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1

Transport of condensing droplets in Taylor-Green vortex flow in the presence of

thermal noise

Anu V. S. Nath∗ and Anubhab Roy†

Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036

Rama Govindarajan‡

International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089

S. Ravichandran§

Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden, SE-10691

We study the role of phase change and thermal noise in particle transport in turbulent flows.We employ a toy model to extract the main physics: condensing droplets are modelled as heavyparticles which grow in size, the ambient flow is modelled as a two-dimensional Taylor-Green (TG)flow consisting of an array of vortices delineated by separatrices, and thermal noise are modelledas uncorrelated Gaussian white noise. In general, heavy inertial particles are centrifuged out ofregions of high vorticity and into regions of high strain. In cellular flows, we find, in agreement withearlier results, that droplets with Stokes numbers smaller than a critical value, St < Stcr, remaintrapped in the vortices in which they are initialised, while larger droplets move ballistically awayfrom their initial positions by crossing separatrices. We independently vary the Peclet number Pecharacterising the amplitude of thermal noise and the condensation rate Π to study their effectson the critical Stokes number for droplet trapping, as well as on the final states of motion of thedroplets. We find that the imposition of thermal noise, or of a finite condensation rate, allowsdroplets of St < Stcr to leave their initial vortices. We find that the effects of thermal noise becomenegligible for growing droplets, and that growing droplets achieve ballistic motion when their Stokesnumbers become O(1). We also find an intermediate regime prior to attaining the ballistic state, inwhich droplets move diffusively away from their initial vortices in the presence of thermal noise.

I. INTRODUCTION

Fluid flows in which solid particles, liquid droplets orgas bubbles of a different material are suspended are therule rather than the exception in natural and industrialsettings [1]. The suspended entities could range froma few micrometres in size (water droplets in clouds) toseveral kilometres (asteroids in the interstellar medium).Such suspended ‘particles’ are advected by the flow, but,due to their finite size, do not necessarily follow fluidstreamlines. As a result, the dynamics of the suspended(‘inertial’) particles could be qualitatively different fromthat of the carrier fluid. For instance, particles muchdenser than the fluid (‘heavy’ inertial particles) are cen-trifuged out of vortical regions and cluster in strain-dominated regions in the flow [2]. Turbulent flows are,in general, chaotic tangles of vortex tubes and sheets [3],and heavy inertial particles suspended in turbulent floware known to cluster onto fractal attractors [4]. The clus-tering of heavy inertial particles in turbulent flow hasbeen studied theoretically, experimentally, and numeri-cally [5–14], and has been reviewed in, e.g., ref. [15].The dynamics of individual inertial particles can give

rise to multivalued particle velocities at a given spatial

∗ am18d701@smail.iitm.ac.in† anubhab@iitm.ac.in‡ rama@icts.res.in§ ravichandran@su.se

location and time, even when they are suspended in anincompressible fluid. These events give rise to folds inparticle-velocity space, commonly referred to as ‘caustics’[16–20]. Caustics lead to enhanced clustering, and areknown to lead to higher collision rates between inertialparticles [21, 22].

Numerical studies of turbulent suspensions model thesuspended phase either as a continuum (the Eulerian ap-proach) or as discrete units which need to be tracked in-dividually (the Lagrangian approach) [23]. These modelscan then be coupled to a suitable solver for the Navier-Stokes equations governing the dynamics of the fluid.

When the volume or mass fraction of the suspendedphase is sufficiently small, the feedback from the particleson the flow can be neglected. This allows the dynamics ofsuch particles to be studied separately from the flow, forexample, by using publicly available data sets of turbu-lent flow (like the Johns Hopkins Turbulence Database,JHTDB. See, e.g. [24] who study the orientation dynam-ics of asymmetric particles). However, when the volumefractions are not negligible, the feedback from the parti-cles on the flow cannot be neglected, and the dynamics ofthe flow and the suspended particles have to be studiedsimultaneously [25, 26].

When the suspended phase is made of liquid dropletsrather than solid particles, these droplets can also quali-tatively change the flow through the exchange of mass orenergy. For example, the evaporation of water dropletsformed by wave-breaking at the ocean surface generatescloud condensation nuclei [27], while the latent heat re-

2

lease accompanying the growth of water droplets drivesthe dynamics in clouds [28]. The fluid dynamics of respi-ratory events, relevant to the ongoing global pandemic,also involve evaporating droplets suspended in turbulentflows (e.g. [29–32]).

It is reasonable to assume that small suspendeddroplets are spherical. The mass transfer may be as-sumed to occur diffusively, leading to an analytical ex-pression for the rate of growth of the droplets (see [33] fora detailed derivation). Higher order corrections account-ing for advective mass transfer have also been proposedand used (e.g. [27, 29, 30, 34, 35]). These higher ordercorrections may be neglected when the droplets are suf-ficiently small. In addition, if the suspended droplets aresmall but larger than a critical radius called the Kohlerradius, effects due to the curvature and salt concentra-tion on the rate of growth may also be neglected, and thegrowth rate takes on a simple form ([36–38]; see sectionII).

In addition to the systematic forces described above,suspended particles may also experience stochastic forcesfrom collisions with fluid molecules due to thermalnoises/fluctuations. For sufficiently small particles, thisleads to the well known phenomenon of ‘Brownian mo-tion’, a diffusive motion. For larger particles, the effectsof the thermal noise are negligible. Studies of the dynam-ics of particles in the size range where both systematicinertial effects as well as the effects of stochastic forces arerelevant are relatively rare, and include studies in simpleshear flows ([39]), Taylor-Green vortices ([40, 41]), andturbulent flows ([42]).

Renaud & Vanneste [40] quantified the thermal dif-fusion of particles using an effective diffusivity Deff, forheavy and light inertial particles for various ranges ofthe Stokes number St and the Peclet number Pe. TheStokes number, St = τp/τf , where τp and τf are particlerelaxation time scale and flow time scale respectively, isa measure of the inertia of a particle. The Peclet numberis given by Pe = τd/τf , where τd = L2

f/DE is the dif-fusion time scale, Lf is the length scale of the flow andDE is the Einstein diffusivity of a particle due to thermalnoise (see section II), and is related to the strength of thethermal noise. Both St and Pe are increasing functionsof the particle size. Thus in flows where the suspendeddroplets grow or shrink due to phase change, St and Peare functions of time. The effects of thermal diffusionare important in the early evolution of droplets growingby condensation, and become negligible as the dropletsbecome sufficiently large.

Here, we study the combined effects of growth by con-densation and thermal diffusion on water droplets inclouds. We use an array of TG vortices as a ‘toy model’for the highly turbulent flow in clouds [43]. This ap-proach is in the same vein as studies using model flowsas proxy for turbulent environments (e.g. [17, 44] wherethe turbulence is modelled as a superposition of Fouriermodes).

