Experiments in Fluids
(2024) 65:69
https://doi.org/10.1007/s00348-024-03800-5
RESEARCH ARTICLE
Micro‑PIV study on the influence of viscosity on the dynamics
of droplet impact onto a thin film
Stefan Schubert1 · Jonas Steigerwald1 · Anne K. Geppert1 · Bernhard Weigand1 · Grazia Lamanna1
Received: 31 January 2024 / Revised: 14 March 2024 / Accepted: 15 March 2024
© The Author(s) 2024
Abstract
This work presents a systematic experimental study of droplet impact onto a wet substrate. Four different silicone oils are
used, covering a range of Reynolds number between 116 < Re < 1106 at two different initial wall film heights. The objective is to characterize the temporal and radial evolution of the velocity field within the crown crater by means of micro-PIV.
Our findings show that the velocity field has the structure of an axisymmetric stagnation point flow with decaying strength
a(t). The latter exhibits an exponential decay and can be explained in terms of the exponential decay of the pressure force
exerted by the impacting droplet onto the wall film. In this context, the commonly accepted functional dependence a(t) ∝ t−1
represents only the first-order Taylor approximation of the exponential decay and has therefore only a limited temporal validity. The analysis also corroborates the existence of an inertial regime concerning the velocity field for Re > 270 . This is
not observed at lower Re numbers due to the increased pressure losses caused by the extensional (normal) strain during the
radial spreading of the lamella. To validate these findings a holistic approach is chosen, which combines numerical results,
analytical solutions and experimental data from literature. In particular, by using the continuity equation, it is shown that
the experimental decay of the wall film height can be reconstructed from the velocity measurements. Consilience of results
from different approaches provides a robust validation of the micro-PIV data obtained in this work.
Graphical abstract
Extended author information available on the last page of the article
Vol.:(0123456789)
69
Page 2 of 23
Experiments in Fluids
1 Introduction
Droplet impact onto a thin liquid layer (film) is a well-known
phenomenon occurring in our everyday life. Even though an
impact of a single droplet appears mundane at first sight, it is
a fundamental process that plays an important role in many
natural and industrial processes such as soil erosion, application of pesticides, spray coating and printing technologies,
spray cooling of micro-electronic devices or in propulsion
systems in air and spacecrafts (Yarin 2006). Optimizing such
applications requires a detailed knowledge of the droplet
impact process.
The complexity of a droplet impact onto a thin liquid
layer is shown in Fig. 1, which shows the rapid displacement and deformation of the liquid droplet and wall film.
A variety of parameters affect the outcome of the droplet
impact, such as the initial droplet diameter D0, impact velocity U0 , initial film height Hf ,0 and the liquid properties (surface tension 𝜎 , dynamic viscosity μ, density 𝜌). In this work,
we investigate single-component, normal droplet impacts,
where the Reynolds number Re = 𝜌U0 D0 ∕μ , the Weber
number We = 𝜌U02 D0 ∕𝜎 and the dimensionless film height
𝛿 = Hf ,0 ∕D0 define the impact conditions. The dimensionless
time 𝜏 is defined as 𝜏 = tU0 ∕D0 with time t.
This wide parameter space leads to a manifold of possible impact outcomes, such as deposition, transition and
crown splashing. A comprehensive overview can be found
in Liang and Mudawar (2016). In most cases, researchers
investigated the macroscopic impact morphology and kinematics, i.e., the transformation of the primary droplet into a
crown, its propagation and the formation of secondary droplets. Moreover, in order to predict certain macroscopic flow
features, e.g., the crown wall propagation in radial direction,
great effort has been put into simplified descriptions of the
whole process via analytic modeling. In this context, Yarin
and Weiss (1995) proposed an asymptotic solution for the
axisymmetric inviscid flow within the crown crater, which
covers the area from the impact point to the crown wall as
shown in Fig. 1 (on the right). For sufficiently large times
t, they modeled the radial velocity as ur (t) = Br∕(1 + Bt),
where r denotes the radius and B is a positive constant. Note
that this velocity profile automatically satisfies the inviscid
momentum equation under the assumption of incompressible flow and negligible pressure gradient. These assumptions lead to a constant radial velocity over the film thickness. A rearranged version of this one-dimensional inviscid
and incompressible model was given by Kittel et al. (2018),
which reads
ur (t) =
1
D
c U0
0
r = aYW (t) r
(1)
+t
with a dimensionless constant c. Due to this reformulation
the constant B is now linked to at least some of the initial
impact parameters, yielding B = U0 ∕(cD0 ). Lamanna et al.
(2022) instead set the constant to c = 1∕𝜆 = f (We, Re, 𝛿),
resulting in the estimation B = 𝜆U0 ∕D0 . The parameter 𝜆
was derived empirically by Gao and Li (2015). Following
the approach of Yarin and Weiss (1995), Lamanna et al.
(2022) employed the parameter 𝜆 to estimate the kinetic
energy transmitted
√ to the crown wall, thereby recovering
the well-known t-dependence to describe its propagation.
Finally, the inclusion of viscous losses led to a very good
agreement with experimental data over longer time periods
(i.e., 5 < 𝜏 < 30) and a wide range of impact conditions and
test fluids (Lamanna et al. 2022). Philippi et al. (2016) performed a detailed numerical simulation of a single droplet
impact onto a dry wall ( Re = 5000, We = 250) by employing an incompressible Navier–Stokes solver. Due to adaptive
mesh refinement procedure, they were able to fully capture
the features of the boundary layer, leading to a very accurate
Dp , ρp
D0
g, U0
(2024) 65:69
ρd , µd , σd
crown
wall
crown
crater
ROI
z
Hf,0
ρf , µf , σf
Sapphire
Fig. 1 Schematic of a single droplet ( D0 , U0 , 𝜌d , μd , 𝜎d ) impact onto
a thin liquid film ( Hf ,0 , 𝜌f , μf , 𝜎f ). The droplet is seeded with tracer
particles ( Dp , 𝜌p). The crown crater consists of the liquid layer of the
droplet, denoted as droplet lamella, as well as the liquid of the ini-
Sapphire
r
tial film, denoted as film lamella. The Region of Interest (ROI) represents the measurement volume, which is limited by the Depth of Field
(DOF) and the Field of View (FOV)
Experiments in Fluids
(2024) 65:69
description of the short-time dynamics of the impact process
(i.e., 𝜏 ≤ 1). During this time span, their numerical findings
showed that the impinging droplet induces an unsteady stagnation point flow. The inviscid self-similar impact pressure
and velocities were
√ shown
√ to depend solely on the self-similar variables (r∕ t, z∕ t). Roisman et al. (2009) developed
an analytical solution for an unsteady laminar viscous flow
in a spreading liquid film. As asymptotic (outer) boundary
condition, Eq. 1 was chosen with c = 0. This approach represents one of the first examples of including boundary layers
effects in droplet impact problems. Building upon this analysis, Roisman (2009) concluded that viscous effects influence
the evolution of the lamella thickness only at late stages of
the spreading.
Despite the noteworthy progress, it remains unclear
whether the temporal velocity decay in Eq. 1 (i.e., the
1/t-dependence) remains valid beyond the short-time
dynamics of the impact process. This is due to a number
of concomitant factors. First, the available analytical solutions all rely on Eq. 1 as boundary condition for the outer
flow (see e.g., Roisman et al. 2009; Roisman 2009). Second,
the absence of reliable experimental data prevents a clearer
picture of the long-time evolution (i.e., 𝜏 > 2 ) of the flow
velocity within the crown crater. Such measurements are
extremely challenging due to the difficult optical access to
the sub-millimeter measurement volume of the spreading
lamella, the unsteadiness and the short duration (O(10−3
s)) of the overall impact processes. Third, accurate velocity
data from highly resolved numerical simulations can only
be obtained with great computational effort (see e.g., FestSantini et al. 2021). This explains why the available studies,
reporting a detailed asymptotic analysis of the velocity and
pressure field, are extremely rare and limited to the shorttime dynamics (i.e., 𝜏 < 1) (Philippi et al. 2016).
More recently and thanks to the rapid development of
high-speed imaging technologies over the last years (Cheng
et al. 2022), experimental investigations have become
increasingly more capable to resolve the relevant spatio-temporal scales, thereby offering a great time benefit if a large
parameter space has to be investigated. The most common
experimental technique for flow visualization is the particle
image velocimetry (PIV), which employs tracer particles
to obtain velocity information of a flow field (Raffel et al.
2018). In classical PIV, the tracer particles are illuminated
by a planar laser light-sheet (typically 1–2 mm thick). The
particles outside of the light-sheet are not illuminated and
thus the measurement volume is well defined as a cross-sectional cut of the flow. In microfluidic applications, instead,
the measurement volume itself is in the sub-millimeter range
and, therefore the so-called volume illumination is typically
applied, leading to the concept of micro-PIV (Santiago et al.
1998). In this case, the measurement depth is defined by
the depth of focus of the microscope objective rather than
Page 3 of 23
69
by the thickness of the laser sheet (Lindken et al. 2009;
Wereley and Meinhart 2010). In both PIV approaches, the
measurement procedure is similar and consists in making
consecutive recordings (image frames) of the tracers with a
time separation Δt in between. The image frames are then
divided into interrogation windows and one velocity vector
is obtained for each interrogation window by performing a
cross-correlation of the image frames. More details on the
post-processing procedure and the strategies to increase its
accuracy can be found in Raffel et al. (2018).
With reference to droplet impact scenarios, both classical
PIV and micro-PIV have been applied with minor differences, since the measurement volume is essentially determined by the thickness of the radially spreading lamella,
which is in all cases in the sub-millimeter range. Hereafter,
a short overview is presented on velocity measurements
within a radially spreading lamella performed with both
PIV approaches. The overview includes studies on droplet
impact onto both dry and wet surfaces, covering the regimes
of deposition and splashing. A very consistent picture can
be derived from these measurements, thus corroborating the previous statement on the equivalence of velocity
measurements with the classical and micro-PIV optical
configuration.
Smith and Bertola were the first to apply time-resolved
micro-PIV to a droplet impact experiment by coupling a
high-speed laser into an inverted epifluorescent microscope (Smith and Bertola 2010, 2011). The objective was
to investigate the effect of polymer additives on the rebound
of impacting droplets onto a hydrophobic surface in the deposition regime. They studied pure water (Newtonian) and
aqueous solutions of poly-ethylene oxide (non-Newtonian)
over time period of 0.5 < 𝜏 < 4 . For the spreading phase
they found that, apart from considerable scatter, the radial
velocity distribution for both fluids fulfills the inviscid solution of Yarin and Weiss (1995) (i.e., Eq. 1) and the constant
c should be zero. Lastakowski et al. (2014) conducted PIV
measurements of droplet impact onto hot (above the Leidenfrost point) and cold (room temperature) surfaces with
ethanol and isopropanol-glycerol mixtures for 2 < 𝜏 < 6. In
the case of a hot surface, they found a good agreement of
the radial velocity with the analytically solution by Yarin
and Weiss (1995). Following their nomenclature, this is
equivalent to setting the constant c = 0.5. In contrast, for
the cold case they observed that the radial velocity is much
lower and decreases faster with time, while preserving the
linear radial dependence. The authors explained this difference in terms of viscous decoupling (hot case) or coupling
(cold case) between the drop and the substrate, respectively.
