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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 869070, 19 pages
http://dx.doi.org/10.5402/2012/869070
Research Article

Generalizing the Multimodal Method for the Levitating Drop Dynamics

1Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3 St., 01601 Kiev, Ukraine
2CeSOS, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received 23 April 2012; Accepted 30 July 2012

Academic Editors: S. Ansoldi, K. Netocny, K.-E. Thylwe, and G. F. Torres del Castillo

Copyright © 2012 M. O. Chernova et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The present paper extends the multimodal method, which is well known for liquid sloshing problems, to the free-surface problem modeling the levitating drop dynamics. The generalized Lukovsky-Miles modal equations are derived. Based on these equations an approximate modal theory is constructed to describe weakly-nonlinear axisymmetric drop motions. Whereas the drop performs almost-periodic oscillations with the frequency close to the lowest natural frequency, the theory takes a finite-dimensional form. Periodic solutions of the corresponding finite-dimensional modal system are compared with experimental and numerical results obtained by other authors. A good agreement is shown.

1. Introduction

Drops levitating in an ullage gas appear in chemical industry [13] and space technology [46]. The levitation is provided by weightless conditions or/and acoustic and/or electromagnetic fields created in the gas. When these external fields do not affect the static spherical drop shape, namely, there is no flattening caused by the fields as described, for example, in [79], one can assume that the drop dynamics is primary determined by the surface tension as if the drop levitates in zero-gravity conditions.

The relative drop dynamics with respect to the static spherical shape driven by the surface tension has been studied, experimentally [1012] and theoretically [13]. Small-amplitude (linear) drop motions were analytically described by Lord Rayleigh [14, 15] in 1879. He found the corresponding natural (linear eigen) modes and frequencies in terms of the spherical harmonics. Theoretical studies of the nonlinear drop dynamics were mainly done numerically by employing different discretization schemes [1618]. An alternative approach to numerical simulations could be theoretical methods developed, for example, in [1921] where a Fourier approximation by the Rayleigh natural modes was combined with variational and asymptotic methods. This approach looks similar to nonlinear multimodal methods elaborated in the 70 s for liquid sloshing dynamics.

The multimodal methods took their canonical form in the pioneering papers by Lukovsky [22] and Miles [23] and, furthermore, were generalized by others. An extensive review on the multimodal methods can be found in the book by Lukovsky [24] and Faltinsen and Timokha [25]. The Lukovsky-Miles version of the multimodal methods makes it possible to derive a well-structured infinite-dimensional (modal) system of nonlinear ordinary differential (modal) equations with respect to generalized coordinates and velocities which, under certain circumstances, is fully equivalent to the original free-surface problem. Naturally, the generalized coordinates in sloshing correspond to the natural sloshing modes. Being “asymptotically-detuned” to a class of liquid sloshing phenomena the Lukovsky-Miles modal equations reduce to a rather compact, finite-dimensional form. The detuning suggests postulating a series of asymptotic relations between the generalized coordinates. Representative examples of such “asymptotic modal equations” can be found in the aforementioned books [24, 25] as well as in [2630] and references therein.

The present paper generalizes the Lukovsky-Miles multimodal method for the free-surface problem describing the nonlinear dynamics of a levitating drop. This generalization includes derivation of general modal equations of Lukovsky-Miles’ type as well as examples of asymptotic modal equations. The paper plan is as follows. In Section 2,we present both differential and variational formulations of the problem. Following Lukovsky and Miles as well as recalling [19], the variational formulation is based on the Bateman-Luke principle (see, also, [25], Ch. 7). In Section 3, we rederive the Rayleigh-type eigensolution to show that, from a mathematical point of view, the set of natural modes is not complete and extra four spherical harmonics should be included into the modal solution. In Section 4, we derive the general modal equations analogous to those in [24, 25, 31]. An approximate form of these equations describing the weakly-nonlinear axisymmetric drop dynamics is constructed in Section 5. These equations keep up to third-order polynomial terms as it has been in [27, 32] for sloshing problems. Based on these approximate equations, we derive in Section 6 a finite-dimensional system of “asymptotic” modal equations modeling the weakly-nonlinear almost-periodic drop motions with the frequency close to the lowest natural frequency. Periodic solutions of these equations are compared with experimental [12] and numerical [19, 33, 34] results. A good agreement is shown.

2. Statement of the Problem

We consider a levitating drop 𝑄(𝑡) of an ideal incompressible liquid that performs oscillatory motions as illustrated in Figure 1. Due to the surface tension, the drop takes spherical shape in its hydrostatic state. We choose the radius 𝑅0 of the sphere as the characteristic length and introduce the characteristic time 𝑡=𝜌𝑅30/𝑇𝑠 (𝑇𝑠 is the surface tension coefficient). The nondimensional drop dynamics is considered in the spherical coordinate system 𝑥=𝑟sin𝜃cos𝜑, 𝑦=𝑟sin𝜃sin𝜑, 𝑧=𝑟cos𝜃, (𝑟0, 0𝜃𝜋, 0𝜑2𝜋) so that the free surface Σ(𝑡) is described by the equation 𝑟=𝜁(𝜃,𝜑,𝑡)=1+𝜉(𝜃,𝜑,𝑡).(2.1)

869070.fig.001
Figure 1: Geometric notations.

According to (2.1), perturbations of Σ(𝑡) relative to the static spherical shape are subject to the volume conservation condition 𝑉𝑙=𝑄(𝑡)4𝑑𝑄=3𝜋02𝜋𝜋013𝜉3+𝜉2+𝜉sin𝜃𝑑𝜃𝑑𝜑=0(2.2) playing the role of a holonomic constraint.

The free-surface problem describing the nonlinear drop dynamics couples the function 𝜁 and the velocity potential Φ (see, e.g., [19]):21Φ=𝑟2𝜕𝑟𝜕𝑟2𝜕Φ+1𝜕𝑟𝑟2𝜕sin𝜃𝜕𝜃sin𝜃𝜕Φ+1𝜕𝜃𝑟2sin2𝜃𝜕2Φ𝜕𝜑2=0in𝑄(𝑡),(2.3a)𝜁Φ𝑟Φ𝜃𝜁𝜃Φ𝜑𝜁𝜑sin𝜃=𝜁𝜁𝑡onΣ(𝑡),(2.3b)𝜕Φ+1𝜕𝑡2(Φ)2+𝜁2+𝜃/𝜁2+𝜁𝜑/(𝜁sin𝜃)2𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃21𝜁2𝜕sin𝜃𝜕𝜃𝜁𝜁𝜃sin𝜃𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃21𝜁2sin2𝜃𝜕𝜕𝜑𝜁𝜁𝜑𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃2+𝑝0(𝑡)=0onΣ(𝑡),(2.3c)

subject to the volume conservation condition (2.2). Here, the Laplace equation (2.3a) and the Neumann boundary condition (2.3b) constitute together the kinematic subproblem and (2.3c) is the so-called dynamic boundary condition in which the square bracket term is the sum of the principal curvatures [𝑘1+𝑘2]. The dynamic boundary condition expresses the pressure balance on the free surface assuming that the ullage pressure is a constant value and using the Bernoulli equation written for an incompressible inviscid liquid with irrotational flow. The time-dependent function 𝑝0(𝑡) implies the difference of the mean pressure between liquid and gas domains caused by the surface tension.

