# Generalizing the Multimodal Method for the Levitating Drop Dynamics

Authors: M. O. Chernova; I. A. Lukovsky; A. N. Timokha
Journal: ISRN Mathematical Physics
 (2012)

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## 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.


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## Body

## 1. 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]. 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 [16–18]. An alternative approach to numerical simulations could be theoretical methods developed, for example, in [19–21] 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 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 Section2,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 dropQ(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*=ρR03/Ts (Ts is the surface tension coefficient). The nondimensional drop dynamics is considered in the spherical coordinate system x=rsinθcosφ, y=rsinθsinφ, z=rcosθ, (r≥0, 0≤θ≤π, 0≤φ≤2π) so that the free surface Σ(t) is described by the equation
(2.1)r=ζ(θ,φ,t)=1+ξ(θ,φ,t).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
(2.2)Vl=∫Q(t)dQ=43π⟹∫02π‍∫0π‍(13ξ3+ξ2+ξ)sin⁡θdθdφ=0
playing the role of a holonomic constraint.The free-surface problem describing the nonlinear drop dynamics couples the functionζ and the velocity potential Φ (see, e.g., [19]):(2.3a)∇2Φ=1r2∂∂r(r2∂Φ∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂Φ∂θ)+1r2sin⁡2θ∂2Φ∂φ2=0inQ(t),(2.3b)ζΦr-Φθζθ-Φφζφsin⁡θ=ζζton⁡Σ(t),(2.3c)(∂Φ∂t+12(∇Φ)2)+[2+(ζθ/ζ)2+(ζφ/(ζsin⁡θ))2ζ2+ζθ2+(ζφ/sin⁡θ)2-1ζ2sin⁡θ∂∂θ(ζζθsin⁡θζ2+ζθ2+(ζφ/sin⁡θ)2)-1ζ2sin⁡2θ∂∂φ(ζζφζ2+ζθ2+(ζφ/sin⁡θ)2)]+p-0(t)=0on⁡Σ(t),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 p-0(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
(2.4)ζ(θ,φ,0)=ζ0(θ,φ),Φ(r,θ,φ,0)=Φ0(r,θ,φ)
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
(2.5)A(Φ,ζ)=∫t1t2‍BL(Φ,ζ)dt
within arbitrary instants t1 and t2 (t1<t2) and independent variables ζ and Φ restricted to
(2.6)δΦ|t1,t2=0,δζ|t1,t2=0,
where the Lagrangian reads as
(2.7)BL(Φ,ζ)=-∫Q(t)‍(∂Φ∂t+12(∇Φ)2)dQ-|Σ(t)|-p-0(∫Q(t)‍dQ-Vl).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 multiplierp-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].