Erkan and Okamoto performed similar studies of impinging water droplets on unheated (Erkan and Okamoto 2014)
and heated (Erkan 2019) surfaces (25–250◦ C) with focus on
the early spreading phase (0.15 < 𝜏 < 1). They analyzed the
69
Page 4 of 23
effect of Weber number ( We = 4 − 5, 10 − 15, 27.6) on the
radial spreading velocity and fitted their data to Eq. 1 with
two different values c = 0.3 for We = 4.9 and c = 0.58 for
We = 27.6. Moreover, the experimental radial velocity distributions demonstrated linear behavior for the inner radial
positions, whereas a nonlinear behavior was observed for
the outer radial positions (i.e., r > D0 ∕2) owing to the vertically upward flow (see also Gultekin et al. 2023). For the
unheated surface, Erkan (2019) compared the radial velocity
distributions from the PIV measurements with the viscous
analytical solution from Roisman et al. (2009) and found
a significant disagreement in the temporal decay. Specifically, the temporal decay predicted by the viscous solution
is faster than in the experiment. The authors attributed this
disagreement to the action of an additional force, such as a
radial pressure gradient, that persistently overwhelms the
viscous forces, thus retarding the deceleration process. This
is, indeed, an interesting remark since all available analytical
models neglect the influence of the pressure gradient, even
though it is well-known that an impacting droplet exerts a
transient force on the target surface (Mitchell et al. 2019).
Gultekin et al. (2020) used PIV measurements to investigate the spreading velocity within the droplet lamella for
0.5 < 𝜏 < 2 . They covered both single and double droplet
impacts onto dry heated and unheated surfaces (20–250◦ C)
for moderate Weber numbers ( 40 < We < 190 ). They
observed that the surface temperature has little effect on
the radial velocity distribution in the inner radial position
(r̃ = r∕D0 ≤ 0.7) and proposed c = 0.48 and c = 0.53 as fitting constants for Eq. 1.
In a recent study, Gultekin et al. (2023) investigated
droplet impacts on dry solid surfaces over a wide range
of We numbers ( 5 < We < 183) and associated Re in the
range of 882 < Re < 5570 . In all cases, the radial velocity
exhibited a linear dependence for all inner radial positions,
albeit the initial slope varied with We for 𝜏 = 0.5. This
indirectly implies that the estimation of the constant c must
depend upon the impact conditions and fluid properties (i.e.,
c = f (We, Re).
Even less studies have been conducted for droplet impact
onto wet surfaces. The first attempt dates back to the work
of Ninomiya and Iwamoto (2012). They measured the velocity of the crater surface of an impacting milk droplet with
black urethane foam, shaped like flakes of size 100-400 μm
as tracer particles. The first example of time and height
resolved velocity measurements on dry and wet surfaces was
presented by Frommhold et al. (2015). As test fluids water
and ethanol were used. The authors employed particle streak
photography, where streak images of fluorescent tracers in
the drop and in the liquid film (if present) were recorded by
a high-speed camera. The latter was attached to an inverted
microscope with a narrow depth of field (< 3 μm), that could
be shifted vertically by a precise piezo-driven z-stage. By
Experiments in Fluids
(2024) 65:69
changing the focal plane, it was possible to scan through the
flow with an effective spatial resolution of 5 μm , because
tracer particles out of focus become blurred and hardly visible. Several experiments were performed for each impact
condition in order to measure the radial velocity stepwise
within separate layers from the substrate up to 40 μm . This
procedure was justified by the high reproducibility of deposition experiments and by the high precision of the dispenser mechanism. The measurements took place during the
elongation phase of the droplet. High-speed imaging of the
impact process were obtained separately. They found that
the maximum wall shear stress by impact on liquid films
is significantly lower (up to one order of magnitude lower)
than for the dry case. This finding is of great importance and
implies that for impact on wet substrates deviations from the
inviscid solution of Yarin and Weiss (1995) (i.e., Eq. 1) may
not necessarily be caused by viscous losses.
This brief excursus on literature studies allows us to
draw the following conclusions: First, most of the velocity measurements have been performed during deposition
experiments, while hardly any data can be found for droplet
splashing on a wet substrate. Second, all available data confirm that the radial velocity increases linearly with distance
from the impact point (stagnation point). No general consensus, instead, is found on the temporal decay of the radial
velocity. In this context, the remark from Erkan (2019) on
the role of the pressure gradient in controlling the velocity decay requires further evaluation. Third, mainly low
viscosity fluids were used (e.g., water or alcohols) with no
systematic variation of the dynamic viscosity. These open
questions motivate the present study, which aims to acquire
reliable radial velocity distributions for droplet impact onto
a wet surface also during a splashing event. In this case,
due to the higher fall height, the impact point varies within
a circular area of a few square millimeters. Consequently,
the macroscopic visualization of a splashing event must
occur synchronously with the velocity measurements on
the microscopic scale, in order to assure that the small scale
features are correctly embedded in the overall dynamics of a
splashing event. The feasibility of this approach was evaluated in two preliminary studies from this group. As a first
step, Vaikuntanathan et al. (2020) investigated the possibility to perform time-resolved micro-PIV by employing a
high-intensity light emitting diode (LED) to constantly illuminate the measurement volume from above. Even though
LED illumination has been already successfully applied in
microfluidic devices (Hagsäter 2008), its application to a
splashing scenario is not straightforward. For this purpose,
a high-speed camera in combination with an inverted microscope was employed to record shadowgrams of the impact
zone with a backlight optical configuration. As a second
step, Bernard (2020) added an additional high-speed camera to capture the macroscopic view of the impact process.
Experiments in Fluids
(2024) 65:69
This enabled a precise determination of the actual impact
conditions and dynamics. The overarching goal of these preliminary studies was to perform a proof-of-concept study
on the feasibility of simultaneous micro and macro flow
visualization of a splashing event. The present work builds
upon the previous studies and performs a trade-off analysis
for the optimal choice of mechanical and optical parameters
(e.g., magnification, light intensity, tracer size) to assure a
high reproducibility of the experiments and maximize the
measurement time. Moreover, it includes a systematic variation of the impact Reynolds number by one order of magnitude (O(102 ) < Re < O(103 )) to cover both the regimes
of deposition and splashing. The Weber number, instead, is
kept approximately constant ( We ≈ 800 ). The objective is
to investigate how the transfer of specific momentum and
its radial distribution are affected by gradually increasing
the relative importance of viscous forces. The term specific
momentum refers to the momentum per unit mass which
represents a velocity e.g., ur (t) (Lamanna et al. 2022). In
this context, a holistic approach to the analysis of the PIV
data is undertaken, which foresees three parallel evaluation
paths to verify the plausibility of the measurements. First,
the experimental data are employed to understand the influence of fluid properties and wall film thickness on the deceleration experienced by the spreading lamella. Second, a single experiment is compared to the predictions from a direct
numerical simulation (DNS). The comparison is not only
limited to the radial velocity distribution, but includes also
the decay of the film height. For the conservation of mass,
the radial spreading of the liquid lamella is associated to a
decrease in film height. Hence, its functional dependence
upon time can be directly derived from the measured radial
velocity profiles. A similar analysis can be also performed
with theoretical models. Consilience of results between
different approaches provides an indirect validation of the
experimental measurements and associated conclusions.
The present paper is organized as follows. Section 1.1 discusses the link between the radial velocity and temporal evolution of the wall film height. It also shows how the choice of
the initial velocity distributions (and associated simplifying
assumptions) constrains the evolution of all other variable
in the flow. Section 2 describes the experimental test rig
and post-processing procedure of the PIV data. Section 3
briefly introduces the in-house numerical solver. Finally,
Sect. 4 discusses the experimental data and the comparison
with numerical simulations and theoretical models.
1.1 Fundamentals of modeling the crown crater
dynamics
This section reviews important fundamentals on the dynamics of the thin liquid lamella within the crown crater during
the impact process. Starting point is the assumption of an
Page 5 of 23
69
axisymmetric stagnation point flow. In the potential flow
region, the radial velocity component is constant over the
height (block profile) and reads as
ur (r, t) = a(t)r.
(2)
The radial velocity increases linearly with the radius but
decreases with time, as confirmed in several experimental
studies, see e.g., Smith and Bertola (2010), Gultekin et al.
(2023). The time decay is controlled by the parameter a(t),
also denoted as the strength of the stagnation point flow.
As shown later, this relation is also observed in the present
study and Eq. 2 holds for the investigated time interval
1 < 𝜏 < 5 and radii 0 < r̃ < 0.5, where r̃ = r∕D0 denotes the
dimensionless radius. Assuming an incompressible liquid
(i.e., 𝜌 = const ) and a vanishing angular velocity component (u𝜙 = 0 ), the height of the film h(t) can be derived by
substituting Eq. 2 into the continuity equation (∇ ⋅ u = 0 ),
yielding
0=
1 𝜕( ) 1 𝜕 ( ) 𝜕
r ur +
u + (w).
r 𝜕r
r 𝜕𝜙 𝜙
𝜕z
(3)
Solving for the vertical velocity component w(t) leads to the
well-known expression for the outer solution of a stagnation
point flow
w(z, t) =
𝜕z
= −2a(t)z.
𝜕t
(4)
With the choice of the coordinate system as shown in Fig. 1
(on the right), we can set h = z and the temporal evolution
of the height is thus given by
𝜕h
= −2a(t)h(t).
𝜕t
(5)
Note that within the time interval, we are looking at, the crater surface can be considered to be flat (h(t, r) ≈ hc (t)) with
the subscript c denoting the position on the symmetry axis.
Equation 5 can be integrated provided the temporal decay
of the strength of the stagnation point flow a(t) is known.
The latter can be derived from the momentum equation in
radial direction
[ (
)
]
𝜕2u
𝜕u
𝜕u
𝜕ur
𝜕 1 𝜕rur
1 𝜕p
+ 2r −
.
+ ur r + w r = 𝜈
𝜕t
𝜕r
𝜕z
𝜕r r 𝜕r
𝜌 𝜕r
𝜕z
(6)
Taking into account that ur does not depend upon z (see
Eq. 2) and neglecting viscous forces, surface tension, gravity
and pressure gradient, Eq. 6 simplifies to
𝜕ur
𝜕u
+ ur r = 0.
𝜕t
𝜕r
(7)
69
Page 6 of 23
Experiments in Fluids
(2024) 65:69
The potential flow of an axisymmetric stagnation point
flow enables the use of the separation ansatz ur = a(t)r (see
Eq. 2), which inserted in Eq. 7 yields
𝜕a(t)
+ a2 (t) = 0.
𝜕t
(8)
The latter admits the unique algebraic solution
a(t) =
1
,
const. + t
(9)
which coincides with the solution proposed by Yarin and
Weiss (1995) for the radial velocity distribution (see Eq. 1,
const. = cD0 ∕U0). By inserting Eq. 9 into Eq. 5 and integrating, the solution of Yarin and Weiss (1995) for the height
profile can also be recovered
h(t) =
𝛽
D
(c U0
0
,
+ t)2
(10)
where 𝛽 is the constant of integration. In literature, Eqs. 1
and 10 are often referred to as the inviscid solutions. Based
on the previous derivation, it is clear that both inviscid solutions can have only a limited spatio-temporal validity. In this
work, they are referred to as 1/t and 1∕t2 dependence. This is
because alternative inertia-driven solutions can be derived
by including the effect of the pressure gradient, as shown
in more detail in Sect. 4.4. This is necessary because both
in the classical stagnation point flow and for an impinging
droplet, the re-direction of velocity from the vertical to the
radial direction results in a pressure force that pushes the liquid lamella radially outwards. For the wall film height, corrections due to boundary layer effects have been proposed. In
particular, it has been shown that the liquid film approaches
a residual film thickness hres . For impacts on dry substrates,
Roisman (2009) proposed the following estimation for hres
hres = 0.79 D0 Re−2∕5 .
