1. Introduction
The impact of liquid droplets on solid surfaces is a common and vital process in both nature and industry, not only displaying fascinating fluid dynamics but also significantly influencing numerous applications (Yarin Reference Yarin2006; Quéré Reference Quéré2013; Josserand & Thoroddsen Reference Josserand and Thoroddsen2016; Lohse Reference Lohse2022). For instance, raindrop impacts contribute to soil erosion (Soto et al. Reference Soto, De Larivière, Boutillon, Clanet and Quéré2014), while the effectiveness of spray cooling is profoundly affected by drop impact dynamics (Breitenbach, Roisman & Tropea Reference Breitenbach, Roisman and Tropea2018). Among the various intriguing impact outcomes, such as rebound, splashing, jetting and atomisation (Rioboo, Tropea & Marengo Reference Rioboo, Tropea and Marengo2001; Tsai et al. Reference Tsai, Pacheco, Pirat, Lefferts and Lohse2009, Reference Tsai, C. A. van der Veen, van de Raa and Lohse2010; Thoroddsen, Takehara & Etoh Reference Thoroddsen, Takehara and Etoh2012; Burzynski, Roisman & Bansmer Reference Burzynski, Roisman and Bansmer2020; Jha et al. Reference Jha, Chantelot, Clanet and Quéré2020; García-Geijo et al. Reference García-Geijo, Quintero, Riboux and Gordillo2021; Liu et al. Reference Liu, Zhang, Cai and Tsai2022; Ma et al. Reference Ma, Aldhaleai, Liu and Tsai2024), the most common outcome is droplet spreading (Rioboo et al. Reference Rioboo, Tropea and Marengo2001; Lagubeau et al. Reference Lagubeau, Fontelos, Josserand, Maurel, Pagneux and Petitjeans2012; Lolla et al. Reference Lolla, Ahmadi, Park, Fugaro and Boreyko2022; Liu et al. Reference Liu, He, Wang, Cai and Tsai2023).
Droplet spreading is typically quantified by the maximum spreading factor (
$\beta _m = D_m/D_0$
) (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014; Liu et al. Reference Liu, Zhang, Gao, Lu and Ding2018; Gordillo, Riboux & Quintero Reference Gordillo, Riboux and Quintero2019), defined as the ratio of the maximum spreading diameter (
$D_m$
) to the initial droplet diameter (
$D_0$
). This factor is critical in applications such as printing, coating, and cooling. Due to its importance, various models have been developed to better understand and predict
$\beta _m$
. These models rely on theoretical analyses using energy or momentum conservation (Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996; Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Lee et al. Reference Lee, Derome, Guyer and Carmeliet2016b
; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016; Yonemoto & Kunugi Reference Yonemoto and Kunugi2017; Huang & Chen Reference Huang and Chen2018; Gordillo et al. Reference Gordillo, Riboux and Quintero2019; Du et al. Reference Du, Wang, Li, Min and Wu2021; Aksoy et al. Reference Aksoy, Eneren, Koos and Vetrano2022), mass balance (Roisman Reference Roisman2009) and empirical fits to experimental data (Scheller & Bousfield Reference Scheller and Bousfield1995; Bayer & Megaridis Reference Bayer and Megaridis2006; Sen, Vaikuntanathan & Sivakumar Reference Sen, Vaikuntanathan and Sivakumar2014; Lee et al. Reference Lee, Laan, de Bruin, Skantzaris, Shahidzadeh, Derome, Carmeliet and Bonn2016c
; Tang et al. Reference Tang, Qin, Weng, Zhang, Zhang, Li and Huang2017; Liang et al. Reference Liang, Chen, Chen and Shen2019; Singh et al. Reference Singh, Hodgson, Sen and Das2021), incorporating dimensionless numbers such as the Weber (
$\textit{We}$
), Reynolds (
$\textit{Re}$
) and Ohnesorge (
$\textit{Oh}$
) numbers, defined as
\begin{equation} {\textit{We}} = \frac {\rho D_0 U_0^2}{\sigma }, \quad \textit{Re} = \frac {\rho D_0 U_0}{\mu } \quad \text{and} \quad {\textit{Oh}} = \frac {\mu }{\sqrt {\rho D_0 \sigma }} = \frac {\sqrt {{\textit{We}}}}{\textit{Re}}, \end{equation}
where
$U_0$
is the impact velocity;
$\rho$
,
$\sigma$
and
$\mu$
are the droplet’s density, surface tension and dynamic viscosity, respectively. Here,
$\textit{We}$
and
$\textit{Re}$
compare inertial forces with capillary and viscous forces, respectively.
$\textit{Oh}$
relates viscous forces to inertial-capillary forces (Josserand & Thoroddsen Reference Josserand and Thoroddsen2016), representing the dimensionless viscous effect (Sanjay & Lohse Reference Sanjay and Lohse2025), and also characterises the ratio of the viscous-capillary time scale
$(\tau _\mu \sim \mu D_0 / \sigma )$
to the inertial-capillary time scale
$(\tau _R \sim \sqrt {\rho D_0^3 / \sigma }$
) (Bartolo, Josserand & Bonn Reference Bartolo, Josserand and Bonn2005; Lin et al. Reference Lin, Zhao, Zou, Guo, Wei and Chen2018). The high-
$\textit{Oh}$
regime typically corresponds to
$\textit{Oh}\,\gtrsim 1$
, where viscous forces dominate over inertial and capillary effects, strongly influencing the spreading dynamics.
Based on the concept of energy conservation, the droplet’s initial kinetic and surface energies are converted into final surface energy at
$\beta _m$
and viscous dissipation. Pasandideh-Fard et al. (Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996) proposed a model for
$\beta _m$
, suggesting that viscous dissipation occurs primarily in the boundary layer (BL) at the droplet–surface interface, with the BL thickness approximated as
$\delta = 2D_0/\sqrt {\textit{Re}}$
. When
${\textit{We}} \gt 12$
and
${\textit{We}} \gg \sqrt {\textit{Re}}$
, this model simplifies to the scaling law
$\beta _m \sim \textit{Re}^{1/4}$
. Our experimental data for ionic liquids with viscosities around 30 mPa s are consistent with this scaling law (Liu et al. Reference Liu, He, Wang, Cai and Tsai2023). However, in the high-viscosity regime, the BL approximation would suggest a physically inconsistent BL thickness, exceeding
$D_0$
. To address this, different BL thickness scales have been proposed for low- and high-viscosity droplets (Mao, Kuhn & Tran Reference Mao, Kuhn and Tran1997; Sanjay & Lohse Reference Sanjay and Lohse2025). For high-viscosity liquids, the height of the deformed pancake-shaped droplet at
$\beta _m$
, estimated from the mass balance as
$h \sim D_m^3/D_0$
, is used as the velocity-gradient depth to approximate the dissipation term. This approach yields the scaling law
$\beta _m \sim \textit{Re}^{1/5}$
(Madejski Reference Madejski1976; Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004), which is consistent with the experimental data for high-viscosity liquids (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Liu et al. Reference Liu, He, Wang, Cai and Tsai2023). Recently, Jørgensen (Reference Jørgensen2024) proposed that the maximum value of the droplet contact diameter (
$D_c$
) on the surface scales with
$\textit{Re}^{1/3}$
at
$\textit{Re} \lt 1$
, based on the energy conservation principles and the assumption that the BL thickness scales with
$D_c$
in this regime.