The dispersion of inertial particles in cellular flows

has been studied without [45–50] and with gravity [51–53]. These studies find that, depending on their Stokesnumber and density ratio, inertial particles can displaychaotic dynamics even in non-chaotic flows. In fact,in time-periodic flows, even tracer particles can displaychaotic dynamics [54].Wang et al. [47] found that large−St inertial particles

suspended in a TG flow undergo periodic zig-zag motionalong open trajectories in the long-time limit. Here wecall this kind of motion ‘ballistic’ (see sections IV and V).In contrast, Renaud et al. [40] found that inertial parti-cles in TG flow with thermal noise behave diffusively atlong times, when the initial conditions are forgotten. Weexamine the competition between these two non-additiveeffects on droplets. We provide a supersaturated envi-ronment in which our droplets can condense, so both Stand Pe increases with time.The remainder of this paper is organised as follows. In

Sec. II we set down the general formulation used in thisstudy. We then revisit the dynamics of inertial particlesin the TG flow in Sec. III, and the role of thermal diffu-sion in inertial particle dynamics in Sec. IV. We examinethe effects of condensation growth of droplets on theirdispersion in Sec. V. We study the combined effects ofcondensation growth and thermal diffusion in Sec. VI.We conclude in Sec. VII.

II. PROBLEM FORMULATION

The motion of suspended droplets is governed by theexchange of momentum, mass and heat between thedroplets and the ambient fluid. Here, we model the mo-mentum transfer using the simple form of the Langevinequation,

dvp

dt=

u(xp)− vp

τp+

√2DE

τpη(t) , (1)

where vp is the Lagrangian velocity of the droplet, uis the ambient flow velocity at the droplet location xp

and η is the stochastic forcing due to thermal noise,whose form will be discussed later. The relaxation timeτp = 2 r2 ρ/(9µf), is the time scale on which the veloc-ity of the droplet relaxes to the fluid velocity, where ρand r are the instantaneous density and radius of thedroplet, and µf is the dynamic viscosity of the ambientfluid (air). The Einstein diffusivity (DE) depends on theinstantaneous size of droplets, and so its value evolvesover time. In Eq. (1), it is assumed that the dominantbalance in the droplet dynamics is between the acceler-ation of the droplet and the Stokes drag and stochasticforces on the droplet, and the effects of added mass, theSaffman lift force and the Basset history force are ne-glected (see [55]). This is justified in the heavy-particlelimit, i.e. when the density ratio of droplet to air islarge (ρ/ρf ∼ O(103) ≫ 1) (see [53]). We have also ne-glected gravity, hydrodynamic interactions and collisions

3

between the droplets. These may not in general be neg-ligible, but are fair assumptions on a horizontal plane,for particle sizes much smaller than flow length scales,and dilute suspensions respectively. Moreover, our focusis on the effects of condensation and thermal diffusion.We consider small droplets to begin with, which are inStokes flow relative to the ambient fluid, so we do nothave included drag corrections based on Reynolds num-ber. Additionally, we assume that the droplets remainspherical at all times and, therefore, that their angulardynamics need not be considered.In supersaturated ambients, suspended water droplets

grow by the diffusion of water molecules towards theirsurfaces, and their subsequent adsorption. An expressionfor the diffusive growth rate of water, accounting for theeffects of solutes present in the water droplet, as well asthe effects of the finite radius of curvature may be foundin [33]. For droplets that are sufficiently large (r > 5µm)for these effects to be neglected, the expression for thegrowth rate takes on a simple form (see, e.g., [36–38, 56]),viz

dr2

dt= 2 ξ . (2)

The parameter ξ is proportional to the vapour pres-sure difference between droplet surface and the ambient,which is assumed to be constant here. This is a fair as-sumption in a dilute suspension, since the ambient tem-perature and water vapour concentration will not changesignificantly due to condensation events. Eq. (2) maybe then integrated to obtain the instantaneous radius asr(t) =

√

r20 + 2 ξ t, where r0 = r(t = 0) is the initial ra-dius of the droplet, and we refer to this as the ‘parabolicgrowth model’. The parameter ξ can be written as ξ1 s,where s is the ambient supersaturation, and ξ1 is propor-tional to the mass transfer coefficient.Particles suspended in a quiescent ambient which is in

thermal equilibrium can nevertheless experience randomcollisions with molecules of the fluid, leading to stochas-tic motion of the particle. This was first observed byRobert Brown in 1827 for pollen grains in water. In1905, Albert Einstein used a molecular approach to de-rive an expression (called the Einstein-Smoluchowski re-lation) for the diffusivity (called the Einstein diffusiv-ity or the Brownian diffusivity), DE = kB T/(6 π µ a)of such particles, where kB is the Boltzmann constant,T is the temperature of the system at equilibrium, µ isthe dynamic viscosity of quiescent ambient fluid and a isthe radius of the spherical particles. Ornstein & Uhlen-beck [57] showed that by modelling the stochastic ther-mal noise (η) as a simple Gaussian ‘white-noise’, in thevanishing limit of particle inertia, Einstein diffusivity isrecovered. The white-noise is an uncorrelated randomsignal which has zero mean (〈η(t)〉 = 0) and an auto-correlation 〈ηi(t) ηj(t′)〉 = δij δ(t − t′), where δij is theKronecker delta, δ(.) is the Dirac delta function and 〈·〉represents the average over ensembles. The white noisecan be naively said to be the differential of a Wienerprocess (W).

Renaud & Vanneste [40] have used the white-noisemodel for particles suspended in a TG flow. They re-visited Childress’ classic calculation of O(Pe−1/2) effec-tive diffusivity of a passive scalar in a cellular flow withthe inclusion of particle inertia [58]. They showed that,in the St ≪ 1 limit, the effective diffusivity increases(decreases) for heavy (light) particles with increasing St.Here we follow the same approach to model the thermaldiffusion of droplets in a TG flow.The TG flow is a doubly periodic array of counter-

rotating cellular vortices. The stream function forthe TG flow with length scale Lf and velocity scaleVf is ψ = Vf Lf sin(x/Lf ) sin(y/Lf). The cor-responding nondimensional velocity field is u∗ =[sin(x∗) cos(y∗),− cos(x∗) sin(y∗)]. We use the flowlength scale (Lf ) and flow time scale (τf ) to nondimen-sionalise the Langevin equation Eq. (1) to get