(11)
Van Hinsberg et al. (2010) extended the previous correlation to impacts on wet surfaces by introducing a dependence
upon 𝛿 , yielding
[
]
hres = 0.98 𝛿 4 + 0.79 D0 Re−2∕5 .
(12)
As a matter of fact, a variation in the initial film thickness
for 𝛿 ≪ 1 has almost no effect on the residual film thickness
hres . This finding was recently corroborated by Stumpf et al.
(2022), who measured hres for droplet impacts onto a very
thin liquid film (0.02 ≤ 𝛿 ≤ 0.06). In the present work, initial film heights of 𝛿 = 0.1 and 0.2 are employed so that both
estimations can be used without any appreciable difference.
Fig. 2 Schematic setup of the experimental facility. It consists of
the droplet generation system, the optical recording system and the
impact area. The fall height H = 0.57 m is defined by the distance
between the needle and the film surface. The mirror angle 𝛼 = 21.2◦
is held constant
Table 1 Properties1 of polystyrol seeding particles
]
[
Particles density 𝜌p kg∕m3
Particle diameter Dp [μm]
Standard deviation of particles [μm]
1050
9.98
0.1
1
Supplier data micro Particles GmbH
2 Experimental method
2.1 Experimental facility
The main components of the facility are the droplet
generation system, the optical recording system and the
impact area. The experimental rig is schematically shown
in Fig. 2 and allows the simultaneous visualization of the
macroscopic impact morphology and the microscopic flow
field within the crown crater. An example of the macroscopic and microscopic recordings is shown in Fig. 3. The
radial velocity is measured with the micro-PIV method
by employing only the microscopic view. As tracers polystyrol particles are used, whose properties are shown in
Table 1.
In order to enable a precise micro-PIV analysis during post-processing, a homogeneous distribution of particles within the droplet and a good reproducibility of the
Experiments in Fluids
(2024) 65:69
seeding is of crucial importance. As the generation of such
a homogeneous mixture can be challenging, the droplet
seeding procedure is described in more detail hereafter.
A syringe pump (0.01 ml/min, Legato 2110, kdScientific)
feeds pure silicon oil through a tubing system towards a
Drifton 1/2" needle. The seeding particles are injected into
this flow at a connector in the vicinity of the needle (see
Fig. 2) by using an additional syringe containing a prepared particle-oil mixture (0.5 g particles added to 40 ml
oil).
This two-needle system not only enables a targeted dosing
of seeding particles, but also prevents particle sedimentation
in the tubing system. Note that the particle concentration
might vary slightly for each droplet impact experiment. This
slight variation, however, does not affect the results of the
micro-PIV analysis and is, therefore, assumed to be negligible. When the weight of the oil-particle mixture, accumulated at the tip of the needle, exceeds the force caused by
surface tension, a droplet detaches. During the free fall, the
droplet passes a light barrier, which triggers the imaging system. The light barrier consists of a 635 nm continuous laser
(1 mW, Laser Components), whose circular beam profile is
transformed into a laser sheet by employing a cylindrical
lens.
The main components of the imaging system are two fully
synchronized Photron Fastcam SA-X2 high-speed cameras
(Macro SA-X2 and Micro SA-X2) and a microscope Axio
Observer Z1 with a 5x objective of the company Carl Zeiss.
The cameras have a CMOS chip (1024 x 1024) with a pixel
size of 20 μ m. Both cameras are set to record 12-bit grayscale images at a frame rate of 12,500 fps. The macroscopic
perspective operates in backlit mode and records the droplet
impact from an inclined top view. As shown in Fig. 2, the
Macro SA-X2 is positioned horizontally next to the impact
area and observes the impact by means of a tilted mirror,
in order to avoid blocking the droplets falling path. A high
power LED (Cree, color temperature of 6500 K, viewing
angle 120◦ , 1,827 lm) is used to illuminate the impact area
(pool). The LED is placed underneath the pool and shines
diagonally upwards onto the tilted mirror. The Macro SA-X2
is equipped with a magnification zoom lens system (NAVITAR). The resulting optical resolution is 19.2 μm/px in the
horizontal direction and 28.5 μm/px in the vertical direction
due to the tilted mirror. The shutter time of the camera is
set to 1 × 10−5 s. The microscopic perspective also operates
in backlit mode and records the droplet impact from a bottom view through the liquid layer. A Constellation 160B
mini 28◦ from the manufacturer Veritas (color temperature
of 5000 K, luminous flux of 16,000 lm) is used as the light
source. The light passes from top to bottom through the
measurement volume, the liquid film and into the inverted
microscope. The optical resolution of the microscopic perspective is 2.44 μm/px and the shutter time of the camera
Page 7 of 23
69
is set to 3.75 × 10−6 s. The impact area consists of a pool
construction, in which the liquid wall film can be placed.
The pool has an inner diameter of 44 mm and its bottom
consists of a sapphire glass disk with a thickness of 1 mm.
The disk surface is considered as plane (S/D-20/10, 𝜆/4)
(Bernard 2020). Sapphire glass was chosen to enhance the
confocal-chromatic film thickness measurements. This technique benefits from larger differences in refractive index 𝜃
between the solid substrate (sapphire glass: 𝜃1 = 1.755) and
the investigated silicon oils (𝜃2 ≈ 1.4).
2.2 Experimental parameter space
The experimental parameter space consists of eight points,
which result from investigating four different silicone oils
(B5, B10, B20, B50) at two different film thicknesses,
respectively. Droplet diameter D0 and impact velocity U0
are kept constant. The parameter space and the liquid properties are shown in Table 2. For each test condition of the
parameter space, each experiment was repeated five times.
An overview of all test conditions is shown in Table 3.
2.3 Analyses of the droplet impact phases
As shown in Fig. 3, the droplet impact process is divided
here into four phases: 𝜏 < 0 , 0 < 𝜏 < 1, 𝜏 > 1 and 𝜏 ≫ 1.
The pre-impact phase (𝜏 < 0 ) is used to determine the initial droplet diameter D0 and impact velocity U0 . The last
ten images of the macroscopic view, preceding the droplet impact onto the film, are used to calculate the impact
velocity. The droplet shadow in the microscopic view is only
sharp and clear in the last two images before impact. The
reason is that the focus plane is set at the interface between
Table 2 Physical properties of test liquids and impact parameters.
The latter have a sample standard deviation of 4.4 %
B5
B10
B20
B50
[
]
Density2 𝜌 kg∕m3
920
945
955
960
Dynamic viscosity2 μ[mNs∕m2 ]
Surface tension2𝜎[mN∕m]
Droplet diameter D0 [mm]
Droplet velocity U0 [m/s]
Film thickness h0 [mm]
4.60
9.45
19.1
48.0
19.2
1.84
3.01
0.182
0.366
0.1
0.20
798
20.2
1.83
3.02
0.184
0.365
0.1
0.20
780
20.6
1.85
3.02
0.184
0.365
0.1
0.20
786
20.8
1.91
3.04
0.184
0.365
0.09
0.19
815
1106
553
280
116
Normalized thickness 𝛿 = Hf ,0 ∕D0[−]
Weber number We = 𝜌U02 D0 ∕𝜎 [−]
Reynolds number Re = 𝜌U0 D0 ∕μ[−]
2
Supplier data QUAX®, T = 298.15 K
69
Page 8 of 23
Fig. 3 Time sequence of the
drop impact of exp. No. 18
(B10, We = 803, Re = 566,
𝛿 = 0.2) recorded with combined macroscopic and microscopic imaging techniques, as
shown in Sect.2.1. (a) Macroscopic image sequence of the
droplet impact onto a thin wall
film resulting into a splashing
outcome and the corresponding
microscopic images of the fluid
flow within the crown base.
(b) and (c) Schematics of the
surface contour at two different
times, representing the early
appearance of the shadowgram
(b) and the growing radius of
the crater area (c), delimited by
the crown wall
Experiments in Fluids
(2024) 65:69
Macroscopic view
2 mm
Microscopic view
2 mm
t = −0.8 ms
τ = −1.30
(a)
t = 0.0 ms
τ =0
shadow
t = 0.8 ms
τ = 1.30
corona wall
corona wall
(b) air bubble
t = 0.4 ms
τ = 0.65
rmax
t = 0.4 ms
τ = 0.65
the pool bottom and film. This shadow is used to determine
the droplet diameter.
The phase shortly after impact (0 < 𝜏 < 1) describes the
initial deformation of the droplet and the formation of the
liquid lamella. During this period the hemispherical outer
shape of the droplet contour reflects the incoming light from
the top LED. Thus, areas with high slope cast a shadow in
the microscopic view, as shown in Fig. 3. In contrast, areas
parallel to the pool surface transmit light, which visualizes
the seeding particles as black dots (≈ 7 px, blurred by the
little out of focus position). The synchronized macroscopic
and microscopic visualization allows to connect the microscale processes to the overall splashing dynamics and facilitates the interpretation of the shadowgrams. During the third
phase ( 𝜏 > 1), the droplet shape flattens and a kinematic
discontinuity (alias the crown wall) is formed, which surrounds a crater with a smooth and plane surface. Finally,
in the late stage of the droplet impact process (𝜏 ≫ 1), the
crown spreads radially and then recedes.
The impacting droplet entraps air that forms a bubble
at the impact center (Chandra and Avedisian 1991; MehdiNejad et al. 2003), which is present in all experiments. As
(c) air bubble
t = 7.2 ms
τ = 11.68
particle cluster
rmax
t = 0.8 ms
τ = 1.30
shown in Fig. 3, it is represented by a larger circular shadow
of approximately 26 px in diameter. Moreover, clusters of
particles may also form, as shown in Fig. 3c. The effect of
these clusters on the velocity field can be neglected. The
microscopic images are used to perform a micro-PIV analysis and extract a two-dimensional map of the radial velocity
within the crown crater. The analysis covers the time interval
1 < 𝜏 < 5. Earlier times cannot be analyzed due to the presence of the droplet shadow.
2.4 Post processing of the experimental data
This section briefly discusses the extraction of the radial
velocity field ur (r, t), its slope a(t) as well as the procedure
to derive the temporal decay of the film height h(t) from the
velocity distribution. In general the velocity of each tracer
particle depends on its radial, angular, and vertical position
within the liquid layer as well as on time, i.e., u(r, 𝜑, z, t).
An in-house Matlab® routine determines the velocity field
ur (r, t) from the microscopic images. Hence, the measured
velocity is averaged over the line-of-sight of the microscopic
view, i.e., in z-direction u(r, 𝜑, t). The depth of field (DOF)
Experiments in Fluids
(2024) 65:69
is of the order of 280 μ m (Bernard 2020) and sufficient
to detect all particles in the liquid layer. In addition, the
velocity field is averaged over the azimuthal angle 𝜑. The
following paragraphs describe how the routine determines
the velocity field. First, a 2D flow field is extracted with
the open source software PIVlab (Thielicke and Stamhuis
2014; Thielicke 2014; Thielicke and Sonntag 2021). PIVlab determines the velocity field in a Cartesian grid with
63 × 63 velocity vectors u(x, y, t) with the following settings.
For image pre-processing, the CLAHE filter (window size
20 px), Wiener filter (window size 3 px) and auto contrast
stretch are enabled. The PIV algorithm uses Fast Fourier
Transformation (FFT) and four passes are chosen, namely,
256, 128, 64, 32 with a step size of 50%. A subset of the
resulting vector field is shown in Fig. 4a. Second, the inhouse routine transforms the velocity field into a cylinder
coordinate system u(r, 𝜑, t). To perform the coordinate transformation the coordinates of the center of impact are used.
