійні коливання. — 2015. — Т. 18, № 3. — С. 413-428 — Бібліогр.: 26 назв. — англ.

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spelling	irk-123456789-1772242021-02-13T01:25:49Z Differential and variational formalism for an acoustically levitating drop Chernova, M.O. Lukovsky, I.O. Tymokha, O.M. Starting with the most general problem on interface waves between two ideal fluids, treated here as an ullage gas and a liquid, respectively, and separating fast and slow time scales, differential and variational formalism for an acoustically levitating drop and its time-averaged shape (the drop vibroequilibrium) are developed. The drop vibroequilibria can differ from spherical shape; stable vibroequilibria are associated with local minima of the quasipotential energy whose analytical form is also derived in the present paper Починаючи з найбiльш загальної задачi про iнтерфейснi хвилi мiж двома iдеальними рiдинами, що розглядаються як газ та рiдина вiдповiдно, i вiддiляючи швидкi та повiльнi часовi шкали, розвинено диференцiальний та варiацiйний формалiзм для акустично левiтуючої краплi та знайдено її усереднену за часом форму (вiброрiвноважний стан). Вiброрiвноважнi стани краплi можуть вiдрiзнятися вiд сферичної форми; стiйкi вiброрiвноважнi стани пов’язанi з локальними мiнiмумами квазiпотенцiальної енергiї, для якої також побудовано аналiтичне зображення. 2015 Article Differential and variational formalism for an acoustically levitating drop / M.O. Chernova, I.O. Lukovsky, O.M. Tymokha // Нелінійні коливання. — 2015. — Т. 18, № 3. — С. 413-428 — Бібліогр.: 26 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177224 532.595 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Starting with the most general problem on interface waves between two ideal fluids, treated here as an ullage gas and a liquid, respectively, and separating fast and slow time scales, differential and variational formalism for an acoustically levitating drop and its time-averaged shape (the drop vibroequilibrium) are developed. The drop vibroequilibria can differ from spherical shape; stable vibroequilibria are associated with local minima of the quasipotential energy whose analytical form is also derived in the present paper
format	Article
author	Chernova, M.O. Lukovsky, I.O. Tymokha, O.M.
spellingShingle	Chernova, M.O. Lukovsky, I.O. Tymokha, O.M. Differential and variational formalism for an acoustically levitating drop Нелінійні коливання
author_facet	Chernova, M.O. Lukovsky, I.O. Tymokha, O.M.
author_sort	Chernova, M.O.
title	Differential and variational formalism for an acoustically levitating drop
title_short	Differential and variational formalism for an acoustically levitating drop
title_full	Differential and variational formalism for an acoustically levitating drop
title_fullStr	Differential and variational formalism for an acoustically levitating drop
title_full_unstemmed	Differential and variational formalism for an acoustically levitating drop
title_sort	differential and variational formalism for an acoustically levitating drop
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/177224
citation_txt	Differential and variational formalism for an acoustically levitating drop / M.O. Chernova, I.O. Lukovsky, O.M. Tymokha // Нелінійні коливання. — 2015. — Т. 18, № 3. — С. 413-428 — Бібліогр.: 26 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT chernovamo differentialandvariationalformalismforanacousticallylevitatingdrop AT lukovskyio differentialandvariationalformalismforanacousticallylevitatingdrop AT tymokhaom differentialandvariationalformalismforanacousticallylevitatingdrop
first_indexed	2025-07-15T15:15:40Z
last_indexed	2025-07-15T15:15:40Z
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fulltext	UDC 532.595 DIFFERENTIAL AND VARIATIONAL FORMALISM FOR AN ACOUSTICALLY LEVITATING DROP ДИФЕРЕНЦIАЛЬНИЙ ТА ВАРIАЦIЙНИЙ ФОРМАЛIЗМ ДЛЯ АКУСТИЧНО ЛЕВIТУЮЧОЇ КРАПЛI M. O. Chernova Bogomolets Nat. Med. Univ. Pushkinska str., 22, Kyiv, 01004, Ukraine e-mail: maria@aquarelle.biz.ua I. A. Lukovsky Inst. Math. Nat. Acad. Sci. Ukraine Tereshchenkivska str., 3, Kyiv, 01601, Ukraine e-mail: lukovsky@imath.kiev.ua A. N. Timokha Inst. Math. Nat. Acad. Sci. Ukraine Tereshchenkivska str., 3, Kyiv, 01601, Ukraine and Centre of Excellence “AMOS” & Norweg. Univ. Sci. and Technology Otto Nielsens veg. 10, Trondheim, NO-7491, Norway e-mail: atimokha@gmail.com Starting with the most general problem on interface waves between two ideal fluids, treated here as an