In the capillary regime, where viscous dissipation is low and negligible, the initial kinetic energy is converted into surface energy, leading to
$\beta _m \sim \textit{We}^{1/2}$
(Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004). Alternatively, by balancing impact and capillary forces (Cheng Reference Cheng1977) or applying momentum and volume conservation (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004), the scaling law
$\beta _m \sim {\textit{We}}^{1/4}$
is obtained, which agrees with the experimental results for millimetre-sized water drops on flat surfaces (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Tsai et al. Reference Tsai, Hendrix, Dijkstra, Shui and Lohse2011; Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014; Liu, Cai & Tsai Reference Liu, Cai and Tsai2020; Wang et al. Reference Wang, Wang, He, Zhang, Yang, Wang and Lee2022b
). As a result, Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004) proposed
$\beta _m \sim {\textit{We}}^{1/4}$
for the capillary regime and
$\beta _m \sim \textit{Re}^{1/5}$
for the viscous regime, with a critical transition point defined as
$P^\star = {\textit{We}} \textit{Re}^{-4/5} = 1$
(Bartolo et al. Reference Bartolo, Josserand and Bonn2005). In response, Wang et al. (2022a) proposed a universal expression of
$\beta _m \textit{Re}^{-1/5} = ({P^\star }^{1/4}+ A_2{P^\star }^{1/2}) /(A_1+{P^\star }^{1/4}+ A_2{P^\star }^{1/2})$
, where
$A_1$
and
$A_2$
are fitting parameters. Differently, Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) suggested a different transition boundary at
$P = {\textit{We}} \textit{Re}^{-2/5}$
, where
$\beta _m \sim \textit{We}^{1/2}$
applies in the capillary regime. A universal expression of
$(D_m/D_0) \textit{Re}^{-1/5} = P^{1/2}/(A+P^{1/2})$
(where
$A$
is a fitting constant) was then proposed (Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). However, recent experimental (Lee et al. Reference Lee, Derome, Dolatabadi and Carmeliet2016a
; Liu et al. Reference Liu, He, Wang, Cai and Tsai2023) and numerical (Sanjay & Lohse Reference Sanjay and Lohse2025) data, across various
$\textit{We}$
,
$\textit{Re}$
and
$\textit{Oh}$
values, show that these universal formulas (Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014; Wang et al. 2022a) effectively describe
$\beta _m$
in low-
$\textit{Oh}$
regimes and at high-impact velocities, indicating the need for further investigation into
$\beta _m$
under a broader range of impact conditions.
In terms of empirical fitting, Scheller & Bousfield (Reference Scheller and Bousfield1995) introduced a regression model correlating
$\beta _m$
with
$\textit{Re}^2 {\textit{Oh}}$
, which fits experimental data well with the expression
$\beta _m \sim (\textit{Re}^2 {\textit{Oh}})^{0.166}$
for droplets with a viscosity range of
$2.5 \leqslant \mu \leqslant 25$
mP s. Since
$\textit{Re}^2 {\textit{Oh}}$
encompasses kinetic, surface and viscous energies, the formula of
$\beta _m = a(\textit{Re}^2 {\textit{Oh}})^b$
, or equivalently
$\beta _m = a({\textit{We}}/{\textit{Oh}})^b$
, has been frequently used to fit experimental data, where
$a$
and
$b$
are fitting parameters dependent on liquid properties (Sen et al. Reference Sen, Vaikuntanathan and Sivakumar2014; Seo et al. Reference Seo, Lee, Kim and Yoon2015; Liang et al. Reference Liang, Chen, Chen and Shen2019), surface roughness (Tang et al. Reference Tang, Qin, Weng, Zhang, Zhang, Li and Huang2017; Singh et al. Reference Singh, Hodgson, Sen and Das2021) and surface wettability (Bayer & Megaridis Reference Bayer and Megaridis2006).
Our previous experiments on viscous ionic liquid droplets impacting flat surfaces revealed distinct fits of
$\beta _m$
scaling:
$\beta _m \sim \textit{Re}^{1/4}$
for low-viscosity droplets and
$\beta _m \sim \textit{Re}^{1/5}$
for high-viscosity droplets (Liu et al. Reference Liu, He, Wang, Cai and Tsai2023). Given the partial good agreement between experimental data and existing universal scaling laws, we hypothesise that
$\beta _m$
behaviour for viscous droplets may vary across different regimes. In this study, we experimentally and theoretically investigate the maximum spreading of viscous droplets on flat surfaces upon impact under a broad range of parameters, including wide liquid viscosities (
$1 \leqslant \mu \leqslant 1,216$
mPa s) and impact velocities (
$0.2 \leqslant U_0 \leqslant 4.5$
m s−1).
2. Experimental
To investigate the influence of liquid viscosity on droplet spreading, silicone oil (Sigma–Aldrich) and mixtures of glycerol (Aladdin) and Milli-Q® water at different mass ratios were prepared, with pure water included for comparison. All experiments were conducted at atmospheric pressure and room temperature. The surface tension and dynamic viscosity of the solutions were measured using a surface tensiometer (Shanghai Fangrui, BZY100) and a rotational viscometer (Shanghai Fangrui, NDJ-5s), respectively. Liquid density was calculated from separate measurements of mass and volume. The measured properties are given in table 1.
The density (
$\rho$
), dynamic viscosity (
$\mu$
), surface tension (
$\sigma$
) and the corresponding Ohnesorge number (
$\textit{Oh}$
) of the glycerol–water (G-W) mixtures and silicone oil at room temperature. The percentage shown in the table is the weight percentage of the glycerol.
Droplets were generated using a blunt needle connected to a syringe pump. Each droplet, with an initial diameter
$D_0$
, detached from the needle once its gravitational force exceeded surface tension. The droplet then fell freely and impacted the solid surface with an impact velocity
$U_0$
, measured from high-speed camera snapshots using image analysis. The value of
$U_0$
was varied by adjusting the free-fall height and ranged from 0.20 to 4.52 m s−1. The dimensionless parameters explored here include:
$2 \leqslant {\textit{We}} \leqslant 1,194$
,
$0.5 \leqslant \textit{Re} \leqslant 13,476$
and
$ 0.002 \leqslant {\textit{Oh}} \leqslant 2.738$
. A flat glass with an average roughness of
$0.5$
nm was used as the solid surface. A high-speed camera (Photron SA-X2), operating at 10 000 frames per second, recorded the impact dynamics from the side view, with backlighting provided by an LED light (Phlox LEDW). All data were extracted from high-speed images using a customised ImageJ (Schindelin et al. Reference Schindelin2012) code. The pixel resolution in our experiments was
$19.2\,\unicode{x03BC}$
m pixel−1. The maximum error in the measurement of
$\beta _m$
is approximately 5 %, based on the estimated errors from image binarisation and three repeated measurements. Since the error bars are too small to be visible, they are not included in the figures.