The center of impact (see Fig. 4b) is derived by calculating
the intersection point of the streamlines, in analogy to the
particle streak approach by Smith and Bertola (2011).
As shown in Fig. 4a, the detected center of impact is
located closely to the entrapped air bubble, which forms
at the point of droplet impact. Since the droplet spreading
is point symmetric with respect to the center of impact the
velocity field is shown by ur (r, t) . For data reduction the
velocity data is filtered with the following three criteria:
• Vectors located within the crown wall or at larger dis-
tance from the center are excluded.
• Vectors with angular deviation from the radial direction
(> 10◦) are excluded.
(a) vector field
Fig. 4 Post-processing of the velocity field. The white circle in both
images marks the center C of the velocity field. It is derived by calculating the intersection point of the streamlines. (a) Example of the
extracted vector field. The length of each vector represents the magnitude. (b) Two-dimensional map of the velocity magnitude field. Here,
Page 9 of 23
69
• Vectors within a discrete annulus Δr are averaged.
The resulting magnitude field for one time step is shown in
Fig. 4b. The outer (red) circular segment marks the inner
border of the shadow cast by the crown. All velocity vectors
beyond this border are excluded and therefore the magnitude is color-coded in dark blue. The angular deviation for
each velocity vector is calculated by performing the scalar
product with its own position vector from the center. As
an example, the magnitude field shown in Fig. 4b has dark
blue regions at the right and left border, where vectors with
high angular deviation were excluded. The discrete annulus ( Δr = 10 px) is represented by the two black circles
in Fig. 4b. All vectors with the origin inside the discrete
annulus are averaged over the azimuthal angle 𝜑 . Vectors
with a magnitude deviation greater than three standard
deviations (SD) are excluded. The location of the averaged
maximum velocity ur,max is indicated by the black circular
arc in Fig. 4b. For further analysis, all velocity vectors with
radius larger than the radius of ur,max are also excluded, as
they are located in the crown wall and experience a vertical
upward flow. This data reduction procedure is applied to all
time steps.
In Fig. 5, the spatially averaged velocities
(ur (r, t) < ur,max (t)) are shown as a function of radial position
and time, respectively. In agreement with previous studies,
the velocity data exhibit a linear increase with radius and
a nonlinear decrease with time. This representation allows
also to identify the early time limit of our measurement technique. Note that the maximum attainable velocity in r-direction ur,max (t) depends upon the spreading of the droplet itself,
which is a function of time and impact conditions (i.e.,
(b) magnitude field
the annulus Δr is exemplary plotted for one radial step. Moreover, the
radius with the maximum velocity ur,max as well as the inner border
cast by the crown wall are labelid. Test case: exp. No. 6 (B5, 𝛿 = 0.2)
at 𝜏 = 1.03
69
Page 10 of 23
Experiments in Fluids
(a) velocity over radius
(2024) 65:69
(b) velocity over time
Fig. 5 Example of a radial velocity field ur (r, t) extracted from the micro-PIV data. The different representations highlight (a) the linear increase
of the velocity with radius and (b) the nonlinear decrease with time for fixed radii. Test case: exp. No. 6 (B5, 𝛿 = 0.2) for the range 1 < 𝜏 < 5
Re, We, 𝛿). For a specific time (e.g., 𝜏 = 1), this dependency
of the maximum radial velocity upon impact conditions was
also reported by Roisman et al. (2009) for different Weber
numbers and constant Re.
Based on the above considerations, a region of interest
(ROI) is introduced to assure an adequate comparison among
the different experiments. Specifically, the analysis of the
velocity field is restricted to the area extending from the impact
point to the radius r̃ ≤ 0.5, as shown in Fig. 6a. This definition
of the ROI guarantees that even the experiment with the lowest
ur,max provides sufficient data points to achieve statistical significance for the determination of the slope a(t) = 𝜕ur (r, t)∕𝜕r.
The latter is derived by performing a linear fit of the velocity
data at a given t, as shown in Fig. 6b. The parameter a(t) [s−1]
represents the decay rate of the radial velocity. Its functional
(a) linear fit for 6 time steps
Fig. 6 Example of data reduction to extract the decay rate a(t) from
the radial velocity data. (a) Linear fit within the region of interest. (b)
Derived function a(t) and comparison with an exponential fit. The
dependence upon time enables a meaningful comparison with
numerical simulations and theoretical models.
As shown in Fig. 6b, the experimental decay of the parameter a(t) can be well approximated by fitting an exponential
function to the experimental data, yielding
a(t) ≈ c1 exp(−c2 t).
(13)
Following the analysis shown in Sect. 1.1, Eq. 13 can be
employed to derive the decay of the wall film. Specifically,
by inserting the above mentioned exponential function in
the continuity equation (Eq. 5) and integrating yields the
following relation for the film height
[
]
hc (t) = c3 exp 2c1 c−1
(14)
2 exp (−c2 t) .
(b) a(t) over time
coefficients c1 and c2 are shown in Table 3. Test case: exp. No. 6 (B5,
𝛿 = 0.2) for the range 1 < 𝜏 < 5
Experiments in Fluids
(2024) 65:69
Page 11 of 23
The constant of integration c3 is determined from Eq. 12 by
setting the residual film thickness hres as asymptotic boundary condition, yielding
lim hc (t) = c3 = hres .
t→∞
(15)
A list of the coefficients c1, c2 and c3 can be shown in Table 3
for each experiment.
3 Numerical simulation
This section briefly describes the in-house multiphase
flow solver Free Surface 3D (FS3D), employed for the
direct numerical simulation (DNS) included in this work.
A more detailed description and its validation for droplet
film interactions can be found in Fest-Santini et al. (2021),
Rieber and Frohn (1999), Steigerwald et al. (2021). FS3D
solves the equations for mass and momentum conservation
𝜕t 𝜌 + ∇ ⋅ (𝜌u) = 0,
(16)
𝜕t (𝜌u) + ∇ ⋅ (𝜌u ⊗ u) = ∇ ⋅ (S − Ip) + 𝜌g + f𝛾
(17)
on finite volumes, where 𝜌 denotes the density, u denotes the
velocity vector, p the static pressure, g the acceleration due
to gravity, S the shear stress tensor and I the identity matrix.
The term f𝛾 models surface tension forces at the interface
between the gaseous and the liquid phase. The interface is
captured by using the volume-of-fluid (VOF) method, which
introduces an additional scalar variable f with 0 ≤ f (x, t) ≤ 1,
representing the liquid volume fraction in each control volume (Hirt and Nichols1981). The VOF-variable f is advected
within the computational domain by solving an additional
transport equation.
69
In the present investigation, we are interested in a comparison of the height-averaged radial velocity within the
lamella between the simulation and experimental data. At
this point one has, however, to keep in mind that only the
droplet is seeded with particles and that the experimentally
evaluated velocities thus only belong to the layer of droplet liquid. As the traditional VOF-variables cannot be used
to distinguish between different liquids, we use the multicomponent framework of FS3D, in which additional VOF
variables 𝜓i = Vi ∕V are introduced, representing the volume
fraction of species i inside the liquid phase. The scalars 𝜓i
are advected simultaneously and in a closely coupled way
to f. In this way we can identify both droplet and film liquid.
For more details about FS3D the reader is referred to Eisenschmidt et al. (2016) and Steigerwald et al. (2021).
The computational setup is identical to the one shown
in Steigerwald et al. (2021) and is therefore only shortly
described. We simulate a quarter of the impact scenario
within a cubic computational domain with an edge length of
7D0. A perfect spherical droplet exhibiting an initial velocity
U0 towards the film is initialized in a distance of 2D0 above a
quiescent film. The domain is discretized rectilinearly with
10243 grid cells. In the impact region, the grid resolution
corresponds to 256 grid cells per droplet diameter D0 . The
simulated impact scenario corresponds to exp. No. 18, which
uses silicon oil B10. The impact conditions are shown in
Table 3. The properties of the surrounding medium are set
to those of ambient air. Figure 7 shows a comparison of the
impact morphology between the exp. No. 18 (top) and the
numerical result (bottom) for several points in time.
The growth and the shape of the numerical reproduced
crown is very similar to the experimentally observed one.
The experimental figures show that the impact scenario lies
within the deposition regime as no liquid fingers occur at
the crown rim. Furthermore, the seeded particles within the
droplet liquid are visible as tiny dots within the crown crater
Fig. 7 Comparison of the impact morphology between exp. No. 18 (B10, 𝛿 = 0.2) and the DNS for several points in time. The minimal temporal
mismatch is due to the finite sampling rate in the experiments and numerical simulations
69
Page 12 of 23
and the crown wall. The impact regime is well reproduced
by the numerical simulation. The only visible difference
between the simulation and the experiment is the shape of
the crown rim as it does not show the bulges observed in the
experiment. The reason for their absence in the simulation
is a premature disintegration of the crown rim, which takes
place slightly before 𝜏 = 1.0 . At this point, the crown wall
is too thin for a proper interface reconstruction. This leads
to an artificial ejection of mass, which is then missing in the
rim. However, this difference is irrelevant, since the crown
top, which is the most difficult feature of drop film interactions to reproduce numerically, is not of interest in this
study. As we will show in the following, the grid resolution
is indeed sufficiently fine to extract precisely velocity data
from the lamella with respect to both the wall film and the
droplet liquid layer.
4 Results and discussion
This section discusses the PIV experimental results in the
framework of a holistic approach that includes a comparison
with predictions from DNS, theoretical models and experimental measurements of the wall film thickness. Consistency
of results from different approaches provides a robust and
indirect validation of the PIV measurements. The analysis
focuses on the temporal evolution of the central film height
h(t), the radial velocity ur (t) and its decay rate a(t). The
implications of our experimental findings on the required
improvements of theoretical models are also discussed.
4.1 Comparison of the total film height
This section presents a comparison between the experimentally derived film height, numerical predictions and
literature data that include both experiments and theoretical
results. Experimental data mainly exist for droplet impact
onto dry walls, where the decay of the droplet central peak
was measured by means of the Fourier transform profilometry technique. A detailed description of the experiments can be found in Lagubeau et al. (2012). Similarly
to this work, the authors systematically varied the dynamic
viscosity of the test fluids, resulting in a variation of the
impact Reynolds number by two orders of magnitude (i.e.,
O(10) < Re < O(103 )). Lagubeau et al. (2012) identified a
self-similar inertial regime, which is independent from Re ,
followed by a viscous regime, where the minimal thickness
is limited by the growth of the boundary layer. Consequently,
the residual film thickness increases with decreasing Re , in
agreement with the findings from van Hinsberg et al. (2010).
Our hypothesis is that these findings should be valid also for
droplet impact onto a wet surface, since the latter experiences significantly smaller wall shear stresses (Frommhold
Experiments in Fluids
(2024) 65:69
et al. 2015). To verify this assumption, Fig. 8a compares
the studies by Lagubeau et al. (2012) ( Re = 2690 ), Roisman et al. (2009) ( Re = 1068 ) and Eggers et al. (2010)
( Re = 400 ) on droplet impacts on dry surfaces with one
test case from this work (B10, 𝛿 = 0.2 , Re ≈ 550 ). For the
B10 test case, both DNS and the experimentally derived
film height (i.e., Eq. 14) are included. To compare the data
adequately, each height shown in Fig. 8a is non-dimensionalized with D0 + Hf ,0 . Additionally, Fig. 8a shows also the
theoretical line hfreefall of the droplet apex of a free fall.