ullage gas and a liquid, respectively, and separating fast and slow time scales, differential and variational formalism for an acoustically levitating drop and its time-averaged shape (the drop vibroequilibrium) are developed. The drop vibroequilibria can differ from spherical shape; stable vibroequilibria are associated with local minima of the quasipotential energy whose analytical form is also derived in the present paper. Починаючи з найбiльш загальної задачi про iнтерфейснi хвилi мiж двома iдеальними рiдинами, що розглядаються як газ та рiдина вiдповiдно, i вiддiляючи швидкi та повiльнi часовi шка- ли, розвинено диференцiальний та варiацiйний формалiзм для акустично левiтуючої краплi та знайдено її усереднену за часом форму (вiброрiвноважний стан). Вiброрiвноважнi стани краплi можуть вiдрiзнятися вiд сферичної форми; стiйкi вiброрiвноважнi стани пов’язанi з локальни- ми мiнiмумами квазiпотенцiальної енергiї, для якої також побудовано аналiтичне зображення. 1. Introduction. The acoustic levitation [1 – 3] has been developing from the 70 – 90’s as a con- tactless technology in chemical and pharmaceutical industry [4, 5] of ultrapure materials. The technology facilitates preventing the liquid contamination and intensifying the chemical reacti- ons. The acoustic levitators are also used in physical measurements of the surface tension and the liquid viscosity [6 – 8]. A typical design of an acoustic levitator is schematically shown in Fig. 1. The levitator consists of an acoustic vibrator and a spheric reflector which create, altogether, an almost planar standing acoustic wave of the length λ. The acoustic wave yields c© M. O. Chernova, I. A. Lukovsky, A. N. Timokha, 2015 ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 413 414 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA (a) (b) Fig. 1. A schematic design of acoustic levitators in which an almost planar standing acoustic wave of the length λ is created along the gravity acceleration vector g and, therefore, there exists the time-inde- pendent acoustic radiation pressure, pL ( x1 + 1 2 λ ) = pL(x1), which counteracts the gravity provi- ding the drop levitation. In the zero-gravity, the drop locates at zeros of pL(x1) but a downward drift d occurs when the gravitation vector is not zero moving the drop into the ‘+’ zone of pL(x1). The long standing acoustic wave (a) does not deform the levitating drop, but when the one-fourth of the wave length is comparable with the drop size,the time-averaged drop shape becomes flattened (b). the acoustic radiation pressure [9] which is a time-independent λ/2-periodic function along the vertical axis. Periodically changing positive (marked by ‘+’) and negative (‘−’) radiation pressure zones enforce droplets to be located in a vicinity of a radiation pressure node with a possible downward shift d into the ‘+’-zone due to the vertical gravity force. As long as the equivalent drop diameterD0 = 2R0 (the spherical drop diameter of the same volume) is much lower of the acoustic standing-wave length (see Fig. 1 (a)), the acoustic radiati- on pressure does not deform the drop shape so that the drop oscillates relative to its spherical shape as if it levitates in the zero gravity. Those nonlinear drop oscillations have been extensi- vely studied by many authors and we refer interested readers to [10 – 14] in which theoretical results are reported utilizing the Lagrange variational formalism. In the contrast, when the vertical drop size and λ/4 are of the comparable order, the acoustic radiation pressure deforms the drop shape so that its averaged, visually observed geometry is far from a sphere as schematically illustrated in Fig. 1 (b). Those acoustically deformed drop shapes and their stability were investigated, experimentally and theoretically, for instance, in [9, 15 – 17]. The employed applied mathematical model in these references has been at a physical level of rigor. It empirically involves the free surface problem on the weightless drop dynamics in which the pressure (dynamic) boundary condition includes an extra quantity responsible for the acoustic radiation pressure generated by an external standing acoustic wave in gas. A feedback of the levitating drop shapes on the external acoustic field has been neglected, — the acoustic field is assumed to be the same as for a solid levitating sphere. Appearance of the acoustic radiation pressure in this empirical model can be interpreted as a so-called vibrati- onal force well-known