The free-surface problem (2.3a), (2.3b), and (2.3c) requires either the initial conditions 𝜁(𝜃,𝜑,0)=𝜁0(𝜃,𝜑),Φ(𝑟,𝜃,𝜑,0)=Φ0(𝑟,𝜃,𝜑)(2.4) defining initial drop shape and velocity field or the periodicity condition 𝜁(𝜃,𝜑,𝑡)=𝜁(𝜃,𝜑,𝑡+𝑇), Φ(𝑟,𝜃,𝜑,𝑡)=Φ(𝑟,𝜃,𝜑,𝑡+𝑇), where 𝑇=2𝜋/𝜎 is a fixed period.

Following Lukovsky and Miles [24, 25], we employ the Bateman-Luke variational formulation which states that the free-surface problem (2.3a), (2.3b), and (2.3c) follows from the necessary extrema condition of the action 𝐴(Φ,𝜁)=𝑡2𝑡1BL(Φ,𝜁)𝑑𝑡(2.5) within arbitrary instants 𝑡1 and 𝑡2 (𝑡1<𝑡2) and independent variables 𝜁 and Φ restricted to ||𝛿Φ𝑡1,𝑡2||=0,𝛿𝜁𝑡1,𝑡2=0,(2.6) where the Lagrangian reads as BL(Φ,𝜁)=𝑄(𝑡)𝜕Φ+1𝜕𝑡2(Φ)2||||𝑑𝑄Σ(𝑡)𝑝0𝑄(𝑡)𝑑𝑄𝑉𝑙.(2.7)

Here, || defines the area and 𝑝0 is the Lagrange multiplier (a time-dependent function) caused by the holonomic constraint (2.2).

The Bateman-Luke variational principle is based on the sum of the pressure-integral and potential energy associated with the surface tension. In addition, there is the Lagrange multiplier 𝑝0 which is the same as the mean pressure difference. Equivalence of the Bateman-Luke variational formulation and free-surface problems in fluid dynamics is, for instance, proven in [24, 25] and the book by Berdichevsky [35].

3. Linear Eigensolution and Natural Modes

Let us consider small-amplitude drop oscillations with respect to its static spherical shape by linearizing the volume conservation as well as kinematic (2.3b) and dynamic (2.3c) boundary conditions in terms of Φ and 𝜉. The linearized volume conservation condition (2.2) takes the form 02𝜋𝜋0𝜉sin𝜃𝑑𝜃𝑑𝜑=0,(3.1) but the linearized boundary conditions 𝜕Φ=𝜕𝑟𝜕𝜉,𝜕𝑡𝜕Φ+1𝜕𝑡2𝜉𝜕sin𝜃𝜕𝜃sin𝜃𝜕𝜉+1𝜕𝜃sin2𝜃𝜕2𝜉𝜕𝜑2=0(𝑟=1)(3.2) can be combined to exclude 𝜉 as follows: 𝜕2Φ𝜕𝑡22𝜕Φ+𝜕𝜕𝑟1𝜕𝑟𝜕sin𝜃𝜕𝜃sin𝜃𝜕Φ+1𝜕𝜃sin2𝜃𝜕2Φ𝜕𝜑=0(𝑟=1).(3.3)

Postulating Φ(𝑟,𝜃,𝜑,𝑡)=𝜙(𝑟,𝜃,𝜑)exp(𝑖𝜎𝑡) where 𝜎 is the so-called natural (linear eigen) frequency leads to the spectral boundary problem 2𝜙=0(𝑟<1),02𝜋𝜋0𝜕𝜙|||𝜕𝑟𝑟=1sin𝜃𝑑𝜃𝑑𝜑=0,𝜎22𝜙=𝜕𝜙+𝜕𝜕𝑟1𝜕𝑟𝜕sin𝜃𝜕𝜃sin𝜃𝜕𝜙+1𝜕𝜃sin2𝜃𝜕2𝜙𝜕𝜑2(𝑟=1)(3.4) with respect to spectral parameter 𝜎2 and eigenfunction 𝜙.

The spectral boundary problem (3.4) can be solved by separating the spatial variables 𝜙(𝑟,𝜃,𝜑)=𝑌𝑙𝑚(𝑟,𝜃,𝜑)=𝑟𝑙𝑌𝑙𝑚(𝜃,𝜑), 𝑙0 which leads to the equation 1𝜕sin𝜃𝜕𝜃sin𝜃𝜕𝑌𝑙𝑚+1𝜕𝜃sin2𝜃𝜕2𝑌𝑙𝑚𝜕𝜑=𝑙(𝑙+1)𝑌𝑙𝑚.(3.5)

The analytical eigensolution follows from (3.5) and consists of the eigenfrequencies 𝜎2=𝜎2𝑙𝑚=𝑙(𝑙1)(𝑙+2),𝑙=0,1,,𝑚=0,,𝑙(3.6) and the eigenfunctions 𝜙𝑙𝑚=𝑌𝑙𝑚(𝑟,𝜃,𝜑)=𝑁𝑙𝑚𝑟𝑙𝑃𝑙(𝑚)(𝑁cos𝜃)cos𝑚𝜑,sin𝑚𝜑,𝑙𝑚=(2𝑙+1)(𝑙𝑚)!4𝜋(𝑙+𝑚)!,𝑚=0,(2𝑙+1)(𝑙𝑚)!2𝜋(𝑙+𝑚)!,𝑚1,(3.7) where 𝑃𝑙(𝑚) are the associated Legendre polynomials.

Four eigenfunctions with 𝑙=0 and 1 imply the zero eigenfrequencies and, from the physical point of view, these eigenfunctions do not belong to the set of natural modes by Lord Rayleigh [14, 15]. The case 𝑙=1 with 𝑚=0 gives 𝜙10=𝑧=𝑟cos𝜃 that describes a translatory drop motion (as a solid body) along 𝑂𝑧, but 𝑙=1 and 𝑚=1 yield𝑦=𝑟sin𝜃sin𝜑 and 𝑥=𝑟sin𝜃cos𝜑 describing the same translatory motions but along 𝑂𝑦 and 𝑂𝑥, respectively. The case 𝑙=0 corresponds to 𝜙00=1/2𝜋. Excluding these four eigenfunctions, that is, concentrating on the Rayleigh solution, makes the functional basis (3.7) incomplete from the mathematical point of view [36, 37].