As expected, despite the large variation in Re numbers, all
data collapse into a single curve in the inertial regime. Only
towards the end of the measurements do small differences
become visible, as the viscous regime is slowly entered. As
shown in the inset of Fig. 8a, the simulation data from Eggers et al. (2010) ( Re = 400) are the first to depart from the
inertial solution. This is in agreement with the experimental
findings of Lagubeau et al. (2012), who found that the highest the Re number the later the viscous regime is entered. In
particular, for Re ≈ 2100, Lagubeau et al. (2012) found that
the viscous regime is entered in the range of 5 < 𝜏 < 6 for
impact onto dry walls. For impact onto a wet surface, due to
the reduced wall shear stress (Frommhold et al. 2015), the
start of the viscous regime is delayed. This explains why our
B10 test case ( Re ≈ 550 ) matches so well the high Reynolds data ( Re = 2690) from Lagubeau et al. (2012) for dry
wall impact both with respect to the DNS and the empirically derived composite exponential function (i.e., Eq. 14).
This agreement among experiments and numerical simulations from different authors indirectly corroborates not only
the accuracy of Eq. 14, but also the procedure to recover
the temporal evolution of the central film height from the
empirically determined decay rate a(t). The small deviations
observed in the time interval 1 < 𝜏 < 2 are due to the different evaluation of h(t) between the DNS and the experiment.
The height h(t) of the DNS is defined directly by the reconstructed surface, whereas the height from the experiments is
derived with Eq. 14. In addition, as discussed in section 4.2,
the simulation can evaluate both liquid layers, unlike the
experiment where only the droplet liquid is measured.
Having established that the film height decay occurs predominantly in the inertia-controlled regime, it is interesting
to compare the widely accepted 1∕𝜏 2-dependence with our
DNS and experimentally derived results. This comparison is
shown in Fig. 8b, which includes two variants of the Yarin
and Weiss solution and additionally a non-dimensionlized
composite exponential fit of the DNS data, defined as
h(t)∕D0 = ĉ 3 exp (̂c1 exp (−̂c2 𝜏))
(18)
All three curves are fitted for the same period of time
1 < 𝜏 < 5 to the DNS data. As shown by the bold dashdotted line, the numerical data can be fitted very accurately
Experiments in Fluids
(2024) 65:69
(a) dry wall literature data vs exp. No. 18
Page 13 of 23
69
(b) different functions fitted to exp. No. 18
Fig. 8 Temporal decay of the top central point of the drop surface.
Reference test case: B10, 𝛿 = 0.2, Re ≈ 550. For the reference test
case, the predictions from DNS and the experimental correlation
(Eq. 14) are compared to (a) literature data on dry wall impact and to
(b) semi-empirical models, based on Eq. 10 or Eq. 18 and fitted to the
DNS data. The respective fitting parameters (ĉ 1 , ĉ 2 , ĉ 3 , 𝛽̂1 , 𝛽̂2 , ĉ 4) are
calculated with the least square method. For Eq. 14, the constants c1,
c2 and c3 are averaged from exp. No. 16-20 (Table 3). Experimental
dry wall data are from Lagubeau et al. (2012) (Re = 2690) as well as
simulation data from Roisman et al. (2009) (Re = 4010) and Eggers
et al. (2010) (Re = 400). The dashed line hfreefall indicates the free fall
propagation
by using Eq. 18, thus corroborating the proposed functional
dependence upon time. If no constrains are attached to the
fitting parameters ( ĉ 1 , ĉ 2 , ĉ 3), the corresponding L2 norm
evaluated by means of the Matlab® function ’lsqcurvefit’
is small ‖residual‖2 = 4.938 ⋅ 10−3. The other two fits were
obtained by using the Yarin and Weiss solution of the film
height with one free parameter 𝛽̂1 (c = 0) or two free parameters 𝛽̂2 and ĉ 4 , respectively. Even though the latter (i.e.,
ĉ 4 ≠ 0) provides a better fit to the DNS data, the validity of
the (𝜏 −2) dependence is temporally limited. Indeed, for 𝜏 > 3
a significant deviation from the DNS data is observed. As
pointed out already in Sect. 1.1, the ( 𝜏 −2 ) dependence for
the height decay was derived by assuming an unbounded,
asymptotic velocity ( ur = a(t)r ) constrained only by the
requirements of mass and momentum conservation. Consequently, as the liquid lamella expands radially without any
limitation, the film height must rapidly decrease to zero, in
order to fulfill the above mentioned requirements. In reality,
the liquid lamella is accelerated by the pressure force created
by the impinging droplet. This pressure force is not constant,
but decreases in time after having reached its peak value.
The viscous forces, on the other hand, become increasingly
dominant with decreasing wall film height and eventually
bring the spreading liquid lamella to rest. The composite exponential function, found empirically in this work,
reflects this complex interplay between decaying pressure
force and increasing relative importance of viscous losses.
To conclude this section, we explicitly point out that both
time dependencies, namely 𝜏 −2 or the composite exponential
function, are not valid for the early times (i.e., 𝜏 < 1), where
the droplet central peak follows the freefall regime as shown
in Fig. 8a.
4.2 Analysis of the radial velocity field
In the following we compare the experimentally measured
radial velocity field with results from direct numerical simulations. In the experiments, only the liquid droplet is seeded
with tracer particles. Hence, for a meaningful comparison,
the numerical data are also averaged over the portion of the
lamella that corresponds to the thickness of the liquid drop
layer. Figure 9 shows the radial velocity ur (r, t) as function
of r for selected points in time and a very good agreement
is found among the corresponding numerical predictions.
This demonstrates that two-dimensional velocity fields
can be measured with great accuracy with the described
micro-PIV setup. The PIV results show a low level of scattering and the DNS accurately reproduce the observed
temporal decay of the radial velocity. The latter increases
linearly with radius and its slope remains almost constant
as long as the velocity field is not influenced by the flow at
the crown base. In such a case, the slope starts to deviate
69
Page 14 of 23
Fig. 9 Comparison between numerical and experimental radial velocity distributions ur (r, t) for selected time steps. Test case: exp. No. 18
(B10, 𝛿 = 0.2). The DNS data are averaged over a portion of the
lamella corresponding to the thickness of the liquid drop layer and
written out at a time interval of Δ𝜏 = 0.1. The experimental data,
instead, are obtained at a fixed frame rate, which corresponds to a
inter frame time interval of Δ𝜏exp ≈ 0.13. This explains the small time
difference between the numerical and experimental data
owing to the vertical upward flow into the crown wall. This
phenomenon is also observed in the experiments, where the
deviations from the linear profile start at slightly smaller
radial distances in comparison to the DNS data, especially
during early times. These differences are caused by the
evaluation routine, since the extraction of the radial velocity becomes more difficult near the crown wall as shown in
Fig. 4b.
To evaluate the effect of the wall film thickness on the
averaging of the radial velocities, the DNS data have been
additionally averaged over the total thickness of the lamella
(both liquids) and over the thickness of the wall film only
(denoted as film liquid). The results of this exercise are
shown in Fig. 10 for the range 0 ≤ r̃ ≤ 0.5. As reference
test case, exp. No. 18 is chosen (B10, 𝛿 = 0.2 ). As can be
seen, the slopes obtained by averaging on the droplet-liquid
lamella and on both liquids lamella decrease in a very similar way, because the lamella consists mainly of droplet liquid. The decrease of the slope of the film liquid lamella is
stronger than that of the droplet liquid lamella due to wall
effects. For comparison, the evaluated slopes from exp.
No. 16–20 are also shown. The experimental data nestle
closely to the slope of the droplet liquid up to 𝜏 ≈ 3. For
later times, in the experiments the temporal evolution of the
slope approaches and follows closely the slope of the full
lamella. This occurs due to the height decay of the liquid
lamella so that even the droplet layer weakly experiences the
retarding effect caused by viscous losses. Apart from these
small deviations, the overall good agreement between the
numerical and experimental data confirms once again the
Experiments in Fluids
(2024) 65:69
Fig. 10 Evaluation of the slope a(t) from DNS data by employing
different portions of the liquid lamella: droplet liquid, film liquid and
both liquids (i.e., total height). Test case: exp. No. 18 (B10, 𝛿 = 0.2).
For comparison, the (𝜏 −1) dependence (i.e., Eq. 1) with two different c values (c = 0, c = 0.25) is plotted together with experimental
slopes, evaluated from exp. No. 16–20
accuracy of the velocity measurements. For comparison, the
decay rate predicted by the (𝜏 −1) dependence (i.e., Eq. 9) are
also shown in Fig. 10 for two different values of the constant
c. The direct comparison of the decay rate a(t) highlights
even more clearly the limited validity of the (𝜏 −1) dependence, alias an inviscid solution without pressure gradient.
The latter can provide reasonable estimations only in the
early times (i.e., 1 < 𝜏 < 2), as shown in Sect. 4.1.
4.3 Comparison of a(t) with analytical models
The preceding discussion highlighted the complex interplay between the pressure force, generated by the impinging droplet, and the shear stresses in the near-wall region.
This system of counteracting forces de facto controls the
temporal decay of both the radial velocity ur (t) and the wall
film height h(t). However, it is still unclear which of the two
components plays the dominant role and essentially dictates
the exponential decay of the slope a(t) in the first place and
subsequently of all other variables. The answer to this question is not straightforward, since both the pressure force for
droplet impact on wet substrates and the viscous forces are
not known a priori. Moreover, the viscous forces are not
only dependent upon the fluid viscosity, but also upon the
velocity gradient normal to the wall, which varies with the
impact conditions. In order to obtain an estimation of the
viscous losses and evaluate their effect on the decay of the
slope a(t), the following procedure is applied. As a first step,
Experiments in Fluids
(2024) 65:69
the theoretical (inviscid) model from Yarin and Weiss (1995)
is taken as a reference (i.e., Eq. 1). As a second step, a viscous correction is applied to the inviscid solution, in order
to evaluate how viscous losses affect the temporal decay of
the function a(t). The details of this evaluation procedure
are explained hereafter.
As a starting point, we point out that Eq. 1, the 𝜏 −1
dependence, has been widely used in literature. Many
authors simply adjusted the constant c to obtain an improved
match to their own experimental or numerical data. Hereafter, three different options are considered, namely the values
c = 0, c = 0.25 and c = 2.49, respectively. The corresponding decay rate is denoted with aYW (t) as a reminder that they
refer to the inviscid solution from Yarin and Weiss (1995).
The corresponding profiles are shown in Fig. 11 together
with the experimentally derived decay rate a(t) for the test
case exp. No. 6 (B5, 𝛿 = 0.2). As can be seen, higher values
of the c parameter lead to a less steep decay of the function a(t) due to the fact that the start value is significantly
quenched. For completeness, the reasons for selecting the
values of the constant c are summarized hereafter. The
value c = 0, for instance, was chosen by Smith and Bertola
(2010, 2011), who studied Newtonian and non-Newtonian
droplets impacting onto a dry hydrophobic surface. The
value c = 0.25 was proposed by Roisman et al. (2009), who
investigated inertia dominated axisymmetric drop collisions onto a dry substrate and onto another sessile droplet.
The correlation with c = 0.25 was validated in the range
0.7 < 𝜏 < 𝜏viscous. The value for 𝜏viscous can be estimated with
Page 15 of 23
correlations from Roisman (2009) and Stumpf et al. (2022).
The value c = 1∕𝜆 = 2.49 is obtained from the empirical
correlation, proposed by Gao and Li (2015) and applied to
the test conditions of exp. No. 6 (B5, 𝛿 = 0.2)
𝜆 = 0.26(Re0.05 We−0.07 𝛿 −0.34 ).