3. Viscous droplets impact on flat surfaces
3.1. Spreading on a flat surface
In the range of experimental parameters explored, we observed two types of impact outcomes: spreading, where the droplet spreads (while simultaneously oscillating) upon contact, reaches its maximum spreading diameter of the lamella,
$D_m$
(see figure 1), and eventually adheres to the surface; and splashing, where secondary droplets are ejected from the advancing lamella during spreading. For a concise analysis of
$\beta _m$
, data with splashing outcomes were excluded. Figure 1 shows sequential snapshots of glycerol–water droplets spreading on a flat surface at the same
$U_0$
. Phenomenologically, while undergoing simultaneous droplet oscillation (Banks et al. Reference Banks, Ajawara, Sanchez, Surti and Aguilar2014), low-viscosity droplets spread and flatten into circular disks (figures 1
a and 1
b), whereas high-viscosity droplets rapidly form ellipsoids and reach
$D_m$
(figure 1
c). Here, we focus on the analysis of early-time
$\beta _m$
, while simultaneously the drop naturally oscillates upon impact for an extended period (Bechtel, Bogy & Talke Reference Bechtel, Bogy and Talke1981; Fukai et al. Reference Fukai, Zhao, Poulikakos, Megaridis and Miyatake1993; Banks et al. Reference Banks, Ajawara, Sanchez, Surti and Aguilar2014; Das et al. Reference Das, Mohammed, Gibson, Weerasiri, McDonnell, Xiang and Yeo2019; McCarthy, Reid & Walker Reference McCarthy, Reid and Walker2023). As expected,
$D_m$
decreases with increasing liquid viscosity (i.e.
$\textit{Oh}$
) at the same
$U_0$
due to enhanced viscous dissipation.
Sequential snapshots of glycerol–water droplets impacting a flat surface at the same impact velocity of
$U_0 =$
1.52 m s−1 for (a)
${\textit{Oh}} = 0.006$
, (b)
${\textit{Oh}} = 0.215$
and (c)
${\textit{Oh}} = 0.858$
. The maximum spreading diameter (
$D_m$
) occurs at
$\tau = 2.58$
,
$\tau = 1.32$
and
$\tau = 0.95$
for (a), (b) and (c), respectively. Here, the dimensionless time
$\tau = t/\tau _i$
, where
$t$
is the time, and
$\tau _i = D_0/U_0$
represents the characteristic impact time. The inset scale bars are 2 mm.
Variations of the experimental maximum spreading factor,
$\beta _m$
, with (a) Reynolds number (
$\textit{Re}$
) and (b) impact parameter
$P = {\textit{We}} \textit{Re}^{-2/5}$
for different glycerol–water droplets. The dashed and dotted lines in (a) represent
$\beta _m \sim \textit{Re}^{1/5}$
and
$\beta _m \sim \textit{Re}^{1/4}$
, respectively. The solid line in (b) refers to the expression of
$\beta _m \textit{Re}^{-1/5} = P^{1/2}/(A+P^{1/2})$
, with
$A = 1.24$
(Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014).
Figure 2 shows our experimental maximum spreading factor (
$\beta _m = D_m/D_0$
) varying with
$\textit{Re}$
and impact number of
$P = {\textit{We}} \textit{Re}^{-2/5}$
(Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). The results reveal good agreement with
$\beta _m \sim \textit{Re}^{1/4}$
in the medium-viscosity regime and
$\beta _m \sim \textit{Re}^{1/5}$
in the high-viscosity regime. However, neither scaling law fully captures
$\beta _m$
across the entire viscosity range examined.
A comparison of our experimental data with the universal expression proposed by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) shows that the model is consistent with our data in the low-
$\textit{Oh}$
and high-
$U_0$
ranges (see figure 2
b). Our findings are also consistent with recent numerical investigations of
$\beta _m$
(see figure 1b reported by Sanjay & Lohse (Reference Sanjay and Lohse2025)), spanning a broad range of
$\textit{We}$
(
$1 \leqslant \textit{We} \leqslant 10^3$
) and
$\textit{Oh}$
(
$10^{-3} \leqslant \textit{Oh} \leqslant 10^{2}$
). Additional comparisons between our experimental
$\beta _m$
and existing scaling laws from the literature (Cheng Reference Cheng1977; Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996; Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Ukiwe & Kwok Reference Ukiwe and Kwok2005; Roisman Reference Roisman2009; Wang et al. 2022a) are provided in figures https://doi.org/10.1017/jfm.2025.10550S1–S3 of the supplementary material (available at https://doi.org/10.1017/jfm.2025.10550), showing that these models align with the data in specific (partial) parameter ranges.
3.2. Modelling of maximum spreading factor (
$\beta _m$
)
This study focuses on viscous droplets, where viscosity dominates the entire drop impact process. By analysing total energy conservation,
$\beta _m$
can be estimated or predicted by converting the initial total energy of the impacting droplet (comprising its initial kinetic and surface energies) into the final energy of the deformed droplet at
$\beta _m$
(covering viscous dissipation and final surface energy, with negligible liquid velocity or kinetic energy). The viscosity dissipation can arise from bulk liquid dissipation (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Attané et al. Reference Attané, Girard and Morin2007; Sanjay & Lohse Reference Sanjay and Lohse2025) and moving contact-line dissipation (Attané et al. Reference Attané, Girard and Morin2007; Snoeijer & Andreotti Reference Snoeijer and Andreotti2013; Carlson et al. Reference Carlson, Bellani and Amberg2012a
,
Reference Carlson, Bellani and Ambergb
; Wang, Amberg & Carlson Reference Wang, Amberg and Carlson2017). However, we assume that the latter can be neglected for high-speed and viscous droplets, as discussed in Appendix A.2. In addition, the change in the droplet–surface energy (
$\Delta E_s$
) compared with the initial kinetic energy (
$E_k$
) is assumed to be insignificant for
${\textit{We}} \gt 10$
(see Appendix A.1 for detailed analyses).
In this context, with our simplified energy conservation, the initial kinetic energy (
$E_k$
) of the impacting droplet is dissipated by viscosity at
$\beta _m$
, where
\begin{equation} E_k = m U_0^2/2 \sim \rho D_0^3 U_0^2, \end{equation}
where
$m$
is the droplet mass. The energy dissipation due to viscosity (
$E_\mu$
) upon reaching
$\beta _m$
can be approximated as (Chandra & Avedisian Reference Chandra and Avedisian1991; Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996):
\begin{equation} E_\mu = \int _0^{t_m} \int _V \phi \,\mathrm{d}V\,\mathrm{d}t, \end{equation}
where
$V$
is the viscous fluid volume where significant viscous dissipation occurs; the duration to reach
$\beta _m$
is approximated as
$t_m \sim R_m/U_o$
, which shows good consistency with our experimental observations (see detailed discussion in Appendix B); here,
$R_m = D_m/2$
is the maximum spreading radius;
$\phi$
is the dissipation function that represents the rate of viscous dissipation per unit volume, expressed as (Kundu & Cohen Reference Kundu and Cohen2002)
\begin{equation} \phi = \frac {\mu }{2} \left (\frac {\partial v_r}{\partial z} + \frac {\partial v_z}{\partial r} \right )^2 \!, \end{equation}
where
$\partial v_r/\partial z$
and
$\partial v_z/\partial r$
represent the gradients of the radial and vertical velocity components,
$v_r$
and
$v_z$
, respectively. Numerical results by Lee et al. (Reference Lee, Derome, Dolatabadi and Carmeliet2016a
); Sanjay et al. (Reference Sanjay, Zhang, Lv and Lohse2025) have shown that the velocity field within a spreading droplet varies both spatially and temporally, and so does the viscous dissipation function
$\phi$
. To simplify the analysis, in this study
$\partial v_z /\partial r = 0$
is assumed (Roisman Reference Roisman2009), and (3.3) is reduced to
$\phi \approx \mu (\partial v_r/\partial z)^2$
.