4. Nonlinear Modal Equations

The Lukovsky-Miles multimodal method suggests the modal solution of the free-surface problem (2.3a), (2.3b), and (2.3c) as follows: 𝜁(𝜃,𝜑,𝑡)=1+𝐼𝛽𝐼(𝑡)𝑓𝐼(𝜃,𝜑),Φ(𝑟,𝜃,𝜑,𝑡)=𝑁𝐹𝑁(𝑡)𝜙𝑁(𝑟,𝜃,𝜑),(4.1) where {𝑓𝐼} and {𝜙𝑁} are the complete sets of functions to define admissible shapes 𝑄(𝑡) satisfying the volume conservation condition and approximating the velocity field, respectively. Dealing with the star-shaped domains 𝑄(𝑡), the solid harmonics (3.7) provide the completeness [36, 37] so that we can write down𝜙𝑙=𝑁𝑙0𝑟𝑙𝑃𝑙𝜙(cos𝜃),𝑙0,𝑙𝑚,𝑐=𝜙𝑙𝑚(𝑟,𝜃)cos𝑚𝜑=𝑁𝑙𝑚𝑟𝑙𝑃𝑙(𝑚)𝜙(cos𝜃)cos𝑚𝜑,𝑙1,𝑚=1,,𝑙,𝑙𝑚,𝑠=𝜙𝑙𝑚(𝑟,𝜃)sin𝑚𝜑=𝑁𝑙𝑚𝑟𝑙𝑃𝑙(𝑚)𝑓(cos𝜃)sin𝑚𝜑,𝑙1,𝑚=1,,𝑙,(4.2a)𝑙=𝑁𝑙0𝑃𝑙(𝑓cos𝜃),𝑙0,𝑙𝑚,𝑐=𝑓𝑙𝑚(𝜃)cos𝑚𝜑=𝑁𝑙𝑚𝑃𝑙(𝑚)𝑓(cos𝜃)cos𝑚𝜑,𝑙1,𝑚=1,,𝑙,𝑙𝑚,𝑠=𝑓𝑙𝑚(𝜃)sin𝑚𝜑=𝑁𝑙𝑚𝑃𝑙(𝑚)(cos𝜃)sin𝑚𝜑,𝑙1,𝑚=1,,𝑙.(4.2b)

This transforms the modal solution (4.1) to the form𝜁(𝜃,𝜑,𝑡)=1+𝑙=0𝛽𝑙(𝑡)𝑓𝑙(𝜃)+𝑙𝑙=1𝑚=1𝛽𝑐,𝑙𝑚(𝑡)cos𝑚𝜑+𝛽𝑠,𝑙𝑚𝑓(𝑡)sin𝑚𝜑𝑙𝑚(𝜃),(4.3a)Φ(𝑟,𝜃,𝜑,𝑡)=𝑙=0𝐹𝑙(𝑡)𝜙𝑙(𝑟,𝜃)+𝑙𝑙=1𝑚=1𝐹𝑐,𝑙𝑚(𝑡)cos𝑚𝜑+𝐹𝑠,𝑙𝑚𝜙(𝑡)sin𝑚𝜑𝑙𝑚(𝑟,𝜃).(4.3b)

Accounting for (4.3a) in (2.2) gives the holonomic constraint 2𝜋𝛽0+𝑖=0𝛽2𝑖+𝑙𝑙=1𝑚=1𝛽2𝑐,𝑙𝑚+𝛽2𝑠,𝑙𝑚+𝐺3𝛽𝑖,𝛽𝑐,𝑙𝑚,𝛽𝑠,𝑙𝑚=0,(4.4) where 𝐺3 implies the cubic, fourth, and so forth polynomial terms. Using the implicit function theorem resolves 𝛽0 as follows: 𝛽0𝛽=𝐺𝑖,𝛽𝑐,𝑙𝑚,𝛽𝑠,𝑙𝑚1,𝑖1,𝑙1=2𝜋𝑖=1𝛽2𝑖+𝑙𝑙=1𝑚=1𝛽2𝑐,𝑙𝑚+𝛽2𝑠,𝑙𝑚12𝜋𝐺3𝛽𝑖,𝛽𝑐,𝑙𝑚,𝛽𝑠,𝑙𝑚,𝑖1,𝑙1(4.5)

(𝐺3 also denotes the cubic and other higher-order polynomial terms in 𝛽) and transforms (4.3a) to the form 1𝜁(𝜃,𝜑,𝑡)=14𝜋𝑖=1𝛽2𝑖+𝑙𝑙=1𝑚=1𝛽2𝑐,𝑙𝑚+𝛽2𝑠,𝑙𝑚+𝐺3+𝑙=1𝛽𝑙(𝑡)𝑓𝑙+(𝜃)𝑙𝑙=1𝑚=1𝛽𝑐,𝑙𝑚(𝑡)cos𝑚𝜑+𝛽𝑠,𝑙𝑚𝑓(𝑡)sin𝑚𝜑𝑙𝑚(𝜃)(4.6) defining the free surface as a function of 𝛽𝑖, 𝛽𝑐,𝑙𝑚, 𝛽𝑠,𝑙𝑚, 𝑖1, 𝑙1. Representation (4.6) automatically satisfies the volume conservation condition and, as a consequence, the Lagrange multiplier in (2.7) should equal to zero, that is, 𝑝0=0.

The generalized velocity 𝐹0 can also be excluded from consideration due to the identity 23𝜋𝑡2𝑡1𝛿̇𝐹02𝑑𝑡=3𝜋𝛿𝐹0𝑡2𝛿𝐹0𝑡1=0,(4.7) provided by (2.6) or, more precisely, by 𝛿𝛽𝑖𝑡1=𝛿𝛽𝑖𝑡2=𝛿𝛽𝑐,𝑙𝑚𝑡1=𝛿𝛽𝑐,𝑙𝑚𝑡2=𝛿𝛽𝑠,𝑙𝑚𝑡1=𝛿𝛽𝑠,𝑙𝑚𝑡2=𝛿𝐹𝑖𝑡1𝑡=𝛿𝐹2=𝛿𝐹𝑐,𝑙𝑚𝑡1=𝛿𝐹𝑐,𝑙𝑚𝑡2=𝛿𝐹𝑠,𝑙𝑚𝑡1=𝛿𝐹𝑠,𝑙𝑚𝑡2=0.(4.8)

Substituting (4.3b) into (2.7) yields the Lagrangian as a function of generalized coordinates and velocities BL=𝑖=1𝐴𝑖̇𝐹𝑖𝑙𝑙=1𝑚=1𝐴𝑐,𝑙𝑚̇𝐹𝑐,𝑙𝑚𝑙𝑙=1𝑚=1𝐴𝑐,𝑙𝑚̇𝐹𝑐,𝑙𝑚12𝑛,𝑘=1𝐴𝑛,𝑘𝐹𝑛𝐹𝑘12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐴(𝑐,𝑙1𝑚1),(𝑐,𝑙2𝑚2)𝐹(𝑐,𝑙1𝑚1)𝐹(𝑐,𝑙2𝑚2)12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐴(𝑠,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝐹(𝑠,𝑙1𝑚1)𝐹(𝑠,𝑙2𝑚2)𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐴(𝑐,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝐹(𝑐,𝑙1𝑚1)𝐹(𝑠,𝑙2𝑚2)𝑛=1𝑙𝑙=1𝑚=1𝐴𝑛,(𝑠,𝑙𝑚)𝐹𝑛𝐹(𝑠,𝑙𝑚)𝑛=1𝑙𝑙=1𝑚=1𝐴𝑛,(𝑐,𝑙𝑚)𝐹𝑛𝐹(𝑐,𝑙𝑚)𝑇𝑆=0,(4.9) where𝐴𝑛=𝑄(𝑡)𝜙𝑛𝑑𝑄=02𝜋𝜋0𝜁0𝜙𝑛𝑟2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.10a)𝑐,𝑙𝑚=𝑄(𝑡)𝜙𝑙𝑚cos(𝑚𝜑)𝑑𝑄=02𝜋𝜋0𝜁0𝜙𝑙𝑚cos(𝑚𝜑)𝑟2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.10b)𝑠,𝑙𝑚=𝑄(𝑡)𝜙𝑙𝑚sin(𝑚𝜑)𝑑𝑄=02𝜋𝜋0𝜁0𝜙𝑙𝑚sin(𝑚𝜑)𝑟2sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.10c)

