J. Fluid Mech. (2014), . 760, pp. …jbostwi/wp-content/uploads/...with hysteresis (solid) and without (dashed). Here, a and r are the advancing and receding static contact angles

J. Fluid Mech. (2014), vol. 760, pp. 5–38. c© Cambridge University Press 2014doi:10.1017/jfm.2014.582

5

Dynamics of sessile drops. Part 1. Inviscid theory

J. B. Bostwick1,† and P. H. Steen2,3

1Department of Engineering Science and Applied Mathematics Northwestern University,Evanston, IL 60208, USA

2Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA3School of Chemical and Biomolecular Engineering and Center for Applied Mathematics,

Cornell University, Ithaca, NY 14853, USA

(Received 30 January 2014; revised 20 July 2014; accepted 1 October 2014)

A sessile droplet partially wets a planar solid support. We study the linear stabilityof this spherical-cap base state to disturbances whose three-phase contact line is(i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact anglethat is a smooth function of the contact-line speed. The governing hydrodynamicequations for inviscid motions are reduced to a functional eigenvalue problemon linear operators, which are parameterized by the base-state volume throughthe static contact angle and contact-line mobility via a spreading parameter. Asolution is facilitated using inverse operators for disturbances (i) and (ii) to reportfrequencies and modal shapes identified by a polar k and azimuthal l wavenumber.For the dynamic contact-line condition (iii), we show that the disturbance energybalance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ thatencompasses the dynamic effects associated with (iii). The effect of the contact-linemotion on the dissipation mechanism is illustrated. We report an instability ofthe super-hemispherical base states with mobile contact lines (ii) that correlateswith horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davisdissipation from the dynamic contact-line condition (iii) can suppress the instability.The remainder of the spectrum exhibits oscillatory behaviour. For the hemisphericalbase state with mobile contact line (ii), the spectrum is degenerate with respectto the azimuthal wavenumber. We show that varying either the base-state volume orcontact-line mobility lifts this degeneracy. For most values of these symmetry-breakingparameters, a certain spectral ordering of frequencies is maintained. However, becausecertain modes are more strongly influenced by the support than others, there areinstances of additional modal degeneracies. We explain the physical reason for theseand show how to locate them.

Key words: capillary waves, contact lines, drops

1. Introduction

Lord Rayleigh (1879) predicted the oscillation frequencies and mode shapes for aninviscid free drop held by surface tension. The Rayleigh mode shapes are given by

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6 J. B. Bostwick and P. H. Steen

Legendre polynomials Pk with scaled frequencies λk ordered by polar wavenumber k,

λ2k = k(k− 1)(k+ 2), k= 0, 1, . . . . (1.1)

Here, shapes are scaled by R, the radius of the undisturbed spherical drop, andfrequencies by a capillary time scale (σ/ρR3)−1/2, where σ is the surface tensionand ρ is the density of the liquid. Rayleigh’s predictions for the free drop havebeen verified experimentally for immiscible drops by Trinh & Wang (1982) and forfree drops in microgravity by Wang, Anilkumar & Lee (1996), both using acousticexcitation.

Lamb (1932) and others (e.g. Chandrasekhar 1961) noted that additional modeshapes also solve the Rayleigh drop equations. The Rayleigh–Lamb (RL) spectrumincludes azimuthal (l 6= 0) mode shapes Y l

k(θ, ψ) with scaled frequencies, λ2k,l =

k(k − 1)(k + 2), k, l = 0, 1, . . . , restricted by l 6 k. Here, θ and ψ are polarand azimuthal angles in spherical coordinates, respectively, and Y l

k is the sphericalharmonic of degree k and order l. One should note the degeneracy of the frequencies:for every k > 0, there are distinct modes l 6 k corresponding to the same λk,l. Themodes l= 0 recover the Rayleigh modes, Pk= Y0

k , a subset of the spherical harmonicswhich are solutions of the Laplace equation in three dimensions (e.g. Courant &Hilbert 1953).

The RL spectrum continues to see widespread use even in situations where the dropis not completely free, as for a levitated drop (Brunet & Snoeijer 2011) or drops incontact with a solid (Noblin, Buguin & Brochard-Wyart 2005; Dong, Chaudhury &Chaudhury 2006; Chebel, Risso & Masbernat 2011) or with another liquid (Dorboloet al. 2008). Modifications to (1.1), ad hoc or otherwise, are often invoked to accountfor the influence of substrate contact (Yoshiyasu, Matsuda & Takaki 1996; Celestini &Kofman 2006; Sharp, Farmer & Kelly 2011). Recent experiments have demonstratedthe inadequacy of the RL spectrum for many sessile drop vibrations (Chang et al.2013).

Situations where drops contact a solid support and behave nearly inviscidlyare ubiquitous in traditional applications (printing, spraying, foaming, coating andcondensation heat transfer), in emerging applications (digital microfluidic systems,flexible printed circuits, gradient-energy substrates) and in nature (lotus-leaf surfaces,evaporative transpiration and splashing). The need for a comprehensive theory forcontacting drop and bubble oscillations has been recognized (Milne et al. 2014). Incontrast to the free drop, which has been studied extensively since the time of LordRayleigh (1879), the existing literature on the sessile drop is sparse, and particularlyso for analytical-based solution methods. This occurs because supported drops possessless symmetry than free drops. Many authors have extended the pioneering workof Lord Rayleigh (1879). Typical extensions include, but are not limited to, theeffects of (i) viscosity (Lamb 1932; Miller & Scriven 1968; Prosperetti 1980), (ii)large-amplitude perturbations (Tsamopoulos & Brown 1983; Lundgren & Mansour1988; Basaran & DePaoli 1994) or (iii) constrained geometries (Strani & Sabetta1984; Bostwick & Steen 2009; Ilyukhina & Makov 2009; Ramalingam & Basaran2010; Bostwick & Steen 2013a,b).

Our focus is on the wetting and spreading properties of the solid substrate and theireffect on fluid motion. We study the natural oscillations of an inviscid sessile dropwith mobile contact lines on a planar support. Our extension embeds the RL predictionin a two-parameter family of spectra. (In this paper, ‘spectrum’ will refer to a set ofeigenfrequencies and eigenmodes.) The two parameters are the volume (or equilibriumcontact angle) and the mobility of the contact line. In the companion to this paper

Dynamics of sessile drops. Part 1. Inviscid theory 7

(Chang et al. 2014), the focus is experiment, and the inviscid results presented hereare extended using viscous potential theory to the forced oscillation problem.

Substrate contact has three main influences on the spectrum: (i) the solid presents abarrier to liquid motion (no penetration); (ii) the solid introduces a contact line whereliquid spreading is resisted (restricted contact-line mobility); (iii) the solid boundaryenhances relative liquid motion, increasing the role of bulk viscosity. It should berecalled that, in the context of inviscid theory, the no-slip condition at any solid isabandoned and hence the classic viscous singularity at the moving contact line neednot enter our discussion (Dussan 1979).

Recently, there has been a growing interest in the dynamics of sessile drops underexternal forcing, with an emphasis on the wetting conditions at the three-phasecontact line and their motion. In drop atomization experiments, James et al. (2003b)and Vukasinovic, Smith & Glezer (2007) observed a hierarchy of instabilities fora sessile drop (vertically) forced by a piezoelectrically driven diaphragm. Dropejection has been studied experimentally by DePaoli et al. (1995) and numericallyusing either the finite-element method (Wilkes & Basaran 2001) or a Navier–Stokessolver (James, Smith & Glezer 2003a). In another study of mechanically vibratedsessile drops, Noblin, Buguin & Brochard-Wyart (2004) focused on the transitionfrom a pinned to a free contact line and the acceleration necessary to overcomecontact-angle hysteresis. Extending their previous work, Noblin et al. (2005) showedthat at sufficiently high accelerations an azimuthal instability called the ‘triplon mode’is generated in large fluid puddles. This instability is triggered once the contact-anglehysteresis is overcome and the contact line depins. These modes, as well as theinstability observed in Vukasinovic et al. (2007), exhibit subharmonic resonances,which are characteristic of Faraday waves (Faraday 1831; Miles & Henderson1990). Chang et al. (2013) characterized the dynamics of mechanically oscillatedsessile water drops by cataloguing the first 37 mode shapes. They also observedmode mixing, or the simultaneous coexistence of multiple mode shapes within theresonating sessile drop driven by one sinusoidal signal of a single frequency.

A sessile drop subject to in-plane (horizontal) forcing exhibits an additional set ofoscillatory modes that are odd or anti-symmetric about the vertical mid-plane (Sharpet al. 2011; Sharp 2012). Daniel et al. (2004) observed translational motion of dropson chemically treated surfaces that had overcome the hysteretic barrier or had mobile(unpinned) contact lines. A study coupling both horizontal and vertical forcing byNoblin, Kofman & Celestini (2009) has shown that this translational motion canbe directionally controlled. Brunet, Eggers & Deegan (2009) have shown that underexternal forcing a sessile drop can overcome the influence of gravity and be drivenup a sloped incline against gravity.