St dv∗p = (u∗(x∗

p)− v∗p) dt

∗ +

√

2

PedW∗ ,with (3)

dx∗p = v∗

p dt∗,

where ‘∗’ indicates that the parameters are nondimen-sional. Hereafter, we only deal with nondimensionalquantities and drop the ‘∗’. The Stokes number isSt = τp/τf = 2 r2 ρ/(9µf τf ) and the Peclet number isPe = τd/τf = Vf Lf 6 π µf r/(kB T ). The first term onthe right hand side of Eq. (3) is the ‘drift term’ and thesecond one is the ‘diffusion term’. Note that St and Peare particular for each droplet, and for growing droplets,they increase with time. The parabolic growth modelgiven by Eq. (2) can be used to obtain their instanta-neous values as

St = St0 +Π t , (4)

Pe = Pe0√

1 + (Π/St0) t , (5)

where Π = τp/τc = 4 ρ ξ/(9µf) is the nondimensionaldroplet growth-rate, τc = r2/(2 ξ) is the condensa-tion time scale while St0 = 2 r20 ρ/(9µf τf ), Pe0 =Vf Lf 6 π µf r0/(kB T ) are the Stokes number and Pecletnumber based on initial droplet size.The temperature and pressure of the atmosphere at

the approximate height where cumulus clouds form areT ≈ 0◦ C and P ≈ 80 kPa. This yields ξ1 ≈ 68.2µm2/s[56] for water droplets. The typical supersaturation ina cloud is s ≈ 0.5%. Thus the estimated value of thegrowth rate Π for water droplets is around 1.4 × 10−5.Typical Kolmogorov scales for a cloud are Lη = 0.8mm,τη = 0.04s and Vη = 2cm/s [43]. Using these scales, theinitial Stokes number and Peclet number for 5µm waterdroplets are St0 ≈ 8.15×10−3 and Pe0 ≈ 6.84×106. Westudy the dynamics for wider ranges of St0, P e0 and Πthan are typical in clouds, in order to better understandthe effects of particle inertia, diffusion and growth.To study the dynamics of the droplets, we integrate

Eqs. (3) in time for each droplet. Since the droplets areinitially micron sized, they have a small Stokes number

4

at initial times. Eq. (3) is singular in the limit of St≪ 1and its overdamped form [40],

dxp =

(

u− StDu

Dt

)

dt+

√

2

PedW , (6)

where D/Dt = ∂/∂t+u·∇ represents the material deriva-tive, may be used instead. Eq. (6) is valid in the limitof St ≪ 1, Pe ≫ 1 and St · Pe = O(1). While we referto Eq. (6) to aid in understanding, the results presentedhere are obtained by integrating Eqs. (3) directly withsmall enough time steps.We study numerically the dispersion and clustering of

identical droplets randomly distributed over a selectedregion of the TG flow. As time progresses, the dropletsget advected and diffused by the flow and the thermalnoise respectively, during which they may also grow insize by condensation. The instantaneous Stokes num-ber and Peclet number are calculated as per Eqs. (4)and (5). A fourth-order Runge-Kutta scheme (RK4) isused to integrate the deterministic cases (Pe−1 = 0) ofEqs. (3), while the Euler–Maruyama method is used tointegrate the stochastic cases (Pe−1 6= 0) of Eqs. (3). Thetime step for integration, dt ≤ 0.1min(St, St/Π, P e), isa small fraction of the relevant time scales in the prob-lem. We validate our numerical scheme by comparingour results with those of [40] (see Fig. 6).The statistics of the distribution of droplets is anal-

ysed using the time evolution of the mean-square-displacement (hereafter referred to as MSD) plots. TheMSD is the ensemble average of the mean square distanceeach droplet covered from its respective initial location

σ2(t) = 〈 1

N

N∑

i=1

||xi(t)− xi(0)||2 〉 . (7)

Here, the angle brackets (〈·〉) represent an averageover many realisations of the initial distribution ofdroplets/many realisations of the thermal noise and weuse the symbol σ2 to represent the MSD. The nature ofthe MSD versus time curve can reveal the behaviour ofthe collective motion of particles/droplets: MSD curvesproportional to t2 indicate ballistic motion, while a con-stant MSD indicates that the droplets have attainedsteady states, i.e., they are all pinned at different sad-dle (stagnation) points, approaching them along the at-tractive manifolds, as discussed below. A measure ofparticles’ ballistic velocity can be calculated from theexpression d

√

σ2(t)/dt in the ballistic regime. How-ever, if the MSD is proportional to t, then the parti-cles/droplets are in diffusive motion with an effective dif-

fusivity Deff = 14dσ2(t)

dt (for two-dimensional flows).

III. REVISITING THE ROLE OF PARTICLEINERTIA (St 6= 0, Π = 0, Pe−1 = 0)

Previous studies of the dispersion of finite density iner-tial particles in TG flow find that the particle trajectories

FIG. 1. Dispersion of 103 identical inertial particles in a TGflow. The Stokes number is set at the beginning of each sim-ulation (a) A representative initial random distribution ofparticles within the cell 0 < x < π and 0 < y < π, withvx = vy = 0 at t = 0. (b) St = 0.1 particles at t = 100, (c)St = 0.5 particles at t = 100 and (d) St = 1.15 particles att = 100. Note that the axes in (b-d) have different scales.

can be periodic or chaotic depending on the values of Stand ρ/ρf [46, 47, 51, 52]. Here we revisit the problemof dispersion of heavy inertial particles in the TG flow(following [47] but considering the heavy particle limit).In the absence of thermal noise, and in the limit of largeρ/ρf but finite St, Eqs. (3) simplify to

dx

dt= vx ,

dvxdt

=−vx + sin(x) cos(y)

St, (8a)

dy

dt= vy ,

dvydt

=−vy − sin(y) cos(x)

St. (8b)

Particles are initially distributed randomly within oneTG vortex cell (0 < x < π, 0 < y < π) as shown inFig. 1(a). The initial velocity of the particles is eitherset to zero or set to the local fluid velocity, with similarresults. All particles with St > 0 are centrifuged awayfrom the vortex centre at (π/2, π/2) and spiral outwards.Whether these particles remain within, or leave, the celldepends on St. In the long-time limit, particles withSt < Stcr remain within the area bounded by the theseparatrices x = 0, x = π, y = 0 and y = π (see Fig. 1(b))whereas particles with St > Stcr leave the cell. We alsosee that particles with St < Stcr are ultimately absorbedby the stagnation points (hereafter referred to as SPs) atthe corners of the cell (at later times than those shownhere).The critical Stokes number Stcr = 1/4 is identified by

plotting the fraction of particles that exit the initial cell(see Fig. 2). This critical St has been previously reported

5

0.1 0.2 0.25 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

FIG. 2. The fraction of particles that leave the TG vortex cellin which they start, plotted for various times as a function ofthe particle Stokes number St. Continuous lines representparticles starting with zero initial velocity, and dashed onesrepresent particles starting with the local fluid velocity. Thecurves are plotted using an ensemble average over 500 simu-lations with Ntotal = 103 particles each. The critical Stokesnumber Stcr is known to be 1/4.