𝐴𝑛,𝑘=𝐴𝑘,𝑛=𝑄(𝑡)𝜙𝑛𝜙𝑘𝑑𝑄=02𝜋𝜋0𝜁0𝜙𝑛𝜙𝑘𝑟2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11a)𝑛,(𝑐,𝑙𝑚)=𝐴(𝑐,𝑙𝑚),𝑛=𝑄(𝑡)𝜙𝑛𝜙𝑙𝑚=cos𝑚𝜑𝑑𝑄02𝜋𝜋0𝜁0𝜙𝑛𝜙𝑙𝑚𝑟cos𝑚𝜑2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11b)𝑛,(𝑠,𝑙𝑚)=𝐴(𝑠,𝑙𝑚),𝑛=𝑄(𝑡)𝜙𝑛𝜙𝑙𝑚=sin𝑚𝜑𝑑𝑄02𝜋𝜋0𝜁0𝜙𝑛𝜙𝑙𝑚𝑟sin𝑚𝜑2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11c)(𝑐,𝑙1𝑚1),(𝑠,𝑙2𝑚2)=𝐴(𝑠,𝑙2𝑚2),(𝑐,𝑙1𝑚1)=𝑄(𝑡)𝜙𝑙1𝑚1cos𝑚1𝜑𝜙𝑙2𝑚2sin𝑚2𝜑=𝑑𝑄02𝜋𝜋0𝜁0𝜙𝑙1𝑚1cos𝑚1𝜑𝜙𝑙2𝑚2sin𝑚2𝜑𝑟2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11d)(𝑐,𝑙1𝑚1),(𝑐,𝑙2𝑚2)=𝐴(𝑐,𝑙2𝑚2),(𝑐,𝑙1𝑚1)=𝑄(𝑡)𝜙𝑙1𝑚1cos𝑚1𝜑𝜙𝑙2𝑚2cos𝑚2𝜑=𝑑𝑄02𝜋𝜋0𝜁0𝜙𝑙1𝑚1cos𝑚1𝜑𝜙𝑙2𝑚2cos𝑚2𝜑𝑟2𝐴sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11e)(𝑠,𝑙1𝑚1),(𝑠,𝑙2𝑚2)=𝐴(𝑠,𝑙2𝑚2),(𝑠,𝑙1𝑚1)=𝑄(𝑡)𝜙𝑙1𝑚1sin𝑚1𝜑𝜙𝑙2𝑚2sin𝑚2𝜑=𝑑𝑄02𝜋𝜋0𝜁0𝜙𝑙1𝑚1sin𝑚1𝜑𝜙𝑙2𝑚2sin𝑚2𝜑𝑟2sin𝜃𝑑𝑟𝑑𝜃𝑑𝜑,(4.11f)𝑇𝑆=Σ(𝑡)𝑑𝑆=02𝜋𝜋0𝜁𝜁2+𝜁2𝜃+𝜁2𝜑sin2𝜃sin𝜃𝑑𝜃𝑑𝜑.(4.12)

Performing a variation of independent generalized velocities 𝐹𝑖, 𝐹𝑐,𝑙𝑚, 𝐹𝑠,𝑙𝑚, 𝑖1, 𝑙1 in the action (2.5) within the Lagrangian (4.9) leads to the equations𝑑𝐴𝑛=𝑑𝑡𝑘=1𝐴𝑛,𝑘𝐹𝑘+𝑘𝑘=1𝑚=1𝐴𝑛,(𝑐,𝑘𝑚)𝐹𝑐,𝑘𝑚+𝐴𝑛,(𝑠,𝑘𝑚)𝐹𝑠,𝑘𝑚,𝑛1,(4.13a)𝑑𝐴𝑐,𝑙𝑚=𝑑𝑡𝑘=1𝐴(𝑐,𝑙𝑚),𝑘𝐹𝑘+𝑘𝑘=1𝑛=1𝐴(𝑐,𝑙𝑚),(𝑐,𝑘𝑛)𝐹𝑐,𝑘𝑛+𝐴(𝑐,𝑙𝑚),(𝑠,𝑘𝑛)𝐹𝑠,𝑘𝑛,(4.13b)𝑑𝐴𝑠,𝑙𝑚=𝑑𝑡𝑘=1𝐴(𝑠,𝑙𝑚),𝑘𝐹𝑘+𝑘𝑘=1𝑛=1𝐴(𝑠,𝑙𝑚),(𝑐,𝑘𝑛)𝐹𝑐,𝑘𝑛+𝐴(𝑠,𝑙𝑚),(𝑠,𝑘𝑛)𝐹𝑠,𝑘𝑛,(4.13c)

(𝑙1, 𝑚=1,,𝑙). Derivations leading to (4.13a), (4.13b), and (4.13c) are quite tedious but, under certain circumstances, these are similar to those in [24, 25] for sloshing problems.

The differentiation rule𝑑𝐴𝑛=𝑑𝑡𝑖=1𝜕𝐴𝑛𝜕𝛽𝑖̇𝛽𝑖+𝑙𝑙=1𝑚=1𝜕𝐴𝑛𝜕𝛽𝑐,𝑙𝑚̇𝛽𝑐,𝑙𝑚+𝜕𝐴𝑛𝜕𝛽𝑠,𝑙𝑚̇𝛽𝑠,𝑙𝑚,(4.14a)𝑑𝐴𝑐,𝑙𝑚=𝑑𝑡𝑖=1𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑖̇𝛽𝑖+𝑗𝑗=1𝑛=1𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑐,𝑗𝑛̇𝛽𝑐,𝑗𝑛+𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑠,𝑗𝑛̇𝛽𝑠,𝑗𝑛,(4.14b)𝑑𝐴𝑠,𝑙𝑚=𝑑𝑡𝑖=1𝜕𝐴𝑠,𝑙𝑚𝜕𝛽𝑖̇𝛽𝑖+𝑗𝑗=1𝑛=1𝜕𝐴𝑠,𝑗𝑛𝜕𝛽𝑐,𝑗𝑛̇𝛽𝑐,𝑗𝑛+𝜕𝐴𝑠,𝑙𝑚𝜕𝛽𝑠,𝑗𝑛̇𝛽𝑠,𝑗𝑛,(4.14c)

shows that (4.13a), (4.13b), and (4.13c) is a system of nonlinear ordinary differential equations with respect to generalized coordinates where the mass-matrix depends on 𝛽. On the other hand, relations (4.13a), (4.13b), and (4.13c) can be considered as a system of algebraic equations with respect to generalized velocities 𝐹𝑖, 𝐹𝑐,𝑙𝑚, 𝐹𝑠,𝑙𝑚, 𝑖1, 𝑙1, where 𝐴𝑛,𝑘 are nonlinear functions of generalized coordinates 𝛽𝑖, 𝛽𝑐,𝑙𝑚, 𝛽𝑠,𝑙𝑚, 𝑖1, 𝑙1 but the left-hand side 𝑑𝐴𝑛/𝑑𝑡, 𝑑𝐴𝑐,𝑙𝑚/𝑑𝑡, 𝑑𝐴𝑠,𝑙𝑚/𝑑𝑡 implies expressions with respect to generalized coordinates 𝛽𝑖, 𝛽𝑐,𝑙𝑚, 𝛽𝑠,𝑙𝑚, 𝑖1, 𝑙1 and their first derivative. Equations (4.13a), (4.13b), and (4.13c) are interpreted as kinematic equations or a nonholonomic constraint.