We consider a sessile drop sitting on a planar solid in a passive gas ambient, takinga spherical-cap shape which models a partially wetting drop with equilibrium contactangle α, as shown in figure 1(a). Spherical caps are scaled with the contact footprint∂Ds radius r and volume,

Vr3= π

3(2− 3 cos α + cos3 α)

sin3 α. (1.2)

Accordingly, the contact angle α and drop volume V are interchangeable parameters.We adopt α as the base-shape parameter, where α < π/2 is a lens-like shape andα > π/2 is a drop-like shape. It should be noted that the principal curvatures ofthe undisturbed free surface ∂D f are simply κ1 = κ2 ≡ sin α in our scaling. The


(a) (b)

(c)

r

D(V )

Solid

Liquid

Gas

uCL

1

ls

lg

sg

FIGURE 1. (Colour online) Definition sketch: (a) equilibrium spherical-cap surface of baseradius r, volume V and static contact angle α, which is defined by (b) the Young–Dupréequation (1.3) schematically illustrated as a mechanical (horizontal) force balance. Thedynamic contact-line condition (c) relates the contact angle to the contact-line speed uCLwith hysteresis (solid) and without (dashed). Here, αa and αr are the advancing andreceding static contact angles (uCL→ 0), respectively. Limiting cases of the ‘continuous’contact-line law (dashed) include the fixed contact-angle Λ = 0 and pinned contact-lineΛ=∞ conditions.

contact angle α is related to the liquid/gas σlg, liquid/solid σls and solid/gas σsg

interfacial tensions through the Young–Dupré equation (Young 1805; Dupré 1869), inthe standard way,

σsg − σls = σlg cos α. (1.3)

Figure 1(b) illustrates (1.3) as the (horizontal) balance of forces at the contact line.Disturbances that probe the drop stability displace the contact line along the solid

and, to complete the problem description, a constitutive relation for the moving contactline is needed (cf. figure 1c). Well-accepted contact-line models include hysteresisof the non-equilibrium contact angle in order to capture the stick–slip behaviourobserved in experiments, as illustrated by the solid curve in figure 1(c) (Dussan1979). Contact-line speeds uCL > 0 correspond to liquid displacing gas; uCL < 0to gas displacing liquid. However, nonlinear constitutive relations with stick–sliphysteresis are incompatible with linear analyses (Fayzrakhmanova & Straube 2009).To allow for contact-line movement in our linear analysis, we invoke the conditionsketched by the dashed line in figure 1(c). This CL condition was introduced byDavis (1980), although it is sometimes referred to as the ‘Hocking condition’ eventhough Hocking (1987) attributes it to Davis. This constitutive relation has been usedpreviously in rivulet (Davis 1980; Weiland & Davis 1981; Young & Davis 1987)and droplet (Lyubimov, Lyubimova & Shklyaev 2006; Alabuzhev & Lyubimov 2007;Fayzrakhmanova & Straube 2009; Alabuzhev & Lyubimov 2012) stability analyses.


The model assumes a smooth dependence of contact-angle deviations 1α on uCL, sayα +1α = f (uCL), with f (0)= α and f ′(0)=Λ, and linearizes to

1α =ΛuCL. (1.4)

The resistance to mobility, Λ, or the mobility, Λ−1, characterizes the contact-linemotion. Here, Λ = 0 corresponds to a fully mobile contact line and Λ = ∞ toa pinned contact line. With this understanding, henceforth we shall refer to Λ asthe ‘mobility’ parameter. Davis (1980) showed that contact-line motion is purelydissipative for Λ 6= 0. The mobility parameter may be viewed as a way to ‘deformthe problem’ continuously from the mobile to the immobile contact line, and issometimes called a homotopy or problem-deformation parameter.

We report the spectra for the sessile drop, as it depends upon (α, Λ), in terms offrequencies λk,l and shapes yk,l(x) cos(lψ), where x≡ cos(s) and s is an arclength-likecoordinate defined on the undeformed surface. Our results are conveniently framedagainst the RL modes. In particular, a subset of the RL modes with k+ l= even aremirror-symmetric about the equatorial plane and therefore satisfy no penetrationon that mid-plane. Hence, these modes are also sessile drop solutions for thehemispherical base state (α = 90), provided that the contact line is free (Λ = 0).The associated spectrum is shown in figures 2(a,c). One should note the degeneratespectrum: distinct mode shapes y2,2 and y2,0 have identical frequencies and, similarly,for y3,1 and y3,3, and so forth. The degeneracy for the sessile drop is inherited fromthe free drop problem.

In contrast, figures 2(b,d) plot the spectrum for (α, Λ) = (90,∞), a preview ofresults in this paper. Contact-line pinning splits the frequency degeneracy; contrastfigure 2(a) against figure 2(b). The splitting occurs in a way that leads to higherfrequencies for higher azimuthal modes (l 6= 0) (cf. figure 2b). Mode shapes aresuperficially similar from the top view but differences can been seen near the contactline. As suggested by the [5, 1] vertical cuts with reflection about the plane of contactshown in figures 2(c,d), the (90, 0) azimuthal modes have non-circular footprintswhile the (90,∞) modes remain circularly pinned. It should be noted that, from thetop view, the contact lines will not appear circular when overhanging shapes obstructthe contact line from sight. Our goal is to follow spectra as they deform from figures2(a,c) to 2(b,d).

Mode [1, 1] is the only mode that is not azimuthally degenerate (cf. figure 2).Therefore, it is not subject to frequency splitting. However, it is subject to breakinganother symmetry, as we shall now explain. For (α,Λ)= (90, 0), the (Rayleigh drop)problem is invariant under translation, consistent with absence of the substrate. ByNoether’s theorem, there is a corresponding first integral of the motion which turns outto correspond to the mode [1, 1], as reflected in its zero frequency. Accordingly, weshall refer to the [1, 1] mode as the ‘Noether mode’. It tends to be ignored, with fewexceptions (Lyubimov, Lyubimova & Shklyaev 2004). For Λ= 0, the Noether modecorresponds to a rigid-body translation at constant velocity (Galilean translation). ForΛ > 0, the substrate begins to exert an influence and the symmetry of translationalinvariance is broken. The laboratory frame comes into play. For Λ> 0, the Noethermode becomes time-dependent and, for Λ=∞, its frequency approaches λ1,1∼ 2.5, asshown in figure 2(b). The situation is quite different for the unpinned Noether mode(Λ 6=∞), which can have imaginary frequencies and hence can exhibit instability.

A few special cases of natural oscillations for a sessile drop have been previouslystudied analytically. All studies have been restricted to hemispherical base states.


(a) (b)

(c) (d )

1 2 3 4 5

k k

5

10

15

1 2 3 4 5

5

10

15

l

10

SC

2 3 4 5

1

[5, 1]

0

2

3

4

5

10

SC

2 3 4 5

1

[5, 1]

0

2

3

4

5

FIGURE 2. (Colour online) Symmetry breaking of modal degeneracy yields spectralsplitting: the spectral lines (a) λk,l for the hemispherical drop (α = 90) exhibita degeneracy with respect to the azimuthal wavenumber l (c) for the (a,c) naturaldisturbance, which breaks for the (b,d) pinned disturbance into split spectral lines (b).Tables of mode shapes for the (c) natural and (d) pinned disturbances, as a function ofpolar k and azimuthal l wavenumbers (all with same deviation amplitude). The insets to(c) and (d) show the symmetric extension of the [5, 1] mode to the full sphere for eachdisturbance.

Lyubimov et al. (2004) introduced a homotopy parameter, like Λ, and reportedfrequencies for the first eight broken modes (up to k= 4 in figure 2b), along with theattenuation introduced by the contact-line motion. The Noether mode is referred to asa ‘bending’ oscillation. Lyubimov et al. (2006) reported natural frequencies of the firstthree axisymmetric modes, along with attenuation. Fayzrakhmanova & Straube (2009)showed how to maintain Hocking stick–slip hysteresis while proceeding analytically;their analysis is necessarily nonlinear. The restriction to hemispherical base statesis related to the need to solve Laplace’s equations analytically. The key innovationthat allows non-hemispherical base states here is solution of the Laplace equation inhybrid coordinates. This approach could be extended to bridges, rivulets and otherheretofore intractable base-state shapes.

We begin this paper with the hydrodynamic equations associated with smalldisturbances to the spherical-cap interface. We consider disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moveswith a contact angle that is a smooth function of the contact-line speed. The governing


(a) (b) (c)

xy

z

s

D

y

x

z

x

y

FIGURE 3. Definition sketch with the unperturbed Γ (dashed) and perturbed interface η(solid) in (a) polar cross-section and three-dimensional (b) perspective and (c) top views.

equations are reduced by a normal-mode expansion to a functional eigenvalue problemon linear operators. We use inverse operators to construct a solution for disturbances(i) and (ii) and report frequencies and modal structures distinguished by a polark and azimuthal l wavenumber. We illustrate the spectral dependence on α andcompare disturbances (i) and (ii). For the dynamic contact-line condition (iii), weuse an alternative solution method to derive an operator equation in the form ofa damped-harmonic oscillator with dissipation controlled by Λ. The relationshipbetween the wetting properties of the substrate, the modal structure and dissipationis illustrated. Lastly, some concluding remarks are offered.

2. Mathematical formulation

The sessile drop is a surface of constant mean curvature H and equivalently a staticequilibrium configuration according to the Young–Laplace equation,

pσ= κ1 + κ2 ≡ 2H, (2.1)

which relates the principal curvatures, κ1 and κ2, to the pressure p there. Theequilibrium surface Γ is defined parametrically as

X(s, ϕ; α)= sin(s)sin(α)

cos(ϕ), Y(s, ϕ; α)= sin(s)sin(α)

sin(ϕ), Z(s; α)= cos(s)− cos(α)sin(α)

,

(2.2a−c)using arclength-like s ∈ [0, α] and azimuthal angle ϕ ∈ [0, 2π] as generalized surfacecoordinates. The interface is given a small perturbation η(s, ϕ, t) (cf. figure 3). Nodomain perturbation is needed for small deformations, thus the droplet domain

D≡ (x, y, z) | 0 6 x 6 X(s, ϕ; α), 0 6 y 6 Y(s, ϕ; α), 0 6 z 6 Z(s; α) (2.3)

is bounded by a free surface ∂Df (≡ Γ ) of constant surface tension σ (nothermocapillary effects), and a planar surface-of-support ∂Ds,

∂Df ≡ (x, y, z) | x= X(s, ϕ; α), y= Y(s, ϕ; α), z= Z(s; α), ∂Ds ≡ (x, y, z) | z= 0.(2.4a,b)

The droplet is immersed in a passive gas and the effect of gravity is neglected.


