The University of Manchester Research
Viscous drops on a layer of the same fluid: from sinking,
wedging and spreading to their long-time evolution
DOI:
10.1017/jfm.2018.127
Document Version
Accepted author manuscript
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Citation for published version (APA):
Bergemann, N., Juel, A., & Heil, M. (2018). Viscous drops on a layer of the same fluid: from sinking, wedging and
spreading to their long-time evolution. Journal of Fluid Mechanics, 843, 1-28. https://doi.org/10.1017/jfm.2018.127
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Journal of Fluid Mechanics
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1
Viscous drops on a layer of the same fluid:
from sinking, wedging and spreading to their
long-time evolution
Nico Bergemann
1,2
, Anne Juel
2
and Matthias Heil
1
†
1
School of Mathematics and Manchester Centre for Nonlinear Dynamics, The University of
Manchester, Oxford Road, Manches ter M13 9PL, United Kingdom.
2
Manchester Centre for Nonlinear Dynamic s and School of Physics & Astronomy, The
University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom.
(Final version of Author Accepted Manuscript; submitted to JFM 24 Jan 2018)
We study the axisymmetric spreading of drops d eposited on a pre-existing horizontal
layer of the same viscous fluid. Using a combination of expe r ime nts, numerical modelling
based on the axisymmetric free-surface Navier–Stokes equations, and scaling analys es , we
explore the drops’ behaviour in a regime where the flow is driven by gravitational and/or
capillary forces while inert ial effects are small. We find that durin g the early stages of the
drops’ evolution there are three distinct spreading behaviours depending on the thickness
of the liquid layer. For thin layers the fluid ahead of a clearly defined spreading front is at
rest and the ove r all behaviour resembles that of a drop spreading on a dry substrate. For
thicker films, the spr eadin g is characterised by an advancing wedge which is sustained
by flu id flow from the drop into the layer. Finally, for thick layers the drop sinks into
the layer, accompanied by significant flow within the layer. As the d r op keeps spreading,
the evolution of its shape becomes self-similar, with a power-law behaviour for its radius
and its excess height above the undisturbed fluid layer. We employ lubrication theory to
analyse the drop’s ultimate long-term behaviour and show that all drops ultimately enter
an asymptotic regime which is reached when their excess height falls below the thickness
of t he undisturbed layer.
1. Introduction
The deposition of a drop onto a film of the same liquid is an important process in
applications ranging from spray painting (Cormier et al. 2012) and ink-jet printing of
solution-processed organic electronics (Thompson et al. 2014), on the mi cr os cale, to the
3D printing of food (Godoi et al. 2016), on the macroscale. In spray painting, a film is
rapidly formed through the coalescence of the first few droplets impacting an initially
dry surface, so that subsequent drops spread on a liquid layer whose thickness increases
with the deposited volume. Similarly, the manufacture of organic electronics relies on the
sequential deposition of p ar tiall y overlapping microdroplets which coalesce upon impact
and spr e ad due to capillary pressure differences to form a liquid line (and eventually
a solid film after evaporation of the solvent). By contrast, in food print in g, the larger
deposited volumes te nd to spread due to gravity. In this paper, we investigate the effect
of the thickness of the underlying liquid film on the spreading of both “small” and “large”
drops using a combination of experiments, numerical modelling and scaling analyses.
While the spreading of droplets on an existing layer of fluid is of interest in its own
right, much previous work on this problem has been motivated by the fact that the
† Email address for co rr espo n dence: M.Heil@maths.manchester.ac.uk
2
presence of a thin pre cu r sor film regularises the contact line singularity that arises when
a drop spreads on a perfectly dry substrate; see, e.g., De Gennes (1985); Yarin (2006);
Bonn et al. (2009); Samsonov (2011); Popescu et al. (2012); Snoeijer & Andreotti (2013).
The existence of such films was confirmed in early experiments by Quincke (1877) and
Hardy (1919). Nanoscale liquid polymer droplets spreading on thin films of the same
fluid have been studied computationally (Milch ev & Binder (2002); Heine et al. (2003);
see also Pierce et al. (2009) for spreading on permeable surfaces). The thickness of the
precursor films tends to be in t he range from h
∞
= 10 to 100 nm (Kavehpour et al.
2003). This is much thinner than the films we consider in the c urrent study within which
we focus on a regime in which the spreading is dr iven by gravity and/or surface tension,
with gravity dominating for “large” drops and surface tension dominating for “small”
ones. We aim to characterise th e spreading of such drops and to contrast their behaviour
to that observed when they spread on dry substrates.