For scaling purposes, the radial velocity
$v_r$
is approximated by the speed of the spreading lamella, denoted
$V_s = {\rm d}D_s/{\rm d}t$
, where
$D_s$
is the spreading lamella diameter (see figure 3
a schematic). Our experimental data show that the evolution of
$D_s$
closely follows that of the contact diameter,
$D_c$
; thus, we approximate
$V_s \approx {\rm d}D_c/{\rm d}t$
(see supplementary material for detailed discussion).
Figure 3 shows the evolution of drop spreading dynamics for various
$\textit{Oh}$
. Initially, droplets spread rapidly within the characteristic impact (or inertial) time (
$\tau _i = D_0/U_0$
) and then saturate to the maximum value. The contact diameter can be expressed as
$D_c(t) = \sqrt {c_d U_0 R_0 t}$
(where
$R_0$
is the initial radius of the droplet) based on geometrical considerations (Mongruel et al. Reference Mongruel, Daru, Feuillebois and Tabakova2009) or Wagner’s theory (Riboux & Gordillo Reference Riboux and Gordillo2014), where
$c_d$
is a prefactor. Our empirical fits of
$D_c(t)$
agree with this scaling law for
$t/\tau _i \lt 1$
(see figure 3). Differentiating
$D_c(t)$
with respect to time provides the approximation of
$V_s \approx {\rm d}D_c/{\rm d}t \sim \sqrt {U_0 D_0/t}$
, which, when evaluated at
$t \sim \tau _i$
, leads to the approximation of
$V_s \sim U_0$
.
(a) Schematic of the spreading diameter of the advancing lamella (
$D_s$
) and the contact-line diameter (
$D_c$
). (b) Variation of normalised contact diameter,
$D_c(t)/D_0$
, with a dimensionless impact time,
$t/\tau _i$
, at
$U_0 = 1.52$
m s–1 for various Ohnesorge (
$\textit{Oh}$
) numbers; here,
$\tau _i = D_0/U_0$
. The measurement error of
$D_c$
is
$\approx 7.7\,\%$
(see supplementary material for more details on the measurement error). The solid lines represent the empirical fits of
$D_c(t)/D_0 \sim \sqrt {t/\tau _i}$
or
$D_c(t) \sim \sqrt {U_0 D_0 t}$
, with varying prefactors.
Before detailedly analysing
$E_\mu$
(3.2), we briefly review the drop impact process based on recent studies of impact force dynamics (Gordillo, Sun & Cheng Reference Gordillo, Sun and Cheng2018; Cheng, Sun & Gordillo Reference Cheng, Sun and Gordillo2021; Zhang et al. Reference Zhang, Sanjay, Shi, Zhao, Lv, Feng and Lohse2022; Sanjay et al. Reference Sanjay, Zhang, Lv and Lohse2025). When a viscous droplet impacts a wetted solid surface, such as glass in this study, the impact force rapidly increases, peaking at time
$\tau _{\rho }$
. The force then decays over a longer time scale as the droplet spreads radially, eventually reaching
$D_m$
as the impact force diminishes to zero. The impact force decelerates the droplet’s vertical motion, redirecting vertical momentum into radial flow and, hence, playing a critical role in viscous dissipation. The drop impact process can hence be characterised into two phases: the impact phase and the spreading phase, separated by the time boundary
$\tau _{\rho }$
, and the duration of the impact phase is typically shorter compared with the spreading period (Sanjay & Lohse Reference Sanjay and Lohse2025). Furthermore, numerical studies have shown that viscous dissipation during the impact phase can account for a proportion of the total energy dissipation (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). More recently, a unifying theoretical model has been developed based on numerical data, which analyses the temporal evolution of viscous dissipation contributions during both the impact and spreading phases (Sanjay & Lohse Reference Sanjay and Lohse2025). These dissipation contributions are modelled to be localised within the BL or distributed throughout the drop bulk, depending on the governing parameter regimes of
$\textit{We}$
and
$\textit{Oh}$
(Sanjay & Lohse Reference Sanjay and Lohse2025).
Our current experimental results, focusing on side-view measurements of spreading dynamics, do not provide time-resolved internal velocity fields or impact force measurements. We are currently unable to directly track the evolution or spatial distribution of viscous dissipation within the droplet. Future experimental studies using techniques such as particle image velocimetry (PIV) or other flow diagnostic tools would be valuable for quantitatively characterising the velocity field and identifying the regions and timing of dissipation. Given this limitation, we adopt a simplified framework that considers two limiting regimes.
In our theoretical framework, we analyse two limiting cases: low-viscosity (low-
$\textit{Oh}$
) and high-viscosity (high-
$\textit{Oh}$
) regimes. In the low-viscosity regime, the BL remains thin, scaling as
$\delta \sim \sqrt {\nu t}$
, where
$\nu = \mu/\rho$
is the drop’s kinematic viscosity (Schlichting & Gersten Reference Schlichting and Gersten2016), and viscous dissipation is assumed to occur within a thin BL adjacent to the liquid–solid interface, where the BL thickness is smaller than the droplet’s spreading height. In particular, we assume that the length scale of the BL remains the same for both the impact and spreading phases in this regime. In contrast, in the high-viscosity regime, BL thickness increases, and viscous dissipation is expected to occur throughout the bulk upon impact until reaching
$\beta _m$
.
3.2.1. High-viscosity (high-Oh) regime
In the high-viscosity regime, the viscous dissipation is assumed to occur across the droplet’s spreading height during both phases, as the experimental observations showed the viscous droplets quickly deform into an ellipsoid shape, reaching
$D_m$
. The corresponding spreading height,
$h_m$
, satisfies mass conservation,
$D_m^2 h_m \sim D_0^3$
, assuming the droplet adopts a cylindrical shape during spreading. The viscous dissipation is then approximated as
$E_\mu \approx ({\mu }/{2}) (V_s/h_m)^2 V t_m \sim \mu (U_0/h_m) D_m^3$
using (3.2) and (3.3), where
$V_s \sim U_0$
,
$V \approx \unicode{x03C0} D_m^2 h_m/4$
and
$t_m \sim R_m/U_0$
.
Energy conservation, by balancing the initial kinetic energy (
$E_k$
) (3.1) with this viscous dissipation, gives rise to
$\beta _m \sim \textit{Re}^{1/5}$
as previously derived by Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004). As shown in figures 2(a) and figure 8 in Appendix D, the prediction
$\beta _m \sim \textit{Re}^{1/5}$
agrees with experimental trends for
${\textit{Oh}} \gt 1$
and
$\textit{Re} \gt 2$
, indicating that the high-viscous regime is applicable for
${\textit{Oh}} \gt 1$
empirically. However, as listed in table 1, only 100 % glycerol and silicon oil have
${\textit{Oh}} \gt 1$
, suggesting that additional studies with
${\textit{Oh}} \gt 3$
are encouraged to further validate this power law.