from the vibrational mechanics [18]. The papers [19 – 23] considered the vibrational hydrodynamic problems of compressible liquids partly filling a container as an ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 415 Fig. 2. The “ullage gas-liquid drop” mechanical system located in a rigid box Q. The acoustic vibrator is marked by S0, but S1 is the box surface appearing as a reflector of the acoustic wave. object of the applied functional analysis. They introduced the so-called vibroequilibria, the time- averaged liquid shapes occurring due to high-frequency vibrational loads. Furthermore, a series of theorems were proved on the spectral boundary problems describing the linear eigenoscillati- ons relative to the vibroequilibria as well as papers developing the Lagrangian formalism for the contained liquid vibromechanics. The present paper follows the applied mathematical studies in [19, 20 23] to construct a new, mathematically justified model that describes slow-time motions of an acoustically levi- tating drop. The analysis starts with the “ulage gas-liquid drop” interface problem formulated within the framework of ideal compressible fluids with irrotational flows. Furthermore, fast and slow time scales are separated in both differential and variational statements. The fast-time averaged interface problem yields a free-surface problem in which the Langevin acoustic radi- ation pressure appears, in a natural way, in the dynamic boundary condition. The kinematic boundary condition of this problem implies that the free surface reflects the acoustic wave. Whereas there are no slow drop oscillations, the derived free-surface problem transforms to a static problem whose solution describes a visually-observed, acoustically deformed drop shape. The shape is called the drop vibroequilibrium. In contrast to the mathematical model from [9, 15 – 17], the drop vibroequilibria change the external vibrational field. This is the first main result of the present paper. Another main result consists in developing the averaged Lagrange variational formalism and deriving a functional which can be interpreted as a quasipotential energy of the drop vibroeqilibria. The forthcoming studies should deal with generalizing the spectral theorems on the linear natural oscillations of the acoustically levitating drops relative to the drop vibroequilibria. 2. Statement of the problem. Fig. 2 schematically shows the “ullage gas-liquid drop” mechani- cal system confined in a closed rigid box Q = {x ∈ R3 \| W (x) < 0} (acoustic levitator), where W (x) = 0 determines a piecewise smooth box boundary and x = (x1, x2, x3) ∈ R3 is the Cartesian coordinate system. The domain Q consists of the ullage gas Q1(t) and liquid Q2(t) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 416 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA time-dependent domains (Q = Q1(t) ∪ Q2(t)) so that the interface Σ(t) = ∂Q2(t) = {x ∈ ∈ Q2 \| ξ(x, t) = 0} is defined by the unknown function ξ(x1, x2, x3, t) = 0 so that ∇ξ/\|∇ξ\| is the exterior normal vector with respect to the drop domain Q2(t). Both gas and liquid are compressible ideal and barotropic fluids with irrotational flows. The box boundary S = ∂Q falls into a reflecting surface S1 ⊂ S and acoustic vibrator S0 ⊂ S, i.e., S = S0 ∪ S1. The gravity acceleration vector is directed downward, against the Ox3 axis. We introduce the velocity potentials ϕi = ϕi(x, t), the pressure pi = pi(x, t) and the density field ρi = ρi(x, t) defined in Qi(t), i = 1, 2. The governing equations for the ideal barotropic fluids [19] read as ρ̇i + div(ρi∇ϕi) = 0, (1a) ρi∇ ( ϕ̇i + 1 2 \|∇ϕi\|2 + gx1 ) = −∇pi, (1b) ρi = ρ0i ( pi p0i )1/γi in Qi(t), (1c) where g is the gravity acceleration, ρ0i are the mean densities, p0i are the mean (static) pressures in the fluids (i = 1, 2), and γi, i = 1, 2, are the adiabatic indexes for barotropic (by definition, the pressure is uniquely a function of the density) ullage gas and liquid, respectively. The time derivative is denoted by the dot. The two fluid domains should also satisfy the mass conservati- on condition ∫ Qi(t) ρi dQ = mi, i = 1, 2, (2) where m1 and m2 are the constant masses of gas and liquid, respectively. The kinematic boundary conditions are ∂ϕi ∂n = − ξ̇ \|∇ξ\| , i = 1, 2, on Σ(t), (3a) ρ1 ∂ϕ1 ∂n = ρ01V0(x) sin(νt) on S0, (3b) ∂ϕ1 ∂n = 0 on S1. (3c) These conditions imply that fluid particles remain on the interface Σ(t) (kinematic condition (3a)), define the normal velocity on the acoustic vibrator S0 (condition (3b)) so that ν is the acoustic frequency and V0(x) 6≡ 0 determines the vibrator (S0) shape, and (3c) implies that S1 is a reflecting surface. Finally, the compressible fluid interface problem requires the dynamic boundary condition p2 + Ts(k1 + k2) = p1 on Σ(t) (4) expressing the pressure balance between the drop and the ullage gas, where the surface tension is associated with the Ts(k1 + k2) quantity in which ki, i = 1, 2, are the principal curvatures of Σ(t) and Ts is the surface tension coefficient. ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 417 The problem (1) – (4) needs the initial conditions ξ(x, 0) = ξ̃0(x), ξ̇(x, 0) = ξ̃1(x), (5) ϕi(x, 0) = υi(x), ϕ̇(x, 0) = υ1i(x), i = 1, 2. 3. The drop vibroequilibrium. 3.1. Nondimensional statement. Henceforth, the free-interface problem (1) – (4) is considered in the nondimensional statement assuming the characteristic si- ze D0 = 2R0 (the equivalent drop diameter) and the characteristic time ν−1 (ν is the circular acoustic frequency). The normalization suggests xnew = D−1 0 x, ξnew = D−1 0 ξ, ϕi(new) = D−2 0 ν−1ϕi, pi(new) = ρ0iD −2 0 ν−2pi, (6) p0i(new) = ρ0iD −2 0 ν−2p0i, ρi(new) = ρi0/ρi, mi(new) = miD −3 0 /ρ0i, i = 1, 2, and introduces the following nondimensional parameters: b = gD2 0ρ02 Ts , δ = ρ01 ρ02 , ν2 ∗ = D3 0ρ02ν 2 Ts , k = νD0 cg , and k∗ = νD0 cl , (7) where δ is the “gas-liquid” mean densities ratio, ν∗ is the nondimensional acoustic frequency, b is the Bond number, and k and k∗ are the wave numbers of compressible wave motions in gas and liquid, respectively; cg and cl are speeds of sound in the corresponding media. After omitting the subscript new, (1) – (4) transforms to the nondimensional form ρ̇i + div(ρi∇ϕi) = 0, (8a) ρi∇ ( ϕ̇i + 1 2 \|∇ϕi\|2 + ν−2 ∗ bx1 ) = −∇pi, (8b) ρi = ( pi p0i )1/γi in Qi(t), (8c) ∫ Qi(t) ρi dQ = mi, i = 1, 2, (8d) ∂ϕ1 ∂n = 0 on S1, (8e) ρ1 ∂ϕ1 ∂n = sup \|V0\| cg︸︷︷︸ ε V0(x) sup \|V0\|︸︷︷︸ V (x)=O(1) 1 k sin t on S0, (8f) ∂ϕi ∂n = − ξ̇ \|∇ξ\| , i = 1, 2, (8g) p2 + ν−2 ∗︸︷︷︸ µµ1ε3 (k1 + k2) = p1 δ︸︷︷︸ µ1ε on Σ(t). (8h) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 418 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA A set of small nondimensional parameters is introduced that are marked by the underbraces. First, the primary, main small parameter is ε = sup \|V0\| cg � 1. (9) It gives the ratio between the maximum acoustic vibrator velocity and the sound speed in the ullage gas. Secondly, the density ratio ρ01 ρ02 = δ = µ1ε, µ1 ∼ 1, (10) is assumed to be of the same order as ε (µ1 = O(1) is the proportionality coefficient). Thirdly, the nondimensional acoustic frequency is chosen as high as to provide the asymptotic relation ν−2 ∗ = µµ1ε 3, µ = O(1). (11) Fourthly, the wave numbers are O(ε) = k2 ∗ � k2 = O(1) (12) implying (from the physical point of view) that the acoustic frequency may be close to lower acoustic resonant frequencies in the ullage gas, k = O(1), but, because speed of sound in the liquid is higher than that in the ullage gas, the compressible liquid motions are far from the resonant condition and, in the first approximation, the drop can be considered as an incompres- sible liquid. 3.2. Introducing slow and fast time variables. As it is usually accepted in the vibrational mechanics [18], the fast and slow time scales can be introduced so that the fast time is associated with the nondimensional time t appearing in the nonhomogeneous condition (8f) expressing the input vibrational signal, but the slow time scale τ should be proportional to the square-root of the nondimensional potential type forces. The latter forces are contributed by the surface tension and the gravity. The related quantities appear in the dynamic interface condition (8h) accompanied by the O(ε3)-multiplier and, therefore, the slow time variable can be defined as τ = ε3/2t. The nondimensional solution of (8) takes the form ϕi = ϕi(x, t, τ), pi = pi(x, t, τ), ρi = ρi(x, t, τ), and ξ = ξ(x, t, τ). (13) The nondimensional problem (8) contains the small parameters ε, ε3 and, because of the slow-time component in (13), ε3/2. The standard assumption of the asymptotic method emplo- ying the fast and slow time separation is that (13) can be posed in the asymptotic series ϕi = ∞∑ k=0 εk/2ϕ (k/2) i (x, t, τ), pi = ∞∑ k=0 εk/2p (k/2) i (x, t, τ), (14) ρi = ∞∑ k=0 εk/2ρ(k/2)(x, t, τ), ξ = ∞∑ k=0 εk/2ξk/2(x, t, τ), where the coefficients are smooth functions of their variables. Specifically, the rational-number superscript indexes are introduced to link the functional coefficients with the small parameter powers. The sequence of the indexes are 0, 1/2, 1, 3/2, 2, 5/2, 3, . . . . ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 419 3.3. Separating slow and fast time variables. Substituting (14) into (8) leads to the k-family of boundary-value problems with respect to ϕ(k/2) i , p (k/2) i , ρ (k/2) i , ξk/2, i = 1, 2, starting with k = 0. The starting point implies the O(1)-order approximation which comes from the homogeneous problem ( ρ (0) i ) t + div ( ρ (0) i ∇ϕ (0) i ) = 0, ρ (0) i ∇ (( ϕ (0) i ) t + 1 2 ( ∇ϕ(0) i )2 ) = −∇p(0) i , ρ (0) i = ( p (0) i p0i )1/γi in Q (0) i , ∂ϕ (0) 1 ∂n = 0 on S1, ∂ϕ (0) 1 ∂n = 0 on S0, ∂ϕ (0) i ∂n = − (ξ0)t \|∇ξ0\| , i = 1, 2, −p(0) 2 = 0 on Σ(0), where (·)t is the fast-time derivative. The last pressure condition on Σ(0) shows that the liquid motions are dynamically uncoupled with the compressible gas flows and, moreover, the drop is not affected by the surface tension. From physical point of view, this means that the O(1)-order drop motions can only slowly deform on the τ -scale and the zero-order solution takes the form ξ0 = ξ0(x, τ), ∫ Q (0) 2 (τ) dQ = m2, ∇ϕ(0) i = 0, i = 1, 2, p (0) 1 = p01, ρ (0) 1 = 1, ρ (0) 2 = p (0) 2 = 0, where ξ0(x, τ) = 0 defines the O(1)-order interface motions Σ(0) = Σ(0)(τ) which, in turn, defines the slowly-deforming domains Q(0) i (τ), i = 1, 2. Henceforth, the O(1)-order drop motions are associated with the fast-time averaged drop shape, i.e., by definition, Σ0(τ) = Σ0(τ) = 〈Σ(t, τ)〉t, Q (0) i (τ) = 〈Qi(t, τ)〉t, i = 1, 2. (15) Furthermore, the higher-order asymptotic problems with respect to ϕ (k/2) i , p (k/2) i , ρ (k/2) i , ξk/2, k ≥ 1, would be formulated in the fast-time averaged domains Q(0) 1 (τ) and Q(0) 2 (τ) separated by Σ0(τ). The problem (8) contains three small input parameters of the orderO(ε), O(ε3/2) andO(ε3), but there are no the O(ε1/2)-order input quantities. This means that the O(ε1/2)-order approxi- ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 420 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA mation is zero. The O(ε)-order approximation (k = 2) comes from the problem k2 ( ϕ (1) 1 ) tt −∇2ϕ (1) 1 = 0 in Q (0) 1 (τ), ∂ϕ (1) 1 ∂n = − (ξ1)t \|∇ξ0\| on Σ0(τ), ∂ϕ (1) 1 ∂n = 0 on S1, ∂ϕ (1) 1 ∂n = V (x) sin t k on S0, ∂ϕ (1) 2 ∂n = − (ξ1)t \|∇ξ0\| , p (1) 2 = µ1p01 on Σ0(τ), ρ̇ (1) 2 +∇2ϕ (1) 2 = 0, ρ (1) 2 = 0 in Q (0) 2 (τ), where the last condition is due to ρ(1) 2 = k2 ∗p (1) 2 and (12). As it happened in the zero-order approximation, the dynamic interface condition (here, p (1) 2 = µ1p01 = const) on the fast-time averaged interface Σ0(τ) decouples the interface problem into two independent boundary-value problems in Q (0) 2 (τ) and Q (0) 1 (τ), respectively. Analyzing the first boundary-value problem in Q(0) 2 (τ) shows that this approximation can only contribute a slow-time drop deformation which, due to definition (15), is already accounted for by the O(1)-order component. As a consequence, ξ1 = 0, ∇ϕ(1) 2 = 0, ρ (1) 2 = 0, p (1) 2 = µ1p01. The second boundary-value problem in Q(0) 1 (τ) has the solution ϕ (1) 1 = Φ1(x, τ) sin t, p (1) 1 = Φ1(x, τ) cos t, (16) where Φ1(x) is the so-called wave function of the linear acoustic field in the ullage gas governed by the Neumann boundary-value problem ∇2Φ1 + k2Φ1 = 0 in Q (0) 1 (τ), ∂Φ1 ∂n = 0 on S1 ∪ Σ0(τ), ∂Φ1 ∂n = V (x) k on S0 (17) and stated in the slowly-deforming gas domain; the fast-time averaged drop surface Σ0(τ) plays the role of a reflector. The interface problem remains decoupled in the O(ε3/2)-order approximation. For the gas domain Q(0) 1 (τ), the homogeneous τ -dependent boundary problem takes the form ∇2ϕ (3/2) 1 = 0, p (3/2) 1 = ( ϕ (3/2) 1 ) t in Q (0) 1 (τ), ∂ϕ (3/2) 1 ∂n = 0 on S0 ∪ S1, ∂ϕ (3/2) 1 ∂n = − (ξ0)τ + ( ξ3/2 ) t \|∇ξ0\| on Σ0(τ), but ∇2ϕ (3/2) 2 = 0, p (3/2) 2 = ( ϕ (3/2) 2 ) t , ρ (3/2) 2 = 0 in Q (0) 2 (τ), (18) ∂ϕ (3/2) 2 ∂n = − (ξ0)τ + ( ξ3/2 ) t \|∇ξ0\| , p (3/2) 2 = 0 on Σ0(τ) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 421 describes the O(ε3/2)-contribution to the drop motions which also is τ -dependent. This means that ϕ(3/2) i = ϕ (3/2) i (x, τ), i = 1, 2. Summarizing all asymptotic quantities obtained from the constructed approximations gives ϕ2(x, t, τ) = ε3/2 ϕ (3/2) 2 (x, τ)︸︷︷︸ ϕ(x,τ) +o(ε3/2), (19a) ξ(x, t, τ) = ξ0(x, τ)︸︷︷︸ ζ(x,τ) +o(ε3/2), (19b) ϕ1(x, t, τ) = εΦ1(x, τ)︸︷︷︸ Φ(x,τ) sin t+ ε3/2ϕ (3/2) 1 (x, τ) + o(ε3/2). (19c) This