Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 52-56

EFECTOS GRAVITATORIOS EN LA RETRACCIÓN DE FILAMENTOS LÍQUIDOS

APOYADOS SOBRE UN PLANO INCLINADO

GRAVITY EFFECTS IN THE RETRACTION OF LIQUID FILAMENTS RESTING ON AN

INCLINED PLANE

P. D. Ravazzoli*1, A. G. González1y J. A. Diez1

1Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos Aires, and

CIFICEN-CONICET-CICPBA, Pinto 399, 7000, Tandil, Argentina.

Recibido: 06/09/2021; Aceptado: 16/11/2021

En este trabajo estudiamos la retracción de ﬁlamentos líquidos apoyados sobre sustratos sólidos inclinados en condi-

ciones de mojabilidad parcial. Debido a esto, cada extremo del ﬁlamento retrae longitudinalmente dando lugar a la

formación de una región de acumulación de masa (cabeza) cerca del extremo, donde subsecuentemente se desarrolla

un cuello en su parte posterior. Luego, este cuello se rompe formando una gota separada, mientras que el resto del

ﬁlamento reinicia la secuencia de formación de cabeza con cuello y su posterior ruptura. En el caso horizontal, este

proceso es simétrico y conduce a un arreglo regular de gotas equiespaciadas. Cuando el sustrato se encuentra inclinado,

la gravedad actúa de manera diferente en cada extremo, modiﬁcando la distancia retraída por cada extremo hasta el mo-

mento de la ruptura, como así también la velocidad de retracción y la distancia ﬁnal entre gotas la cual, además, deja de

ser uniforme. Encontramos que, en el caso del plano inclinado, la distancia máxima que retrae el extremo que recede en

contra (a favor) de la gravedad disminuye (aumenta) respecto del caso horizontal. Este efecto resulta ser un mecanismo

útil para el auto–posicionamiento de las gotas resultantes al ﬁnal del proceso. Nuestros resultados experimentales, tales

como posiciones de los extremos, perﬁles del espesor en el corte longitudinal, etc., son contrastados con simulaciones

numéricas de las ecuaciones de Navier–Stokes, obteniéndose un buen acuerdo. Éste se ha logrado empleando como

condición de contorno en la línea de contacto la relación de Cox–Voinov–Blake (CVB), la cual incluye tanto los efectos

de la hidrodinámica macroscópica como así también de la dinámica molecular.

Palabras Claves: mojabilidad, ﬁlamentos, ángulo de contacto.

We study the retraction of liquid ﬁlaments resting on inclined solid substrates under partially wetting condition. This

one causes each extreme of the ﬁlament to retract and form a region of mass accumulation (head) that subsequently

develops a neck at its rear part. This neck then breaks up into a separate drop, while the rest of the ﬁlament restarts the

sequence. In the horizontal case, this process is symmetric and leads to a regular arrangement of evenly spaced drops.

When the substrate is inclined, gravity acts differently at each end, thus modifying the distance retracted by each end,

the retraction speed and the ﬁnal distance between drops, which also ceases to be uniform. We ﬁnd that, in the case

of the inclined plane, the maximum distance retracted by the end that recedes against (in favor of) gravity decreases

(increases) with respect to the horizontal case. This effect turns out to be a useful mechanism for the self–positioning

of the resulting drops at the end of the process. Our experimental results, such as end positions, thickness proﬁles in the

longitudinal section, etc., are contrasted with numerical simulations of the Navier–Stokes equations, obtaining a good

agreement. This is achieved by using in the contact line the boundary condition known as Cox–Voinov–Blake (CVB)

relationship, which includes both the effects of macroscopic hydrodynamics as well as molecular dynamics.

Keywords: wettability, ﬁlaments, contact angle.

https://doi.org/10.31527/analesafa.2022.ﬂuidos.52 ISSN 1850-1168 (online)

I. INTRODUCTION

In this work, we are concerned with the retraction and

fragmentation of a long liquid ﬁlament sitting on an incli-

ned substrate under partial wetting conditions. As schema-

tically presented in Fig. 1(a), we focus our attention on the

evolution of both extremes of the ﬁlament when the subs-

trate inclination angle with respect to the horizontal is α.

Once the liquid thread is placed on the incline, with its axis

along the gravity component parallel to the plane, its ends

start to recede along the longitudinal direction and advance

in the transverse one. This process leads to the formation of

a bulged region, here referred to as head. As a consequen-

* pravazzoli@ifas.exa.unicen.edu.ar

ce of these combined motions a thinner region, called neck,

is developed in the connecting region between the head and

the rest of the ﬁlament that rapidly breaks up thus leading to

a sessile drop that separates from the rest of the ﬁlament [1].