3.2.2. Low-viscosity (low-Oh) regime
For the low-viscosity regime, viscous dissipation is assumed to be localised within a thin BL during the entire
$t_m$
. Within this BL, a velocity gradient
${\partial v_r}/{\partial z}$
exists, and the relevant dissipation function
$\phi$
in (3.3) for this regime is approximated as
\begin{equation} \phi \approx \mu \left ( \frac {\partial v_r}{\partial z} \right )^2 \sim \mu \left (\frac {U_0}{\delta } \right )^2, \end{equation}
where
$\delta$
represents the BL thickness.
In the literature, various expressions for
$\delta$
have been considered to estimate viscous dissipation analytically. Pasandideh-Fard et al. (Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996) proposed
$\delta = 2 D_0/\sqrt {\textit{Re}}$
, based on stagnation-point flow. Alternatively, Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) suggested
$\delta = \sqrt {\nu t}$
, where
$\nu = \mu / \rho$
is the kinematic viscosity. Numerical studies by Lee et al. (Reference Lee, Derome, Dolatabadi and Carmeliet2016a
) and Philippi, Lagrée & Antkowiak (Reference Philippi, Lagrée and Antkowiak2016) further indicated that
$\delta$
varies both spatially and temporally. For simplicity, Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) approximated
$\delta = \sqrt {\nu {t_{{sp}}}}$
, where
$t_{{sp}}$
represents the time to reach
$\beta _m$
and is estimated from their numerical data.
Both experimental studies (Banks et al. Reference Banks, Ajawara, Sanchez, Surti and Aguilar2014; McCarthy et al. Reference McCarthy, Reid and Walker2023) and numerical simulations (Bechtel et al. Reference Bechtel, Bogy and Talke1981; Fukai et al. Reference Fukai, Zhao, Poulikakos, Megaridis and Miyatake1993; Das et al. Reference Das, Mohammed, Gibson, Weerasiri, McDonnell, Xiang and Yeo2019) show that an inviscid droplet (e.g. water) oscillates upon impacting a solid surface, with the oscillation frequency (
$\omega$
) proportional to the natural free oscillation frequency (
$\omega _0$
), i.e.
$\omega \approx \omega _0 \sim \sqrt {\sigma / (\rho D_0^3)}$
, with surface tension
$\sigma$
acting as the restoring force (Fukai et al. Reference Fukai, Zhao, Poulikakos, Megaridis and Miyatake1993; Marmanis & Thoroddsen Reference Marmanis and Thoroddsen1996). For viscous droplets, this frequency is damped to
$\omega _{\mu } \sim \omega _0 \sqrt {1-25 {\textit{Oh}}^2/4}$
(Prosperetti Reference Prosperetti1980; Tonini & Cossali Reference Tonini and Cossali2024) due to viscous dissipation. However, in the low-viscosity (low-
$\textit{Oh}$
) regime, the influence of viscous dissipation on
$\omega$
is negligible, i.e.
$\omega \approx \omega _0$
, consistent with similar experimental findings for
${\textit{Oh}} \leqslant 0.174$
(Banks et al. Reference Banks, Ajawara, Sanchez, Surti and Aguilar2014).
As a major assumption, we assume that an oscillatory BL forms during droplet impact, as our (see figure 7 later) and others (Banks et al. Reference Banks, Ajawara, Sanchez, Surti and Aguilar2014; McCarthy et al. Reference McCarthy, Reid and Walker2023) experiments show that droplet oscillations are initiated upon impact. This behaviour may be analogous to the classical second Stokes BL problem (Schlichting & Gersten Reference Schlichting and Gersten2016), where a BL develops in response to an oscillating wall. Similarly, in an oscillating droplet, an oscillatory BL may form near the substrate (Bechtel et al. Reference Bechtel, Bogy and Talke1981; Kim & Chun Reference Kim and Chun2001). Based on this analogy, we approximate the BL thickness using the classical Stokes formulation,
$\delta \sim \sqrt {2\mu /(\rho n)}$
(Bechtel et al. Reference Bechtel, Bogy and Talke1981; Batchelor Reference Batchelor2000; Schlichting & Gersten Reference Schlichting and Gersten2016), where
$n = 2 \unicode{x03C0} \omega = 2 \unicode{x03C0} /t_{\textit{osc}}$
is the angular frequency, resulting in
$\delta \sim \sqrt {\nu t_{\textit{osc}}}$
. By substituting
$t_{\textit{osc}} \sim 1/\omega _0$
for low-viscosity droplets, we obtained an expression for the BL thickness:
\begin{equation} \delta \sim \sqrt {\nu t} \sim D_0 {\textit{Oh}}^{1/2}. \end{equation}
With energy conservation (
$E_k \sim E_{\mu }$
) at
$\beta _m$
, using (3.1), (3.2), (3.4), (3.5),
$t_m$
-approximation of
$R_m/U_0$
, and the viscous fluid volume of
$V = \unicode{x03C0} D_m^2 \delta /4$
, we derive
\begin{equation} \beta _m = \frac {D_m}{D_0} \sim ( \textit{Re}^{2} \textit{Oh} )^{1/6} = ( {\textit{We}}/{\textit{Oh}} )^{1/6}, \end{equation}
for the
$\beta _m$
scaling in the low-viscosity (low-
$\textit{Oh}$
) regime. This theoretical expression of (3.6) interestingly matches the empirical regression model fitted by Scheller & Bousfield (Reference Scheller and Bousfield1995) for glycerol–water mixtures with a viscosity range of
$2.5 \leqslant \mu \leqslant 25$
mPa s.
Figure 4(a) shows experimental
$\beta _m$
varying with
${\textit{We}}/ {\textit{Oh}}$
, demonstrating good agreement with the scaling law (3.6) for
${\textit{Oh}} \lesssim 0.1$
under the
${\textit{We}}/{\textit{Oh}}$
range explored, as represented by the solid line in figure 4(a). However, since surface energy is not considered in (3.6),
$\beta _m$
for Milli-Q® water (
${\textit{Oh}} = 0.002$
) and the 30 % glycerol–water mixture (
${\textit{Oh}} = 0.006$
) with the low-viscosity (box and circle in figure 4
a) deviates from (3.6) at lower-
$U_0$
values. For these cases, the energy conservation equation should account for surface energy change:
$E_k - \Delta E_s = E_{\mu }$
, where
$\Delta E_s$
is the surface energy difference between the initial state and at
$\beta _m$
and can be approximated as
$\Delta E_s \sim \sigma (D_m^2 -D_0^2)$
(see Appendix A.1 for details). At the low-
$\textit{Oh}$
and low-
$U_0$
ranges,
$\Delta E_s$
and
$E_k$
become comparable, and excluding
$\Delta E_s$
from the model overestimates
$\beta _m$
.
(a) Variations of the maximum spreading factor,
$\beta _m$
, with
${\textit{We}}/ {\textit{Oh}}$
. (b) Experimental data of
$\beta _m$
extracted from the literature for viscous droplets. The solid and dashed lines represent
$\beta _m = 0.61 ( {\textit{We}}/ {\textit{Oh}} )^{1/6}$
(3.6) and
$\beta _m = 0.61 ( 23.3 + {\textit{We}}/ {\textit{Oh}} )^{1/6}$
(3.7), respectively, where the prefactor 0.61 is obtained empirically.