The Euler-Lagrange equations follow from the extrema condition of the action with respect to generalized coordinates 𝛽𝑖, 𝛽𝑐,𝑙𝑚, 𝛽𝑠,𝑙𝑚, 𝑖1, 𝑙1. They are often called the dynamic modal equations and take the form𝑛=1𝜕𝐴𝑛𝜕𝛽𝜇̇𝐹𝑛+𝑙𝑙=1𝑚=1𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝜇̇𝐹𝑐,𝑙𝑚+𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝜇̇𝐹𝑐,𝑙𝑚+12𝑛,𝑘=1𝜕𝐴𝑛,𝑘𝜕𝛽𝜇𝐹𝑛𝐹𝑘+𝑙𝑛,𝑙=1𝑚=1𝐹𝑛𝜕𝐴𝑛,(𝑐,𝑙𝑚)𝜕𝛽𝜇𝐹𝑐,𝑙𝑚+𝜕𝐴𝑛,(𝑠,𝑙𝑚)𝜕𝛽𝜇𝐹𝑠,𝑙𝑚+𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝜇+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑐,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑐,𝑙2𝑚2)𝜕𝛽𝜇+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑠,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑠,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝜇+𝜕𝑇𝑆𝜕𝛽𝜇=0,𝜇1,(4.15a)𝑛=1𝜕𝐴𝑛𝜕𝛽𝑐,𝜇𝜈̇𝐹𝑛+𝑙𝑙=1𝑚=1𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑐,𝜇𝜈̇𝐹𝑐,𝑙𝑚+𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑐,𝜇𝜈̇𝐹𝑐,𝑙𝑚+12𝑛,𝑘=1𝜕𝐴𝑛,𝑘𝜕𝛽𝑐,𝜇𝜈𝐹𝑛𝐹𝑘+𝑙𝑛,𝑙=1𝑚=1𝐹𝑛𝜕𝐴𝑛,(𝑐,𝑙𝑚)𝜕𝛽𝑐,𝜇𝜈𝐹𝑐,𝑙𝑚+𝜕𝐴𝑛,(𝑠,𝑙𝑚)𝜕𝛽𝑐,𝜇𝜈𝐹𝑠,𝑙𝑚+𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝑐,𝜇𝜈+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑐,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑐,𝑙2𝑚2)𝜕𝛽𝑐,𝜇𝜈+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑠,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑠,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝑐,𝜇𝜈+𝜕𝑇𝑆𝜕𝛽𝑐,𝜇𝜈=0,𝜇1,𝑛=1,,𝜇,(4.15b)𝑛=1𝜕𝐴𝑛𝜕𝛽𝑠,𝜇𝜈̇𝐹𝑛+𝑙𝑙=1𝑚=1𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑠,𝜇𝜈̇𝐹𝑐,𝑙𝑚+𝜕𝐴𝑐,𝑙𝑚𝜕𝛽𝑠,𝜇𝜈̇𝐹𝑐,𝑙𝑚+12𝑛,𝑘=1𝜕𝐴𝑛,𝑘𝜕𝛽𝑠,𝜇𝜈𝐹𝑛𝐹𝑘+𝑙𝑛,𝑙=1𝑚=1𝐹𝑛𝜕𝐴𝑛,(𝑐,𝑙𝑚)𝜕𝛽𝑠,𝜇𝜈𝐹𝑐,𝑙𝑚+𝜕𝐴𝑛,(𝑠,𝑙𝑚)𝜕𝛽𝑠,𝜇𝜈𝐹𝑠,𝑙𝑚+𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝑠,𝜇𝜈+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑐,𝑙1𝑚1𝐹𝑐,𝑙2𝑚2𝜕𝐴(𝑐,𝑙1𝑚1),(𝑐,𝑙2𝑚2)𝜕𝛽𝑠,𝜇𝜈+12𝑙1,𝑙2𝑙=11,𝑙2𝑚1,𝑚2=1𝐹𝑠,𝑙1𝑚1𝐹𝑠,𝑙2𝑚2𝜕𝐴(𝑠,𝑙1𝑚1),(𝑠,𝑙2𝑚2)𝜕𝛽𝑠,𝜇𝜈+𝜕𝑇𝑆𝜕𝛽𝑠,𝜇𝜈=0,𝜇1,𝑛=1,,𝜇.(4.15c)

The derivative by 𝛽 is done assuming that (4.6) accounts for the volume conservation condition so that, for instance,𝜕𝑇𝑆𝜕𝛽𝜇=02𝜋𝜋0𝑘1+𝑘2𝜁2𝑓𝜇1𝛽2𝜋𝜇14𝜋𝜕𝐺3𝜕𝛽𝜇=sin𝜃𝑑𝜃𝑑𝜑Σ(𝑡)𝜁𝑘1+𝑘2𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃2𝑓𝜇1𝛽2𝜋𝜇14𝜋𝜕𝐺3𝜕𝛽𝜇𝑑𝑆,(4.16a)𝜕𝑇𝑆𝜕𝛽𝑐,𝜇𝜈=02𝜋𝜋0𝑘1+𝑘2𝜁2𝑓𝜇𝜈1cos(𝜈𝜑)𝛽2𝜋𝑐,𝜇𝜈14𝜋𝜕𝐺3𝜕𝛽𝑐,𝜇𝜈=sin𝜃𝑑𝜃𝑑𝜑Σ(𝑡)𝜁𝑘1+𝑘2𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃2𝑓𝜇𝜈1cos(𝜈𝜑)𝛽2𝜋𝑐,𝜇𝜈14𝜋𝜕𝐺3𝜕𝛽𝑐,𝜇𝜈𝑑𝑆,(4.16b)𝜕𝑇𝑆𝜕𝛽𝑠,𝜇𝜈=02𝜋𝜋0𝑘1+𝑘2𝜁2𝑓𝜇𝜈1sin(𝜈𝜑)𝛽2𝜋𝑠,𝜇𝜈14𝜋𝜕𝐺3𝜕𝛽𝑠,𝜇𝜈=sin𝜃𝑑𝜃𝑑𝜑Σ(𝑡)𝜁𝑘1+𝑘2𝜁2+𝜁2𝜃+𝜁𝜑/sin𝜃2𝑓𝜇𝜈1sin(𝜈𝜑)𝛽2𝜋𝑠,𝜇𝜈14𝜋𝜕𝐺3𝜕𝛽𝑠,𝜇𝜈𝑑𝑆.(4.16c)

In summary, the Lukovsky-Miles modal equations (4.13a), (4.13b), (4.13c), (4.15a), (4.15b), and (4.15c) constitute an infinite-dimensional system of nonlinear ordinary differential equations with respect to generalized coordinates and velocities. A direct Runge-Kutta simulation with this system (Perko-type method [25, 26]) is possible adopting appropriate initial conditions following from (2.4). However, when the goal consists of analytical studies and/or description of almost-periodic motions, it would be better to reduce the system to a simpler approximate form by postulating asymptotic relationships between generalized coordinates and velocities and neglecting the higher-order terms. The reduced (asymptotic) modal system may in particular cases possess a finite-dimensional form.