2.1. Hydrodynamic field equations

The fluid is incompressible and the flow is assumed to be irrotational. Therefore, thevelocity field may be described as v=−∇Ψ , where the velocity potential Ψ satisfiesLaplace’s equation

∇2Ψ = 0 [D] (2.5)

on the drop domain. Additionally, the velocity potential satisfies the no-penetrationcondition

∇Ψ · z= 0 [∂Ds] (2.6)

on the surface-of-support and a (linearized) kinematic condition

∂Ψ

∂n=−∂η

∂t[∂Df ] (2.7)

on the free surface, which relates the normal velocity to the perturbation amplitudethere. In the limit of small interface deflection and in accordance with potential flowtheory, the pressure field is expressed by the linearized Bernoulli equation

p= %∂Ψ∂t

[D], (2.8)

where % is the fluid density. Finally, disturbances to the equilibrium surface Γ

generate pressure gradients, and thereby flows, according to the Young–Laplaceequation

p/σ =−∆Γ η−(κ2

1 + κ22

)η [∂Df ], (2.9)

where the Laplace–Beltrami operator ∆Γ , which depends on Γ , will be defined below.The hydrodynamic field equations (2.5)–(2.9) govern the motion of an inviscid fluid,

whose spherical-cap interface is given a small disturbance. The field equations mustbe augmented with a boundary condition on the three-phase contact line to yield awell-posed system of partial differential equations, a condition that we discuss later.

2.2. Laplace–Beltrami operator

The Laplace–Beltrami operator ∆Γ , introduced in (2.9), depends on equilibrium-surface curvatures and operates on functions η,

∆Γ η≡ 1√g∂

∂uµ

(√ggµν

∂η

∂uν

), (2.10)

with the surface metric given by

gµν ≡ x,µ x,ν =(

csc2(α) 00 (csc(α) sin(s))2

), g= (sin(s)csc2(α)

)2, (2.11a,b)

and µ, ν = 1, 2, using notation standard to differential geometry (e.g. Kreyszig 1991).


2.3. Normal-mode reductionNormal modes, parameterized by frequency Ω and azimuthal wavenumber l,

η(s, ϕ, t)= y(s)eiΩteilϕ, Ψ (x, t)= φ(ρ, θ)eiΩteilϕ, (2.12a,b)

are applied to the field equations (2.5)–(2.9) to yield

∂

∂ρ

(ρ2 ∂φ

∂ρ

)+ 1

sin θ∂

∂θ

(sin θ

∂φ

∂θ

)− l2

sin2 θφ = 0 [D], (2.13a)

∂φ

∂n= 0 [∂Ds], (2.13b)

∂φ

∂n=−iλy [∂Df ], (2.13c)

y′′ + cot(s)y′ +(

2− l2

sin2(s)

)y=−iλ csc2(α)φ [∂Df ], (2.13d)∫

Γ

∂φ

∂ndΓ = 0, (2.13e)

λ2 ≡ %Ω2r3

σ. (2.13f )

Equation (2.13a) is Laplace’s equation written in spherical coordinates (ρ, θ), (2.13b)is the no-penetration condition on the support surface, (2.13c) is the kinematiccondition and (2.13e) is the integral form of the incompressibility condition or avolume conservation constraint. The dynamic pressure balance across the free surfaceis represented by (2.13d) with differentiation with respect to the arclength coordinate′ = d/ds. Equations (2.13) are an eigenvalue problem in the scaled frequency λ,defined in (2.13f ).

3. Reduction to operator equationThe dynamic pressure balance (2.13d) can be rewritten using (2.13c),(

∂φ

∂n

)′′+ cot(s)

(∂φ

∂n

)′+(

2− l2

sin2(s)

)(∂φ

∂n

)=− λ2

sin2(α)φ [∂Df ]. (3.1)

3.1. Operator formalismThis integrodifferential equation governs the motion of the interface and may beformulated as a operator equation by two alternative representations.

3.1.1. Direct problem: unknown interface deflectionThe first formulation uses the interface deflection ∂φ/∂n as the unknown function,

K[∂φ

∂n; l]=−λ2M

[∂φ

∂n

], λ2 ≡ λ2

sin2(α), (3.2)

which we refer to as the direct problem. Here, M is an integral operator representativeof the fluid inertia and K is a differential operator related to the curvature,

M[∂φ

∂n

]≡ φ, (3.3a)


K[∂φ

∂n; l]≡(∂φ

∂n

)′′+ cot(s)

(∂φ

∂n

)′+(

2− l2

sin2(s)

)(∂φ

∂n

). (3.3b)

To proceed with this formulation, one must construct a sufficiently general solution tothe boundary-value problem

∇2φ = 0 [D], ∂φ

∂n= fk [∂Df ]. (3.4a,b)

More specifically, given a surface deformation fk, one needs to compute thecorresponding velocity potential φk, in accordance with the inertia operator (3.3a).In general, for all but the simplest of geometries, solution of this Neumann-typeboundary-value problem requires a computationally intensive approach.

3.1.2. Inverse problem: unknown velocity potentialAlternatively, one may use the velocity potential φ as the unknown function,

M−1 [φ]=−λ2K−1 [φ; l] , (3.5)

which follows directly from (3.2) and is referred to as the inverse problem. Here, theintegrodifferential nature of the governing equation persists, with

M−1 [φ]≡ ∂φ∂n

(3.6)

the differential operator and K−1 an integral operator, inversely related to the curvatureoperator K defined in (3.3b). As with the direct problem, the primary difficultiesassociated with the inverse problem are related to the integral operator. Specifically,construction of the inverse operator K−1 depends on the parametrization of theequilibrium surface and may or may not be analytically tractable.

3.1.3. Direct versus inverse problemThe two representations defined here are completely equivalent, but each has its

respective solution difficulties. In the direct problem, one has to construct a sufficientlygeneral solution to Laplace’s equation with Neumann boundary conditions. In mostcases, this is analytically intractable. Likewise, with the inverse operator formalism,one must construct the Green’s function to the differential curvature operator. Whilethe inverse problem is seen to be more tractable in general, the direct problem has anattractive variational structure that will be exploited in a subsequent section.

3.2. Contact-line conditionsTo compute the spectrum of the eigenvalue problem (2.13), one needs to specifythe type of disturbance via a boundary condition on the three-phase contact line.We decompose the solutions into two distinct classes, kinematic and dynamic.Kinematic disturbances depend solely upon the interface deflection, whereas dynamicdisturbances are related to dynamic field quantities, such as the contact-line speed.For all contact-line boundary conditions, to ensure that the boundary-value problemis well-posed mathematically, we require

∂φ

∂n

∣∣∣∣s=0

− bounded, (3.7)

a necessary condition which guarantees that the interface disturbance is physical.


3.2.1. Natural (fixed contact angle)The first type of kinematic disturbance preserves the static contact angle α,

∂

∂s

(∂φ

∂n

)+ cos(α)

∂φ

∂n= 0∣∣∣∣

s=α, (3.8)

and corresponds to the ‘natural’ boundary condition related to the linearized Young–Dupré equation. A derivation of (3.8) is given in appendix A.

3.2.2. Pinned (fixed contact line)The second type of kinematic disturbance has an immobile contact line,

∂φ

∂n

∣∣∣∣s=α= 0, (3.9)

and is referred to as the ‘pinned’ contact-line condition. Courant & Hilbert (1953,chap. 6, p. 410) show that the pinned contact-line disturbance is the most restrictivefor the general boundary-value problem.

3.2.3. Dynamic contact lineThe dynamic contact-line condition follows by assuming that the contact angle

depends smoothly on the contact-line speed through the function f (cf. figure 1c).Assuming normal modes for contact-angle deviation, 1α = εαeiΩteilϕ , setting∂η/∂t= uCL, and applying (2.13c) to (1.4) allows one to obtain the reduced variationin contact angle,

α + εα = f(

0+ ε(∂φ

∂n

)); α =Λ

(∂φ

∂n

); Λ≡ f ′(0). (3.10)

Applying (3.10) to the linearized Young–Dupré equation (A 10) gives

∂

∂s

(∂φ

∂n

)+ cos(α)

(∂φ

∂n

)= iλΛ

(∂φ

∂n

)∣∣∣∣s=α. (3.11)

Here, 1/Λ is a measure of the mobility of the contact line, which smoothly ‘deforms’the boundary from a natural (fully mobile) to a pinned (immobile) condition. This issometimes called a ‘homotopy’ parameter. In the limits Λ→ 0,∞, (3.11) reduces to(3.8), (3.9), respectively.

4. Method for kinematic disturbances: natural and pinned contact linesAs stated earlier, the spectrum of the operator (3.1) may be computed from either

the direct or the inverse problem. For the kinematic disturbances, we compute thespectrum from the inverse problem (3.5).

4.1. Inverse operatorThe kinematic disturbances, natural and pinned, are structurally similar, and we derivethe respective inverse operators simultaneously. To use the inverse operator formalism,one must construct the integral operator,

K−1 [φ] (x)=∫ 1

bG (x, y; l) φ(y)dy, (4.1)


or fundamental solution (Green’s function) of the curvature operator (3.3b),

G (x, y; l)=

ξ(l)y1(y; l)

[τ2(l)τ1(l)

y1(x; l)− y2(x; l)], b< x< y< 1,

ξ(l)y1(x; l)[τ2(l)τ1(l)

y1(y; l)− y2(y; l)], b< y< x< 1,

(4.2)

where

x≡ cos(s), b≡ cos(α). (4.3a,b)

The Green’s function is parameterized by the azimuthal wavenumber l and thetransformed contact angle b. The functions y1 and y2 belong to the kernel of thecurvature operator K and are given by

y1(x; 0)= P1(x), y2(x; 0)=Q1(x), y1(x; 1)= P(1)1 (x), y2(x; 1)=Q(1)1 (x),

y1(x; l > 2)= (x+ l)(

1− x1+ x

)l/2

, y2(x; l > 2)= (x+ l)2l(l2 − 1

) (1+ x1− x

)l/2

,

(4.4)

where P1, Q1 and P(1)1 , Q(1)1 are the order 0 and 1 Legendre functions of index 1,

respectively (MacRobert 1967). Similarly, the scale factor is given by

ξ(l)≡

12 , l= 1,1, l 6= 1,

(4.5)

while the parameters τ1 and τ2 are related to the contact-line boundary conditions

τ(n)1 = y′1(b; l)+

b√1− b2

y1(b; l), τ(n)2 = y′2(b; l)+

b√1− b2

y2(b; l) (4.6a,b)

for the natural disturbance (superscript n) and

τ(p)1 = y1(b; l), τ

(p)2 = y2(b; l) (4.7a,b)

for the pinned contact-line disturbance (superscript p), respectively.