## 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
(3.1)∫02π‍∫0π‍ξsin⁡θdθdφ=0,
but the linearized boundary conditions
(3.2)∂Φ∂r=∂ξ∂t,∂Φ∂t+{-2ξ-(1sin⁡θ∂∂θ[sin⁡θ∂ξ∂θ]+1sin⁡2θ∂2ξ∂φ2)}=0(r=1)
can be combined to exclude ξ as follows:
(3.3)∂2Φ∂t2-{2∂Φ∂r+∂∂r[1sin⁡θ∂∂θ(sin⁡θ∂Φ∂θ)+1sin⁡2θ∂2Φ∂φ]}=0(r=1).PostulatingΦ(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
(3.4)∇2ϕ=0(r<1),∫02π‍∫0π‍∂ϕ∂r|r=1sin⁡θdθdφ=0,-σ2ϕ={2∂ϕ∂r+∂∂r[1sin⁡⁡θ∂∂θ(sin⁡⁡θ∂ϕ∂θ)+1sin⁡2⁡θ∂2ϕ∂φ2]}(r=1)
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)=Y-lm(r,θ,φ)=rlYlm(θ,φ), l≥0 which leads to the equation
(3.5)1sin⁡θ∂∂θ(sin⁡θ∂Ylm∂θ)+1sin⁡2θ∂2Ylm∂φ=-l(l+1)Ylm.The analyticaleigensolution follows from (3.5) and consists of the eigenfrequencies
(3.6)σ2=σlm2=l(l-1)(l+2),l=0,1,…,m=0,…,l
and the eigenfunctions
(3.7)ϕlm=Y-lm(r,θ,φ)=NlmrlPl(m)(cos⁡θ){cos⁡mφ,sin⁡mφ,Nlm={(2l+1)(l-m)!4π(l+m)!,m=0,(2l+1)(l-m)!2π(l+m)!,m≥1,
where Pl(m) are the associated Legendre polynomials.Four eigenfunctions withl=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:
(4.1)ζ(θ,φ,t)=1+∑I‍βI(t)fI(θ,φ),Φ(r,θ,φ,t)=∑N‍FN(t)ϕN(r,θ,φ),
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 down(4.2a)ϕl=Nl0rlPl(cos⁡θ),l≥0,ϕlm,c=ϕlm(r,θ)cos⁡mφ=NlmrlPl(m)(cos⁡θ)cos⁡mφ,l≥1,m=1,…,l,ϕlm,s=ϕlm(r,θ)sin⁡mφ=NlmrlPl(m)(cos⁡θ)sin⁡mφ,l≥1,m=1,…,l,(4.2b)fl=Nl0Pl(cos⁡θ),l≥0,flm,c=flm(θ)cos⁡mφ=NlmPl(m)(cos⁡θ)cos⁡mφ,l≥1,m=1,…,l,flm,s=flm(θ)sin⁡mφ=NlmPl(m)(cos⁡θ)sin⁡mφ,l≥1,m=1,…,l.This transforms the modal solution (4.1) to the form(4.3a)ζ(θ,φ,t)=1+∑l=0∞‍βl(t)fl(θ)+∑l=1∞‍∑m=1l‍(βc,lm(t)cos⁡mφ+βs,lm(t)sin⁡mφ)flm(θ),(4.3b)Φ(r,θ,φ,t)=∑l=0∞‍Fl(t)ϕl(r,θ)+∑l=1∞‍∑m=1l‍(Fc,lm(t)cos⁡mφ+Fs,lm(t)sin⁡mφ)ϕlm(r,θ).Accounting for (4.3a) in (2.2) gives the holonomic constraint
(4.4)2πβ0+∑i=0∞‍βi2+∑l=1∞‍∑m=1l‍(βc,lm2+βs,lm2)+G~3(βi,βc,lm,βs,lm)=0,
where G~3 implies the cubic, fourth, and so forth polynomial terms. Using the implicit function theorem resolves β0 as follows:
(4.5)β0=G(βi,βc,lm,βs,lm,i≥1,l≥1)=-12π[∑i=1βi2+∑l=1∞∑m=1l(βc,lm2+βs,lm2)]-12πG3(βi,βc,lm,βs,lm,i≥1,l≥1)(G3 also denotes the cubic and other higher-order polynomial terms in β*) and transforms (4.3a) to the form
(4.6)ζ(θ,φ,t)=1-14π(∑i=1∞‍βi2+∑l=1∞‍∑m=1l‍(βc,lm2+βs,lm2)+G3)+∑l=1∞βl(t)fl(θ)+∑l=1∞‍∑m=1l‍(βc,lm(t)cos⁡mφ+βs,lm(t)sin⁡mφ)flm(θ)