shows that the lowest-order component of the velocity field in the drop domain is of the order O(ε3/2); the velocity field does not depend on the fast time t. In the contrast, the lowest- order component of the velocity field in the gas domain describes the linear acoustic standing wave for which the slowly-varying drop surface Σ0(τ) : ζ(x, τ) = 0 is a reflector. Because the right-hand side of the dynamic interface condition (8h) has theO(ε)-multiplier, the drop oscillates on the fast-time scale caused by the linear acoustic field (16) so that ϕ(2) 2 = = sin t F1(x, τ). The velocity potential in Q (0) 1 (τ) takes the form ϕ (2) 1 = sin(2t)F2(x, τ)+ + cos(2t)F3(x, τ). However, due to quadratic terms, the second-order pressure component in Q (0) 1 (τ) contains the fast-time averaged quantity 〈p(2) 1 〉t(x, τ) = 1 4 ( k2(Φ1)2 − (∇Φ1)2 ) + const (20) expressing the so-called Langevin acoustic radiation pressure. The O(ε5/2)-order component is of more complicated structure, but it does not affect the O(ε3)-order approximation which yields the fast-time averaged dynamic boundary condition( ϕ (3/2) 2 ) τ + 1 2 ( ∇ϕ(3/2) 2 )2 − µµ1(k1 + k2) + µ1µbx1+ + 1 4 µ1 ( k2(Φ1)2 − (∇Φ1)2 ) = const on Σ0(τ). (21) 3.4. Slow-time oscillations with respect to the drop vibroequilibrium. Accounting for the asymptotic solution (19), the fast-time averaged dynamic condition (21) as well as the governi- ng boundary-value problems for the lowest-order quantities in (19), we arrive, finally, at the following free-interface problem with respect to ζ(x, τ) = ξ0(x, τ), ϕ(x, τ) = ϕ (3/2) 2 (x, τ) and Φ(x, τ) = Φ1(x, τ): ∇2ϕ = 0 in Ω2(τ), ∂ϕ ∂n = − ζτ \|∇ζ\| on Γ(τ), ∫ Ω2(τ) dΩ = m2, (22a) ϕτ + 1 2 (∇ϕ)2 − µµ1(k1 + k2) + µµ1bx1 + 1 4 µ1 ( k2(Φ)2 − (∇Φ)2 ) = const on Γ(τ), ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 422 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA ∇2Φ + k2Φ = 0 in Ω1(τ), ∂Φ ∂n = 0 on S1 ∪ Γ(τ), ∂Φ ∂n = V (x) k on S0 (22b) where Ω1(τ) = Q (0) 1 (τ),Ω2(τ) = Q (0) 2 (τ), and Γ(τ) = Σ0(τ). In fact, we have proved the following proposition. Proposition 1. If the original interface problem (8) has the asymptotic solution (14), the lowest order terms in (19) depend only on the slow time τ = ε3/2t and these terms are governed by the free-surface problem (22). The free-interface problem (22) is the announced mathematical model for the acoustically levitating drops. It describes slow-time oscillations of an acoustically levitating drop. The problem (22) is similar to the earlier empirical mathematical model in [9, 15 – 17] and should, perhaps, theoretically clarify the vertical vibrations and shape oscillations of droplets [25]. A difference consists of an extra term in the dynamic interface condition on Γ(τ) expressing the Langevin radiation pressure which becomes now parametrically depending on the τ -instant drop shape (due to the zero-Neumann boundary condition (22b) on Γ(τ)). The latter boundary condition means that the slowly-oscillating drop surface is, in the lowest-order approximation, a reflector for the linear acoustic field in the ullage gas. When assuming that the fast-time averaged drop shape does not oscillate, we arrive at the static free-interface problem −µ(k1 + k2) + bµx1 + 1 4 ( k2(Φ)2 − (∇Φ)2 ) = const on Γ0, ∫ Ω20 dΩ = m2, (23a) ∇2Φ + k2Φ = 0 in Ω10, ∂Φ ∂n = 0 on S1 ∪ Γ0, ∂Φ ∂n = V (x) k on S0. (23b) The drop shape Γ0 is called the drop vibroequilibria. The drop vibroequilibria shape is what one can see in acoustic levitators but the evoluti- on problem (22) describes, in fact, nonlinear motions with respect to the vibroequilibria. The drop vibroequilibria can be stable or not depending on input parameters. The stability analysis should normally involve the spectral problem on linear natural (eigen) oscillations with respect to Γ0, or, alternatively, the extremal problem on the quasipotential energy as in Section 4. The aforementioned spectral problem has the classical exact Rayleigh solution [26] for the weightless drop when the acoustic field is absent. The acoustically-deformed levitating drops are not the case and dedicated studies are required on the natural (eigen) modes and frequencies which differ from those in [26]. 