5. Weakly-Nonlinear Modal Equations for Axisymmetric Drop Motions

For the axisymmetric drop dynamics, the velocity potential takes the form Φ(𝑟,𝜃,𝜑,𝑡)=𝑙=1𝐹𝑙(𝑡)𝜙𝑙(𝑟,𝜃)(5.1) and the free-surface equation is as follows: 1𝜁(𝜃,𝜑,𝑡)=14𝜋𝑖=1𝛽2𝑖+13𝑖,𝑗,𝑘=1Λ(3)𝑖𝑗𝑘𝛽𝑖𝛽𝑗𝛽𝑘+𝐺5+𝑙=1𝛽𝑙(𝑡)𝑓𝑙(𝜃),(5.2) where Λ(3)𝑖𝑗𝑚=2𝜋𝜋0𝑓𝑖𝑓𝑗𝑓𝑚1sin𝜃𝑑𝜃=2(2𝑖+1)(2𝑗+1)𝐶𝜋(2𝑚+1)𝑚0𝑖0,𝑗02(5.3) and 𝐶𝑚0𝑖0,𝑗0 are the Clebsch-Gordan coefficients [38].

Henceforth, adopting ideas from [27, 32], we will construct a weakly-nonlinear, third-order modal equations by postulating the relationships 𝛽𝑙𝐹𝑙𝜖=𝑂1/3,𝜖1,(5.4) and neglecting the 𝑜(𝜖)-terms in the Lukovsky-Miles modal equations (4.13a), (4.13b), (4.13c), (4.15a), (4.15b), and (4.15c).

Accounting for (4.14a), kinematic modal equations (4.13a) read as 𝑖=1𝜕𝐴𝑛𝜕𝛽𝑖̇𝛽𝑖=𝑘=1𝐴𝑛𝑘𝐹𝑘,𝑛1,(5.5) where neglecting the 𝑜(𝜖)-terms implies that 𝜕𝐴𝑛/𝜕𝛽𝑖 and 𝐴𝑛𝑘 keep only the second-order polynomial quantities, that is, 𝜕𝐴𝑛𝜕𝛽𝑖=𝛿𝑛𝑖+(2+𝑛)𝑗=1Λ(3)𝑛𝑖𝑗𝛽𝑗+(𝑛+1)(𝑛+2)2𝑗,𝑘=1Λ(4)𝑛𝑖𝑗𝑘𝛽𝑗𝛽𝑘2+𝑛𝛿4𝜋𝑛𝑖𝑗=1𝛽2𝑗+2𝛽𝑖𝛽𝑛=𝛿𝑛𝑖+𝑗=1𝜒(1)𝑛,𝑖,𝑗𝛽𝑗+𝑗,𝑘=1𝜒(2)𝑛,𝑖,𝑗𝑘𝛽𝑗𝛽𝑘,𝐴(5.6)𝑛𝑘=𝑛𝛿𝑛𝑘+𝑗=1𝑛𝑘Λ(3)𝑘𝑛𝑗+Λ(3)𝑛𝑘,𝑗𝛽𝑗+𝑛+𝑘2𝑖,𝑗=1𝑛𝑘Λ(4)𝑘𝑛𝑖𝑗+Λ(4)𝑛𝑘,𝑖𝑗𝛽𝑖𝛽𝑗𝑛(𝑛+𝑘+1)𝛿4𝜋𝑛𝑘𝑗=1𝛽2𝑗=𝑛𝛿𝑛𝑘+𝑗=1(1)𝑛𝑘,𝑗𝛽𝑗+𝑖,𝑗=1(2)𝑛𝑘,𝑖𝑗𝛽𝑖𝛽𝑗.(5.7)

Here, Λ(3)𝑖𝑗𝑘 is defined by (5.3) and Λ(4)𝑖𝑗𝑘𝑚 is also expressed via the Clebsch-Gordan coefficients Λ(4)𝑖𝑗𝑘𝑚=2𝜋𝜋0𝑓𝑖𝑓𝑗𝑓𝑘𝑓𝑚=sin𝜃𝑑𝜃(2𝑖+1)(2𝑗+1)(2𝑘+1)(2𝑚+1)4𝜋min(𝑖+𝑗,𝑘+𝑚)||||,||||𝑛=max𝑖𝑗𝑘𝑚1𝐶2𝑛+1𝑛0𝑖0,𝑗0𝐶𝑛0𝑘0,𝑚02.(5.8)

Furthermore, Λ(2)𝑛𝑘=2𝜋𝜋0𝜕𝑓𝑛𝜕𝜃𝜕𝑓𝑘𝜕𝜃sin𝜃𝑑𝜃=𝑛(𝑛+1)𝛿𝑛𝑘,Λ(3)𝑖𝑛,𝑘=2𝜋𝜋0𝜕𝑓𝑖𝜕𝜃𝜕𝑓𝑛𝑓𝜕𝜃𝑘1sin𝜃𝑑𝜃=2𝑖(𝑖+1)(2𝑖+1)𝑛(𝑛+1)(2𝑛+1)𝜋𝐶(2𝑘+1)𝑘0𝑖0,𝑛0𝐶𝑘0𝑖(1),𝑛1,Λ(4)𝑖𝑛,𝑘𝑗=2𝜋𝜋0𝜕𝑓𝑖𝜕𝜃𝜕𝑓𝑛𝑓𝜕𝜃𝑘𝑓𝑗1sin𝜃𝑑𝜃=4𝜋×𝑖(𝑖+1)(2𝑖+1)𝑛(𝑛+1)(2𝑛+1)𝑘(𝑘+1)(2𝑘+1)𝑗(𝑗+1)(2𝑗+1)min(𝑖+𝑛,𝑘+𝑗)||||𝑚=max|𝑖𝑛|,𝑘𝑗1𝐶2𝑚+1𝑚0𝑖0,𝑛0𝐶𝑚0𝑖(1),𝑛1𝐶𝑚0𝑘0,𝑗0𝐶𝑚0𝑘(1),𝑗1.(5.9)