While the original head becomes a sessile drop, the rest

of the ﬁlament repeats this process till a ﬁnal arrangement

of aligned drops is reached. A main feature of these drops

is that the shape of their footprints is noncircular [2]. For

the horizontal case, these drops are equally distanced [3],

but this is not the case when the substrate is inclined (see

Fig. 1(b)).

The physical mechanisms involved in the evolution of the

head and its retraction in the horizontal case have been pre-

FIG. 1: (a) Scheme of the liquid ﬁlament placed on an inclined

substrate with the axis of the ﬁlament along the gravity compo-

nent parallel to the plane. (b) Final drop arrangement after the

complete breakup of a ﬁlament of length L =3.36 cm and width

w=0.6mm on a substrate inclined α=12.5◦. The segments na-

med as Li(i =1,...,7) are the retraction distances travelled by

the contact line to form the drop separated from the remaining ﬁ-

lament: L1, L2and L3(L4, L5, L6and L7) correspond to the uphill

(downhill) extreme.

viously reported [4]. Here, the main goal is to determine the

effects of the component of gravity along the incline and

parallel to the axis of the ﬁlament on its dynamic evolution

and the ﬁnal drops arrangement. We measure the evolution

of the height proﬁles of the ﬁlament in the xz–plane (lon-

gitudinal direction) and from that information we are able

to determine the contact angle at the ﬁlament tips, θx, and

their velocities, vcl . When the breakup processes have come

to an end, we measure the distance between drops for both

uphill and downhill extremes.

Our results are contrasted with numerical simulations,

which solve the Navier–Stokes equations with appropria-

te boundary conditions at the contact line. To do so, we

employ a combination between the hydrodynamic model

proposed by Cox [5] and Voinov [6] (CV model) and a

molecular–kinetics one from Blake [7]. This combined mo-

del was ﬁrst proposed by Petrov [8,9] and more recently

adjusted to the liquid–solid combination used here [4] and

named as CVB model.

II. EXPERIMENTAL RESULTS

We place a liquid ﬁlament on an inclined substrate with

the axis of the ﬁlament along the gravity component pa-

rallel to the plane. The liquid partially wets the substrate

(glass microscope slide) that was coated with a ﬂuorinated

solution (EGC-1700 of 3M). We focus our interest on the

axial dewetting of the tips of the ﬁlament, which is made

of a silicone oil (polydimethylsiloxane, PDMS), with den-

sity ρ=0.97 g/cm3, viscosity µ=21.7 Poise and surface

tension γ=21.0 dyn/cm. The ﬁlaments are generated from

a controlled impact of the substrate against a vertical jet

of PDMS, thus obtaining a liquid thread on the substrate

with uniform width and parallel straight contact lines [4].

The whole system is placed on a base inclined α=12.5◦

and the experiments where perform at room temperature

(a)

(b)

(c)

FIG. 2: Height proﬁles at the ﬁlament extremes for α=0(both

extremes are identical) and α=12.5◦(uphill and downhill extre-

mes) at different times: (a) t =0s, (b) t =50 s and (c) t =100 s.

(T≈20◦C).

The evolution of an inclined ﬁlament is similar to that of

a horizontal one. While in that case, the dewetting and brea-

kup mechanism repeats itself until a regular arrangement of

drops is formed, in the inclined plane case, the retraction

distances of the tips of the ﬁlament, xf, are different becau-

se of gravity effects (xf(t)≥0 for t≥0). Consequently, the

space between the resulting drops also differs. It should be

mentioned that the inclination angle, α, needs to be small

enough so that the formed drops remain static and do not

slide over the substrate leading to drop coalescence [10].

From the lateral observation of the ﬁlament evolution (si-

de view) we are able to obtain the height proﬁles of the head

region at any time. In Fig. 2we show these proﬁles for both

the horizontal and inclined cases at some selected times. We

ﬁrstly observe that the downhill (uphill) tip recedes slower

(faster) for α=12.5◦than for α=0. From these proﬁles

we measure the position of the tip, xf, as a function of time,

whose results are shown in Fig. 3. At early times (t<20 s),

the gravity effects are not yet important. For later times, the

case with inclined plane shows that the downhill (uphill) tip

recedes slower (faster) than the horizontal case. This im-

plies that the maximum retraction distance (achieved when

the receding processes stop) becomes smaller (larger) for

the downhill (uphill) extreme.