4.1.1. Axisymmetric operator (l= 0)To satisfy the conservation of volume constraint (2.13e), we recall that the velocity

potential φ is defined up to an arbitrary constant C, which allows one to write (3.5)as

∂φ

∂n(x)=−λ2

[C∫ 1

bG(x, y; 0)dy+

∫ 1

bG(x, y; 0)φ(y)dy

]. (4.8)

Integrating (4.8) along the equilibrium surface and enforcing (2.13e) determines

C=−

∫ 1

b

∫ 1

bG(x, y; 0)φ(y)dydx∫ 1

b

∫ 1

bG(x, y; 0)dydx

. (4.9)


Finally, applying (4.9) to (4.8) results in the following functional eigenvalue equation:

∂φ

∂n(x)= λ2

∫ 1

b

∫ 1

bG(x, y; 0)φ(y)dydx∫ 1

b

∫ 1

bG(x, y; 0)dydx

∫ 1

bG(x, y; 0)dy−

∫ 1

bG(x, y; 0)φ(y)dy

,(4.10)

with volume conserved for all functions φ.

4.1.2. Azimuthal operator (l 6= 0)The volume conservation constraint (2.13e) is naturally satisfied for disturbances

with azimuthal wavenumber l> 1. Thus, the corresponding operator equation is givenby

∂φ

∂n(x)=−λ2

[∫ 1

bG(x, y; l)φ(y)dy

]. (4.11)

4.2. Computational evaluation: Rayleigh–RitzThe eigenvalue spectrum of (4.10) and (4.11) can be evaluated in any number of ways.We reduce the operator equations to a truncated set of linear algebraic equations usingthe Rayleigh–Ritz variational procedure. The stationary values of the functional

λ2 =(−M−1 [φ] , φ

)(K−1 [φ] , φ

) , φ ∈ S, (4.12)

are the characteristic oscillation frequencies, where S is a predetermined functionspace. We choose this function space to satisfy Laplace’s equation (2.13a) and theno-penetration condition (2.13b). In this case, the minimizers of (4.10) and (4.11) arealso solutions to the eigenvalue problem (2.13). We sketch the method here, while amore thorough illustration can be found in Segel (1987).

The necessary input for the Rayleigh–Ritz procedure is a solution series

φ =N∑

j=1

ajφ(l)j , (4.13)

constructed from basis functions φ(l)j . These functions are applied to (4.10) and (4.11)and inner products are taken to generate a set of linear algebraic equations

N∑j=1

(m(l)

ij − λ2κ(l)ij

)aj = 0i, (4.14)

with

m(l)ij ≡

∫ 1

b

(∂φ

(l)i

∂n

)φ(l)j dx, κ

(l)ij ≡

(Gi0G0j

G00δl,0 −Gij

), (4.15a,b)


where δl,0 is the Kronecker delta, δ0,0 = 1 and δl,0 = 0 if l 6= 0, and

Gij

ξ(l)≡[∫ 1

by1(x; l)φ(l)i (x) dx

] [∫ 1

b

(τ2

τ1y1(x; l)− y2(x; l)

)φ(l)j (x) dx

]+∫ 1

by1(x; l)φ(l)i (x)

∫ 1

xy2(y; l)φ(l)j (y)dydx

−∫ 1

by2(x; l)φ(l)i (x)

∫ 1

xy1(y; l)φ(l)j (y)dydx. (4.16)

Allowable solutions of the operator equation (4.10), (4.11) must satisfy thehydrodynamic field equations (2.13). We recall that volume conservation (2.13e) hasbeen satisfied by proper selection of the constant C and the contact-line conditionsare incorporated into the Green’s function (4.2), but the no-penetration condition(2.13b) has yet to be satisfied. This can be accomplished through proper selection ofthe basis functions,

φ(l)j (ρ, θ)= ρ jP(l)j (cos θ) , (4.17)

written here in spherical coordinates, ρ and θ , and chosen to be harmonic, as requiredby (2.13a). Here, P(l)j is the Legendre function of degree j and order l. The followingrestriction on the polar j and azimuthal l wavenumbers is sufficient to ensure that theno-penetration condition (2.13b) is satisfied: l+ j= even. Additionally, a consistencycondition requires l 6 j.

The normal derivatives of the basis functions (4.17), evaluated on the equilibriumsurface, are expressed as

∂φ(l)j

∂n≡ ∇φ(l)j · n= jP(l)j (cos θ) (−sin s sin θ + cos s cos θ) (ρ)j−1

+ sin θP′(l)j (cos θ) (sin s cos θ + cos s sin θ) (ρ)j−1 , (4.18)

using mixed coordinates for efficiency in presentation. To compute the matrix elements(4.15a,b), the basis functions (4.17) and their normal derivatives (4.18) are evaluatedon the equilibrium surface using the following coordinate transformation:

ρ≡√

X2 + Y2 + Z2, cos θ ≡ Z√X2 + Y2 + Z2

, sin θ ≡√

X2 + Y2

X2 + Y2 + Z2, (4.19a−c)

which relates spherical coordinates, whose origin is centred on the surface-of-support,to the arclength coordinate. Here, X = X(s), Y = Y(s), Z = Z(s) have been definedin (2.2). Finally, the basis functions (4.17) are applied to (4.15a,b) to generate a setof algebraic equations (4.14), from which the characteristic oscillation frequencies arecomputed.

The eigenvalues λk,l of (4.14) have been computed using a resolution of N = 10basis functions in the solution series (4.13) for both natural and pinned disturbances,as they depend upon the contact angle α and azimuthal wavenumber l. Theeigenfunctionφk,l associated with the eigenfrequency λk,l and corresponding eigenvectora(k,l)j is given by

φk,l(x)=N∑

j=1

a(k,l)j φ(l)j (x). (4.20)


Here, eigenfrequencies/eigenvectors are distinguished by polar k and azimuthal lwavenumbers. Similarly, the corresponding interface deformation yk,l is expressed as

yk,l(x)=N∑

j=1

a(k,l)j

(∂φ

(l)j

∂n

)(x). (4.21)

5. Validation of computations

Prior computations for special limiting cases provide validation of our model. Oursolutions naturally reduce to the RL spectrum in the limit of (α, Λ)→ (90, 0). Theaxisymmetric pinned [2, 0] mode has been simulated by the finite-element method fora range of α by Basaran & DePaoli (1994); the comparison with our results is shownin figure 6(b) of § 7.1. Comparison with observation provides further corroboration.Frequencies of water droplets excited by a puff of air have been reported by Sharpet al. (2011) for a range of α. The reported mode shape has the character of the[1, 1] rocking mode; comparison against our prediction is shown in figure 7(b) of§ 7.2. Further comparison against experiment is the subject of Chang et al. (2013),where reasonable agreement is reported. Chang et al. (2014) report further tests oftheory against experiment.

6. Mode-shape terminology and overview of results

Figures 2(c,d) suggest a spherical harmonic classification by wavenumber pair[k, l] inherited from the RL modes, even though, for (α, Λ) 6= (90, 0), the sphericalharmonic shapes are not solutions to the governing equations (3.2). According to thespherical harmonic terminology, mode shapes are categorized as zonal (l= 0), sectoral(k= l) and tesseral (l 6= 0, k 6= l) (e.g. MacRobert 1967). This classification scheme isequivalent to identifying nodes (zeros) of the interface shape yk,l, the geometric curveformed by the intersection of the disturbed shape with the undisturbed shape. Nodesmay be composed of circles of both latitude and longitude. One can determine thenumber of nodal lines by counting the number of zeros of yk,l in the latitudinal andlongitudinal directions, independently. Given a natural mode with wavenumber pair[k, l], there are precisely l longitudinal and (k − l)/2 latitudinal intersections (zeros)of the undisturbed shape. Zonal modes (l = 0) are axisymmetric and intersect theundisturbed shape only along latitudinal lines. In contrast, sectoral modes (k= l) haveonly longitudinal intersections. Mode shapes with both types of nodes are referred toas tesseral modes. The pinned mode shapes have an additional latitudinal intersection,because the support plane lies along a constant latitude.

Figure 4 shows example modes with mobile contact lines for α = 75 to illustrate(a–d) interface shapes, (e–h) streamlines and (i–l) velocity potentials (pressure).Interface shapes are shown in longitudinal section (a–d), perspective view (e–h) andtop view (i–l). The nodal lines for the [8, 0] zonal mode (a) consist of four latitudinalcircles, whereas the [2, 2] sectoral mode (c) has two longitudinal circles (most readilyseen in latitudinal section). The remaining tesseral modes, (b) [5, 1] and (d) [7, 3],respect the prescription for (k− l)/2 latitudinal and l longitudinal nodal circles.

It turns out that the l= 1 modes are especially easy to identify in experiments sincetheir one sector rocks from side to side. These ‘rocking’ modes are illustrated by the[5, 1] mode in figure 4(b). During an oscillation, its peak moves down, across and upagain, as it rocks from one side to the other side, much like a seventies disco move.


(a) (b) (c) (d )

(e) ( f ) (g) (h)

(i) ( j) (k) (l )

FIGURE 4. (Colour online) Eigenmodes: (a,e,i) zonal [k, l] = [8, 0], (b, f,j) lateral [k, l] =[5, 1], (c,g,k) sectoral [k, l] = [2, 2] and (d,h,l) tesseral [k, l] = [7, 3] modes illustratingthe (a–d) disturbed interface, (e–h) streamlines and (i–l) velocity potential (pressure) for asubhemispherical drop α= 75 with mobile contact lines. Here, the dashed (dotted) curvesin (e–h) denote points of zero horizontal (vertical) velocity. The drop surface in (e–l) isunperturbed.