Kinematic equations (5.5) can be considered as linear algebraic equations with respect to 𝐹𝑘 whose asymptotic solution (neglecting 𝑜(𝜖)) should admit the form 𝐹𝑙=̇𝛽𝑙𝑙+𝑖,𝑗=1𝑉(2)𝑙,𝑖,𝑗̇𝛽𝑖𝛽𝑗+𝑖,𝑗,𝑘=1𝑉(3)𝑙,𝑖,𝑗,𝑘̇𝛽𝑖𝛽𝑗𝛽𝑘,𝑙1.(5.10)

Substituting (5.10) into (5.5) and gathering all the similar polynomial terms give 𝑉(2)𝑛,𝑖,𝑗=𝜒(1)𝑛,𝑖,𝑗Π(1)𝑛𝑖,𝑗/𝑖𝑛,𝑉(3)𝑛,𝑖,𝑗,𝑘=𝜒(2)𝑛,𝑖,𝑗,𝑘Π(2)𝑛𝑖,𝑗𝑘/𝑖𝑙=1𝑉(2)𝑙,𝑖,𝑗Π(1)𝑛𝑙,𝑘𝑛.(5.11)

For the axisymmetric drop dynamics, the dynamic modal equations (4.15a), (4.15b), and (4.15c) take the form 𝑛=1𝜕𝐴𝑛𝜕𝛽𝜇̇𝐹𝑛+12𝑛,𝑘=1𝜕𝐴𝑛,𝑘𝜕𝛽𝜇𝐹𝑛𝐹𝑘+𝜕𝑇𝑆𝜕𝛽𝜇=0,𝜇1.(5.12)

Pursuing the announced weakly-nonlinear theory, we should employ (5.6), (5.7) and expand (4.16a) up to the third-order polynomial terms, that is, 𝜕𝑇𝑆𝜕𝛽𝜇=2𝜋𝜋0𝜁2𝜁sin𝜃2+𝜃/𝜁2𝜁2𝜁2𝜃1𝜁2𝜕sin𝜃𝜕𝜃𝜁𝜁𝜃sin𝜃𝜁2𝜁2𝜃×𝑓𝜇1𝛽2𝜋𝜇14𝜋𝑖,𝑗=1Λ𝑖𝑗𝜇𝛽𝑖𝛽𝑗𝑑𝜃=2𝜋𝜋0𝑓(2+2𝜉)𝜇1𝛽2𝜋𝜇14𝜋𝑖,𝑗=1Λ𝑖𝑗𝜇𝛽𝑖𝛽𝑗sin𝜃𝑑𝜃+2𝜋𝜋0𝜉𝜃12𝜉3𝜃𝜕𝑓𝜇𝜕𝜃sin𝜃𝑑𝜃=(𝜇+2)(𝜇1)𝛽𝜇+𝑖,𝑗=1𝑇(2𝜇)𝑖𝑗𝛽𝑖𝛽𝑗+𝑖,𝑗,𝑘=1𝑇(3𝜇)𝑖,𝑗,𝑘𝛽𝑖𝛽𝑗𝛽𝑘,(5.13) where 𝑇(2𝜇)𝑖𝑗=2Λ(3)𝑖𝑗𝜇,𝑇(3𝜇)𝑖,𝑗,𝑘1=2Λ(4𝜃)𝑖𝑗𝑘𝜇+1𝜋𝛿𝑖𝜇𝛿𝑗𝑘,Λ(4𝜃)𝑖𝑗𝑘𝜇=2𝜋𝜋0𝜕𝑓𝑖𝜕𝜃𝜕𝑓𝑗𝜕𝜃𝜕𝑓𝑘𝜕𝜃𝜕𝑓𝜇𝜕𝜃sin𝜃𝑑𝜃.(5.14)

Substituting (5.6), (5.7), (5.10), and (5.13) into (5.12) and neglecting the 𝑜(𝜖)-terms yield the following weakly-nonlinear modal equations: 𝑖=1𝛿𝜇𝑖+𝑗=1𝑑1,𝜇𝑖,𝑗𝛽𝑗+𝑗,𝑘=1𝑑2,𝜇𝑖,𝑗,𝑘𝛽𝑗𝛽𝑘̈𝛽𝑖+𝑛,𝑘=1𝑡0,𝜇𝑛,𝑘+𝑚=1𝑡1,𝜇𝑛,𝑘𝛽𝑚̇𝛽𝑛̇𝛽𝑘+𝜎2𝜇𝛽𝜇+𝑖,𝑗=1𝜇𝑇2𝜇𝑖𝑗𝛽𝑖𝛽𝑗+𝑖,𝑗,𝑘=1𝜇𝑇3𝜇𝑖,𝑗,𝑘𝛽𝑖𝛽𝑗𝛽𝑘=0,𝜇1,(5.15) where 𝜎𝜇=𝜎𝜇0 are the nondimensional frequencies (3.6) and 𝑑1,𝜇𝑖,𝑗𝜒=𝜇(1)𝑖,𝜇,𝑗𝑖+𝑉(2)𝜇,𝑖,𝑗,𝑑2,𝜇𝑖,𝑗,𝑘𝜒=𝜇(2)𝑖,𝜇,𝑗,𝑘𝑖+𝛼=1𝜒(1)𝛼,𝜇,𝑗𝑉(2)𝛼,𝑖,𝑘+𝑉(3)𝜇,𝑖,𝑗,𝑘,𝑡0,𝜇𝑛,𝑘𝑉=𝜇(2)𝜇,𝑛,𝑘+Π(1)𝑛𝑘,𝜇,𝑡𝑛𝑘1,𝜇𝑛,𝑘,𝑚=𝜇𝑉(3)𝜇,𝑛,𝑘,𝑚+Π(2)𝑛𝑘,𝜇𝑚+𝑛𝑘𝛼=1𝜒(1)𝛼,𝜇,𝑚𝑉(2)𝛼,𝑛,𝑘+Π(1)𝛼𝑘,𝜇𝑉(2)𝛼,𝑛,𝑚𝑘+Π(1)𝛼𝑛,𝜇𝑉(2)𝛼,𝑘,𝑚𝑛.(5.16)

We see that the weakly-nonlinear modal equations (5.15) constitute an infinite-dimensional system of ordinary differential equations (with respect to generalized coordinates 𝛽𝑛) that are not resolved relative to the highest derivative. Again, one can use (5.15) for direct numerical simulations (see, e.g., [32]). However, as in [27] and other analytical papers on sloshing, the derived weakly-nonlinear modal equations can reduce to a finite-dimensional form by employing a Duffing-type asymptotics. Using this asymptotics in sloshing problems implicitly suggests that we look for almost-periodic solutions with the frequency close to the lowest natural frequency of the mechanical system. For the studied case within no external excitations, these periodic solutions mean the nonlinear eigenoscillations of a freely-levitating drop.