Ravazzoli et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 52-56 53

α=0○Exp

α=12.5○(uphill)Exp

α=12.5○(downhill)Exp

α=0○Num

α=12.5○(uphill)Num

α=12.5○(downhill)Num

xf(cm)

0,1

0,2

0,3

0,4

0,5

t(s)

0 25 50 75 100 125 150

FIG. 3: Time evolution of the ﬁlament tips. The symbols corres-

pond to the experimental data: black circles to α=0, while red

squares and blue triangles to uphill and downhill tips, respectivley,

for α=12.5◦. The lines correspond to the results of the numerical

simulation: black solid line to α=0, while red and blue dashed

lines to uphill and downhill tips, respectively, for α=12.5◦.

TABLE 1: Retraction lengths: Distances travelled by the ﬁlament

tip up to neck breakup and separated drop formation.

α=12.5◦

Uphill

L10.412 cm

L20.398 cm

L30.405 cm

Downhill

L40.216 cm

L50.211 cm

L60.208 cm

L70.210 cm

After the complete evolution of the ﬁlament, a conﬁgu-

ration of separated drops is reached (Fig. 1(b)), so that we

measure the distance, Li, between them. These results are

presented in Table 1. As expected, the distances between

drops at the uphill extreme are larger than the ones for the

horizontal case (≈0.295 cm), while they are shorter at the

downhill extreme. For a given inclination angle, each extre-

me shows a typical separation distance, within the experi-

mental error. Therefore, the variation of αcan be used as a

method to control the distance between ﬁnal drops.

III. NUMERICAL SIMULATIONS

Wettability model

In order to perform the numerical simulations we need to

use a suitable boundary condition to describe the contact li-

ne dynamics. As mentioned above, we use a combined mo-

del ﬁrst proposed by Petrov [8], that includes both the hy-

drodynamic and molecular aspects of the local ﬂuid–solid

interaction (CVB model). The model yields the following

relationship between the contact angle, θ, and the velocity

of the contact line, vcl ,

θ3=arccos3cosθ0−1

Γsinh−1vcl

v0

+9µvcl

γlnac

ℓ,(1)

where θ0is the microscopic equilibrium contact angle, ac=

pγ/(ρg)is the capilary length (macroscopic length scale),

and gis gravity. The microscopic length scale, ℓ, the cha-

racteristic velocity, v0and the dimensionless parameter, Γ,

are of molecular origin and depend on both the frequency of

molecular displacements at equilibrium and the average dis-

tance between adsortion sites. These parameters were mea-

sured and reported elsewhere [4] for the PDMS and glass

previously coated with the ﬂuorinated solution EGC-1700,

which is the same liquid–substrate conﬁguration used here.

Their values are:

θ0=50.57◦,v0=6.212 ×10−7cm/s,

ℓ=8.302 ×10−4ac,Γ=95.455.(2)

In Fig. 4we compare the experimental results of the con-

tact angle at the tip, θx, as function of the contact line velo-

city, vcl with the CVB model. The dots represent the ex-

perimental results for α=0 and α=12.5◦(uphill and

downhill tips), while the solid line corresponds to Eq. (1)

with the parameters in Eq. (2). The experimental error in θ

is ∆θ=±1◦, and in vcl it is ∆vcl =±1.1×10−3cm/s. This

agreement between experiments over an inclined substrate

and the CVB model, conﬁrms that the relationship in Eq. (1)

effectively represents a local phenomenon, non affected by

the presence of gravity.

α=0○

α=12.5○(uphill)

α=12.5○(downhill)

Model:CVB

θx(○)

vcl×10-3(cm/s)

−6 −5 −4 −3 −2 −1 0

FIG. 4: Tip contact angle, θx, as function of the contact line velo-

city, vcl . The symbols correspond to the experimental cases, α=0

(black) and α=12.5◦(red squares for uphill and blue triangles

for downhill). The solid line shows the theoretical model (Eq. 1)

and the arrow indicates the time evolution direction.

Time evolution

We obtain the time evolution of a liquid ﬁlament by nu-

merically solving the dimensionless Navier–Stokes equa-

tion,

Re∂v

∂t+ (v·

∇)v=−

∇p+∇2v−zsinα,(3)

where Re =ρacγ/µ2is the Reynolds number. In our ca-

se, we have Re =0.14, so that some slight inertial effects

are considered in the simulations. Besides, we impose the

boundary condition at the contact line as given by Eq. (1)

by using the speciﬁc parameters for our liquid–solid conﬁ-

guration (see Eq. (2)). Since we have chosen acas the cha-

racteristic length of the problem, the corresponding Bond

number is Bo =sinα.