Moreover, our experimental data show that the scaling law
$\beta _m \sim ({\textit{We}}/{\textit{Oh}})^{1/6}$
, derived under the assumption of viscous dissipation occurring within a thin BL, remains applicable for the moderate-
$\textit{Oh}$
(
$0.1 \lt {\textit{Oh}} \lt 1$
) and high-
$\textit{We}$
(
${\textit{We}} \gt 155$
) regime. We interpret this as follows: although viscosity is non-negligible, the maximum spreading time
$t_m$
becomes shorter with increasing
$U_0$
, as shown later in figure 6. Consequently, the dissipation region (
$\delta \sim \sqrt{\nu t}$
) may not have sufficient time to grow beyond the boundary layer thickness before maximum spreading is reached. This suggests that, even at the moderate-
$\textit{Oh}$
and high-
$\textit{We}$
range, viscous dissipation remains confined to the BL, supporting the applicability of the boundary-layer-based scaling in this regime.
3.2.3. Semi-empirical scaling model
Returning to (3.6), this theoretical scaling law predicts
$\beta _m = 0$
as
${\textit{We}}/ {\textit{Oh}} = 0$
. However, in practice, the maximum spreading diameter (
$D_m$
) tends to be
$D_0$
as the droplet deposits gently onto a surface with negligible velocity (
$U_0 \approx 0$
). Physically,
$\beta _m$
should converge to a finite value (i.e. 1) as
${\textit{We}}/ {\textit{Oh}} \rightarrow 0$
. To incorporate this finite spreading as
$U_0 \rightarrow 0$
, (3.6) is mathematically modified to be
\begin{equation} \beta _m \sim ( A+ {\textit{We}}/ {\textit{Oh}})^{1/6}, \end{equation}
where
$A$
is a constant. By applying
$ \lim _{{\textit{We}}/ {\textit{Oh}} \rightarrow 0} \beta _m = 1$
, we obtain
$A = 23.3$
. Remarkably, we found that the semi-empirical model in (3.7) consistently aligns with our experimental data for the wide-viscosity range explored, as revealed by the dashed line in figure 4(a). We further validate our models with the existing experimental data from the literature, as presented in figure 4(b). The good agreement between our semi-empirical model (3.7) and various data under wide-ranging parameters (
$10^{-3} \leqslant {\textit{Oh}} \leqslant 10^{0}$
and
$10^{1} \leqslant {\textit{We}} \leqslant 10^{3}$
) highlights the good application of our model for viscous droplet spreading on a flat solid surface.
We further compare our experimental data and semi-empirical model with the universal prediction proposed recently by Sanjay & Lohse (Reference Sanjay and Lohse2025), as shown in figure 5. The comparative results show that the universal prediction by Sanjay & Lohse (Reference Sanjay and Lohse2025) agrees well with our experimental data across most of the parameter space (figures 5
a–e). Our semi-empirical model (3.7) also yields consistent results with those of Sanjay & Lohse (Reference Sanjay and Lohse2025) for most range of
$\textit{Oh}$
from 0.01 to 10 (see figure 5
a–f), except in two regimes: (i) at low
$\textit{Oh}\, (= 0.01)$
, relatively low
$\textit{We}\ ( 1-20)$
(figure 5
a) and (ii) at high
$\textit{Oh}\,(= 10)$
, relatively high
$\textit{We}$
(
${\gt } 100$
) (figure 5
f).
In the low-
$\textit{Oh}$
and relatively low-
$\textit{We}$
regime (see figure 5
a), the droplet’s surface energy difference (
$\Delta E_s$
) can play a non-negligible role. While the model of Sanjay & Lohse (Reference Sanjay and Lohse2025) explicitly incorporates
$\Delta E_s$
in the energy balance analysis, our semi-empirical formulation does not. Consequently, our model (dashed line) tends to overestimate the maximum spreading factor
$\beta _m$
in this regime.
Comparison among our experimental data (symbols), our semi-empirical model (dashed line), and the universal prediction (solid lines) proposed by Sanjay & Lohse (Reference Sanjay and Lohse2025), across various
$\textit{Oh}$
and
$\textit{We}$
. The universal prediction is obtained at a specific
$\textit{Oh}$
while varying
$\textit{We}$
from 1 to 1000. Here, the experimental data points are the same as in figure 4(a). The dashed line represents our semi-empirical model
$\beta _m = 0.61 ( 23.3 + {\textit{We}}/ {\textit{Oh}} )^{1/6}$
(3.7).
In the high-
$\textit{Oh}$
regime (see figure 5
f), at
$\textit{Oh}= 10$
the prediction by Sanjay & Lohse (Reference Sanjay and Lohse2025) exhibits a flattening trend, with
$\beta _m \approx 1$
indicating that most of the initial kinetic energy is dissipated primarily during the impact phase (Sanjay & Lohse Reference Sanjay and Lohse2025). Under such conditions, increases in
$\textit{We}$
no longer contribute significantly to spreading. Our experimental data are currently limited in these two regimes, and future experimental studies are needed to investigate
$\beta _m$
variation in the low-
$\textit{Oh}$
and low-
$\textit{We}$
regime and to confirm the limiting spreading as
$D_m \approx D_0$
at high
$\textit{Oh}$
(
$\gtrsim$
10).
4. Conclusions
The maximum spreading factor of viscous droplets impacting solid surfaces has been systematically investigated across a wide range of
$\textit{We}$
,
$\textit{Re}$
and
$\textit{Oh}$
. For the broad viscosity range of
$\mu = 1{-}1216$
mPa s explored, the spreading dynamics is predominantly governed by liquid viscosity. We theoretically consider two limiting regimes: viscous dissipation occurring within a BL adjacent to the liquid–solid interface for low-viscosity droplets and throughout the entire bulk of droplet spreading height for high-viscosity droplets. By applying energy conservation principles, where the initial kinetic energy is fully dissipated due to liquid viscosity, scaling laws for
$\beta _m$
are derived:
$\beta _m \sim ( {\textit{We}}/ {\textit{Oh}})^{1/6}$
for the low-viscosity regime and
$ \beta _m \sim \textit{Re}^{1/5}$
for the high-viscosity regime. Our experimental data indicate that the former scaling law is valid for
${\textit{Oh}} \lesssim 0.1$
(low-viscosity range), whereas the latter is applicable for
${\textit{Oh}} \gt 1$
(high-viscosity range) and
$\textit{Re} \gt 2$
. These validity ranges of
$\textit{Oh}$
delineating the low- and high-viscosity regimes are determined empirically. Experimental data further show that
$\beta _m \sim (\textit{We} / {\textit{Oh}})^{1/6}$
remains applicable at high impact velocities (
${\textit{We}}/{\textit{Oh}} \gt 155$
) in the intermediate-viscosity range (
$0.1 \lt \textit{Oh} \lt 1$
). To account for finite spreading as
$U_0 \rightarrow 0$
, a prefactor was introduced, yielding a universal scaling law:
$\beta _m \sim ( A + {\textit{We}}/ {\textit{Oh}})^{1/6}$
, with
$A = 23.3$
, determined from the fitting the experimental data. This modification effectively models
$\beta _m$
across the broad range of
$\textit{Oh}$
(
$0.002 \leqslant {\textit{Oh}} \leqslant 2.738$
) explored in this study.