(19)
As explained in Sect. 1, Lamanna et al. (2022) proposed
to use this empirical correlation (i.e., Eq. 19) to estimate
the kinetic energy effectively transmitted to the radially
spreading lamella, deprived of impact losses. The latter are
associated to dissipative effects induced by the strain rate
during the deformation of the impinging droplet and by the
conversion of kinetic energy into surface (potential) energy.
As a second step, the viscous correction developed by
Lamanna et al. (2022) is applied to two inviscid formulations, corresponding to the potential flow solutions obtained
by setting c = 0.25 and c = 2.49 in Eq. 1, respectively. The
viscous correction is obtained by solving numerically the
unsteady momentum balance equation in the radial direction
for an axisymmetric flow. The boundary layer flow admits a
self-similar solution and enables the calculation of a profileaveraged velocity ū vis . The latter encompasses the momentum losses in the boundary layer. In non-dimensional terms,
this can be expressed by the ratio 𝜆1 (t) = ū vis ∕ur∞, which
measures the specific momentum reduction compared to the
asymptotic velocity. Thanks to the self-similarity of the solution, this approach can be applied to any position within the
impact zone, yielding
ur = aYW (t) 𝜆1 (t) r = aL (t) r.
Fig. 11 Comparison among the experimentally derived function a(t)
and the predictions from different formulation of the inviscid model
of Yarin and Weiss, obtained by specifying different value for the
constant c, specifically c = 0, c = 0.25 and c = 2.49. In addition, the
viscous correction from Lamanna et al. (2022) (Eq. 20) is applied to
two different formulation of the inviscid model, namely c = 0.25 and
c = 2.49. Test case: exp. No. 6 (B5, 𝛿 = 0.2)
69
(20)
Here, the parameter aL (t) is introduced as a reminder that the
potential flow solution has been adjusted to include the effect
of viscous losses. The results of this exercise are also shown
in Fig. 11 and lead to the following conclusions: First, as
stated above, the relative importance of the boundary layer
correction is strongly dependent upon the velocity gradient
in the normal direction to the wall. For the case c = 2.49, the
strongly quenched values of the potential flow ur lead to very
reduced viscous losses. A stronger effect is observed for the
test case c = 0.25 due to the higher values of the parameter
a(t). Second, the inclusion of viscous losses provides only a
correction to the potential flow (inviscid) solution. In other
words, they do not dictate the functional dependence of the
parameter a(t) upon time. The validity of this statement
is verified in Sect. 4.4. Based on these findings, it follows
that the exponential decay, observed experimentally, for the
parameter a(t) must be associated to an exponential decay of
the pressure force. First confirmations of this statement are
provided by the numerical simulations of Roisman (2009)
and by the experiments of Mitchell et al. (2019). In both
works, the authors investigated the transient force profile of
a low-speed droplet impinging on a dry wall at the center of
69
Page 16 of 23
symmetry. In particular, Mitchell et al. (2019) modeled the
transient force as follows
√
F(t) = A1 A2 𝜏 e−A2 𝜏 .
(21)
Here, A1 and A2 are constants that depends upon the impact
conditions. Mitchell et al. (2019) found that, for early times
√
(i.e., 𝜏 ≤ 0.1), Eq. 21 behaves like 𝜏 . This phase corresponds to the creation of a peak pressure force. For 𝜏 > 0.1,
the transient force exhibits an exponential decay with time.
This finding is consistent with our measurement of the decay
rate a(t) for 𝜏 > 1. The interdependence between pressure
and decay rate a(t) is further explored in Sect. 4.4.
4.4 Parametric study on the effects of Re number
We vary systematically the dynamic viscosity of the liquid
μ and analyze the effect on the evolution of the slope a(t).
For this purpose, the radial velocity distribution has been
measured for eight different impact scenarios by varying the
Reynolds number Re ∈ {116, 280, 554, 1106} and the film
Experiments in Fluids
(2024) 65:69
thickness 𝛿 ∈ {0.1, 0.2}, while keeping the Weber number
constant (i.e., We ≈ 800 ). For each impact scenario, the
experiment was repeated five times, which resulted in 40
experiments. Details about each experiment can be found
in the appendix (see Table 3). In Fig. 12, the radial velocity of the four most different test cases is presented, corresponding to the silicon oils B5 and B50 for 𝛿 = 0.1 and
𝛿 = 0.2. In all cases, the radial velocity ur increases linearly
with the radius regardless of the initial film thickness and
Reynolds number. In this representation, the variations in the
radial velocity distribution with the initial film thickness 𝛿
are hardly discernible. This is because the investigated film
thicknesses 𝛿 induce small wall shear stress (Frommhold
et al. 2015). In some cases, the effect of decreasing the
Reynolds number can be visible even at earlier times (i.e.,
0.6 < t < 2.5 m/s), resulting in a faster temporal decay of
the radial velocity. This is clearly visible when comparing
the B50 ( Re = 115) and B5 ( Re = 1142 ) test cases (e.g.,
for 𝛿 = 0.2 ). At r̃ = 0.5 and t = 1.56 ms (pink marker), the
radial velocity has already fallen below ur = 0.25 m/s for
B50, whereas for B5 the radial velocity is still ur ≈ 0.4 m/s.
(a) δ = 0.2, B5, Re = 1142 (exp. No. 6)
(b) δ = 0.1, B50, Re = 115 (exp. No. 36)
(c) δ = 0.2, B5, Re = 1088 (exp. No. 3)
(d) δ = 0.1, B50, Re = 117 (exp. No. 33)
Fig. 12 Radial velocity fields for two different viscosities (B5, B50) and two different initial film heights 𝛿 = 0.2 and 𝛿 = 0.1
Experiments in Fluids
(2024) 65:69
This behavior is shown even more clearly in Fig. 16 and is
rather unexpected. Indeed, in the inertial regime (typically
valid for 1 < 𝜏 < 2), the velocity decay should be independent of Re (see e.g., Eq. 1), because the shear strain rate does
not play a significant role yet.
In order to better visualize the influence of Re and initial film thickness 𝛿 , the experimentally derived slopes a(t)
of all 40 experiments are shown in Figs. 13 and 14 in a
normal and in a semi-logarithmic scale, respectively. From
both plots, the following main conclusions can be drawn.
First, at 𝜏 ≈ 1, the starting value for the slope a(t) seems
to be independent of Re and 𝛿 . Second, the decrease of a(t)
with time is stronger for lower Re , albeit all curves tend
towards zero. Third, the experimental results can be reasonably fitted using Eq. 1 (i.e., 𝜏 −1 dependence) only for
a short time period, e.g., with c = 0 for B5 and 1 < 𝜏 < 2 .
Fourth, with increasing fluid viscosity, the period of validity
of Eq. 1 decreases rapidly, as reported also by Bakshi et al.
(2007). Fifth, starting from 𝜏 = 1 the slope a(t) shows a linear trend in the semi-log plot of Fig. 14, which indicates an
exponential decrease. Moreover, the influence of the initial
thin film height (𝛿 = 0.1, 0.2) on the slope a(t) is negligible.
These findings are consistent with the inception of an inertial
regime, where the velocity decay rate is independent upon
Re and 𝛿 . However, around 𝜏 = 2 all curves exhibit a slight
kink. The latter is associated to the inception of the viscous
regime, which leads to stronger deviations for thinner films
due to increased relative importance of viscous losses. The
only noteworthy exception to this behavior is observed for
Page 17 of 23
69
Fig. 14 Semi-logarithmic plot of the slope a(t) from Fig. 13. The
𝜏 −1 dependence (i.e., Eq. 1 with c = 0) is plotted as a dashed black
curve. The 𝜏 values are calculated by averaging D0 and U0 over the
exp. No. 1-40
the B50 test cases, where a clear dependence upon Re in the
otherwise inertial regime is detected.
In the following, a detailed analysis is presented, aiming to provide a sound physical explanation to the above
mentioned experimental findings. As a first step, Fig. 15
illustrates the variation of the fitting coefficients c1, c2 (see
Eq. 13) and their ratio as function of the initial Reynolds
number. The larger scatter of c1 for small Re is due to the
significant scatter of a(t) at early times, as shown in Fig. 13.
The fitting coefficients for each experiment are shown in
Table 3. As can be seen, both empirical parameters decrease
1
7000
0.8
6000
5000
0.6
4000
0.4
3000
2000
0.2
1000
0
Fig. 13 Influence of viscosity and non-dimensional film thickness on
the slope a(t). All eight test cases are plotted (exp. No. 1-40). Results
for different Reynolds numbers are indicated by color and for different film heights by the line style. In addition, the 𝜏 −1 dependence (i.e.,
Eq. 1 with c = 0) is plotted as a dashed black curve. The 𝜏 values are
calculated by averaging D0 and U0 over the exp. No. 1-40
200
400
600
800
1000
0
1200
Fig. 15 Mean values of the empirical coefficients c1, c2 and their ratio
as function of Re for the eight impact scenarios (exp. No. 1-40). The
error bars of c1 and c2 represent the min and max value of each test
case
69
Page 18 of 23
Experiments in Fluids
with increasing Re . The decay is nonlinear and seems to
evolve towards an asymptotic value with increasing Re. Both
coefficients are almost insensitive to variations in the initial
film thickness 𝛿 and their ratio remains basically constant
(c2 ∕c1 ≈ 0.4) over the parameter space investigated in this
work. These functional dependencies are already an indicator that the empirically derived exponential function is
indeed suited to describe the inertial regime, particularly for
Re > 300. The constancy of the ratio c2 ∕c1 seems to suggest
a direct dependence upon the Weber number, which is held
constant in the present work.
As a second step, the reciprocal of the empirically derived
exponential function (i.e., Eq. 13) is approximated using a
Taylor’s expansion, yielding
a(t)−1 =
c
(c t)2
1
1
exp (c2 t) =
+ 2t + 2 + ⋯.
c1
c1 c1
c1 2!
(22)
Likewise the reciprocal of Eq. 1 reads
aYW (t)−1 = const. + t.
(23)
By comparing Eq. 23 with the Taylor series (Eq. 22), it is
clear that the (t−1)-dependence for the inertial regime represents the first-order approximation of the exponential function. Moreover, the slope of the first-order approximation is
c2 ∕c1 ≈ 0.4 for the investigated cases and not one, as commonly assumed in Eq. 23 (or alternatively in Eq. 1). Note
that Smith and Bertola (2011) found c2 ∕c1 = 0.81 for a dry
impact with We = 75. Based on the previous remarks, this
would imply that the slope of the first-order model for a(t)
depends upon the Weber number only, alias upon the relative importance of inertial forces and surface tension. It is
also clear that the use of a single constant in Eq. 23 cannot
simultaneously satisfy the Re and We dependency observed
experimentally in Eq. 22. This also explains why, in literature, the constant (c) in Eq. 1 was continuously adapted to
fit different datasets. The higher order derivatives can only
be neglected when (c2 t) ≪ 1. In the considered time interval
t = (0.6 − 3) ms, the terms (c2 t)n with ( n > 2 ) are not negligible for our parameter space, thus explaining the faster
decay of a(t) compared to the (t−1)-dependence. As a third
step, the momentum balance equation in direction normal
to the wall (i.e., z-axis) is considered. Pressure is integrally
transmitted through the boundary layer, so that the analysis
can be simply restricted to the potential flow region, yielding
(
)
𝜕p
𝜕w
𝜕w
𝜕w
=− .
+ ur
+w
𝜌
(24)
𝜕t
𝜕r
𝜕z
𝜕z
For an unsteady and axisymmetric stagnation point flow,
the velocity components for the potential flow region are
ur (r, t) = a(t)r and w(z, t) = −2a(t)z (see Eqs. 2 and 4).