defining the free surface as a function of βi, βc,lm, βs,lm, i≥1, l≥1. 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, p-0=0.The generalized velocityF0 can also be excluded from consideration due to the identity
(4.7)23π∫t1t2‍δF˙0dt=23π[δF0(t2)-δF0(t1)]=0,
provided by (2.6) or, more precisely, by
(4.8)δβi(t1)=δβi(t2)=δβc,lm(t1)=δβc,lm(t2)=δβs,lm(t1)=δβs,lm(t2)=δFi(t1)=δF(t2)=δFc,lm(t1)=δFc,lm(t2)=δFs,lm(t1)=δFs,lm(t2)=0.Substituting (4.3b) into (2.7) yields the Lagrangian as a function of generalized coordinates and velocities
(4.9)BL=-∑i=1∞‍AiF˙i-∑l=1∞‍∑m=1l‍Ac,lmF˙c,lm-∑l=1∞‍∑m=1l‍Ac,lmF˙c,lm-12∑n,k=1∞‍An,kFnFk-12∑l1,l2=1∞‍∑m1,m2=1l1,l2‍A(c,l1m1),(c,l2m2)F(c,l1m1)F(c,l2m2)-12∑l1,l2=1∞‍∑m1,m2=1l1,l2‍A(s,l1m1),(s,l2m2)F(s,l1m1)F(s,l2m2)-∑l1,l2=1∞‍∑m1,m2=1l1,l2‍A(c,l1m1),(s,l2m2)F(c,l1m1)F(s,l2m2)-∑n=1∞‍∑l=1∞‍∑m=1l‍An,(s,lm)FnF(s,lm)-∑n=1∞‍∑l=1∞‍∑m=1l‍An,(c,lm)FnF(c,lm)-TS=0,
where(4.10a)An=∫Q(t)‍ϕndQ=∫02π‍∫0π‍∫0ζ‍ϕnr2sin⁡θdrdθdφ,(4.10b)Ac,lm=∫Q(t)‍ϕlmcos⁡(mφ)dQ=∫02π‍∫0π‍∫0ζ‍ϕlmcos⁡(mφ)r2sin⁡θdrdθdφ,(4.10c)As,lm=∫Q(t)‍ϕlmsin⁡(mφ)dQ=∫02π‍∫0π‍∫0ζ‍ϕlmsin⁡(mφ)r2sin⁡θdrdθdφ,(4.11a)
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d
φ
.Performing a variation of independent generalized velocitiesFi, Fc,lm, Fs,lm, i≥1, l≥1 in the action (2.5) within the Lagrangian (4.9) leads to the equations(4.13a)dAndt=∑k=1∞‍An,kFk+∑k=1∞‍∑m=1k‍(An,(c,km)Fc,km+An,(s,km)Fs,km),n≥1,(4.13b)dAc,lmdt=∑k=1∞‍A(c,lm),kFk+∑k=1∞‍∑n=1k‍(A(c,lm),(c,kn)Fc,kn+A(c,lm),(s,kn)Fs,kn),(4.13c)dAs,lmdt=∑k=1∞‍A(s,lm),kFk+∑k=1∞‍∑n=1k‍(A(s,lm),(c,kn)Fc,kn+A(s,lm),(s,kn)Fs,kn),(l≥1, 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 rule(4.14a)dAndt=∑i=1∞‍∂An∂βiβ˙i+∑l=1∞‍∑m=1l‍(∂An∂βc,lmβ˙c,lm+∂An∂βs,lmβ˙s,lm),(4.14b)dAc,lmdt=∑i=1∞‍∂Ac,lm∂βiβ˙i+∑j=1∞‍∑n=1j‍(∂Ac,lm∂βc,jnβ˙c,jn+∂Ac,lm∂βs,jnβ˙s,jn),(4.14c)dAs,lmdt=∑i=1∞‍∂As,lm∂βiβ˙i+∑j=1∞‍∑n=1j‍(∂As,jn∂βc,jnβ˙c,jn+∂As,lm∂βs,jnβ˙s,jn),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, i≥1, l≥1, where An,k are nonlinear functions of generalized coordinates βi, βc,lm, βs,lm, i≥1, l≥1 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, i≥1, l≥1 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 coordinatesβi, βc,lm, βs,lm, i≥1, l≥1. They are often called the dynamic modal equations and take the