It should be noted that the Noether mode ([1, 1]), defined in § 1, is a special rockingmode.

In order to more easily identify tesseral modes in experiments, Chang et al.(2013) have introduced an alternative scheme based on ‘layers’ n and ‘sectors’ lwith n = (k − l)/2 + 1. These are most easily seen in top view. For example, themode shown in figure 4(d) has three sectors and three layers, yielding (n, l)= (3, 3).Alternatively, figure 4(b) shows a mode with one sector and three layers, correspondingto (n, l)= (3, 1). Sectoral modes have one layer and zonals have zero sectors. A videotutorial for this classification scheme can be found in Chang (2013). Figures 5(a,b) usethe layer/sector classification and show the first 35 modes, contrasting natural modesfor (a) α = 90 (RL modes) and (b) pinned modes with α = 75. The differencesappear minor, but are most clearly seen near the contact line, where the interfaceshape adjusts itself to accommodate the pinned contact-line condition. The abilityto model wetting properties is one reason why our predictions have been able toaccurately capture recent experiments on sessile drop vibrations (Chang et al. 2013).In Chang et al. (2014), we demonstrate the utility of our model over a wide rangeof contact angles.


(a) (b)1

0

1

2

3

4

5

6

7

8

9

10

SC

2 3 4 5 6 1

0

1

2

3

4

5

6

7

8

9

10

SC

2 3 4 5 6

FIGURE 5. Sector (l)–layer (n) table of rendered mode shapes (top view): (a) RL modes(α = 90, free contact line), which reorganize and extend figure 2, compared with (b) asessile drop with α= 75 and a pinned contact line. Here, l is the azimuthal wavenumberand n ≡ (k − l)/2 + 1 is the number of layers, with k the standard polar wavenumber;SC = spherical cap.

The streamlines, illustrated in figures 4(e–h), show velocities with vertical andhorizontal components. The motions of the sectoral mode (g) are purely horizontalwhile those of the zonal mode (e) are strongly vertical. It turns out that thepredominantly horizontal or vertical character of the motion correlates with theinfluence of support on the modal frequency. That is, the frequencies of the sectoralmode are only slightly affected by the volume (equivalently α) while those of thezonal mode are strongly affected. Vertical motion, alternatively, correlates with thenumber of layers n. In summary, larger n modes are more sensitive to the presenceof support, as measured by the frequency slope.

Additionally correlating with the horizontal/vertical motion of a natural mode isthe displacement at the contact line. Horizontal motion favours, while vertical motioninhibits, contact-line displacement. In figures 4(a–d), top row, the four plots allhave the same maximum deviation amplitude from the spherical cap. However, thatmaximum occurs at different places around the cap. The [2, 2] sectoral mode (c)exhibits a larger excursion than the [8, 0] zonal mode (a), while the rocking [5, 1](b) and tesseral [7, 3] (d) modes have intermediate displacements. In summary, thenumber of layers correlates with both streamline (horizontal/vertical) motion andcontact-line excursion.


(a) (b)

50 70 90 110 130

10

20

30

–0.9 –0.6 –0.3 0 0.3 0.6 0.9

D

1.0

10.0

5.0

2.0

3.0

1.5

7.0 [2, 0] PinnedBasaran (1994)

4

6

FIGURE 6. (Colour online) Zonal modes (l= 0). (a) Frequency λk,0 against contact angleα for the natural N and pinned P disturbances. The RL frequencies are circled. (b)Frequency λB against shape parameter D=−cosα for finite-element simulations (symbols)(Basaran & DePaoli 1994, figure 7) and pinned [2, 0] predictions (solid line) (this work).

7. Results for kinematic disturbances: α dependence7.1. Axisymmetric (l= 0) modes

Figure 6(a) plots the frequency λk,0 against the contact angle α for the axisymmetric(l = 0) modes. Frequencies for the natural (N ) and pinned (P) modes are shownpaired. For the hemispherical drop, the natural modes are precisely the RL modeswith associated frequency, identified by circles, while the corresponding pinned modesshift to higher frequency. These frequencies reproduce the summary in figures 2(a,b),respectively. The fact that the pinned frequency is always higher than the naturalfrequency is a result of the restriction that pinning places on the function spaceof disturbances. As with many classical problems in mathematical physics, such asthe vibrating drum head, more constrained function spaces lead to uniformly higherfrequencies (Courant & Hilbert 1953).

Figure 6(a) shows that frequencies break from the hemisphere by increasing forlens-like and decreasing for drop-like caps. This is a consequence of the lesser(greater) volume of the lens-like (drop-like) shape which implies lesser (greater)inertia. Another noteworthy feature of figure 6(a) is the convergence of frequenciesas α gets larger. Indeed, all the frequencies limit to zero as α→ 180. This occursbecause, in the limit of the drop touching the support plane at a single point, thenatural disturbance degenerates into the pinned disturbance. Moreover, this limitcorresponds to infinite volume and hence all the modal frequencies approach zero,related to our scaling. It should be noted that the spectral ordering established forthe RL modes is maintained for all α. That is, any zonal mode can be related backuniquely to a RL mode.

The relative frequency difference (λp − λn)/λp between the natural (λn) and pinned(λp) disturbances measures the ‘degree-of-pinning’ of a contact line. A small relativedifference in frequency indicates that the contact line is essentially pinned and onecannot distinguish between the natural and pinned disturbances. For example, considerthe hemispherical drop (α = 90) and compare the relative frequency difference of11 % for the [6, 0] mode with the 42 % difference for the [2, 0] mode (cf. figure 6).One concludes that the [6, 0] mode has a much smaller contact-line displacementthan the [2, 0] mode. For the natural disturbance, the contact-line degree-of-pinningis greatly influenced by the polar wavenumber k.


(a) (b)

50 70 110 130

5

15

25

45 90 135 180

0.005

0.010

0.015

0.020 [1, 1] PinnedSharp (2011)

FIGURE 7. (Colour online) The l= 1 spectrum. (a) Scaled frequency λk,1 against contactangle α for the natural N and pinned P disturbances. Here, Im[λk,1] = 0 in all casesexcept for k = 1N , α > 90 (cf. figure 8a). RL frequencies are denoted by E and theNoether mode by ?. (b) Comparison between experiments (Sharp et al. 2011, figure 3)and the [1, 1] pinned mode by plotting f 2

1,1 ×m against α, where f is the real frequencyand m is the mass of the droplet.

Finally, regarding validation against prior work, figure 6(b) replots the pinned[2, 0] frequencies (solid line) in a form compatible with Basaran & DePaoli (1994).They report finite-element simulations (symbols) for drop vibrations over a range ofparameters that include the variable wetting properties of the solid substrate. Thecomparison is reasonable. We also reproduce the zonal frequencies for the limitingcases Λ = 0, ∞ predicted by Lyubimov et al. (2006, table I) for a hemisphericaldrop. More recently, experiments by Chang et al. (2013) have been reported whichmeasure a range of zonal frequencies, k= 2, 4, 6, 8, 10, 12, 14, for pinned drops withα ∼ 70. They show conclusively that the RL frequency predictions are inadequate,while our predictions are consistent with observation (Chang et al. 2013, figure 6a,d).

7.2. Rocking (l= 1) modesFigure 7(a) plots the frequency λk,1 against the contact angle α for the rocking (l= 1)modes. Again, frequencies of natural (N ) and pinned (P) modes are paired. Forthe hemisphere (α= 90), the [1, 1] natural mode is the Noether mode (?), while the[3, 1] and [5, 1] natural modes are the RL modes (E). As for the zonals, the pinnedfrequencies are shifted upwards from the natural frequencies, and for the same reason.The frequencies all break similarly from the hemisphere, increasing (decreasing) forsmaller (greater) α because of decreasing (increasing) inertia due to volume.

The hemispherical Noether mode (?) has zero frequency. That the oscillationfrequency is exactly zero is important and we sketch the proof. To begin, the functionP(1)1 (x) is a fundamental solution of the curvature operator (3.3b) and uniquely satisfiesthe natural boundary conditions (3.8) for the hemispherical base state. Therefore, thisfunction belongs to the kernel of (3.3b). It is also straightforward to show that the‘inertia’ operator (3.3a) is positive definite. Thus, applying the Fredholm alternativefor linear operators to this particular mode shape delivers λ2

1,1 = 0. The motion ofthe ? mode corresponds to a horizontal translation of the drop’s centre-of-mass. Thismotion is analogous to the zero-frequency mode of the drop pinned on an equatorialcircle-of-contact (Bostwick & Steen 2009) and likewise can be attributed to the


(a) (b)

90.0 115.0 132.5 150.0 180.0

0.0458

0.0300

0.0100

FIGURE 8. The [k, l] = [1, 1] instability: (a) instability growth rate −λ21,1 against contact

angle α and (b) a typical instability mode shape with contact angle α=120 in polar view.

additional symmetry inherent in the hemispherical base state (Field, Golubitsky &Stewart 1991), as outlined in § 1.

The Noether frequency squared, λ21,1, breaks higher for lens-like base states, yielding

a shift upwards as for the other rocking modes. In the direction of drop-like shapes, itbreaks to λ2

1,1<0, also consistent with higher rocking modes. The important differenceis that the break is from zero which puts it in negative territory. Exponential growthresults, according to (2.12). Hence, the walking instability may be anticipated by thesymmetry breaking of the RL Noether mode. It should be noted that figure 7(a) plotsthe real part of λ1,1.

Finally, regarding validation against prior work, figure 7(b) replots the pinned [1, 1]frequencies (solid line) in the form that Sharp et al. (2011) report their experiments.Figure 7(b) reports the observed oscillation frequency f1,1 against the contact angle α(symbols). The pinned drop experiments (α∼70) of Chang et al. (2013) also measurethe frequencies of several rocking modes, k= 1, 3, 5, 7, 9, with reasonable comparison(Chang et al. 2013, figures 6, 13, 14).

