Generalizing the Multimodal Method for the Levitating Drop Dynamics

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.


Introduction
Drops levitating in an ullage gas appear in chemical industry 1-3 and space technology 4-6 .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 7-9 , 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 10-12 and theoretically 13 .Smallamplitude 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 [16][17][18] .An alternative approach ISRN Mathematical Physics to numerical simulations could be theoretical methods developed, for example, in [19][20][21] 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 26-30 and references therein.
The present paper generalizes the Lukovsky-Miles multimodal method for the freesurface 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.

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 R 0 of the sphere as the characteristic length and introduce the characteristic time t * ρR 3 0 /T s T s is the surface tension coefficient .The nondimensional drop dynamics is considered in the spherical coordinate system x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ, r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π so that the free surface Σ t is described by the equation According to 2.1 , perturbations of Σ t relative to the static spherical shape are subject to the volume conservation condition 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 : where the Lagrangian reads as Here, |•| defines the area and p 0 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 p 0 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 .

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 2π 0 π 0 ξ sin θdθdϕ 0, 3.1 but the linearized boundary conditions can be combined to exclude ξ as follows: 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 with respect to spectral parameter σ 2 and eigenfunction φ.
The spectral boundary problem 3.4 can be solved by separating the spatial variables φ r, θ, ϕ Y lm r, θ, ϕ r l Y lm θ, ϕ , l ≥ 0 which leads to the equation The analytical eigensolution follows from 3.5 and consists of the eigenfrequencies and the eigenfunctions

3.7
where P m l 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 φ 10 z r cos θ that describes a translatory ISRN Mathematical Physics drop motion as a solid body along Oz, but l 1 and m 1 yield y r sin θ sin ϕ and x r sin θ cos ϕ describing the same translatory motions but along Oy and Ox, respectively.The case l 0 corresponds to φ 00 1/2 √ π.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 .

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: where {f I } and {φ N } 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 down

4.2b
This transforms the modal solution 4.1 to the form F c,lm t cos mϕ F s,lm t sin mϕ φ lm r, θ .

4.3b
Accounting for 4.3a in 2.2 gives the holonomic constraint where G 3 implies the cubic, fourth, and so forth polynomial terms.Using the implicit function theorem resolves β 0 as follows: G 3 also denotes the cubic and other higher-order polynomial terms in β * and transforms 4.3a to the form A n, c,lm F n F c,lm − TS 0, 4.9 where Performing a variation of independent generalized velocities F i , F c,lm , F s,lm , i ≥ 1, l ≥ 1 in the action 2.5 within the Lagrangian 4.9 leads to the equations

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The Euler-Lagrange equations follow from the extrema condition of the action with respect to generalized coordinates β i , β c,lm , β s,lm , i ≥ 1, l ≥ 1.They are often called the dynamic modal equations and take the form

4.15c
The derivative by β * is done assuming that 4.6 accounts for the volume conservation condition so that, for instance,

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.

Weakly-Nonlinear Modal Equations for Axisymmetric Drop Motions
For the axisymmetric drop dynamics, the velocity potential takes the form and the free-surface equation is as follows: where and C m0 i0,j0 are the Clebsch-Gordan coefficients 38 .
Henceforth, adopting ideas from 27, 32 , we will construct a weakly-nonlinear, thirdorder modal equations by postulating the relationships and neglecting the o -terms in the Lukovsky-Miles modal equations 4.13a , Accounting for 4.14a , kinematic modal equations 4.13a read as where neglecting the o -terms implies that ∂A n /∂β i and A nk keep only the second-order polynomial quantities, that is,

5.8
Furthermore, 5.9 Kinematic equations 5.5 can be considered as linear algebraic equations with respect to F k whose asymptotic solution neglecting o should admit the form Substituting 5.10 into 5.5 and gathering all the similar polynomial terms give

5.11
For the axisymmetric drop dynamics, the dynamic modal equations 4.15a , 4.15b , and 4.15c take the form

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,

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 β n 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.

Nonlinear Axisymmetric Eigenoscillation
We consider almost-periodic oscillations of an axisymmetric drop with the frequency σ close to the lowest linear eigenfrequency natural frequency σ 20 subject to the Duffing-type thirdorder asymptotics implying the dominant character of the primary generalized coordinate β 2 O 1/3 .Analyzing the nonzero coefficients in 5.15 shows that this asymptotics yields

6.3
Other modal equations in 5.15 do not include nonlinear terms.

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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 σ 2 : the nonlinear eigenfrequency σ is unknown .The wanted periodic solution takes then the form where A is the unknown dominant amplitude and substituting 6.5 into 6.2a and 6.2b and accounting for 6.4 give

6.6d
Gathering the A 3 -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 σ − σ 2 /σ 2 O 2/3 and the nondimensional amplitude parameter 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 σ − σ 2 /σ 2 and the maximum ratio H/W between the instant drop height and width were reported.The experimental data for R 0 0.49 cm are denoted by •, but • marks measurements made for R 0 0.62 cm.In the lowest-order approximation, the Legendre polynomials properties deduce that where A is defined by 6.7 .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 R 0 .Our theoretical values are in good agreement with experimental measurements for R 0 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 .

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.

3 ijk is defined by 5 .3 and Λ 4 ijkm
is also expressed via the Clebsch-Gordan coefficients Λ

Figure 2 :
Figure 2: Theoretical 6.8 , solid line and experimental dependence between σ − σ 2 /σ 2 σ 2 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 R 0 0.49 cm • and R 0 0.62 cm • .Theoretical results do not depend on R 0 and T s within the framework of an ideal incompressible liquid model.The figure also shows numerical results from 19 dashed line , 33 and 34 .