by [59, 60] (Stcr was 1/(8 π) in their analysis due to a dif-ferent choice of scaling). We calculate the leak fractionas the fraction of particles that cross the separatrices ofthe initial cell. Some of the particles that leave the ini-tial cell, we note, may eventually be captured by SPsother than those of the initial cell (see Fig. 1(c)). How-ever, when St & 0.77, a fraction of the particles moveoutwards forever with a mean direction parallel to thediagonals of the initial cell, continually crossing TG vor-tex cells (Fig. 1(d)), and exhibiting periodic motion onopen zig-zag trajectories. Similar ‘ballistic’ motion inwhich the MSD scales quadratically with time has previ-ously has been observed for inertial particles with finitedensity ratios in TG flow [45, 47].In [59], St = 1/4 was identified as the critical Stokes

number of escape across the separatrices. Below we uselinear stability analysis at the SPs to describe the changein behaviour across this Stokes number, and to explainthe leakage of particles to neighbouring cells for St > 1/4.

A. Stability properties of inertial particles in TGflow

Eqs. (8) constitute a dynamical system with fourvariables (x, y, vx, vy). The fixed points of the systemare vortex centers ((n + 1/2)π, (m + 1/2)π, 0, 0) andSPs (nπ,mπ, 0, 0) where n,m ∈ Z. The system islinearised about the fixed points, with perturbations(x′, y′, v′x, v

′y) = (x, y, vx, vy) e

λ t and solved for the eigen-values λ to obtain exponential stability characteristics,where (x, y, vx, vy) are perturbation amplitudes. At the

vortex centres, the eigenvalues all have positive real partsand the vortex centres behave as unstable spirals for anyfinite St particle, explaining why particles are centrifugedaway from the centre (π/2, π/2) in the simulations.

The behaviour at the SPs is more complicated andthe eigenvalues are plotted as a function of the Stokesnumber in Fig. 3. For St < 1/4, all the eigenvalues arepurely real, and one of them is positive. Such a fixedpoint is termed a ‘3 : 1 saddle’ [61]. For St > 1/4, twoof the eigenvalues become complex conjugates, while thepositive eigenvalue remains positive; the fixed point isthus a ‘spiral-3 : 1 saddle’ [61]. The change in the four-dimensional phase space behaviour is best shown in thetwo-dimensional projections in Fig. 4. The trajectories ofparticles of St < 1/4, asymptote to the separatrices anddo not cross them (see Fig. 4(a)), whereas particles ofSt > 1/4 can cross separatrices. In phase space the lattersupport spiral trajectories at the Stokes number shown,which is consistent with the ability to cross separatricesin finite time (see Fig. 4(b)).

The existence of a positive eigenvalue indicates thatthe SPs are linearly unstable fixed points for inertial par-ticles of finite St. However, the phase-space behaviour ofthese unstable fixed points changes when the St exceeds1/4, and the phase space trajectories attain a spiral na-ture as well. While this change to unstable spiral-saddlebehaviour does not explain why particles with St < 1/4remain inside the initial cell (see Fig. 2), we expect thatthe agreement between the Stcr found numerically andfrom the linear stability analysis here is not simply coin-cidental. In fact the connection can be clearly explained,as done in the following subsection.

-4

-3

-2

-1

0

1

0 0.25 0.5 1 1.5

-1

0

1

FIG. 3. The (a) real and (b) imaginary parts of the four eigen-values (λ) of the linearised dynamics at a stagnation point,plotted as a function of the particle Stokes number St. Thedotted lines represent asymptotes to eigenvalues for small Stobtained from the slow manifold approximation, Eq. (6).

6

FIG. 4. The projections of phase-space trajectories onto x−y(a,b) and y − vy (c,d) planes near an SP, here placed at theorigin, for particles with St = 0.1 (a,c) and St = 0.5 (b,d).Here n + m is even, the axes represent separatrices in theflow, and each curve shows the trajectory of a particle. Thespiralling of trajectories in the phase plane in (d) is a signatureof the crossing of the flow separatrices by the particles.

B. The threshold to cross a separatrix

Numerical simulations of Eqs. (8) reveal that particleswith St > 1/4 always cross the separatrices in the vicinityof one of the SPs. Therefore, we linearise Eqs. (8) at ageneral SP (nπ,mπ, 0, 0) where n,m ∈ Z, to get

dx′

dt= v′x ,

dv′xdt

=−v′x + (−1)n+m x′

St, (9a)

dy′

dt= v′y ,

dv′ydt

=−v′y − (−1)n+m y′

St. (9b)

where (x′, y′, v′x, v′y) are perturbation quantities. Since

the x and y equations are decoupled, we can combinethem and rewrite Eqs. (9) as follows

Std2x′

dt2+

dx′

dt− (−1)n+m x′ = 0 , (10)

Std2y′

dt2+

dy′

dt+ (−1)n+m y′ = 0 , (11)

which are the equations for damped harmonic oscillatorswith two degrees of freedom. The two oscillators haveopposite stability, since they have oppositely signed stiff-ness coefficients (i.e., if n + m is an even integer, thenthe x−oscillator is unstable while the y−oscillator is sta-ble, and vice-versa). Here, without loss of generality, we

consider SPs with even n+m to explain things unless oth-erwise specified. The behaviour at SPs with odd n +mare obtained by exchanging x and y.Eq. (11), therefore, represents a damped harmonic os-

cillator in the y direction, with a positive stiffness coef-ficient. The damping coefficient for the system is 1, andthe critical damping factor is 2

√St, giving a damping

ratio of 1/√4St. Therefore, the system is overdamped

for St < 1/4, and underdamped for St > 1/4. The oscil-lations in y′ for St > 1/4 are about the horizontal sepa-ratrix connected to the SP, and thus the particle crossesthe separatrix in the y direction near the SP. For SPswith n+m is odd, the identical argument reads: oscilla-tions in x′ for St > 1/4 about vertical separatrices makesthe particle to cross separatrices in x direction near SPs.Since this argument is true at all SPs, we conclude thatparticles can only cross the separatrices if St > 1/4.Eqs. (10) and (11) are, in fact, exactly solvable. For

an initial condition (x′0, y′0, v

′x0, v

′y0), and n+m even, the

exact solutions are

x′ = C1 e(α−1) t

2St + C2 e−(α+1) t

2 St , (12)

y′ = C3 e(β−1) t

2St + C4 e−(β+1) t

2St , (13)

where α =√1 + 4St, β =

√1− 4St, C1 = x′0 (1 +

α)/(2α) + v′x0 St/α, C2 = x′0 (−1 + α)/(2α)− v′x0 St/α,C3 = y′0 (1 + β)/(2 β) + v′y0 St/β and C4 = y′0 (−1 +β)/(2 β)− v′y0 St/β.From these solutions, it can be seen that the nature of

the system changes at Stcr = 1/4. Furthermore, the timetaken by a particle with St > 1/4 to cross the horizontalseparatrix y = mπ and escape the cell (escape time, tesc)is the smallest of the solution for y′(tesc) = 0, and is

tesc ∼2St√4St− 1

π − tan−1

√4St− 1

1 + 2Stv′

y0

y′

0

.(14)