On dry substrates, drops of partially-wetting fluids evolve towards their sessile equilib-
rium configuration which is parametrised by the finite equilibrium contact angle between
the liquid and solid substrate. As the equilibrium contact angle approaches zero th e drops
become perfectly wetting and continue to spread indefinitely. Tanner (1979) analysed
this scenario using a thin-film mode l and showe d that for “small” drops whose motion
is driven by a balance of capillary and v is cous forces, the drop height, H(t), and radius,
R(t), ultimately display a power-law behaviour, with H(t) ∝ t
−1/5
and R(t) ∝ t
1/10
,
respectively. Lopez et al. (1976) and Huppert (1982) considered the c ase of “large” drops
whose motion is driven by a balance of gravity and viscous f orc es . They showed that in
this regime, the large-time behaviour is again described by power-laws but with different
exponents, namely H(t) ∝ t
−1/4
and R(t) ∝ t
1/8
.
When deposited on a pre-existing liquid film, the drop continues to spread and
ultimately approaches a configuration in which the liquid layer is again perfectly level.
For thin precursor films (relative to the size of the drop) the drop h as a cle arl y defined
spreading front whose radius ultimately follows a power -l aw, but with an exponent that
is slightly larger than for spreading on a dr y substrate. This scen ario was studied on
the basis of a thin film mo de l for the case of “small” drops by Tanner (1979) and
later by K ali nin & Starov (1986) and Chebbi (1999). They showed that the scaling
derived for spreading on a dry substrate is not recovered as h
∞
→ 0 because this
limit presents a singular perturbation (see, e.g., Voinov (1976); Hocking (1983); Cox
(1986) for analyses of this problem). Conversely, the drop s p r eadi ng on a thin film only
provides a weak perturbation to the liquid layer ahead of itself, the most prominent
feature being the development of a small dip just ahead of the spr ead ing front. This dip
is, in fact, the first extremum of an exponentially-damped oscillatory perturbation to the
precursor film, remin is ce nt of that observed when a fluid-coated plate is pushed into a
bath of the same viscous fluid; see Landau & Levich (1942) and Derjaguin (1943) for
the classical t he ory, and M aleki et al. (2011) for a recent detailed comparison between
theory and experiments. Similar featur e s are observed in many other flow problems where
a perturb ation propagates into a thin-film region; see, e.g., Gaver et al. (1996); Stillwagon
& Larson (1988); Salez et al. (2012); Pihler-Puzovi´c et al. (2015).
In this paper, we investigate the influence of the thickness of the unde r ly in g liquid
film on the axisymmetric spreading of viscous drops considering a wide range of film
thicknesses up to the size of the deposited drop. We start by analysing the behaviour of
“large” drops by performing experi ments with drops of glucose syrup. These experimental
studies are augmented by finite-element simulations which provide detailed insight into
the flow field and the evolution of th e drop shape. Drops deposited on finite-depth films
are found to spread increasingly rapidly with increasing layer thickness. On thin fluid
3
films, d r ops retain clearly defined spreading fronts. As the film thickness increases, this
“spreading” behaviour is progressively replaced by a “wedging” beh aviour (where the
term refers to the overall shape of the d r op rather than its localised shape near its outer
edge), whil e for even thicker layer s , the drop “sinks” into the layer. For drops spreading
on thin films, our computational model approximately recovers the long-time power-law
predictions based on scaling arguments for drops spreading on dry substrates (Tanner
1979; Cazabat & Cohen-Stuart 1986; Lopez et al. 1976; Huppert 1982). We also c ons ide r
the beh aviour of “small” drops, again covering the range from very thin films to films
that are thicker than the drop itself. The results of our numerical simulations are then
compared to the experimental results of Cormier et al. (2012) who studied the levelling of
shallow microdroplets of molten polystyrene on films of the same mate r ial. The regime
in which drops spread on thin layers (relative to the drops’ exces s height above the
undeformed layer) is inevitably transient because the continued spreading ultimately
reduces the drops’ excess height to become less than the layer thick ne ss . We employ
a lubrication-theory-based model to analyse the transition to this ultimate spreading
regime and extend an approach first introduced by Cormier et al. (2012) (for “small”
drops) to derive an explicit prediction for the evolution of the drops’ excess height (for
“large” and “small” drops) as they approach this regime.