7.2.1. Noether mode: ‘walking’ droplet instabilityThe sessile drop exhibits instability (λ2 < 0) to the natural disturbance for drop-like

base states; that is, those in the range of contact angles 90 < α < 180. Figure 8(a)plots the square of the instability growth rate −λ2

1,1 against the static contact angleα. It shows that the maximum growth rate (−λ2

1,1 = 0.0458) occurs at α = 132.5. Atypical unstable mode shape is shown in figure 8(b).

Insight into the instability mechanism is gained by evaluating the static free energyfunctional U at the mode shape y1,1. The idea is that the instability will occur whenthe perturbed state has a lower energy than the base state. For a fully mobile sessiledrop of contact angle α, we define U for a general capillary surface x(u, v),

U/σlg =∫Γ

|x,u × x,v |dΓ , (7.1)

where u, v are surface coordinates. We then take the second variation of the potentialenergy (Myshkis et al. 1987),

δ2U =−∫ 1

b

[((1− x2

)y′)′ +(2− 1

1− x2

)y]

y dx, (7.2)


(a) (b)

30 60 90 120 150 180

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

E

E3

E2

E1

90.0 120.0 136.8 150.0 180.0

0.050

0.100

0.150

0.211Scaled

FIGURE 9. (Colour online) Energetics for the walking instability: (a) decomposition of thedisturbance energy (E) and (b) scaled walking speed vCM (solid line), evaluated in cm s−1

for a 1 µL drop of water (dashed line) using the velocity scale v√ρr/σ .

and integrate by parts using the natural boundary condition (3.8) to obtain

δ2U =∫ 1

b

[(1− x2

) (y′)2 −

(2− 1

1− x2

)y2

]dx+ b

√1− b2 y2(b). (7.3)

This equivalent form of the total disturbance energy Et may be decomposed into twointerfacial energies (E1, E2) and one contact-line energy (E3),

Et = E1 + E2 + E3, (7.4a)

E1 ≡∫ 1

b

(1− x2

) (y′)2 dx, (7.4b)

E2 ≡ −∫ 1

b

(2− 1

1− x2

)y2dx, (7.4c)

E3 ≡ b√

1− b2 y2(b). (7.4d)

Here, E1 is a positive-definite (stabilizing) measure of the gradients in the meancurvature of the perturbed surface, while E2 represents the tendency of a volumeof liquid to form an isolated spherical drop (minimal energy state). In contrast, thecontact-line energy E3 can either stabilize (E3 > 0) or destabilize (E3 < 0), dependingupon the geometry of the base state. Figure 9(a) plots the decomposition of thedisturbance energy, showing that for the range of base states that exhibit instability,90 < α < 180, the droplet reduces its energy by lowering the interfacial (E1 + E2)and contact-line (E3) energies.

The reduction in energy drives fluid motion, which is shown to correlate withhorizontal motion of the centre-of-mass. Hence, we refer to the instability as the‘walking’ droplet instability. A standard calculation of the horizontal centre-of-massx(t)= ∫ x(t)dV/

∫dV of the disturbed interface shape y(s) shows that the instantaneous

walking speed vCM ≡ ˙x(t= 0) is given by

vCM =√−λ2

3∫ α

0y(s) sin2 sds

2 (1− cos α) sin α. (7.5)


(a) (b) (c)

(d ) (e) ( f )

50 70 90 110 130

2

4

6

50 70 90 110 130

4

8

12

50 70 90 110 130

6

10

14

50 70 90 110 130

6

10

14

18

22

50 70 90 110 130

12

18

24

50 70 90 110 130

10

18

26

34

FIGURE 10. (Colour online) Natural disturbance: frequency λk,l, as it depends uponazimuthal wavenumber l, against contact angle α for fixed polar wavenumber (a) k = 2,(b) k= 3, (c) k= 4, (d) k= 5, (e) k= 6 and (f ) k= 7. It should be noted that the scalingof the frequency is different for (a)–(f ).

Figure 9(b), which plots the walking speed vCM against the static contact angle α,shows that the maximum walking speed (vCM = 0.211) occurs at α = 136.8. Forreference, we evaluate (7.5) for a 1 µL drop of water (H2O) and plot against α.

7.3. Azimuthal (l 6= 0) modesFor natural and pinned disturbances, respectively, figures 10 and 11 show how theazimuthal modes break the hemispherical base-state degeneracy (cf. figures 2a,c).Zonal l = 0 modes are included for reference. For fixed mode shape, in all cases,the frequency decreases monotonically with increasing volume (increasing α) sincehigher inertia (volume) biases toward lower frequencies, just as for the zonal modesdiscussed above. For natural and pinned disturbances, this negative slope decreaseswith higher mode number l relative to the reference mode, zonal l= 0 or rocking l= 1.The least slope coincides with the sectoral mode, the highest azimuthal wavenumberpossible. That is, for the natural disturbances, the frequencies of drop-like volumesbreak upward while those for lens-like volumes break downward, relative to thereference mode (figure 10). The net effect is a plot that looks like a bowtie, onlywith a skew. For the pinned disturbances, the frequencies behave similarly, only withthe centre of the bowtie shifted to lower volumes (figure 11).

In summary, the new feature of this degeneracy breaking is that, for the same polarwavenumber k, higher azimuthal wavenumber l modes can have lower frequencies.This occurs because the influence of the support, as measured by the relative slope offrequency with α, increases with increasing number of layers n, as discussed above.There is one volume for which the shape is indifferent to horizontal and verticalmotions. That is, all frequencies must pass through a centre. For natural disturbances,this centre is the hemispherical shape while, for pinned disturbances, the centre shiftsto a lens-like shape. This shift can be understood as pinning being more inhibiting to


(a) (b) (c)

(d) (e) ( f )

50 70 90 110 130

2468

50 70 90 110 130

6

10

14

50 70 90 110 130

6

10

14

18

50 70 90 110 130

812162024

50 70 90 110 130

16

24

32

50 70 90 110 13012

20

28

36

FIGURE 11. (Colour online) Pinned disturbance: frequency λk,l, as it depends uponazimuthal wavenumber l, against contact angle α for fixed polar wavenumber (a) k = 2,(b) k= 3, (c) k= 4, (d) k= 5, (e) k= 6 and (f ) k= 7. It should be noted that the scalingof the frequency is different for (a)–(f ).

motions of a horizontal than a vertical nature. Together, these two factors explain whythe sectoral mode frequencies are lowest for lens-like and highest for drop-like shapes.Stated differently, the mode with least vertical motion has the lowest frequency forlens-like shapes while that with the greatest vertical motion has the lowest frequencyfor drop-like shapes. The relative frequency difference between the zonal (l = 0)and sectoral (l = k) modes can be large. For example, the difference between the[4, 0] and [4, 4] modes approaches 100 % for the lens-like base state α = 50 (cf.figure 10c). Finally, for a fixed polar wavenumber k, the frequency difference for thepinned disturbance is much smaller than that for the natural disturbance (e.g. figures10a, 11a). Pinning inhibits motion, lessening frequency differences.

Comparison of these predictions against the pinned drop experiments (α ∼ 70)of Chang et al. (2013, figure 14d), who measure the frequencies of several tesseralmodes, [6, 2], [7, 3], [8, 4], [9, 5], [10, 6], are found to be reasonable.

7.4. Frequency crossings: α dependenceUp to now, spectral ordering has been preserved in every case discussed. Acrossall α, pinned and natural zonals respect the spectral ordering of the RL modes (cf.figures 6a,b), as do pinned and natural rocking modes (cf. figure 7a). Moreover,for the other natural tesseral modes, spectral ordering in l (fixed k) is preserved forα > 90 while a reverse spectral ordering is seen for α < 90, as shown in figure 10.For the other pinned tesseral modes, a spectral (or reverse spectral) ordering is nearlyrespected (cf. figure 11). The preservation or not of spectral ordering is related to theSturm–Liouville nature of the operator (3.1). In general, zonal and azimuthal modesmust both be considered in which case spectral ordering will be the exception.

Figure 12 illustrates two examples where spectral ordering is broken, one for lens-like and the other for drop-like shapes. Consistent with the influence of inertia, all


(a) (b)

50 60 70 80

15

20

25

30

110 120 130 140

6

10

14

FIGURE 12. (Colour online) Frequency crossings controlled by base-state volume:frequency λk,l as a function of static contact angle α for (a) natural and (b) pinneddisturbances.

frequencies decay monotonically with α but with different slopes depending on thehorizontal/vertical motion of associated modes. A necessary condition for crossing isthat modes of different k compete. In figure 12(a), polar wavenumbers k = 5, 6 andk = 7 are represented. Three different k = 6 modes are included and none of thesefrequencies can cross, by spectral ordering. On the other hand, the rocking [5, 1]and sectoral [7, 7] could cross the k = 6 modes (and/or each other). The sectoralmode, with its predominantly horizontal motions, has the weakest slope and greatestrelative difference in slopes; this may be viewed as the active mode in crossing theothers, yielding four crossings in total. Figure 12(b) likewise includes five modes: twosectorals, one rocking and three with the same k. Here, the ‘loner’ sectoral [4, 4] hasthe weakest slope and crosses the rocking mode. The other sectoral [5,5] is also activeand crosses the [6,0] zonal. A total of two crossings are seen in the range of α plotted.Experiments reported in Chang et al. (2014) test such predictions.

8. Method for dynamic disturbances: mobile contact linesIn principle, the inverse method could be used to satisfy the dynamic contact-line

condition (3.11), but this formulation gives rise to an integral operator with theeigenvalue parameter in its kernel, which is notorious for computational instability(e.g. Walter 1973). To circumvent this issue, we solve the direct problem by utilizingresults for the natural disturbance and exploiting the variational structure associatedwith (3.1).