4. Lagrangian formalism for (1) – (4). We will follow [19] to prove two theorems providing equivalence of (1) – (4) to the classical Lagrange and the Bateman – Luke variational formulati- ons. The first case is the classical Lagrange principle. Theorem 1. When functions ξ, ϕi and ρi, i = 1, 2, are smooth enough, the free-interface problem (1) – (4) is equivalent to the necessary condition of the extremal points of the action G(ξ, ϕi, ρi) = t2∫ t1 [T − U −Π]dt= t2∫ t1  2∑ i=1 ∫ Qi(t) ρi [ 1 2 (∇ϕi)2 − Ui(ρi)− gx1 ] dQ− Ts\|Σ\|  dt (24) ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 423 subject to the kinematic constraint (1) and assuming the smooth isochronous variations δξ\|t1,t2 = 0, δρi\|t1,t2 = 0. (25) Here, T is the kinetic energy, Π is the potential energy, and Ui(ρi) is the inner energy of gas and liquid, respectively. The area is denoted as \| · \|. The inner energy of barotropic fluids defines the pressure as pi df = ρ2 i dUi dρi . (26) Remark 1. Because of constraint (1), the action is a function of ξ and ρi. Proof. We employ the formula ∫ Ω(t) [ρ̇+ (∇ϕ · ∇ψ)] dQ+ d dt ∫ Ω(t) ρϕ dQ− ∫ S0 ρ01V0ϕ sin(νt) dS = = − ∫ Ω(t) [ρ̇+ div(ρ∇ψ)]ϕdQ+ ∫ S1 ρ ∂ψ ∂n ϕdS + ∫ Σ(t) ρ ∂ψ ∂n ϕdS+ + ∫ Σ(t) ρ ξ̇ \|∇ξ\| ϕdS + ∫ S0 [ ρ ∂ψ ∂n − ρ01V0 sin(νt) ] ϕdS (27) following from the Reynolds transport theorem and the Green formulas when Ω(t), ∂Ω(t) = = Σ(t) ∪ S1 ∪ S0 is an arbitrary domain, Σ(t) (ξ(x, t) = 0) is a piece of the time-dependent boundary, but ϕ(x, t) and ψ(x, t) are smooth functions. Using the kinematic constraint (1) with ϕ = ϕ1, ψ = ψ1 for Ω(t) = Q1(t), the right-hand side of (27) equals to zero. Analogously, when ϕ = ϕ2 and ψ = ψ2 in Ω(t) = Q1(t), ∂Ω(t) = = Σ(t), the right-hand side is also zero. After integration by t from t1 to t2 of the remaining left-hand sides and subtracting the results from the action, we come to G(ξ, ϕi, ρi) = t2∫ t1  2∑ i=1 ∫ Qi(t) ρi [ −ϕ̇i − 1 2 (∇ϕi)2 − Ui(ρi)− gx1 ] dQ − − Ts\|Σ(t)\|+ ∫ S0 ρ01V0ϕ1 sin(νt) dS  dt− 2∑ i=1 (ρiϕi)\|t2t1 . (28) Now, assuming the kinematic constraint (1) is satisfied, one can compute variations of G by ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 424 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA ρi and ξ employing (28). Variations by ρi give δρiG = t2∫ t1  ∫ Qj(t) δρj [ −ϕ̇j − 1 2 (∇ϕj)2 − gx1 − Uj(ρj)− ρj dUj dρj ] dQ − − ∫ Qj(t) ρj [δϕ̇j + (∇ϕj · ∇δϕj)] dQ+ ∫ S0 ρ01V0δϕ1 sin(νt) dS  dt− − [δρjϕj + ρjδϕj ]\|t2t1 = 0, j = 1, 2. (29) Accounting for (27), (25) and (1) leads to −ϕ̇j − 1 2 (∇ϕj)2 − gx1 − Uj(ρj)− ρj dUj dρj = 0. (30) Taking the gradient action and using (26) give (1b). Computing the ξ-variation of (28), accounting for (25) and using formulas [24] for variations of the Σ(t) area by ξ give, altogether, δξG = t2∫ t1  2∑ i=1 ∫ Qi(t) ρi[−δϕ̇i − (∇ϕi · ∇δϕi)] dQ + + ∫ Σ(t) 2∑ i=1 (−1)i δξ \|∇ξ\| ρi [ −ϕ̇i − 1 2 (∇ϕi)2 − gx1 − Ui(ρi) ] dS− − Ts ∫ Σ(t) [−k1 − k2] δξ \|∇ξ\| dS + ∫ S0 ρδϕ1V0 sin(νt) dS  dt− 2∑ i=1 (ρiδϕi)\|t2t1 = 0. (31) Employing the formula (27) within ϕ and δϕ transforms (31) to the form δξG = t2∫ t1 [{ ∫ Σ(t) 2∑ i=1 (−1)iρi [ −ϕ̇i − 1 2 (∇ϕi)2 − gx1 − Ui(ρi) ] + + Ts[k1 + k2] } δξ \|∇ξ\| dS ] dt = 0, (32) which leads to the dynamic condition (4) provided by (30) (following from the condition δρjG = = 0, j = 1, 2). Theorem 1 is proved. Another variational formulation is associated with the so-called Bateman – Luke variational principle [24] for a compressible fluid. Specifically, this variational principle is not restricted to ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 425 the kinematic constraint. The Bateman – Luke action takes the form B(ξ, ϕi, ρi) = t2∫ t1  2∑ i=1 ∫ Qi(t) ρi [ −ϕ̇i − 1 2 (∇ϕi)2 − gx1 − Ui(ρi) ] dQ − − Ts\|Σ(t)\|+ ∫ S0 ρ01V0ϕ1 sin(νt) dS  dt (33) which is the same as expression (28) but without the last summand. Theorem 2. When functions ξ, ϕi and ρi, i = 1, 2, are smooth enough, the free-interface problem (1) – (4) follows from the necessary condition of the extremal points of the action (33) subject to the isochronous smooth variations δξ\|t1,t2 = 0, δϕi\|t1,t2 = 0, δρi\|t1,t2 = 0. (34) Proof. The theorem immediately follows from the already computed variations of (28) by ρj , ξ as well as the formula for variations by ϕj : δϕjB = t2∫ t1 − ∫ Qj(t) ρj [δϕ̇j + (∇ϕj · ∇δϕj)] dQ+ ∫ S0 ρ01δϕjV0 sin(νt) dS  dt = = t2∫ t1  ∫ Qj(t) [ρ̇j + div(ρj∇ϕj)]δϕj dQ − − ∫ S1 ρj ∂ϕj ∂n δϕj dS − ∫ Σ(t) ρj [ ∂ϕj ∂n + ξ̇ \|∇ξ\| ] δϕj dS− − ∫ S0 ( ρ1 ∂ϕ1 ∂n − ρ01V0 sin(νt) ) δϕ1 dS  dt+ ρjδϕj \|t2t1 = 0. (35) We should account for (34) and the fact that S0 = S1 = ∅ for j = 2 in (35). Theorem 2 is proved. 