Supplementary material
Supplementary material is available at https://doi.org/10.1017/jfm.2025.10550.
Acknowledgements
The authors thank the anonymous referees for their constructive and stimulating comments.
Funding
This work was supported by the National Natural Science Foundation of China (L.L., grant number 12202032); the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery (P.A.T., grant number RGPIN-2020-05511) and the Canada Research Chair Program (P.A.T., grant number CRC Tier2 233147).
Declaration of interests
The authors report no conflict of interest.
Appendix A. Analysis of the energy budget upon reaching for viscous droplets
Before impact, the total energy of the droplet comprises the initial surface energy (
$E_s^i$
) and kinetic energy (
$E_k$
) (3.1). During the drop impact process, the viscous dissipation arises due to liquid viscosity and the moving contact line, denoted as
$E_\mu$
and
$E_{cl}$
, respectively. Upon reaching
$\beta _m$
, the kinetic energy (of the central mass) becomes zero as the spreading velocity vanishes. Comparing the initial impacting and the final deformed state of the droplet, energy conservation can be expressed as
\begin{equation} E_k + E_s^i = E_s^f +E_{\mu } + E_{cl}, \end{equation}
where
$E_s^f$
is the surface energy at
$\beta _m$
,
$E_{\mu }$
and
$E_{cl}$
are the viscous dissipation due to the liquid viscosity and the moving contact line, respectively. This equation can be reformulated as
\begin{equation} E_k = \Delta E_{s} +E_{\mu } + E_{cl}, \end{equation}
where
$\Delta E_{s} = E_s^f - E_s^i$
is the change in surface energy.
A.1. Influence of surface energy change on the energy budget
The initial surface energy (
$E_s^i$
) of a spherical droplet is given by (Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996)
\begin{equation} E_s^i = 4 \unicode{x03C0} \sigma R_0^2 \sim \sigma D_0^2. \end{equation}
The surface energy at
$\beta _m$
can be approximated as (Ukiwe & Kwok Reference Ukiwe and Kwok2005)
\begin{equation} E_s^f\, \approx \,\unicode{x03C0} \sigma D_m h_m +\unicode{x03C0} \sigma D_m^2 (1- \cos {\theta _Y})/4 \,\sim \,\sigma D_m^2, \end{equation}
where
$h_m$
is the (minimum) droplet height at
$\beta _m$
, and
$\theta _Y$
is the Young contact angle. The surface energy difference (
$\Delta E_{s}$
) can be approximated as
\begin{equation} \Delta E_{s} = E_{s}^f - E_{s}^i \sim \sigma \big(D_m^2 - D_0^2 \big). \end{equation}
To evaluate the contribution of surface energy difference to the energy budget, we compare
$\Delta E_{s}$
with
$E_k$
, expressed as
\begin{equation} \frac {\Delta E_{s}}{E_k} \sim \frac {\sigma \big(D_m^2 - D_0^2 \big)}{ \rho D_0^3 U_0^2} = \frac {1}{{\textit{We}}} \big(\beta _m^2 -1 \big). \end{equation}
As shown in (A6), the change in surface energy
$\Delta E_{s}$
is smaller than the kinetic energy (
$E_k$
) at the high-
$\textit{We}$
range. Given that
$\beta _m$
decreases with liquid viscosity (see figure https://doi.org/10.1017/jfm.2025.10550S1 in the supplementary material), we hence assume that the ratio of
$\Delta E_{s}$
to
$E_k$
is negligible for
${\textit{We}} \gt 10$
under our conditions, and exclude surface energy contributions from the energy balance in our simplified theoretical analyses.
A.2. Viscous dissipation induced by the moving contact line
The energy dissipation rate at the moving contact line for the drop impact process (Attané et al. Reference Attané, Girard and Morin2007; Wang et al. Reference Wang, Amberg and Carlson2017) (
$\dot {E}_{cl}$
) can be estimated as
\begin{equation} \dot {E}_{cl} \sim \int _0^{R_{cl}} \mu _f U_{cl}^2 \mathrm{d}R_{cl}, \end{equation}
where
$\mu _f$
is the contact-line friction parameter, typically extracted by fitting numerical data to experimental results of drop spreading for each combination of liquid, gas and solid (Wang et al. Reference Wang, Amberg and Carlson2017),
$U_{cl}$
is the contact-line velocity during drop impact, and
$R_{cl}$
is the contact-line radius and can be modelled as
$R_{cl} = D_c/2 \sim \sqrt { U_0 D_0 t}$
(see figure 3).
Wang et al. (Reference Wang, Amberg and Carlson2017) observed that
$\mu _f$
is relatively insensitive to the impact velocity and has similar magnitudes in both inertial drop impact (
$U_0 \gt 0$
) and wetting-driven (
$U_0 \approx 0$
) cases for a given liquid–solid–air system. Their observations are for water and glycerol–water mixture (
$\mu = 10$
mPa s) drops impacting wetting/partial-wetting surfaces (with a static contact angle range of
$52^{\circ } \leqslant \theta _e \leqslant 94^{\circ }$
), under
$0.28 \leqslant U_0 \leqslant 1.86$
m s−1. In our case, lacking the empirical values of
$\mu _f$
for our liquid–surface combination, we approximated its value using results from wetting-driven, dynamic spreading by Yue & Feng (Reference Yue and Feng2011); Carlson et al. (Reference Carlson, Bellani and Amberg2012b
) as a first-order estimate.
From the dynamic wetting model (Yue & Feng Reference Yue and Feng2011; Carlson et al. Reference Carlson, Bellani and Amberg2012b
), the contact-line velocity (
$\overline {u}_{cl}$
) for the purely wetting-driven case can be approximated as
\begin{equation} \overline {u}_{cl} \sim \frac {\sigma }{\mu _f} \frac {\cos \theta _e - \cos \theta }{ \sin \theta }, \end{equation}
where
$\theta _e$
and
$\theta$
are the equilibrium and dynamic contact angles, respectively. Rearranging, we estimate
$\mu _f \sim \sigma / \overline {u}_{cl}$
. Using this form and geometric approximation for the spreading of contact-line diameter (Mongruel et al. Reference Mongruel, Daru, Feuillebois and Tabakova2009; Riboux & Gordillo Reference Riboux and Gordillo2014), we can approximate
$R_{cl}(t) \sim \sqrt {U_0D_0 t}$
and
$U_{cl} = {\rm d} R_{cl}/{\rm d}t \sim \sqrt {U_0D_0/t}$
. Substituting into the integral form (A7), with scaling approximation we estimate the total contact line (CL) dissipation until
$t_m$
:
\begin{equation} E_{cl} = \int _0^{t_m} \dot {E}_{cl}(t) \mathrm{d}t \sim \int _0^{t_m} \mu _f U_{cl}^2 R_{cl} \mathrm{d}t \sim \frac {\sigma U_0 D_0^{3/2} R_m^{1/2}}{\overline {u}_{cl}}, \end{equation}
where
$t_m \sim R_m/U_0$
(see the discussion in Appendix B), and
$R_m$
is the maximum spreading radius.