The initial shape of the ﬁlament is represented by a cylin-

Ravazzoli et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 52-56 54

FIG. 5: Contour lines of the ﬁlament for t =0−130 s every

∆t=10 s as given by the numerical simulation. At t =130 s the

downhill neck is formed and it is close to its breakup. At t =0the

ﬁlament height is h0=0.02 cm, its width is w =0.6cm, and its

length is L =3cm.

drical cap with contact angle θ0along its sides. The extre-

mes of the ﬁlament are shaped as halves of ellipsoidal caps

with a contact angle at the tip, θx=20◦. The ellipsoidal

and cylindrical caps are matched with smooth continuity of

thickness and contact angle.

Top views of the time evolution of the ﬁlament are shown

in Fig. 5. As observed in the experiments, the simulations

also yield that the downhill extreme recedes a shorter dis-

tance than the uphill one, till they break up and form the co-

rresponding drops. Since the numerical method used here

(moving mesh) does not allow for separations of the ﬂuid

into several domains, the code is unable to describe the

complete breakup ﬂow at the necks. Nevertheless, it is able

to simulate the ﬂow up to moments near the breakup, and its

results remain useful for the purposes of the present work.

However, the retraction distances and the retraction velo-

cities are smaller than those measured in the experiments.

As shown in Fig. 3, the numerical simulation underestima-

tes the displacement of the tip in the α=0◦case. This dif-

ference changes when the substrate is inclined. It becomes

larger for the uphill extreme and smaller for the downhill

one. In the latter case, the differences are practically negli-

gible.

In Fig. 6we compare the experimental and numerical

height proﬁles for an inclined ﬁlament. In order to point

out the shape differences, the experimental data have been

shifted till the position of the experimental and numerical

contact lines are coincident. It can be seen that for t=50 s

(Fig. 6(a)) the main differences are behind the heads of both

extremes, in the neck regions, where the experimental proﬁ-

les are a bit lower than the numerical ones. This departure is

more pronounced for t=100 s in the downhill neck, while

the numerical head height is smaller than the experimental

one at the uphill head. We believe that the differences at the

uphill head and neck are related with the slower retraction

of the tip as observed in Fig. 3.

IV. CONCLUSION

The experiments on the retraction of liquid ﬁlaments over

an inclined substrate show that the presence of a longitudi-

nal component of gravity produces an increase (decrease) of

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.01

0.02

0.03

0.04

zExperiment

Numerical Solution

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.01

0.02

0.03

0.04

zExperiment

Numerical Solution

FIG. 6: Comparison between experimental and numerical height

proﬁles. The gray dashed line corresponds to the initial numerical

state, the solid blue line to the numerical solution and the red dots

to the experimental proﬁle. In order to compare the shape of the

proﬁle, the experimental data have been shifted so that the position

of the tip is the same as in the numerical case. (a) t =50 s and (b)

t=100 s.

the retraction distance for the uphill (downhill) extreme ne-

cessary to produce each separated droplet. Moreover, as the

breakup process repeats itself at each new ﬁlament extre-

me, we ﬁnd that these retraction distances remain constant

for all drops formed in both the uphill and downill portions

of the initial ﬁlament (see Table 1).

It is interesting to point out that we conﬁrmed the lo-

cal behavior of the CVB wettability model. As we show

in Fig. 4the relationship vcl (θ)measured for the uphill and

downhill extremes of the ﬁlament is in good agreement with

the model parametrized for the spreading of a circular drop

over a horizontal substrate.

By using the CVB model in the simulations we found

a good agreement with the experiments for early times res-

pect to the breakup time (say t<30 s, see Fig. 3). For α=0

and later times we observe that the simulation predicts lo-

wer retraction velocities of the extremes. This is due to the

fact that the numerical method is unable to model comple-

te breakups (or ﬂuid domain separation), and thus it leaves

a very thin ﬁlament connecting the almost separated drop

from the remaining ﬁlament. The inclusion of a longitudi-

nal volume force (gravity component parallel to the incline)

for α>0 generates an additional downslope ﬂow inside

the ﬁlament such that it decreases (increases) the volume of

the uphill (downhill) head due to the ﬂow along the neck,

which remains spuriously not broken for longer times due

to the numerical ﬂaw. This effect explains the even lower

retraction velocity of the uphill extreme, and the compensa-

tion of this velocity at the downhill extreme, which is now

increased respect to the α=0 case.

We consider that a complete understanding of the who-

le phenomenology still requires a more detailed physical

model which, to our knowledge, is still unavailable in the

literature. This type of ﬂows constitute a simple method of

Ravazzoli et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 52-56 55

technological interest to produce an arrange of self positio-

ning drops with varying spacings.

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