form(4.15a)∑n=1∞‍∂An∂βμF˙n+∑l=1∞‍∑m=1l‍(∂Ac,lm∂βμF˙c,lm+∂Ac,lm∂βμF˙c,lm)+12∑n,k=1∞∂An,k∂βμFnFk+∑n,l=1∞‍∑m=1lFn(∂An,(c,lm)∂βμFc,lm+∂An,(s,lm)∂βμFs,lm)+∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fc,l1m1Fs,l2m2∂A(c,l1m1),(s,l2m2)∂βμ+12∑l1,l2=1∞‍∑m1,m2=1l1,l2Fc,l1m1Fc,l2m2∂A(c,l1m1),(c,l2m2)∂βμ+12∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fs,l1m1Fs,l2m2∂A(s,l1m1),(s,l2m2)∂βμ+∂TS∂βμ=0,μ≥1,(4.15b)∑n=1∞‍∂An∂βc,μνF˙n+∑l=1∞‍∑m=1l‍(∂Ac,lm∂βc,μνF˙c,lm+∂Ac,lm∂βc,μνF˙c,lm)+12∑n,k=1∞∂An,k∂βc,μνFnFk+∑n,l=1∞‍∑m=1lFn(∂An,(c,lm)∂βc,μνFc,lm+∂An,(s,lm)∂βc,μνFs,lm)+∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fc,l1m1Fs,l2m2∂A(c,l1m1),(s,l2m2)∂βc,μν+12∑l1,l2=1∞‍∑m1,m2=1l1,l2Fc,l1m1Fc,l2m2∂A(c,l1m1),(c,l2m2)∂βc,μν+12∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fs,l1m1Fs,l2m2∂A(s,l1m1),(s,l2m2)∂βc,μν+∂TS∂βc,μν=0,μ≥1,n=1,…,μ,(4.15c)∑n=1∞‍∂An∂βs,μνF˙n+∑l=1∞‍∑m=1l‍(∂Ac,lm∂βs,μνF˙c,lm+∂Ac,lm∂βs,μνF˙c,lm)+12∑n,k=1∞∂An,k∂βs,μνFnFk+∑n,l=1∞‍∑m=1lFn(∂An,(c,lm)∂βs,μνFc,lm+∂An,(s,lm)∂βs,μνFs,lm)+∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fc,l1m1Fs,l2m2∂A(c,l1m1),(s,l2m2)∂βs,μν+12∑l1,l2=1∞‍∑m1,m2=1l1,l2Fc,l1m1Fc,l2m2∂A(c,l1m1),(c,l2m2)∂βs,μν+12∑l1,l2=1∞‍∑m1,m2=1l1,l2‍Fs,l1m1Fs,l2m2∂A(s,l1m1),(s,l2m2)∂βs,μν+∂TS∂βs,μν=0,μ≥1,n=1,…,μ.The derivative byβ* is done assuming that (4.6) accounts for the volume conservation condition so that, for instance,(4.16a)∂TS∂βμ=∫02π‍∫0π‍(k1+k2)ζ2[fμ-12πβμ-14π∂G3∂βμ]sin⁡θdθdφ=∫Σ(t)‍ζ(k1+k2)ζ2+ζθ2+(ζφ/sin⁡θ)2[fμ-12πβμ-14π∂G3∂βμ]dS,(4.16b)∂TS∂βc,μν=∫02π‍∫0π‍(k1+k2)ζ2[fμνcos⁡(νφ)-12πβc,μν-14π∂G3∂βc,μν]sin⁡θdθdφ=∫Σ(t)‍ζ(k1+k2)ζ2+ζθ2+(ζφ/sin⁡θ)2[fμνcos⁡(νφ)-12πβc,μν-14π∂G3∂βc,μν]dS,(4.16c)∂TS∂βs,μν=∫02π‍∫0π‍(k1+k2)ζ2[fμνsin⁡(νφ)-12πβs,μν-14π∂G3∂βs,μν]sin⁡θdθdφ=∫Σ(t)‍ζ(k1+k2)ζ2+ζθ2+(ζφ/sin⁡θ)2[fμνsin⁡(νφ)-12πβs,μν-14π∂G3∂βs,μν]dS.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 AxisymmetricDrop Motions

For the axisymmetric drop dynamics, the velocity potential takes the form(5.1)Φ(r,θ,φ,t)=∑l=1∞‍Fl(t)ϕl(r,θ)
and the free-surface equation is as follows:
(5.2)ζ(θ,φ,t)=1-14π(∑i=1∞‍βi2+13∑i,j,k=1∞‍Λijk(3)βiβjβk+G5)+∑l=1∞‍βl(t)fl(θ),
where
(5.3)Λijm(3)=2π∫0π‍fifjfmsin⁡θdθ=12(2i+1)(2j+1)π(2m+1)(Ci0,j0m0)2
and Ci0,j0m0 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
(5.4)βl~Fl=O(ϵ1/3),ϵ≪1,
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
(5.5)∑i=1∞‍∂An∂βiβ˙i=∑k=1∞‍AnkFk,n≥1,