8.1. Contact-line energyTo begin, consider the second variation of the surface energy U, equivalent to, but ina form more general than, (7.1),

δ2U =−∫Γ

((κ2

1 + κ22

) ∂φ∂n+∆Γ

∂φ

∂n

)∂φ

∂ndΓ +

∫ζ

(∂

∂s

(∂φ

∂n

)+ cos(α)

∂φ

∂n

)∂φ

∂ndζ ,

(8.1)with contributions from the free surface Γ and contact line ζ (Myshkis et al. 1987).The corresponding boundary-value problem for the hydrodynamic problem is givenby the integrodifferential equation (3.1) combined with a boundary condition on thethree-phase contact line. As is evident from (8.1), there is no contact-line energy


associated with the natural (3.8) or pinned (3.9) disturbances. In contrast, applyingthe dynamic contact-line condition (3.11) to (8.1) results in an associated contact-lineenergy, iλΛ

∫ζ(∂φ/∂n)2dζ , whose form demonstrates that contact-line motion is

dissipative. To distinguish this effective dissipation from viscous dissipation, we shallrefer to it as ‘Davis’ dissipation, first identified by Davis (1980).

To proceed with our solution technique, we recall two results from the calculus ofvariations: (i) the natural disturbance is the absolute minimizer and (ii) the pinneddisturbance is the absolute maximum of the minimizers of the second variation ofthe surface energy (8.1) (Courant & Hilbert 1953). That is, applying the natural(pinned) boundary condition to the operator (3.1) results in the smallest (largest)possible disturbance energy. We exploit (i) in formulating the solution method for thedynamic contact-line disturbance.

8.2. Computational evaluation: hybrid Rayleigh–RitzWe compute the spectrum for the dynamic contact-line disturbance by solving thedirect problem (3.2). We recall that for the direct problem the unknown functionin the Rayleigh–Ritz procedure is the interface disturbance and one must computethe associated velocity potential. As stated earlier, this proves to be a dauntingtask for most geometries, including the spherical-cap geometry used here. However,judicious choice of the function space in the Rayleigh–Ritz method can circumventthis difficulty associated with an ab initio solution of the direct problem. We choosethe complete set of natural mode shapes yj,l, defined in (4.21), as the basis functionsin the requisite solution series

∂φ(l)

∂n=

N∑j=1

cjyj,l(x), (8.2)

where l is the azimuthal wavenumber.To explain our choice of function space, we recall that solution of the inverse

problem (3.5) results in eigenfunctions (4.20) identified as velocity potentials φ,which are related to the interface disturbance y ∝ ∂φ/∂n (4.21). Hence, this solvesprecisely the difficulty with the direct problem (3.2) in a relatively simple manner.Finally, we choose the natural mode shapes as the basis functions (8.2) becausethey span the least restrictive function space (see Courant & Hilbert 1953; Lanczos1986; Segel 1987) and solve a limiting case (Λ = 0) of the dynamic contact-linecondition (3.11). It should be noted that the pinned contact-line disturbance, limitingcase (Λ→∞) of (3.11), can be recovered from the natural disturbance using anynumber of solution techniques (e.g. Lyubimov et al. 2006; Bostwick & Steen 2010;Prosperetti 2012). In contrast, the natural disturbance cannot be recovered from thepinned disturbance.

Finally, we apply the solution series (8.2) to the direct problem (3.2), resulting inthe following set of algebraic equations:

N∑j=1

[K (l)

ij +iλ√

1− b2Φ(l)ij +

λ2

1− b2M(l)

ij

]cj = 0, (8.3)

with matrix elements

K (l)ij ≡

∫ 1

b

((1− x2

) (yi,l)

xx − 2x(yi,l)

x +(

2− l2

1− x2

)yi,l

)yj,ldx, (8.4a)


Φ(l)ij ≡Λ yi,l(b)yj,l(b), (8.4b)

M(l)ij ≡

∫ 1

bφi,lyj,ldx. (8.4c)

Here, we have applied the dynamic contact-line law (3.11) to the second variation ofthe surface energy (8.1) to yield the contact-line energy (8.4b). As is evident from thedamped-harmonic oscillator structure of (8.3), contact-angle variation controlled by thespreading parameter Λ in (8.4b) is dissipative.

The complex frequencies λ = −γ + iω, as they depend upon the static contactangle α and spreading parameter Λ, are computed from the characteristic equationassociated with the matrix (8.3). Computations show that a truncation of N= 12 termsin the solution series (8.2) is sufficient to generate relative eigenvalue convergenceof 0.1 % for the results presented here. For a given eigenvalue λk,l, the mode shapescorresponding to the eigenvector c(k,l)j with polar k and azimuthal l wavenumbers aregiven by

ψk,l =N∑

j=1

c(k,l)j yj,l. (8.5)

9. Results for dynamic disturbances: Λ dependence9.1. Davis dissipation

The spreading parameter Λ is a measure of the resistance of a contact line tospreading that can change the disturbance from natural (Λ→ 0) to pinned (Λ→∞).Here, Λ is also a measure of the Davis dissipation (8.4b), an irrecoverable lossthat results from the dynamic contact-line condition (3.11) and leads to damping ofoscillations γ > 0. Here, one should note that the decay rate scales with the capillaryfrequency (2.13f ). In comparison, for an isolated spherical drop, the decay rate fromviscous dissipation scales with the viscosity ν as γk,l = ν/R2 (k − 1)(2k + 1) (Lamb1932).

One can use the damped-harmonic oscillator structure of (8.3) and related formalism(e.g. Taylor 2005, § 5.6) to define the Davis dissipation over an oscillation cycle as

Qk,l ≡ 2πγk,l

ωk,l. (9.1)

This is commonly referred to as the Q-factor. For reference, for an isolated sphericaldrop, the effective dissipation due to bulk viscous effects is given by Qk,l ∝

√k.

The influence of Λ on the frequency is illustrated for zonals in figure 13 andfor azimuthal modes in figure 14. The plots show a step-like function increase infrequency from natural to pinned disturbances. This monotonic increase is consistentwith increasing constraint. Figures 13(a,d,g) show that the spectral ordering atΛ = 0 and Λ = ∞ noted for zonals is maintained. The plots in figures 13(b,e,h)illustrate that the decay rate also maintains spectral ordering over the Λ range.Starting from Λ = 0, the decay rate rises to reach a maximum at a finite valueof Λ and then tends back to zero for large Λ. Figures 13(c,f,i) show that thelow-wavenumber modes have the largest dissipation per cycle. This can be directlyattributed to the contact-line displacement, which is more pronounced for the lowerpolar wavenumber modes. Since Davis dissipation (8.4b) is related to contact-linedisplacement, the low-wavenumber modes might be expected to dissipate the most per


(a) (b) (c)

(d ) (e) ( f )

(g) (h) (i)

10–2 10–1 100 101 102

10–2 10–1 100 101 102

10–2 10–1 100 101 102 10–2 10–1 100 101 102 10–2 10–1 100 101 102

10–2 10–1 100 101 102 10–2 10–1 100 101 102

10–2 10–1 100 101 102 10–2 10–1 100 101 10205

10152025

012345

0

1.0

2.0

3.0

0

5

10

15

20

0

1.0

2.0

3.0

00.51.01.52.02.5

0

4

8

12

00.51.01.52.02.5

0

0.5

1.0

1.5

2.0

FIGURE 13. (Colour online) Complex frequency for zonal (l = 0) modes: (a,d,g)oscillation frequency ωk,0, (b,e,h) decay rate γk,0 and (c,f,i) Davis dissipation Qk,0 as afunction of the spreading parameter Λ for an (a–c) sub-hemispherical (α = 75), (d–f )hemispherical (α = 90) and (g–i) super-hemispherical (α = 105) drop.

(a) (b) (c)

0.01 0.10 1.00 5.0012

14

16

0.01 0.10 1.00 5.000

1

2

3

4

0.01 0.10 1.00 5.000

0.5

1.0

1.5

FIGURE 14. (Colour online) Complex frequency for the k = 6 modes: (a) oscillationfrequency ω6,l, (b) decay rate γ6,l and (c) and effective dissipation Q6,l as a function ofthe spreading parameter Λ for contact angle α = 105.

cycle. Furthermore, it has been established in § 7.1 that the maximum displacementof the contact line decreases as the static contact angle increases. Figures 13(c,i)compare the dissipation per cycle for lens-like (α = 75) and drop-like (α = 105)shapes, respectively. Unsurprisingly, the lens-like modes dissipate more energy, astheir contact lines show greater displacement. For these comparisons, recall that thedisturbance kinetic energy is normalized via the denominator in the Rayleigh–Ritzquotient.

For azimuthal modes with α = 105, figure 14(a) shows that spectral ordering ismaintained over the range of Λ. For azimuthal modes with small α, reverse spectral


90 120 150 18010–6

10–4

10–2

100

102

Stable

Unstable

FIGURE 15. Onset of walking for the [k, l] = [1, 1] mode: for fixed α, on decreasing Λ,rocking turns to walking across a critical value (solid line).

ordering is maintained (not shown), consistent with the bowtie structure for natural(cf. figure 10e) and pinned (cf. figure 11e) disturbances. Moreover, for a fixed polarwavenumber k, the mode of maximal dissipation has either the largest or the smallestpossible azimuthal wavenumber for the lens-like and drop-like base states, respectively.However, changes in spectral ordering and the variation in position of the step changein frequency (cf. figure 14a) accommodate Λ-crossings, discussed below in § 9.3. Forsimilar reasons, the mode that dissipates the largest amount of energy per cycle maychange at different values of the spreading parameter. For example, figure 14(c) showsthat for fixed spreading parameters Λ = 0.5 and Λ = 1, the [6, 4] and [6, 2] modeshave the largest effective dissipation, respectively. However, the mode with maximaleffective dissipation over the entire range of spreading parameters is predicted by thegeometry of the base state: high (low) azimuthal wavenumber modes for lens-like(drop-like) base states.