By this time, the particle could typically be sufficientlyfar away from the SP in the x direction so that the lin-earised system no longer governs further dynamics. Thus,Eq. (14) would be only a rough estimate for the escapetime of the particles in the y direction across horizontalseparatrices.From the exact solution Eq. (12), we also see that par-

ticles with a sufficiently large initial velocity, directedspecifically, can cross the vertical separatrices as well.The magnitude of critical velocity can be obtained fromEq. (12) as |v′x0| > vcr = 2 |x′0|/(−1 +

√1 + 4St), and

should be directed towards the vertical separatrix x =nπ. For St & 1/4, the particles can usually have thatmuch velocity; however, that will not be directed towardsthe vertical separatrix, instead directed away from it nearany SP, due to the centrifuging effect of the vortex. Ex-tra forces in the system like gravity, acting towards thevertical separatrix could activate this criterion. Thus, itis not relevant in explaining the leakage of particles withSt & 1/4 from the initial TG vortex cell in the presentsystem.

7

-5 0 5

10 -2

-5

-3

-1

0

110 -2

FIG. 5. The trajectories of inertial particles with differentStokes numbers St and different initial velocities vx (con-tinuous lines) starting from the same location near the SPat the origin in a TG flow represented by the black dot(−0.01,−0.05). These trajectories could cross the horizontaland vertical separatrices depending on the initial conditions.Faded purple dashed or dotted lines represent trajectories ofSt = 0.1 particles with zero initial velocity, for different re-alisations of thermal noise (Pe = 104), and show that evenSt < 1/4 particles could cross the separatrices for individualrealisations of the noise. The thin dashed grey lines are thestreamlines of the flow.

In Fig. 5, we plot the trajectories of inertial particlesstarting near an SP placed at the origin (n+m = 0). Theaxes coincide with separatrices. When the initial velocityis large, the trajectories cross the x = 0 separatrix; whenthe initial Stokes number is large, the trajectories crossthe y = 0 separatrix; when both the initial velocity andthe Stokes number are large, trajectories cross both thex = 0 and the y = 0 separatrices. Examples are shownin the figure.

IV. THE ROLE OF THERMAL NOISE(St 6= 0, Π = 0, Pe−1 6= 0)

In the weak molecular diffusion limit (Pe≫ 1), the ef-fective diffusivity of passive scalars crucially depends onthe flow topology. In shear flows with open streamlinesDeff ∼ Pe - the classical Taylor-Aris dispersion [62–64].For cellular flows, molecular diffusion becomes dominantin a thin boundary layer near the separatrices, assistingmigration across cells, leading to Deff ∼ Pe−1/2 [58]. Tounderstand the enhanced transport due to convection inthe above two scenarios, one should recall that the diffu-sivity in the absence of flow is D ∼ Pe−1. The effectivediffusivity Deff of inertial particles in a TG flow, with theasumption of St ≪ 1, Pe ≫ 1 and StPe = O(1), wasrecently calculated by [40]. For St = 0.1, our simulatedresults find excellent agreement with theirs, as shown inFig. 6. At higher values of St, however, the expressionof [40] is no longer accurate. As St increases, we findthat the effective diffusivity acquires a non monotonic

variation with Pe, and decreases rapidly for large Pe,in qualitative departure from the distinguished limit ofStPe = O(1). The diffusion of inertial particles in pe-riodic, shear, and elongational flows are studied in [65–68]. The study by Rubi & Bedeaux [66] on elongationalflows is of particular interest since, near the SPs, TGflow resembles elongational flow. The linearised govern-ing equations near an SP (nπ,mπ, 0, 0) when Pe−1 > 0read

Std2x′

dt2+

dx′

dt− (−1)n+m x′ =

√

2

Peηx(t) , (15)

Std2y′

dt2+

dy′

dt+ (−1)n+m y′ =

√

2

Peηy(t) . (16)

The MSD of a particle near the elongational flow can becalculated as (for n+m even)

〈x′2〉 = C21 e

(α−1) t

St + 2C1C2 e−tSt + C2

2 e−(α+1) t

St

1

α2 Pe

{

e−tSt

[

cosh

(

α t

St

)

+ α sinh

(

α t

St

)]

−1− 4St (1− e−tSt )

}

, (17)

〈y′2〉 = C23 e

(β−1) t

St + 2C3 C4 e−tSt + C2

4 e−(β+1) t

St

1

β2 Pe

{

− e−tSt

[

cosh

(

β t

St

)

+ β sinh

(

β t

St

)]

+1− 4St (1− e−tSt )

}

, (18)

where α, β and the constants C1, C2, C3 and C4 are asdefined in section III. Eqs. (17) and (18) generalise theexpressions in [66] to arbitrary initial conditions andSt. At short times, the MSD scales as 〈x′2 + y′2〉 ∼4 t3/(3PeSt2). Thus, when Pe−1 > 0, particles of anySt can cross separatrices and escape the initial cell. This

10 1 10 2 10 3 10 4

0.3

0.4

0.5

0.6

0.7

0.8

FIG. 6. Effective diffusivity for heavy particles in TG flowagainst Peclet number Pe at various Stokes numbers. Ourresults (solid lines with markers) are compared with the ex-pression of Renaud & Vanneste [40] (dashed lines).

8

tendency increases for stronger noise (smaller Pe). Fig. 5shows sample trajectories of particles with Pe = 104 andzero initial velocity. For the same initial condition, in-dividual realisations of the thermal noise may lead totrajectories crossing the separatrices.