where neglecting the o(ϵ)-terms implies that ∂An/∂βi and Ank keep only the second-order polynomial quantities, that is,
(5.6)∂An∂βi=δni+(2+n)∑j=1∞‍Λnij(3)βj+(n+1)(n+2)2∑j,k=1∞‍Λnijk(4)βjβk-2+n4π[δni∑j=1∞‍βj2+2βiβn]=δni+∑j=1∞‍χn,i,j(1)βj+∑j,k=1∞‍χn,i,jk(2)βjβk,(5.7)Ank=nδnk+∑j=1∞‍[nkΛknj(3)+Λnk,j(-3)]βj+n+k2∑i,j=1∞‍[nkΛknij(4)+Λnk,ij(-4)]βiβj-n(n+k+1)4πδnk∑j=1∞‍βj2=nδnk+∑j=1∞‍∏nk,j(1)βj+∑i,j=1∞‍∏nk,ij(2)βiβj.Here,Λijk(3) is defined by (5.3) and Λijkm(4) is also expressed via the Clebsch-Gordan coefficients
(5.8)Λijkm(4)=2π∫0π‍fifjfkfmsin⁡θdθ=(2i+1)(2j+1)(2k+1)(2m+1)4π∑n=max⁡(|i-j|,|k-m|)min⁡(i+j,k+m)‍12n+1(Ci0,j0n0⋅Ck0,m0n0)2.Furthermore,(5.9)Λnk(-2)=2π∫0π‍∂fn∂θ∂fk∂θsin⁡θdθ=n(n+1)δnk,Λin,k(-3)=2π∫0π‍∂fi∂θ∂fn∂θfksin⁡θdθ=-12i(i+1)(2i+1)n(n+1)(2n+1)π(2k+1)Ci0,n0k0Ci(-1),n1k0,Λin,kj(-4)=2π∫0π‍∂fi∂θ∂fn∂θfkfjsin⁡θdθ=14πi(i+1)(2i+1)n(n+1)(2n+1)×k(k+1)(2k+1)j(j+1)(2j+1)∑m=max⁡(|i-n|,|k-j|)min⁡(i+n,k+j)‍12m+1Ci0,n0m0Ci(-1),n1m0Ck0,j0m0Ck(-1),j1m0.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
(5.10)Fl=β˙ll+∑i,j=1∞‍Vl,i,j(2)β˙iβj+∑i,j,k=1∞‍Vl,i,j,k(3)β˙iβjβk,l≥1.Substituting (5.10) into (5.5) and gathering all the similar polynomial terms give
(5.11)Vn,i,j(2)=χn,i,j(1)-Πni,j(1)/in,Vn,i,j,k(3)=χn,i,j,k(2)-Πni,jk(2)/i-∑l=1∞‍Vl,i,j(2)Πnl,k(1)n.For the axisymmetric drop dynamics, the dynamic modal equations (4.15a), (4.15b), and (4.15c) take the form
(5.12)∑n=1∞‍∂An∂βμF˙n+12∑n,k=1∞‍∂An,k∂βμFnFk+∂TS∂βμ=0,μ≥1.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,
(5.13)∂TS∂βμ=2π∫0π‍ζ2sin⁡θ(2+(ζθ/ζ)2ζ2ζθ2-1ζ2sin⁡θ∂∂θ(ζζθsin⁡θζ2ζθ2))×[fμ-12πβμ-14π∑i,j=1∞‍Λijμβiβj]dθ=2π∫0π‍(2+2ξ)[fμ-12πβμ-14π∑i,j=1∞‍Λijμβiβj]sin⁡θdθ+2π∫0π‍[ξθ-12ξθ3]∂fμ∂θsin⁡θdθ=(μ+2)(μ-1)βμ+∑i,j=1∞‍Tij(2μ)βiβj+∑i,j,k=1∞‍Ti,j,k(3μ)βiβjβk,
where
(5.14)Tij(2μ)=-2Λijμ(3),Ti,j,k(3μ)=-12Λijkμ(-4θ)+1πδiμδjk,Λijkμ(-4θ)=2π∫0π‍∂fi∂θ∂fj∂θ∂fk∂θ∂fμ∂θsin⁡θdθ.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:
(5.15)∑i=1∞‍[δμi+∑j=1∞‍di,j1,μβj+∑j,k=1∞‍di,j,k2,μβjβk]β¨i+∑n,k=1∞‍[tn,k0,μ+∑m=1∞‍tn,k1,μβm]β˙nβ˙k+σμ2βμ+∑i,j=1∞‍[μTij2μ]βiβj+∑i,j,k=1∞‍[μTi,j,k3μ]βiβjβk=0,μ≥1,
where σμ=σμ0 are the nondimensional frequencies (3.6) and
(5.16)di,j1,μ=μ[χi,μ,j(1)i+Vμ,i,j(2)],di,j,k2,μ=μ[χi,μ,j,k(2)i+∑α=1∞‍χα,μ,j(1)Vα,i,k(2)+Vμ,i,j,k(3)],tn,k0,μ=μ[Vμ,n,k(2)+Πnk,μ(1)nk],tn,k,m1,μ=μ[V-μ,n,k,m(3)+Πnk,μm(2)nk+∑α=1∞‍(χα,μ,m(1)Vα,n,k(2)+Παk,μ(1)Vα,n,m(2)k+Παn,μ(1)Vα,k,m(2)n)].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 frequencyσ 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