(2024) 65:69
Inserting these expressions into the momentum balance
yields
(
)
𝜕p
𝜕w
𝜕w
=− ,
+w
𝜌
(25)
𝜕t
𝜕z
𝜕z
[
]
𝜕p
𝜌 −2a� (t)z + 4a2 (t)z = − .
𝜕z
(26)
Integration of Eq. 26, from the wall ( z = 0) to an arbitrary
height z, by separation of variable gives
∫0
[
p(t,r,z)
z
[−2a� (t) + 4a2 (t)]̃z d̃z = −
1
dp,
𝜌 ∫p0 (t,r)
]
]
1[
−a� (t) + 2a2 (t) z2 = p0 (t, r) − p(t, r, z)
𝜌
(27)
(28)
with the boundary condition p = p0 (t, r) at z = 0 and p0 (t, r)
denoting the stagnation pressure at the wall. Hence, for an
arbitrary height position z, the temporal decay of a(t) must
follow the exponential decay in time of the pressure. This
conclusion is not only consistent with our experimental
findings, but also physically plausible. The finite mass and
momentum of the impinging droplet is inevitably associated to the rapid decay of the driving force for the motion
of the lamella. The exponential decay of the pressure force
has been corroborated both experimentally and numerically
by Mitchell et al. (2019), Yu and Hopkins (2018), Yu et al.
(2022) and Roisman et al. (2009). In physical terms, the
limited temporal validity of the ( t−1)-dependence can be
therefore ascribed to the omission of the pressure term, as
pointed out already in Sect. 1.1.
Equally interesting insights can be derived by solving the
unsteady momentum balance equation in radial direction and
restricting it to the potential flow region. The exact derivation is presented in appendix. Hereafter, only the main findings are summarized. By inserting the potential flow solution
ur = a(t)r with a(t) given by Eq. 13, it is found that the pressure profile in radial direction has the shape of a parabola
with downward concavity. Its curvature decreases in time till
it almost resembles a plateau. These theoretical evaluations,
based on the empirically found exponential decay of a(t),
agree qualitatively very well with the numerical simulations
of Roisman et al. (2009). This indirectly provides an additional confirmation on the plausibility of the experimental
findings and associated analysis.
One last aspect to clarify is the Re dependency of the
slope a(t). As shown in Fig. 14, for Re ≈ 100 this occurs
already at 𝜏 ≈ 1, when the lamella radial spreading is mainly
controlled by pressure and inertial forces even for droplet
impact on a dry wall (Mitchell et al. 2019). The close interdependence between a(t) and pressure established in this
Experiments in Fluids
(2024) 65:69
work enables us to provide a physical explanation for this
behavior. Following Roisman et al. (2009), the pressure in
an expanding liquid lamella can be expressed as
]
[
𝜕ur ur
+
p = p𝜎 − 2μ
,
(29)
𝜕r
r
where p𝜎 represents the additional pressure contribution due
to the surface curvature, as predicted by the Young-Laplace
equation (Roisman et al. 2009). Equation 29 implies that
in presence of extensional (normal) strain due to the radial
spreading of the lamella the effective pressure is reduced.
At high Re numbers, the pressure loss due to the extensional
strain is negligible. Consequently, the slopes a(t) collapse
to a single curve in the inertial regime, as shown in Fig. 14.
Deviations appear as soon as viscous losses are no longer
negligible in presence of significant shear strain rate. At low
Re numbers, instead, the pressure losses caused by the extensional strain are so high that the decrease in a(t) (alias in the
radial velocity ur ) is immediately evident even outside of the
boundary layer flow.
5 Conclusion
In this study, the simultaneous recording of the macroscopic
droplet impact dynamics together with micro-PIV measurements within the crown crater is performed for the first time.
Thanks to this optical configuration, it is possible to investigate the characteristic of the radial velocity distributions
across all impact regimes, namely deposition, transition and
splashing. The study additionally includes a systematic variation of the impact Reynolds number by one order of magnitude (O(102 ) < Re < O(103 )), while We is kept constant
( We ≈ 800). The overarching goal is to gain better physical
insights into the complex interplay between pressure force,
shear and normal stresses on the evolution of the radial
velocity distribution within the crown crater.
In agreement with literature data, our measurements
confirm the linear radial increase of the spreading velocity
within the crown crater. A discrepancy, instead, is found on
its temporal evolution, which exhibits an exponential decay.
To explain this behavior, a holistic approach is chosen which
foresees an integrated analysis among numerical simulations
and analytical models. The following major conclusions are
found: In the lamella’s velocity and height decay, it is possible to identify a self-similar inertial regime and a viscous
regime. In the self-similar inertial regime, the decay is independent of Re and 𝛿 , and is controlled by the exponential
decay of the pressure force. The only noteworthy exception
Page 19 of 23
69
is found at low Reynolds number, where the extensional
(normal) strain leads to enhanced pressure losses even outside of the boundary layer flow. The inception of the viscous
regime corresponds to a kink in the exponential decay. The
strength of this deviation depends upon Re and 𝛿 . The commonly used t−1 dependence for the inertial regime represents
only a first-order approximation of the exponential function
and has therefore limited validity.
Appendix
Radial momentum balance
For an unsteady and axisymmetric laminar stagnation point
flow, the velocity components for the potential flow region
read as: ur (r, t) = a(t)r and w(z, t) = −2a(t)z. Inserting these
expressions into the momentum balance in radial direction
yields
)
(
𝜕ur
𝜕ur
𝜕ur
𝜕p
+ ur
+w
=− ,
𝜌
(30)
𝜕t
𝜕r
𝜕z
𝜕r
[
]
𝜕p
𝜌 a� (t) + a2 (t) r = − .
𝜕r
(31)
Integration of Eq. 31, from the stagnation point (r = 0 ) to
an arbitrary position r, by separation of variables gives the
following functional pressure dependency
p(t, r) =
𝜌 � 2 𝜌 2 2
a (t)r + a (t)r + c4
2
2
(32)
with c4 = p(t, r = 0) = p0 (t) denoting the stagnation pressure
at the wall. Inserting the exponential decay for the slope
a(t) = c1 exp (−c2 t) yields
p(t, r) =
𝜌
𝜌
c c exp (−c2 t)r2 + c21 exp (−c2 t) exp (−c2 t)r2 + p0 (t),
2 1 2
2
(33)
p(t, r) =
[
]
𝜌 2
c r exp (−c2 t) c1 exp (−c2 t) − c2 +p0 (t),
2 1
(34)
�������������������������
<0
p(t, r) = −f (t)2 r2 + p0 (t).
(35)
As can be immediately seen, the pressure profile in radial
direction has the shape of a downward concave parabola. Its
curvature decreases in time, due to the exponential decay of
a(t). This finding is consistent with the numerical simulations of Roisman et al. (2009) and provides a physical explanation for the numerical simulations.
69
Page 20 of 23
Table 3 Experiment impact
parameters
Experiments in Fluids
(2024) 65:69
No
Liquid
Hf ,0 [mm] D0 [mm] U0 [m/s] 𝛿 [−]
Re [−] We [−] c1 [1∕s]
c2 [1∕s]
c3 [mm]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
B5
B5
B5
B5
B5
B5
B5
B5
B5
B5
B10
B10
B10
B10
B10
B10
B10
B10
B10
B10
B20
B20
B20
B20
B20
B20
B20
B20
B20
B20
B50
B50
B50
B50
B50
B50
B50
B50
B50
B50
0.1796
0.1799
0.1813
0.1853
0.1849
0.3678
0.3672
0.3641
0.3634
0.3672
0.1816
0.1850
0.1853
0.1818
0.1850
0.3659
0.3661
0.3666
0.3636
0.3634
0.1867
0.1851
0.1814
0.1842
0.1844
0.3634
0.3680
0.3649
0.3653
0.3642
0.1811
0.1854
0.1877
0.1841
0.1831
0.3626
0.3641
0.3651
0.3648
0.3684
1168
1095
1088
1090
1087
1142
1088
1116
1102
1088
550
550
552
550
551
549
551
566
552
554
289
283
277
285
280
279
274
282
278
277
121
117
117
115
118
115
113
113
115
117
1361
1378
1422
1415
1397
1349
1381
1367
1382
1262
1639
1640
1634
1632
1606
1502
1492
1523
1537
1647
1899
1922
1953
1929
1886
1725
1665
1715
1751
1670
2685
2397
2618
2632
2488
2457
2342
2508
2570
2391
0.09192
0.08780
0.08770
0.08765
0.08727
0.08902
0.08681
0.08783
0.08723
0.08733
0.11561
0.11614
0.11548
0.11622
0.11656
0.11507
0.11553
0.11712
0.11560
0.11524
0.15676
0.15432
0.15268
0.15459
0.15441
0.15349
0.15144
0.15493
0.15342
0.15211
0.23262
0.22581
0.22554
0.22574
0.22942
0.22412
0.22258
0.22132
0.22349
0.22684
1.962
1.826
1.820
1.820
1.810
1.880
1.797
1.838
1.815
1.808
1.825
1.835
1.826
1.835
1.842
1.813
1.822
1.868
1.825
1.822
1.913
1.868
1.832
1.877
1.862
1.845
1.807
1.870
1.840
1.821
2.004
1.921
1.918
1.910
1.960
1.892
1.866
1.850
1.884
1.926
Experimental conditions
In this section, the experimental conditions and empirically derived fitting coefficients are listed for all experiments. In addition, a detailed comparison of the radial
2.977
2.998
2.990
2.995
3.002
3.037
3.026
3.036
3.034
3.008
3.012
3.000
3.021
2.996
2.992
3.031
3.023
3.031
3.025
3.041
3.019
3.029
3.020
3.037
3.010
3.027
3.034
3.015
3.017
3.037
3.016
3.049
3.051
3.024
3.020
3.047
3.038
3.042
3.050
3.038
0.0915
0.0985
0.0996
0.1018
0.1022
0.1957
0.2043
0.1981
0.2002
0.2031
0.0995
0.1008
0.1015
0.0991
0.1004
0.2018
0.2009
0.1962
0.1992
0.1994
0.0976
0.0991
0.0990
0.0982
0.0990
0.1969
0.2037
0.1952
0.1986
0.1999
0.0904
0.0965
0.0979
0.0964
0.0934
0.1917
0.1951
0.1974
0.1937
0.1912
833
787
780
782
782
831
789
812
801
784
775
772
779
771
771
779
779
803
781
788
808
795
774
803
782
784
771
788
776
779
841
825
824
806
825
811
795
790
809
821
3280
3693
3686
3739
3686
3452
3759
3560
3877
3433
4359
4200
4159
4332
3952
4239
3615
3788
3843
4220
4246
4528
4905
4890
4536
4642
4097
4116
4849
4263
6298
5645
7232
7214
5622
6579
5083
6273
7008
5879
velocity distribution between the two most different test
case is shown in Fig. 16.
Experiments in Fluids
(2024) 65:69
(a) B5 (exp. No. 6)
Page 21 of 23
69
(b) B50 (exp. No. 36)
Fig. 16 Selected time steps of the radial velocity ur ≤ ur,max for 𝛿 = 0.2. The vertical black line marks the image boarder, which dependence of
the impact center and varies for each experiment
Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s00348-024-03800-5.
Acknowledgements The authors kindly acknowledge the High Performance Computing Center Stuttgart (HLRS) for support and supply
of computational resources on the HPE Apollo(Hawk) platform under
the Grant No. FS3D/11142.