V. THE ROLE OF CONDENSATION(St 6= 0, Π 6= 0, Pe−1 = 0)

We next study the dynamics of growing (by conden-sation) droplets in the TG flow without thermal noise.Droplets of initial Stokes number St0 = 0.1 and zero ini-tial velocity are distributed randomly in a square patchas shown in Fig. 1(a), and allowed to grow with a growth-rate Π = 10−2. We note that a different choice of initialvelocity (the local fluid velocity) does not change the dy-namics qualitatively. We solve Eqs. (3) with Pe−1 ≡ 0.The advective motion by the flow dominates the ini-tial dynamics of the droplets, where the droplets arethrown out of the vortex center (π/2, π/2). As timeprogresses, the instantaneous Stokes number of droplets,St(t) = St0 +Π t exceeds 1/4, allowing droplets to crossthe separatrices and spread in a manner qualitativelysimilar to that seen in Fig. 1(c). Unlike fixed St par-ticles, all continuously growing droplets eventually en-ter the ballistic regime of motion (qualitatively as willbe seen in Fig. 9(d)). In this phase, the droplets areobserved to travel along 45◦ − 135◦ lines in a zig-zagmanner, which is similar to the open-trajectory periodicmotion identified in Wang et al. [47]. However, for suffi-ciently small growth rates (Π . 0.005), our simulationsshow that growing droplets get trapped at the stagnationpoints instead of attaining ballistic velocities. Once thesedroplets are trapped (to within numerical precision) atthe SPs, their velocities remain zero despite their contin-uous growth in size.We plot the MSD of the droplets, ensemble-averaged

over 103 realisations of initial distributions of 103 par-ticles each, for an initial Stokes number St = 10−2,three different values of growth-rates (Π = 10−3, 10−2

and 10−1) in Fig. 7 (Pe0 = ∞ case). Initially, i.e., fort ≪ 1, the MSD grows independent of Π. In the in-termediate phase (t ∼ O(10) − O(103)), the MSD hasa clear dependence on Π. The curves in this phase dis-play waviness, which could be caused by droplets hop-ping back and forth between neighbouring cells. At largetimes (t > O(103)), the MSD scales as t2/2 and dropletsenter the ballistic regime. The time at which the dynam-ics becomes ballistic is approximately the same time atwhich St(t) ∼ O(1). This time decreases for larger Πas t ∼ 1/Π, as shown in the inset of Fig. 7. The scal-ing Π t was obtained empirically. For the lowest growthrate Π = 10−3 < 0.005 shown in the figure, the MSDis saturated at large times, indicating that the dropletsare trapped at the SPs of the flow even though they arecontinuously growing.For other values of initial Stokes number St0 (not

shown), the MSD plot is qualitatively the same as shownin Fig. 7. We also see that for sufficiently large Π andlong times, the numerical value of the MSD becomes in-dependent of St0 and Π. Empirically we obtained thatthe asymptotic fit in this phase is σ2(t) ∼ t2/2, indicatingthat the measure of the nondimensional ballistic velocityof droplets in this phase asymptotically reaches the value1/

√2 for large St (see Sec. II).

The evolution of the leakage fraction of particlesagainst their initial Stokes number St0 is plotted inFig. 8. Unlike in Fig. 2, there is no critical initial Stokesnumber St0 for growing particles, as one would intuitivelyexpect. In the following subsection, we examine the rea-son using a local analysis near SPs. Also, we obtain ananalytical expression for the escape time of droplets froma vortex cell.

A. Local analysis near SPs

We observe from the numerical simulations that con-densing droplets, like constant-size particles, also crossthe separatrices near SPs. The linearized perturba-tion equations (see section III B) for a droplet with aStokes number St1 > St0 (allowing for droplet growthby the time it reaches the vicinity of an SP) near the SP(nπ,mπ, 0, 0) read

(St1 +Π t)d2x′

dt2+

dx′

dt− (−1)n+m x′ = 0 , (19)

(St1 +Π t)d2y′

dt2+

dy′

dt+ (−1)n+m y′ = 0 . (20)

The system is thus comprised of damped harmonic oscil-lators with two degrees of freedom, but with increasing

FIG. 7. MSD for St0 = 10−2 droplets with (Pe0 = 103)and without (Pe0 = ∞) thermal noise, for different growthrates, plotted against time. The asymptote for the long timeballistic motion is shown in green. The circle within the insetindicates that the switch to the ballistic regime occurs at ascaled time Π t ∼ O(1).

9

mass. Using the transformation t1 = t + St1/Π, theseequations for SPs with even n+m can be written as

Π t1d2x′

dt21+

dx′

dt1− x′ = 0 , (21)

Π t1d2y′

dt21+

dy′

dt1+ y′ = 0 . (22)

The general solutions are in terms of Bessel functions,

x′ = t− γ

21

{

C5 Iγ

(

2

√

t1Π

)

+ C6 I−γ

(

2

√

t1Π

)}

,(23)

y′ = t− γ

21

{

C7 Jγ

(

2

√

t1Π

)

+ C8 J−γ

(

2

√

t1Π

)}

,(24)

where γ = −1 + Π−1. Since these expressions are notparticularly helpful in the limit (Π → 0), because of thesingular nature of arguments of Bessel functions, we usethe Wentzel–Kramers–Brillouin (WKB) method to ob-tain the asymptotic solutions for Π → 0 (see AppendixA)

x′ ∼ t−12π

1 {C9 exp(−χ+) + C10 exp(χ+)}4

√

Π(

1t1

− 12 t21

)

+ 14 t21

, (25)

y′ ∼ t−12 π

1 {C11 sinχ− + C12 cosχ−}4

√

Π(

1t1

+ 12 t21

)

− 14 t21

, (26)

where

χ± =1

Π

∫ t1

St1Π

√

Π

(

1

τ∓ 1

2 τ2

)

± 1

4 τ2dτ . (27)

0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

FIG. 8. The fraction of condensing droplets (Π = 10−2) thatexit their initial cell is plotted against St0 for various simu-lation times. Continuous lines indicate the case of particleswith zero initial velocity, and dashed ones represent particlesinitially at the local fluid velocity. Simulations are performedwith Ntotal = 103 particles over 500 realisations.

The lower limit of the integral is taken as the value oft1 corresponding to t = 0, i.e. St1/Π. The constantsC5...C12 depend on the initial conditions of the pertur-bation (x′0, y

′0, v

′x0, v

′y0). The form of the asymptotic so-

lution Eq. (26) implies that there exists a critical Stokesnumber (1 − 2Π)/4, a modification to Stcr = 1/4 offixed St particles. When St1 > (1 − 2Π)/4, the be-haviour of y′ would be oscillatory, similar to the case ofa fixed St particle, but the time-variation in St is ac-counted for here. In contrast, when St1 < (1 − 2Π)/4the scenario is different from that for constant St. Ata time tTP = (1 − 2Π − 4St1)/(4Π) there is a ‘turningpoint’ by WKB analysis (see Appendix A), close to whichthe WKB solution Eq. (26) will not be valid. However,when t ≫ tTP, this oscilatory solution will be valid evenfor St1 < (1 − 2Π)/4, which could allow the droplet tocross the separatrices. Again the time taken to cross ahorizontal separatrix y = mπ would be the smallest ofthe solution of y′(tesc) = 0. Using Eq. (24), the actualestimate would be the solution tesc of the following tran-scendental equation