We qualitatively analysed the portion of kinetic energy dissipated by the moving contact line, by comparing the contact-line dissipation (
$E_{cl}$
) with the initial kinetic energy (
$ E_k \sim \rho D_0^3 U_0^2$
) of the impacting droplet:
\begin{equation} \frac {E_{cl}}{E_k} \sim \frac {\sigma R_m^{1/2} }{ \overline {u}_{cl} \rho U_0 D_0^{3/2}}. \end{equation}
Due to the lack of a well-characterised friction coefficient (
$\mu _f$
) under our experimental conditions, it is challenging to accurately estimate the energy contribution from contact-line dissipation (
$E_{cl}$
). Our qualitative analysis indicates that the ratio
$E_{cl}/E_k$
decreases with increasing
$U_0$
(or
$\textit{We}$
), as shown by (A10). At fixed
$\textit{We}$
, this ratio also decreases with increasing
$\textit{Oh}$
, owing to the experimentally observed decrease in
$R_m$
with
$\textit{Oh}$
(see figure https://doi.org/10.1017/jfm.2025.10550S1(a) in the supplementary material). The estimation of the ratio
$E_{cl}/E_k$
is based on a scaling argument and thus provides a qualitative rather than quantitative measure. Accordingly, in line with commonly adopted assumptions in the literature (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Attané et al. Reference Attané, Girard and Morin2007; Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014; Sanjay & Lohse Reference Sanjay and Lohse2025), here we treat contact-line dissipation as a minor contributor to the total energy loss in our regime of high-
$\textit{We}$
and high-
$\textit{Oh}$
. Further investigation into the quantitative role of contact-line dissipation (
$E_{cl}$
), particularly under varied surface or fluid conditions, may help clarify its potentially regime-dependent contribution.
In summary, the energy budget (A2) upon reaching
$\beta _m$
for our viscous droplets at
${\textit{We}} \gt 10$
may be approximated as (A11), due to relatively small contributions from
$\Delta E_s$
and
$E_{cl}$
compared with
$E_k$
:
\begin{align} E_k \approx E_{\mu }. \\[-22pt] \nonumber \end{align}
Appendix B. The maximum spreading time (
$\boldsymbol{t}_{\boldsymbol{m}}$
)
Comparison of the experimental maximum spreading time (
$t_m^{\textit{exp}}$
) and the corresponding theoretical estimation (
$t_m = R_m/U_0$
) at various
$\textit{Oh}$
and
$U_0$
. Here,
$t_m^{\textit{exp}}$
is the time duration obtained directly from the high-speed frames upon reaching the maximum spreading diameter;
$R_m$
and
$U_0$
are obtained from the experimental data. The solid line corresponds to
$y\,\sim \,x$
.
Our high-speed recordings reveal that the time scale
$t_m$
varies with viscosity
$\mu$
, even when
$D_0$
and
$U_0$
are held constant, as illustrated in figure 1. Theoretically,
$t_m$
can be approximated as
$t_m \approx R_m / V_s$
, where
$R_m = D_m / 2$
is the maximum spreading radius, and
$V_s \sim U_0$
represents the characteristic radial velocity of the spreading lamella (as discussed in the main text). This leads to the scaling
$t_m \sim R_m / U_0$
, which is consistent with the empirical relation
$t_m = b D_m / U_0$
proposed by Lee et al. (Reference Lee, Derome, Guyer and Carmeliet2016b
), where
$b$
is a fitting coefficient. This expression of
$t_m$
also aligns with the use of
$t_m \sim D_m / U_0$
by Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004) in their derivation of
$\beta _m \sim Re^{1/5}$
.
To evaluate the validity of this scaling, we compare our experimentally measured maximum spreading time
$t_m^{\textit{exp}}$
, extracted from high-speed image recordings, with the theoretical estimate
$t_m = R_m / U_0$
, as shown in figure 6. The data generally follow the expected linear trend, indicating that this scaling serves as a reasonable approximation for the time required to reach
$\beta _m$
across a wide range of conditions. However, an interesting deviation occurs at low
$U_0$
, which shows
$t_m^{\textit{exp}}/\tau _i \gt 1$
, and the contact-line velocity
$U_{cl}$
– and thus the lamella spreading speed
$V_s$
– plateau below
$U_0$
(see figure 3). As a result, the assumption
$V_s \sim U_0$
leads to an underestimation of the actual spreading time in this regime, causing those data points to fall below the linear trend. Despite this deviation, the estimate
$t_m \sim R_m / U_0$
remains a useful and sufficiently accurate approximation for the maximum spreading time for most data in figure 6, and is therefore adopted in this study.
Appendix C. Droplet oscillation upon impact
Figure 7 shows the typical time evolution of spreading lamella diameter (
$D_s(t)$
) and spreading central height (
$h(t)$
). The experimental results suggest that the oscillation is initiated immediately upon contact and persists for up to 60 ms, or possibly longer. The maximum spreading diameter (
$D_m$
) is reached at 9.3 ms. As the droplet oscillates, the oscillation amplitude gradually decreases over time due to viscous dissipation.
Variation of normalised spreading diameter (
$D_s(t)/D_0$
) and normalised spreading central height (
$h(t)/D_0$
) with impact time (
$t$
), at an impact velocity of
$U_0 = 0.28$
m s−1 and
${\textit{Oh}} = 0.006$
. The inset is a schematic of the spreading height.
Variations of the maximum spreading factor,
$\beta _m$
, with
$Re$
on the flat surface for
${\textit{Oh}} \gt 1$
. The solid and dashed lines represent
$\beta _m = 0.98 Re^{1/5}$
and
$\beta _m = 0.92 Re^{1/5}$
, receptively.
Because direct measurements of the BL development and thickness remain experimentally challenging, a definitive characterisation of BL structure is currently unavailable. However, based on the observed oscillatory behaviour in both our experiments and numerical simulations, we infer that droplet oscillations are initiated at the moment of impact. Accordingly, as a major assumption, we adopt the Stokes oscillatory BL thickness as an estimate for the characteristic viscous length scale during this stage. More detailed measurements of the velocity field – such as those obtained via PIV – and high-resolution numerical simulations could provide further insight into the development and evolution of the BL in the future, while such efforts are beyond the scope of the present study.
Appendix D. Experimental data in high-viscosity regime (
${\textit{Oh}} \boldsymbol{\gt} \boldsymbol{1}$
)
Figure 8 presents the variation of
$\beta _m$
with
$Re$
on a flat glass surface for
${\textit{Oh}} \gt 1$
. The results suggest that the theoretical model of
$\beta _m \sim Re^{1/5}$
aligns with the experimental
$\beta _m$
for
${\textit{Oh}} \gt 1$
and
$Re \gt 2$
. However, the energy conservation becomes complex as
$Re \lt 2$
(the low-
$U_0$
range). In this regime, two factors contribute to the deviation from this theoretical model. First, the work done by the uncompensated Young’s force (Carlson et al. Reference Carlson, Bellani and Amberg2012a
) becomes comparable to the initial kinetic energy. Second, dissipation at the moving contact line also plays a significant role and should be considered. As a result, the simple balance between the initial kinetic energy and liquid bulk dissipation (A11) cannot fully capture the
$\beta _m$
behaviour in the very low-
$U_0$
range, causing
$\beta _m \sim Re^{1/5}$
to deviate from the experimental trend when
$Re \lt 2$
.
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