9.2. Dissipative Noether mode: onset of ‘walking’Davis dissipation from the dynamic contact-line condition can suppress the walkingdroplet instability (cf. figure 8). Figure 9 shows that the disturbance energy for the[1, 1] natural mode is lower than the base-state energy, resulting in instability. Theeffective dissipation from the dynamic contact-line condition (8.4b),

Ed =Λ(y1,1(b)

)2, (9.2)

can counteract the net energy gained in going from the base to the disturbed state(Et) (7.4). The base state is neutrally stable when the dissipation exactly balancesthe disturbance energy (Ed = Et). This criterion is used to determine the onset of‘droplet walking’, as plotted in figure 15. In the limit α → 180, the instabilitywindow remains open despite the small instability growth rate observed in this limit(cf. figure 8a). Here, the stabilizing dissipation Ed → 0 because the contact-linedisplacement is small, y1,1(b)→ 0. Hence, a large Λc can be overcome by a smalldestabilizing disturbance energy (Et < 0), as shown in figure 9(a). With regards togrowth rate, the effective dissipation is a monotonically increasing function of thespreading parameter Λ; therefore, the maximum instability growth rate correspondsto the natural disturbance Λ= 0, as shown in figure 8(a).


(a) (b)

0.1 0.2 0.5 1.0 2.0 5.0

16

18

20

22

0.1 0.2 0.5 1.0 2.0 5.0

16

18

20

22

A

BC

D

FIGURE 16. (Colour online) Frequency crossings: frequency ωk,l for (a) the k = 7 and[8, 0] modes and (b) the [7, 7] and [8, 0] modes only, with corresponding mode shapesat Λ= 1(A, B) and Λ= 2.1(C,D), against the spreading Λ for α = 105.

9.3. Frequency crossings: Λ dependenceAs discussed in § 7.1, spectral ordering is maintained for zonals. More generally,spectral ordering is respected for families of fixed k, as seen in figure 16(a) forthe four k = 7 modes. In contrast, crossings become typical when a different k isintroduced, as with [8, 0] in figure 16. We count five crossings amongst the [8, 0]and [7, 3], [7, 5] and [7, 7] modes. Frequency crossings correspond to multiplicity inthe spectrum and are important physically since mode mixing and exchange of modalbehaviour can then occur.

To illustrate the utility of these results, consider the scenario posed in figure 16(b)where two modes from figure 16(a) are isolated. It should be noted that both [8,0] and[7,7] rise from the lower depinned plateau to the upper pinned plateau, but at differentrates. Say that a pinned drop, Λ = 2.1, is driven at the [8, 0] resonance, point C.Now suppose that Λ is slowly decreased. (It is well documented that increasingforcing amplitude can cause depinning, for example, so increasing amplitude can actfunctionally to change Λ.) Following the [8, 0] mode in figure 16(b), the systemhas two choices as the cliff approaches – either depin with a consequent loweringof frequency (C→ A), opposed by the forcing, or remain pinned and change modalidentity to [7, 7] (C→ B), at the cost of reconfiguration.

This scenario may already be relevant to observation. Vukasinovic et al. (2007)drove a small sessile water drop with a piezoelectric actuator. They described thewave patterns corresponding to resonance frequencies on driving harder until thedrop was atomized. As part of this sequence of transitions, they clearly documentedthe transition from zonal, their figure 5(a), to a non-zonal mode, figure 5(d). Withfrequency fixed, as forcing amplitude is increased, the contact line of the zonaldepins, resulting in an azimuthal disturbance generated along the contact line. Asthe amplitude is increased further, the azimuthal instability propagates over the entiredrop surface and finally the drop atomizes. Such a simultaneous change of modes,from pinned to depinned and from zonal to non-zonal, probably occurs by a modeswitching event at frequency crossings (or near-crossings).

10. Concluding remarksThe hydrodynamic stability of an inviscid sessile drop has been studied in this

paper. We consider three types of contact-line disturbance: (i) the natural disturbance


preserves the static contact angle (3.8), (ii) the pinned disturbance fixes the contactline (3.9) and (iii) the dynamic disturbance relates the ‘dynamic’ contact angle to thecontact-line speed via a spreading parameter (3.11). The spectrum is computed foreach disturbance from the linearized hydrodynamic equation, which we formulate asa functional eigenvalue equation on linear operators and solve using the Rayleigh–Ritz variational procedure. A Green’s function solution for the Laplace equation inthe spherical-cap domain enables the approach. Prior analytical treatments have beenlimited to hemispherical drops.

The problem is parameterized by the azimuthal wavenumber l, the base-state volumevia the static contact angle α and the boundary conditions on the three-phase contactline through the spreading parameter Λ. The spreading (homotopy) parameter is ameasure of contact-line mobility and can smoothly change the boundary conditionfrom natural (Λ= 0) to pinned (Λ→∞).

At finite values of the spreading parameter, Davis dissipation associated withcontact-line motion leads to damped oscillation amplitudes, whereas both naturaland pinned motions are without dissipation. Amongst natural modes, for α < 90those with fewest layers show greatest contact-line excursion while, for α > 90,those with fewest sectors show the greatest contact-line excursion. These contact-lineexcursions correlate with the Davis dissipation per cycle when Λ is turned on. Thus,sectorals dissipate most for small volumes and zonals most for large volumes. If allelse is equal, lower polar wavenumbers dissipate more per cycle. Regarding volumeinfluence, for a fixed mode [k, l], we find that more energy is dissipated by thelens-like (α < 90) than the drop-like (α > 90) mode shape (cf. figure 13).

The majority of motions reported here are oscillatory. However, the sessile dropdoes exhibit instability to the natural disturbance for a range of base states of 90 <α < 180. It turns out that the rocking mode becomes a walking mode for drop-like volumes (cf. figure 9a). Davis dissipation can stabilize the instability, dependingon α, and the window where walking can occur is identified in figure 15. Walkingcorresponds to horizontal motion of the droplet centre-of-mass and can be relatedback to the degenerate zero-frequency translational mode of the RL spectrum. Dropletwalking represents a heretofore unidentified pathway of energy conversion of potentialsignificance. Energy stored in the liquid shape and its contact with the solid supportconverts into energy of liquid motion, regardless of viscous dissipation.

Acknowledgements

This work was supported by NASA Grant NNX09AI83G and NSF Grant CBET-1236582. The authors thank C.-T. Chang, E. Wesson, A. Altieri and A. Macner foruseful discussions.

Appendix A. Linearization of Young–Dupré equation

The natural (3.8) and contact-line speed (3.11) boundary conditions are derived here.We follow the development set forth in Myshkis et al. (1987).

Consider a liquid/gas interface with normal curvature k and surface normal nin contact with a surface-of-support described by normal curvature k and normalvector n1, as shown in figure 17(a). The Young–Dupré equation,

n · n1 = σsg − σsl

σlg≡ cos α, (A 1)


(a) (b)

D

n1

e1

n

e

s

y

D

s

FIGURE 17. Contact-line region: (a) the static equilibrium and (b) the disturbedconfiguration.

relates the surface normals n,n1 to the surface tensions of the solid/liquid σsl, solid/gasσsg and liquid/gas σlg ≡ σ interfaces and equivalently the static contact angle α (cf.figure 1b).

We would like to linearize the Young–Dupré equation (A 1). As shown infigure 17(b), we decompose the variation δx to the equilibrium surface into itsperpendicular δ⊥x and parallel δ‖x components,

δx= δ⊥x+ δ‖x= y n+ δ‖x, (A 2)

with the linear surface perturbation y defined as

δ⊥x≡ y n. (A 3)

The variation of the Young–Dupré equation (A 1),

δ (n · n1)=−(sin α)α, (A 4)

relates the geometry of the disturbed interface to the deviation in the static contactangle α. We apply the variation (A 2) to (A 4) to yield

δ (n · n1)=(δ⊥n+ δ‖n

)· n1 + n ·

(δ⊥n1 + δ‖n1

), (A 5)

which can be further simplified with the following identities:

δ⊥n=−∂y∂s

e, (A 6a)

δ‖n= ∂n∂s(e · δx) , (A 6b)

δ⊥n1 = 0, (A 6c)

δ‖n1 = ∂n1

∂s1(e1 · δx) . (A 6d)

Here, (A 6a) follows directly from the definition of the surface perturbation (A 3),(A 6b) and (A 6d) are standard differentials and (A 6c) represents the variation normal


to the surface-of-support, which is trivial because the motion of the fluid interfacethere is tangent e1 to the support. In addition, s and s1 are arclength coordinatesdefined on the equilibrium surface and surface-of-support, respectively.

The Frenet–Serret equations,

∂n∂s=−ke, (A 7)

relate the directional change in the normal vector n with respect to its arclengthcoordinate s to the normal curvature k of that curve in the tangential direction e(see Kreyszig 1991). This equation applies to both the free surface (s, n, e) and thesurface-of-support (s1, n1, e1) and allows one to reduce (A 6b) and (A 6d) to

∂n∂s(e · δx) · n1 =−k (e · n1) (e · δx) , (A 8a)

∂n1

∂s1(e1 · δx) · n=−k (e1 · n) (e1 · δx) . (A 8b)

The vectors n, n1, e, e1 are coplanar and related by the following vector identities:

e · δx= (n · n1) e1 · δx= (cos α) e1 · δx, (A 9a)n · δx= (n · e1) e1 · δx+ (n · n1) n1 · δx, (A 9b)

e1 · n=−e · n1 = sin α. (A 9c)

Finally, one uses (A 8) and the vector identities (A 9) on (A 5) to generate thelinearized Young–Dupré equation,

∂y∂s+(

k cot α − ksin α

)y=−α. (A 10)

If the variation in contact angle α= 0, then the linear surface disturbance y preservesthe static contact angle α. For the sessile drop, k = sin α, k = 0, and the naturalboundary condition (3.8) follows directly. Similarly, application of contact-anglevariation (3.10) to (A 10) results in the contact-line speed condition (3.11).

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J. Fluid Mech. (2014), . 760, pp. …jbostwi/wp-content/uploads/...with hysteresis (solid) and without (dashed). Here, a and r are the advancing and receding static contact angles