Jγ

(

2√Π

√

tesc +St1Π

)

J−γ

(

2√Π

√

tesc +St1Π

) =

J1+γ

(

2√St1Π

)

+v′

y0

y′

0

√St1 Jγ

(

2√St1Π

)

−J−1−γ

(

2√St1Π

)

+v′

y0

y′

0

√St1 J−γ

(

2√St1Π

) . (28)

An approximate estimate can be obtained using theWKB solution Eq. (26) as tesc ∼ tTP + (9 π/(32

√Π))2/3

when St1 < (1−2Π)/4 (see Appendix B), indicating thatas Π decreases, the escape time increases. A more accu-rate expression using WKB is given in the Appendix B,both for St1 < (1 − 2Π)/4 and St1 > (1 − 2Π)/4 cases.As we mentioned earlier, for the case of non-condensing

particles, this exit time is a rough estimate from the lin-ear theory. By this time, the droplet could be sufficientlyaway from SP so that nonlinear effects could alter thisexit time.

VI. COMBINED EFFECTS OFCONDENSATION AND THERMAL NOISE

(St 6= 0, Π 6= 0, Pe−1 6= 0)

We now study the dynamics of condensing droplets inTG flow with thermal noise by solving the full stochas-tic Langevin equation Eq. (3). Since the droplets arecondensing, both St and Pe increases with time as perEqs. (4) and (5). The strength of the thermal noise is in-versely proportional to the Pe. Thus, as time progresses,the influence of thermal noise becomes weaker. Diffusivebehaviour takes a long time to be set up even for dropletswhich are not growing, i.e., when the thermal noise isnot decreasing in strength with time. It would take evenlonger for growing droplets. Similarly, ballistic motion

10

FIG. 9. Dispersion of identical condensing droplets (N =103, St0 = 0.1, Π = 10−2) in TG flow with thermal noise(Pe0 = 103) at (a) t = 0, (b) t = 15, (c) t = 150 and (d)t = 300. Initial velocity of all the droplets are chosen to bezero.

would take less time to be set up for growing droplets.We therefore expect that at long times, ballistic dynamicswill be predominant. The intermediate time behaviour,where the effects of advection and thermal diffusion maybe in competition, is not easy to anticipate.We numerically study the dynamics of initially identi-

cal droplets (St0 = 0.1 and Pe0 = 103, v = 0 distributedrandomly over the TG vortex cell, as shown in Fig. 9(a).The growth rate is chosen as Π = 10−2. For short times,advection and stochastic forcing together make the parti-cles to cross separatrices despite having Stokes numbersSt < 1/4 (see Fig. 9(b)). For larger times, the combinedeffects of condensation growth and thermal noise lead tothe greater diffusion than with condensation alone (seeFig. 9(c)). As expected, in large time limit, the dropletsmove ballistically along 45◦ − 135◦ paths (see Fig. 9(d)).The MSD, ensemble-averaged over 103 realisations

with 103 droplets each with St0 = 10−2 and Pe0 = 103, isplotted in Fig. 7 for three different values of the growthrate Π. We see that thermal noise (finite Pe) signifi-cantly alters the dynamics for small droplet growth ratesΠ = 10−3, leading to ballistic motion instead of dropletstrapped at SPs.In Fig. 10, we plot the MSD for different Pe0 but the

same St0 and Π, showing that the intermediate phasebecomes smoother for smaller Pe0 (and thus larger ther-mal noise). Furthermore, the intermediate regime scaleslinearly with t indicating that the behaviour is diffusive.As Pe0 decreases, this diffusive regime becomes wider.Lastly, we study the behaviour of droplets with param-

eter values representative of atmospheric clouds: St0 =

10 -1 10 0 10 1 10 2 10 3 10 4

10 -2

10 0

10 2

10 4

10 6

10 8

FIG. 10. MSD for droplets with St0 = 10−2 and Π = 10−3, fordifferent Pe0. The O(t) asymptote, indicative of a diffusiveregime, is shown in magenta.

8.15×10−3, Pe0 = 6.84×106, Π = 1.4×10−5 (see Sec. II).The MSD, ensemble-averaged over 100 realisations of 104

particles each, is plotted in Fig. 11. We note that fewerrealisations were possible due to the extremely long timesfor which these simulations need to be run. For compar-ison, the MSD curves corresponding to the special casesstudied in sections III–V are also plotted in Fig. 11. It isobserved that after 106 nondimensional time (11 hours),the enhancement in MSD is of O(1010) by the inclusionof both condensation and thermal noise. We observe thatthe MSD for Pe0 = ∞ cases is independent of whethercondensation occurs, and Π > 0 (still Π < 0.005) leads to

10 -2 10 0 10 2 10 4 10 6

10 -4

10 -2

10 0

10 2

10 4

10 6

10 8

10 10

10 12

FIG. 11. MSD versus time for 5µm droplets with initialStokes number St0 = 8.15 × 10−3, realistic Π and Pe0compared with non-condensing and non-diffusing droplets.Asymptotes for ballistic and diffusive behaviour are shown.For Π = 0 and finite Pe0, the diffusivity Deff = 0.05786matches that calculated from the expression of [40]. Growingdroplets subject to thermal noise transition from diffusive toballistic behaviour at t ∼ 1/Π.

11

a constant MSD indicative of a steady state. Here, thissteady state is achieved due to the capture of droplets atthe SPs of the TG flow, despite the fact that the dropletStokes numbers have increased to St = O(10) by t = 106

(cf. the discussion of figure 7).

With nonzero thermal noise, droplets behave diffu-sively both for Π = 0 and Π > 0, although growingdroplets show departures from the asymptote (4Deff t),eventually transitioning to ballistic motion at t ∼ 1/Π,as expected from section V. The combined effects of con-densation and thermal noise are reflected in the greaterdiffusion at intermediate times.

VII. CONCLUSION

We studied the effects of thermal noise and conden-sation on the dispersion of monodisperse droplets sus-pended in a Taylor-Green vortex flow. In the absence ofthermal noise and condensation, we found, in agreementwith [59], that droplets with St < 1/4 remain trappedin their initial vortices. We showed that the additionof either thermal noise or a finite condensation rate re-moves this condition, increasing droplet dispersion by or-ders of magnitude. We showed that droplets growingby condensation typically attain a state of ballistic mo-tion away from their initial vortices for times t & Π−1,travelling along 45◦ diagonal trajectories with averagenondimensional velocities of 1/

√2, but that sufficiently

small growth rates Π . 0.005 allow droplets to remaintrapped at the stagnation points of the flow. We showedthat, in the presence of thermal noise, this transitionfrom the trapped state to the ballistic state occurs pro-ceeds through an intermediate diffusive regime where themean squared displacement of the droplets grows linearas σ2 ∼ t. Our results with this model flow are encour-aging, and suggest further studies where the effects ofpolydispersity and collision-coalescence of droplets, theirgravitational settling, and the effects of latent heatingand buoyancy are included.