(6.1)β2=O(ϵ1/3),β4=O(ϵ2/3),β6=O(ϵ),βl=o(ϵ),l≠2,4,6
so that neglecting the o(ϵ)-terms in (5.15) leads to the finite-dimensional system of nonlinear modal equations(6.2a)β¨2+σ22β2+d1β¨2β4+d2β¨4β2+d3β˙2β˙4+d4β¨2β22+d5β˙22β2+t1β22+t2β2β4+t3β23+c1β¨2β2+c2β˙22=0,(6.2b)β¨4+σ42β4+d6β¨2β2+d7β˙22+t4β22+t5β2β4+c3β¨4β2+c4β˙4β˙2=0,(6.2c)β¨6+σ62β6+d8β¨2β4+d9β¨4β2+d10β˙4β˙2+d11β¨2β22+d12β˙22β2+t6β2β4+t7β23=0,whereσ2=σ20,σ4=σ40,σ6=σ60 and
(6.3)d1=244π,d2=1514π,d3=7514π,d4=-6798π,d5=-585196π,d6=157π,d7=-97π,d8=10565286π,d9=3065143π,d10=-7565143π,d11=135651001π,d12=135652002π;t1=-457π,t2=-247π,t3=-767π,t4=-247π,t5=-160577π,t6=-18065143π,t7=54065143π;c1=9514π,c2=457π,c3=755154π,c4=1855154π.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:
(6.4)σ-σ2σ2=O(ϵ2/3)
(the nonlinear eigenfrequency σ is unknown). The wanted periodic solution takes then the form
(6.5)β2=Acos⁡(σt)+A2(E1+E2cos⁡(2σt))+O(A3),β4=A2(E3+E4cos⁡(2σt))+O(A3),β6=O(A3),A=O(ϵ1/3),
where A is the unknown dominant amplitude and substituting (6.5) into (6.2a) and (6.2b) and accounting for (6.4) give(6.6a)E1=-t1+(c1-c2)σ22σ22=-t1+(c1-c2)σ222σ22+O(A2),(6.6b)E2=-t1+(c1+c2)σ22(σ22-4σ2)=t1-(c1+c2)σ226σ22+O(A2),(6.6c)E3=-t4+(d6-d7)σ22σ42=-t4+(d6-d7)σ222σ42+O(A2),(6.6d)E2=-t4+(d6+d7)σ22(σ42-4σ22)=-t4+(d6+d7)σ222(σ42-4σ22)+O(A2).Gathering theA3-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-σ2)/σ2=O(ϵ2/3) and the nondimensional amplitude parameter A2=O(ϵ2/3)(6.7)σ-σ2σ2+63477840πA2=0.Secular equation (6.7) determines the so-called “soft-type” spring behavior suggesting that amplitude A increases with decreasing frequency σ.Figure2 compares our asymptotic result (6.7) with experimental data from [12] (see, also [19]) where dependence between (σ-σ2)/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
(6.8)HW=1+5/(4π)A1-(1/2)5/(4π)A,
where A is defined by (6.7).Figure 2
Theoretical ((6.8), solid line)) and experimental dependence between (σ-σ2)/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] (▲).Figure2 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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*Source: 869070-2012-09-06.xml*