6. Nonlinear Axisymmetric Eigenoscillation

We consider almost-periodic oscillations of an axisymmetric drop with the frequency 𝜎 close to the lowest linear eigenfrequency (natural frequency) 𝜎20 subject to the Duffing-type third-order asymptotics implying the dominant character of the primary generalized coordinate 𝛽2=𝑂(𝜖1/3). Analyzing the nonzero coefficients in (5.15) shows that this asymptotics yields 𝛽2𝜖=𝑂1/3,𝛽4𝜖=𝑂2/3,𝛽6=𝑂(𝜖),𝛽𝑙=𝑜(𝜖),𝑙2,4,6(6.1) so that neglecting the 𝑜(𝜖)-terms in (5.15) leads to the finite-dimensional system of nonlinear modal equations̈𝛽2+𝜎22𝛽2+𝑑1̈𝛽2𝛽4+𝑑2̈𝛽4𝛽2+𝑑3̇𝛽2̇𝛽4+𝑑4̈𝛽2𝛽22+𝑑5̇𝛽22𝛽2+𝑡1𝛽22+𝑡2𝛽2𝛽4+𝑡3𝛽32+𝑐1̈𝛽2𝛽2+𝑐2̇𝛽22̈𝛽=0,(6.2a)4+𝜎24𝛽4+𝑑6̈𝛽2𝛽2+𝑑7̇𝛽22+𝑡4𝛽22+𝑡5𝛽2𝛽4+𝑐3̈𝛽4𝛽2+𝑐4̇𝛽4̇𝛽2̈𝛽=0,(6.2b)6+𝜎26𝛽6+𝑑8̈𝛽2𝛽4+𝑑9̈𝛽4𝛽2+𝑑10̇𝛽4̇𝛽2+𝑑11̈𝛽2𝛽22+𝑑12̇𝛽22𝛽2+𝑡6𝛽2𝛽4+𝑡7𝛽32=0,(6.2c)

where 𝜎2=𝜎20,𝜎4=𝜎40,𝜎6=𝜎60 and 𝑑1=244𝜋,𝑑2=1514𝜋,𝑑3=7514𝜋,𝑑4=6798𝜋,𝑑5=585,𝑑196𝜋6=157𝜋,𝑑79=7𝜋,𝑑8=10565286𝜋,𝑑9=3065143𝜋,𝑑10=7565143𝜋,𝑑11=135651001𝜋,𝑑12=135652002𝜋;𝑡14=57𝜋,𝑡2=247𝜋,𝑡3=76,𝑡7𝜋4=247𝜋,𝑡5=160577𝜋,𝑡6=18065143𝜋,𝑡7=54065143𝜋;𝑐1=9514𝜋,𝑐2=457𝜋,𝑐3=755154𝜋,𝑐4=1855154𝜋.(6.3)

Other modal equations in (5.15) do not include nonlinear terms.

To construct a periodic asymptotic solution of (6.2a), (6.2b), and (6.2c) we assume, as usually [26], the asymptotic closeness condition between 𝜎 and 𝜎2: 𝜎𝜎2𝜎2𝜖=𝑂2/3(6.4) (the nonlinear eigenfrequency 𝜎 is unknown). The wanted periodic solution takes then the form 𝛽2=𝐴cos(𝜎𝑡)+𝐴2𝐸1+𝐸2𝐴cos(2𝜎𝑡)+𝑂3,𝛽4=𝐴2𝐸3+𝐸4𝐴cos(2𝜎𝑡)+𝑂3,𝛽6𝐴=𝑂3𝜖,𝐴=𝑂1/3,(6.5) where 𝐴 is the unknown dominant amplitude and substituting (6.5) into (6.2a) and (6.2b) and accounting for (6.4) give𝐸1=𝑡1+𝑐1𝑐2𝜎22𝜎22=𝑡1+𝑐1𝑐2𝜎222𝜎22𝐴+𝑂2,𝐸(6.6a)2=𝑡1+𝑐1+𝑐2𝜎22𝜎224𝜎2=𝑡1𝑐1+𝑐2𝜎226𝜎22𝐴+𝑂2,𝐸(6.6b)3=𝑡4+𝑑6𝑑7𝜎22𝜎24=𝑡4+𝑑6𝑑7𝜎222𝜎24𝐴+𝑂2,𝐸(6.6c)2=𝑡4+𝑑6+𝑑7𝜎22𝜎244𝜎22=𝑡4+𝑑6+𝑑7𝜎222𝜎244𝜎22𝐴+𝑂2.(6.6d)

Gathering the 𝐴3-order terms at the first harmonics in (6.2a) and using (6.4) lead to the secular equation to find the dependence between the normalized nonlinear eigenfrequency (𝜎𝜎2)/𝜎2=𝑂(𝜖2/3) and the nondimensional amplitude parameter 𝐴2=𝑂(𝜖2/3)𝜎𝜎2𝜎2+6347𝐴7840𝜋2=0.(6.7)

Secular equation (6.7) determines the so-called “soft-type” spring behavior suggesting that amplitude 𝐴 increases with decreasing frequency 𝜎.

Figure 2 compares our asymptotic result (6.7) with experimental data from [12] (see, also [19]) where dependence between (𝜎𝜎2)/𝜎2 and the maximum ratio (H/W) between the instant drop height and width were reported. The experimental data for 𝑅0= 0.49 cm are denoted by , but marks measurements made for 𝑅0= 0.62 cm. In the lowest-order approximation, the Legendre polynomials properties deduce that 𝐻𝑊=1+5/(4𝜋)𝐴1(1/2)5/(4𝜋)𝐴,(6.8) where 𝐴 is defined by (6.7).

869070.fig.002
Figure 2: Theoretical ((6.8), solid line)) and experimental dependence between (𝜎𝜎2)/𝜎2 (𝜎2 is the first linear eigenfrequency, 𝜎 is the corresponding nonlinear eigenfrequency) and the maximum ratio between the drop heights and width (H/W). Experimental values [12] are for 𝑅0=0.49 cm () and 𝑅0=0.62 cm (). Theoretical results do not depend on 𝑅0 and 𝑇𝑠 within the framework of an ideal incompressible liquid model. The figure also shows numerical results from [19] (dashed line), [33] (∆) and [34] (▲).

Figure 2 shows that (6.8) provides a good agreement with the numerical values from [19] (dashed line), [33] (∆), and [34] (▲) in which the studies were also based on the incompressible ideal liquid model (the results do not depend on 𝑅0). Our theoretical values are in good agreement with experimental measurements for 𝑅0= 0.63 cm. For lower drop radius, an qualitative agreement with experimental measurements is established. The discrepancy can be related to viscous effects discussed, for example, in [19].

7. Concluding Remarks

The present paper gives a generalization of Lukovsky-Miles’ multimodal method and derives the corresponding general nonlinear modal equations describing the nonlinear dynamics of a levitating drop. The modal equations are a full analogy of the original free-surface problem and look similar to those known for the nonlinear liquid sloshing problem [24, 25, 31].

The derived nonlinear modal equations are used to construct weakly-nonlinear modal equations for axisymmetric drop motions. The weakly-nonlinear equations possess a finite-dimensional form for the case of almost-periodic drop motions with the nonlinear eigenfrequency close to the lowest linear eigenfrequency. The latter case was studied by other authors, experimentally and numerically. To compare our analytical asymptotic results with earlier experimental [11, 12] and numerical [19, 33, 34] results, we constructed periodic solutions of the finite-dimensional modal system. All the results on periodic solutions from different sources are in a good agreement.

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