ACKNOWLEDGMENTS

SR is supported through Swedish Research Councilgrant no. 638-2013-9243. RG acknowledges support ofthe Department of Atomic Energy, Government of India,under project no. RTI4001. AVSN would like to thankthe Prime Minister’s Research Fellows (PMRF) scheme,Ministry of Education, Government of India. AR andAVSN would like to acknowledge the support from Lab-oratory for Atmospheric and Climate Sciences, IndianInstitute of Technology Madras.

Appendix A: WKB analysis for condensing dropletsnear SPs

By eliminating first order terms, Eqs. (21) and (22)can be rewritten as,

Π2 d2x′′

dt21+

{

−Π

t1+

(

Π− 1

2

)

1

2 t21

}

x′′ = 0 , (A1)

Π2 d2y′′

dt21+

{

Π

t1+

(

Π− 1

2

)

1

2 t21

}

y′′ = 0 , (A2)

where x′′ = x′ t1/(2 Π)1 and y′′ = y′ t

1/(2 Π)1 . These equa-

tions resemble the form of differential equations

Π2 d2φ

dt2+ q(t)φ = 0 , (A3)

which can be asymptotically solved using the WKBmethod [69]. For Π → 0, the asymptotic solution is,

φ(t) ∼ 1

q(τ)1/4(A sin θ +B cos θ) , (A4)

where θ = 1Π

∫ t√q(τ) dτ and A,B are constants to be

determined using initial/boundary conditions. The so-lution can be sinusoidal or exponential type dependingon the nature of the ‘potential’ q(t). By substituting re-spective q(t) terms from Eqs. (A1) and (A2) in Eq. (A3)and rearranging, x′(t) and y′(t) can be obtained as inEqs. (25) and (26) respectively.The asymptotic expression Eq. (A3) is valid only away

from the ‘turning point’ (tTP ) at which q(tTP ) = 0.

Thus, from Eq. (A2),{

Πt1

+(

Π− 12

)

12 t21

}

= 0 has a so-

lution at t1 = (1− 2Π)/(4Π), indicates that there existsa turning point time tTP = (1 − 2Π− 4St1)/(4Π) nearwhich the oscillatory solution Eq. (26) is not valid (re-member t1 = t + St1/Π). Neverthless, away from thisturning point time, Eq. (26) will be a good approxima-tion. Thus y′(tesc) ∼ 0 can be asymptotically solved toget escape time estimate when Π → 0 for a SP with n+mis even.

Appendix B: Escape time for condensing dropletsestimated using WKB when Π → 0

To calculate escape time, here we solve y′(tesc) ∼ 0.

1. St1 > (1− 2Π)/4

In this situation, 1 − 2Π − 4St1 < 0 indicates thattTP < 0, i.e. the turning point does not exists in positivetime, thus the solution Eq. (26) is valid in all t > 0. χ−can be evaluated by performing the integral Eq. (27) asχ− = F (t+ |tTP |)−F (|tTP |), where |tTP | = (−1+2Π+

12

4St1)/(4Π) > 0 and

F (τ) =

√1− 2Π

Π

{

2√Π τ√

1− 2Π− tan−1

[

2√Π τ√

1− 2Π

]}

.

(B1)

For the initial position y′0 and initial velocity v′y0, theconstants C11 and C12 can be evaluated as,

C11 =C12

G+(|tTP |), (B2)

C12 = y′0

(

St1Π

)γ/2

Π1/4 |tTP |1/4 , (B3)

where (remember γ = −1 + Π−1)

G±(τ) =4√Π τ

4St1v′

y0

y′

0± St1

τ + 2 (1−Π). (B4)

From Eq. (26), y′(tesc) = 0 thus leads to C11 sinχ− +C12 cosχ− = 0, can be simplified to

tan {F (tesc + |tTP|)− F (|tTP|)} +G+(|tTP|) = 0 .(B5)

The solution tesc of this transcendental expression givesthe asymptotic estimate for escape time when Π → 0 andSt1 > (1− 2Π)/4.

2. St1 < (1− 2Π)/4

In this situation, 1 − 2Π − 4St1 > 0 indicates thattTP = (1−2Π−4St1)/(4Π) > 0, i.e. there exists a turn-ing point in positive time, near by which the oscillatorysolution Eq. (26) is not valid. However, far in time fromthe turning point (i.e. t >> tTP or t << tTP ), the solu-tion Eq. (26) will be valid. From Eq. (27), we can see thatthe numerical value of χ− = i (Fh(tTP − t) − Fh(tTP ))will be purely imaginary for t ∈ (0, tTP ) and complex

number χ− = F (t− tTP )− i Fh(tTP ) for t > tTP , where

Fh(τ) =

√1− 2Π

Π

{

2√Π τ√

1− 2Π− tanh−1

[

2√Π τ√

1− 2Π

]}

.

(B6)Thus, the solution Eq. (26) will behave exponentiallyfor t < tTP and can have oscillations only when t >tTP . Thus, we conclude that the condensing dropletscan hence cross the separatrix by oscillation only whent > tTP . (Note that when Π = 0, this reduces to the caseSt < 1/4 and the corresponding tTP → ∞, indicates thatthe particle will never cross the separatrix.)Using initial position and initial velocity, the constants

C11 and C12 can be evaluated for Eq. (26) as,

C11 =−i C12

G−(tTP ), (B7)

C12 =(1 + i)√

2y′0

(

St1Π

)γ/2

Π1/4 t1/4TP . (B8)

However these constants along with Eq. (26) will beasymptotically valid estimate of y′(t) only when t≪ tTP ,and can not be extrapolated for t ≫ tTP . The con-stants C11 and C12 need to be determined separatelyfor this region using appropriate solution matching tech-niques at t = tTP . However, we observed that thereal part of the solution Eq. (26) along with constantsEq. (B7) and (B8) have oscillatory nature and its ze-ros matches with the zeros of actual asymptote Eq. (26)with appropriate constants. Thus simply Re(y′(tesc)) ∼ 0solved using Eq. (26) along with constants Eq. (B7) and(B8) can give estimate of escape time when Π → 0 andSt1 < (1 − 2Π)/4 as the solution of following transcen-dental equation,

tan {F (tesc − tTP)} =

tanh{

Fh(tTP)− tanh−1G−(tTP)}

. (B9)

By expanding terms in series for Π ≪ 1, the leadingorder approximate solution can be obtained as tesc ∼tTP + (9 π/(32

√Π))2/3.

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