This paper is organised as follows. The experimental methods and results for the
spreading of “large” glucose drops ar e presented in §2. The theoretical model and
numerical methods are described in §3. Results are presented in §4 where we start in
§4.1 with a comparison between our experiments and numerical simulations. In §4.2 we
characterise the effect of variations in liquid layer thickness, spanning three orders of
magnitude, on the early stages of the spreading of a drop of gluc ose syr up . We discuss
scaling laws for spreading at intermediate times in §4.3 and assess the influence of the
drop size on the spreading in §4.4. In section 4.5 we analyse the drops’ evolution towards
its ultimate spreadin g regime which is reached when the excess drop height has fallen
below the layer thickness. Finally, we summarise our results and present our conclusions
in §5.
2. Experiments
2.1. Experimental methods
2.1.1. Experimental setup
A schematic diagram of the experimental apparatus used to examine the sp r ead ing
following depositi on of a drop on a substrate is shown in figure 1. The substrate (a Perspex
plate of dimensions 100× 100×10 mm with surfaces milled to an accuracy of ±0.02 mm)
was secured to the base plate with three finger-tight nylon screws. A featureless, flat
substrate was used for the dry spreading experiments, whereas for spreading on a viscous
layer, centred, circular troughs with a diameter of 60.5 mm and depths of 0.52 ± 0.02
mm, 0.85 ± 0.03 mm, 1.51 ± 0.04 mm and 1.95 ± 0.03 mm we r e milled into the plate into
which a uniform film of liquid was deposited prior to exper i mentation. The s ub s tr ate
was supported on a Perspex base plate, which was adjustably mounted on three vertical,
threaded poles (with a pitch of 1.25 mm) using nuts, thus allowing accurate levelling
to ±0.1
◦
. The fluid was deposited using a standard 10 ml plastic syringe whose inner
diameter was enlarged to 8 ± 0.05 mm to facilitate the manual deposition of the highly
viscous liquid. To ensure reproducible deposition of the fluid, we placed the syringe inside
a tightly-fitting removab le holder which was mounted on the thre e vertical, threaded
poles, allowing its level to be adjusted in a similar way to the base plate. The accurate
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syringe
holder
removable
substrate
nylon
screws
syringe
base plate
threaded poles
camera
Figure 1. Schematic diagram of the experimental apparatus.
levelling of both base plate and syringe holder was essential to ensure axisymmetric
spreading. Side-view images of the backlit drop were captured with a wall-mounted CCD
camera (Pulnix TM-6740CL, 640 × 480 pixels).
We per for med the spreading experime nts with glucose syrup (Cerestar UK Ltd.),
which is a transparent, highly viscous Newtonian liquid. In order to enhance contrast
in the images, the glucose syrup was dyed using green food colouring. Mixing of the dye
entrained air bubbles, which were left to rise out of the fluid overnight. The experiments
were performed by filling the syringe with 5 ml of glucose syrup and wiping any excess
fluid from the outside of the nozzle with a dry paper towel. The filled syringe was then
placed inside the syringe holder, the image acquisition was initiated, and the plunger of
the syringe was displaced manually to empty the syringe barrel within a deposition time
of 2.5 ± 0.5 s.
We measured the density of the glucose syrup at the laboratory temperature of 20.5 ±
0.5
◦
C to be ρ = 1387 ± 1 kg/m
3
by accurately weighing five samples of known volume
between 5 and 20 ml. We determined the viscosity of glucose syrup at the laboratory
temperature, using a Brookfield R/S-Plus (SST) rheometer with a concentric cylinder
CC25 geometry. We performed shear rate measurements with linear i nc r eas e from zero
to a maximum value of 25 s
−1
with increments of 1 s
−1
, applying a cy cl e of incremental
shear rate increase and d ec r eas e. The total experimentation time was 50 s, with one
measurement taken every second. Hence, we recorded 50 viscosity measu r eme nts and
these experiments confirmed that the viscosity of glucose syr u p is independe nt of the
shear rate within the investigated parameter range. The resulting averaged dynamic
viscosity is µ = 119.73 ± 0.86 Pa s. The surface tension of the glucose syrup was taken
from the literature (Monta˜nez-Soto et al. 2013) to be σ = 55.0 ± 0.6 × 10
−3
N/m.
2.1.2. Substrate prepa ration
When performing the experiments in which the drop is deposited on a uniform layer
of the same fluid, we prepared the substrate by slightly overfilling the trough and then
scraping off the excess fluid with a square-edged ruler. The ruler was moved at an angle of
around 30
◦
and with low speed to avoid the washboard instabi lity (Hewitt et al. 2012) at
the free surface. This method had t he advantage of rapid deposition, thus preventing the
formation of a skin due to evaporation and subsequent crystallisation at the surface (Lee s
2012; Edwards 2000). Since a certain amount of the (highly viscous) fluid adhered to the
scraper this proc ed ur e resulted in an underfilled trough, with the surface of the fluid layer