5. Quasipotential energy of the drop vibroequilibrium. In Subsection 3.4, we showed that the nondimensional problem (8) has the asymptotic solution (19) whose lowest-order terms describe slow-time motions with respect to the drop vibroequilibrium. The slow time variable is τ = ε3/2t and the lowest-order terms are governed by (22). In this section, we separate slow and fast time variables in the variational formulations from Section 4 to derive the quasipotential energy of the drop vibroequilibrium governed by (23). Theorem 3. Finding the fast-time averaged solution from the classical Lagrange variational formulation (Theorem 1) is equivalent to description of the extremal points of the nondimensi- onal functional 〈G∗(ξ, ϕi, ρi)〉t = const + ε3/2G(ζ, ϕ) +O(ε2), ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 426 M. O. CHERNOVA, I. A. LUKOVSKY, A. N. TIMOKHA within G(ζ, ϕ,Φ) = τ2∫ τ1  ∫ Ω2(τ) [ 1 2 (∇ϕ)2 − µµ1bx1 ] dQ− µµ1\|Γ(τ)\| + + µ1 4 ∫ Ω1(τ) [ k2Φ2 − (∇Φ)2 ] dQ− µ1 2k ∫ S0 ΦV (x) dS  dτ (36) subject to the kinematic constraint ∇2ϕ = 0 in Ω2(τ), ∂ϕ ∂n = − ζτ \|∇ζ\| on Γ(τ), (37a) ∇2Φ + k2Φ = 0 in Ω1(τ), ∂Φ ∂n = 0 on S1 ∪ Γ(τ), ∂Φ ∂n = V (x) k on S0 (37b) for isochronous smooth variations δζ\|τ1,τ2 = 0 where ζ(x, τ) = 0 governs the slow-time oscillati- ons of the drop surface Γ(τ) (Ω2(τ) and Ω1(τ) are liquid and gas domains, respectively, separated by Γ(τ)) on the slow-time scale. Proof. According to Theorem 1, finding the solution of (1) – (4) (nondimensional statement (8)) is equivalent to description of the extremal points of the action (24). Adopting the nondi- mensional variational statement, substituting (19) into variational and differential formulations of Theorem 1 and choosing \|t2−t1\| > ε−3/2,we get 〈G(ξ, ϕi, ρi)〉t = const+ε3/2G(ζ, ϕ)+O(ε2) and the kinematic constraint (37). Let ζ, ϕ be a local extrema point of the action (36) subject to (37). Obviously, ζ and ϕ satisfy (22). Taking (19) in the nondimensional formulation of Theorem 1 gives, within to higher-order terms, an extremal point of G∗. Theorem 3 is proved. Theorem 4. Finding the fast-time averaged solution from the Bateman – Luke variational formulation (Theorem 2) is equivalent to finding the extremal points of the time-averaged nondi- mensional action 〈B∗(ξ, ϕi, ρi)〉t = const + ε3/2B(ζ, ϕ,Φ) +O(ε2), where B(ζ, ϕ,Φ) = τ2∫ τ1  ∫ Ω2(τ) [ −ϕτ − 1 2 (∇ϕ)2 − µµ1bx1 ] dQ− µµ1\|Γ(τ)\| + + µ1 4 ∫ Ω1(τ) [ k2Φ2 − (∇Φ)2 ] dQ− µ1 2k ∫ S0 ΦV (x) dS  dτ, subject to isochronous smooth variations δζ\|τ1,τ2 = 0, δϕ\|τ1,τ2 = 0, δΦ\|τ1,τ2 = 0. ISSN 1562-3076. Нелiнiйнi коливання, 2015, т . 18, N◦ 3 DIFFERENTIAL AND VARIATIONAL FORMALISM . . . 427 Proof. The proof is similar to that in the previous theorem. Remark 2. The fast-time averaged variational formulation of the Bateman – Luke type leads to Theorem 4 which can be treated as the Bateman – Luke variational formulation for the weightless drop dynamics levitating in the zero-gravity and affected, altogether, by the surface tension and the Langevin radiation pressure. Assuming the τ -independent solutions in Theorems 3 and 4 leads to the quasipotential energy of the mechanical system. This means that: Theorem 5. Finding the stable drop vibroequilibria from (23) is equivalent to finding the local minima of the quasipotential energy functional U = µ\|Γ0\|+ µb ∫ Ω10 x1 dQ− 1 4 ∫ Ω10 ( k2Φ2 − (∇Φ)2 ) dQ+ 1 2k ∫ S0 V (x)Φ dS subject to ∫ Ω20 dQ = m2 = const and ∇2Φ + k2Φ = 0 in Ω10, ∂Φ1 ∂n = 0 on S1 ∪ Γ0, ∂Φ1 ∂n = V (x) k on S0. 6. Conclusions. Employing the differential and variational formulations of an interface prob- lem for two compressible fluids, we studied the fast-time averaged motions of an acousti- cally levitated drop. A new mathematical model is derived describing slow-time motions of the drop with respect to the visually-observed quasi static drop shapes which are called the drop vibroequilibria. The derived mathematical model is qualitatively similar to the physically- postulated models in [9, 15 – 17]. They all introduce the Langevin radiation pressure quantity appearing in the dynamic boundary condition on the drop surface. However, there is a novelty in our new mathematical model — it expresses the important fact that the acoustical field geometry parametrically depends on the drop shape. Along with the differential formulation of the mathematical model, we present a series of theorems on the Lagrange variational formalism and derive a functional responsible for the quasipotential energy of the mechanical system. 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Differential and variational formalism for an acoustically levitating drop

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