Author contributions SS contributed to experimental investigation,
methodology, data analysis, writing-original draft and writing-review;
JS contributed to numerical investigation and was involved in methodology, data analysis, writing-review and editing; AKG contributed
to conceptualization, writing-review as well as editing; GL contributed to the conceptualization, writing, data analysis, project supervision and funding acquisition; BW contributed to conceptualization,
writing-review, editing supervision, project administration and funding
acquisition.
Funding Open Access funding enabled and organized by Projekt
DEAL. The authors thank the Deutsche Forschungsgemeinschaft
(DFG) for financial support in the framework of the project GRK
2160/2 ‘Droplet Interaction Technologies’ (DROPIT), under Project
No. 270852890.
Data availability See https://doi.org/10.18419/darus-3887 for selected
numerical and experimental data to reproduce some of the figures.
Additional data will be made available upon request to the corresponding authors.
Declarations
Conflict of interest The authors declare that they have no conflict of
interest.
Ethical approval Not applicable.
Consent for publication The authors confirm that this work is original
and has not been published elsewhere, nor is it under consideration for
publication elsewhere, and consent for the publication in experiments
in fluids.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
References
Bakshi S, Roisman IV, Tropea C (2007) Investigations on the impact
of a drop onto a small spherical target. Phys Fluids 19(3):032102.
https://doi.org/10.1063/1.2716065
Bernard RA (2020) Macro and micro dynamics of droplet impact onto
wall-films made of different liquids. PhD thesis, Institut für Thermodynamik der Luft- und Raumfahrt an der Universität Stuttgart
Chandra S, Avedisian C (1991) On the collision of a droplet with a
solid surface. Proc R Soc Lond A 432(1884):13–41. https://doi.
org/10.1098/rspa.1991.0002
Cheng X, Sun TP, Gordillo L (2022) Drop impact dynamics: impact
force and stress distributions. Annu Rev Fluid Mech 54(1):57–81.
https://doi.org/10.1146/annurev-fluid-030321-103941
Eggers J, Fontelos MA, Josserand C, Zaleski S (2010) Drop dynamics
after impact on a solid wall: theory and simulations. Phys Fluids
10(1063/1):3432498
Eisenschmidt K, Ertl M, Gomaa H, Kieffer-Roth C, Meister C,
Rauschenberger P, Reitzle M, Schlottke K, Weigand B (2016)
Direct numerical simulations for multiphase flows: an overview of
the multiphase code FS3D. J Appl Math Comput 272(2):508–517.
https://doi.org/10.1016/j.amc.2015.05.095
Erkan N (2019) Full-field spreading velocity measurement inside droplets impinging on a dry solid-heated surface. Exp Fluids 60(5):88.
https://doi.org/10.1007/s00348-019-2735-0
69
Page 22 of 23
Erkan N, Okamoto K (2014) Full-field spreading velocity measurement inside droplets impinging on a dry solid surface. Exp Fluids.
https://doi.org/10.1007/s00348-014-1845-y
Fest-Santini S, Steigerwald J, Santini M, Cossali G, Weigand B (2021)
Multiple drops impact onto a liquid film: direct numerical simulation and experimental validation. Comp Fluids 214:104761.
https://doi.org/10.1016/j.compfluid.2020.104761
Frommhold PE, Mettin R, Ohl CD (2015) Height-resolved velocity
measurement of the boundary flow during liquid impact on dry
and wetted solid substrates. Exp Fluids. https://doi.org/10.1007/
s00348-015-1944-4
Gao X, Li R (2015) Impact of a single drop on a flowing liquid film.
Phys Rev E 92:053005. https:// doi. org/ 10. 1103/ PhysRevE. 92.
053005
Gultekin A, Erkan N, Colak U, Suzuki S (2020) PIV measurement
inside single and double droplet interaction on a solid surface. Exp
Fluids. https://doi.org/10.1007/s00348-020-03051-0
Gultekin A, Erkan N, Colak U, Suzuki S (2023) Investigating the
dynamics of droplet spreading on a solid surface using PIV for a
wide range of Weber numbers. J Visualiz 26(5):999–1007. https://
doi.org/10.1007/s12650-023-00920-8
Hagsäter M (2008) Development of micro-PIV techniques for applications in microfluidic systems. PhD thesis, Department of Microand Nanotechnology, Technical University of Denmark
van Hinsberg NP, Budakli M, Göhler S, Berberović E, Roisman IV,
Gambaryan-Roisman T, Tropea C, Stephan P (2010) Dynamics of the cavity and the surface film for impingements of single
drops on liquid films of various thicknesses. J Colloid Interface
Sci 350(1):336–343. https://doi.org/10.1016/j.jcis.2010.06.015
Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the
dynamics of free boundaries. J Comput Phys 39(1):201–225.
https://doi.org/10.1016/0021-9991(81)90145-5
Kittel HM, Roisman IV, Tropea C (2018) Splash of a drop impacting
onto a solid substrate wetted by a thin film of another liquid. Phys
Rev Fluid 3(7):073601. https://doi.org/10.1103/physr evfluids.3.
073601
Lagubeau G, Fontelos MA, Josserand C, Maurel A, Pagneux V,
Petitjeans P (2012) Spreading dynamics of drop impacts. J Fluid
Mech 713:50–60. https://doi.org/10.1017/jfm.2012.431
Lamanna G, Geppert A, Bernard R, Weigand B (2022) Drop impact
onto wetted walls: an unsteady analytical solution for modelling
crown spreading. J Fluid Mech. https://doi.org/10.1017/jfm.2022.
69
Lastakowski H, Boyer F, Biance AL, Pirat C, Ybert C (2014) Bridging
local to global dynamics of drop impact onto solid substrates. J
Fluid Mech 747:103–118. https://doi.org/10.1017/jfm.2014.108
Liang G, Mudawar I (2016) Review of mass and momentum interactions
during drop impact on a liquid film. Int J Heat Mass Transf 101:577–
599. https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.062
Lindken R, Rossi M, Große S, Westerweel J (2009) Micro-particle
image velocimetry (μPIV): recent developments, applications,
and guidelines. Lab Chip 9:2551–2567. https://doi.org/10.1039/
B906558J
Mehdi-Nejad V, Mostaghimi J, Chandra S (2003) Air bubble entrapment under an impacting droplet. Phys Fluids 15(1):173–183.
https://doi.org/10.1063/1.1527044
Mitchell BR, Klewicki JC, Korkolis YP, Kinsey BL (2019) The transient force profile of low-speed droplet impact: measurements and
model. J Fluid Mech 867:300–322. https://doi.org/10.1017/jfm.
2019.141
Ninomiya N, Iwamoto K (2012) PIV measurement of a droplet impact
on a thin fluid layer. In: AIP Conference Proceedings, AIP, https://
doi.org/10.1063/1.3694683
Philippi J, Lagrée PY, Antkowiak A (2016) Drop impact on a solid surface: short-time self-similarity. J Fluid Mech 795:96–135. https://
doi.org/10.1017/jfm.2016.142
Experiments in Fluids
(2024) 65:69
Raffel M, Willert CE, Scarano F, Kähler C, T WS, Kompenhans J
(2018) Particle image velocimetry - a practical guide, 3rd edn.
Springer, Berlin https://doi.org/10.1007/978-3-319-68852-7
Rieber M, Frohn A (1999) A numerical study on the mechanism of
splashing. Int J Heat Fluid Flow 20(5):455–461. https://doi.org/
10.1016/S0142-727X(99)00033-8
Roisman IV (2009) Inertia dominated drop collisions II. an analytical
solution of the Navier-Stokes equations for a spreading viscous
film. Phys Fluids 21(5):052104. https://doi.org/10.1063/1.31292
83
Roisman IV, Berberović E, Tropea C (2009) Inertia dominated drop
collisions. I. on the universal flow in the lamella. Phys Fluids
21(5):052103. https://doi.org/10.1063/1.3129282
Santiago JG, Wereley ST, Meinhart CD, Beebe DJ, Adrian RJ (1998)
A particle image velocimetry system for microfluidics. Exp Fluids
25(4):316–319. https://doi.org/10.1007/s003480050235
Smith MI, Bertola V (2010) Effect of polymer additives on the wetting
of impacting droplets. Phys Rev Lett 104(15):154502. https://doi.
org/10.1103/physrevlett.104.154502
Smith MI, Bertola V (2011) Particle velocimetry inside Newtonian
and non-Newtonian droplets impacting a hydrophobic surface. Exp Fluids 50(5):1385–1391. https:// doi. org/ 10. 1007/
s00348-010-0998-6
Steigerwald J, Geppert A, Weigand B (2021) Numerical study of drop
shape effects in binary drop film interactions for different density
ratios. In: ICLASS 2021, 15th triennial International Conference
on Liquid Atomization and Spray Systems, vol Edinburgh, UK,
doi: https://doi.org/10.2218/iclass.2021.6038
Stumpf B, Hussong J, Roisman IV (2022) Drop impact onto a substrate
wetted by another liquid: flow in the wall film. Coll Interf 6(4):58.
https://doi.org/10.3390/colloids6040058
Thielicke W (2014) The flapping flight of birds: analysis and application. PhD thesis, University of Groningen
Thielicke W, Sonntag R (2021) Particle image velocimetry for MATLAB: accuracy and enhanced algorithms in PIVlab. J Open Res
Softw 9(1):12. https://doi.org/10.5334/jors.334
Thielicke W, Stamhuis EJ (2014) PIVlab – towards user-friendly,
affordable and accurate digital particle image velocimetry in
MATLAB. J Open Res Softw https://doi.org/10.5334/jors.bl
Vaikuntanathan V, Bernard R, Lamanna G, Cossali GE, Weigand B
(2020) On the measurement of velocity field within wall-film during droplet impact on it using high-speed micro-PIV. In: Fluid
mechanics and its applications, Springer, pp 215–223 https://doi.
org/10.1007/978-3-030-33338-6_17
Wereley ST, Meinhart CD (2010) Recent advances in micro-particle
image velocimetry. Annu Rev Fluid Mech 42(1):557–576. https://
doi.org/10.1146/annurev-fluid-121108-145427
Yarin A (2006) DROP IMPACT DYNAMICS: splashing, spreading,
receding, bouncing. Annu Rev Fluid Mech 38(1):159–192. https://
doi.org/10.1146/annurev.fluid.38.050304.092144
Yarin AL, Weiss DA (1995) Impact of drops on solid surfaces: selfsimilar capillary waves, and splashing as a new type of kinematic
discontinuity. J Fluid Mech 283:141–173. https://doi.org/10.1017/
s0022112095002266
Yu X, Shao Y, Teh KY, Hung DLS (2022) Force of droplet impact on
thin liquid films. Phys Fluids 34(4):042111. https://doi.org/10.
1063/5.0083437
Yu Y, Hopkins C (2018) Experimental determination of forces applied
by liquid water drops at high drop velocities impacting a glass
plate with and without a shallow water layer using wavelet
deconvolution. Exp Fluids 59(5):84. https:// doi. org/ 10. 1007/
s00348-018-2537-9
Publisher's Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Experiments in Fluids
(2024) 65:69
Page 23 of 23
Authors and Affiliations
Stefan Schubert1 · Jonas Steigerwald1 · Anne K. Geppert1 · Bernhard Weigand1 · Grazia Lamanna1
* Stefan Schubert
stefan.schubert@itlr.uni-stuttgart.de
* Jonas Steigerwald
jonas.steigerwald@itlr.uni-stuttgart.de
Anne K. Geppert
anne.geppert@itlr.uni-stuttgart.de
Bernhard Weigand
bernhard.weigand@itlr.uni-stuttgart.de
Grazia Lamanna
grazia.lamanna@itlr.uni-stuttgart.de
1
Institute of Aerospace Thermodynamics, University
of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany
69