# The fluid mechanics of splat painting

Diego Ávila-García  Fluid Mechanics Group, Department of Thermal and Fluid
Engineering, Universidad Carlos III de Madrid, 28911 Leganés (Madrid), Spain
Lucía Lacambra-Asensio  Fluid Mechanics Group, Department of Thermal and Fluid
Engineering, Universidad Carlos III de Madrid, 28911 Leganés (Madrid), Spain
Javier Rodríguez-Rodríguez  Fluid Mechanics Group, Department of Thermal and
Fluid Engineering, Universidad Carlos III de Madrid, 28911 Leganés (Madrid),
Spain  Gregorio Millán Institute for Fluid Dynamics, Nanoscience and
Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés
(Madrid), Spain  Roberto Zenit  Center for Fluid Mechanics, Brown University,
Providence, RI 02912, USA  Lorène Champougny  Fluid Mechanics Group,
Department of Thermal and Fluid Engineering, Universidad Carlos III de Madrid,
28911 Leganés (Madrid), Spain  Gregorio Millán Institute for Fluid Dynamics,
Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid,
28911 Leganés (Madrid), Spain

###### Abstract

In splat painting, a collection of liquid droplets is projected onto the
substrate by imposing a controlled acceleration to a paint-loaded brush. To
unravel the physical phenomena at play in this artistic technique, we perform
a series of experiments where the amount of expelled liquid and the resulting
patterns on the substrate are systematically characterized as a function of
the liquid viscosity and brush acceleration. Experimental trends and orders of
magnitude are rationalized by simple physical models, revealing the existence
of an inertia-dominated flow in the anisotropic, porous tip of the brush. We
argue that splat painting artists intuitively tune their parameters to work in
this regime, which may also play a role in other pulsed flows, like violent
expiratory events or sudden geophysical processes.

##  1 Introduction

Painting is one of the earliest human artistic expressions. Early cave
paintings are considered by many the signature for the emergence of human
behavior [1]. A series of recent papers have argued that the essence of
artistic painting lays within a keen understanding of the physical constraints
imposed by the interaction between the artist, their instruments and materials
[2, 3, 4, 5]. In other words, artists develop a deep physical knowledge on how
to manipulate fluid flows to paint. They skillfully choose tools, paints and
action to cover canvases with certain fluid textures in order to convey their
intention to the painting.

In this paper we study the splat painting technique. It consists in subjecting
a paint-loaded brush to a sudden acceleration, for example by “flicking” the
brush or tapping its handle. As a result, the paint is expelled from the
brush, producing droplets or filaments that then are projected onto a canvas,
leaving fragmented patterns [2]. Figure 1 displays an example of a modern
artwork created with splat painting ; video S1 shows the process in action. In
this technique, perhaps most emblematically used by American painters Jackson
Pollock (1912-1956) [5] and Sam Francis (1923-1994) [6], the brush never
touches the canvas. Splat painting is thus considered an _action painting_
technique. This term, coined by Harold Rosenberg [7], refers to the use of
dribbling, splashing or smearing paint onto a canvas. Its relevance, discussed
by many art historians and critics, emerges from the fact that the creation
process is not entirely under the control of the artist; the deposition of
paint on the canvas is governed by the mechanical constraints of the process
rather than the will of the artist. In this sense, Rosenberg argued that art
could be redefined to include the act rather than just the object, as a
process rather than a product. In the absence of contact between the brush and
the substrate, paint composition becomes one of the main leverages to tune the
production of droplets and their final deposition. The artist therefore
develops a certain intuition on how the physical properties of the paint
(mostly viscosity) should be adjusted to obtain a desired effect.

The splat painting technique can be conceptualized as being the sequence of
the following physical processes. First, the brush is loaded with paint. A
certain volume of paint can be retained inside the bundle of bristles without
dripping due to a capillary effects [8, 9]. Once loaded, the brush is flicked,
either by rapid hand motion or a gentle tap. The sudden motion accelerates the
tip of the brush, causing the fluid trapped within the bristles to flow and
eventually leave the brush. Acceleration-driven ejection of liquid can be
encountered in a variety of contexts. In nature, wet mammals shake [10] and
insects flutter their wings [11] at given frequencies to get rid of excessive
moisture or droplets. In industrial processes, the rotary atomization of
liquids submitted to centrifugal acceleration is commonly used to produce
droplets or sprays (see [12] and references therein). Once the fluid is
ejected from the bristle bundle, it forms filaments that fly away. In most
cases, these filaments fragment into droplets due to the classical Rayleigh-
Plateau instability [13]. The droplets finally land on the canvas to create
the artist composition, as illustrated in Fig. 1. Droplet impact on a
substrate has been extensively studied [14, 15], due to its importance in many
engineering applications and natural phenomena, from inkjet printing [16] to
pathogens transmission [17].

The process of splat painting is clearly a rich fluid mechanical problem.
While some aspects of its mechanics have been addressed by other studies, a
specific investigation that focuses on the parameters relevant to this
artistic painting technique does not exist. To unravel the physical mechanisms
at play in splat painting, we performed experiments on a setup allowing us to
impose a controlled acceleration to a liquid-loaded brush (see Methods
section). Varying the paint viscosity and brush acceleration, we are then able
to rationalize the effect of these parameters on the amount of liquid ejected
(section 2) and on the resulting spotted pattern (section 3).

Figure 1: Top: Painting produced with the splat painting technique. Portrait
of Nadia Cortes by Octavio Moctezuma, 2022 (reproduced with permission).
Bottom left: Zoom of the droplet pattern (Scale bar:
$1~{}\mathrm{c}\mathrm{m}$). Bottom right: snapshot of the point-of-view movie
of the artist painting (see Video S1).

##  2 How much liquid is ejected?

Figure 2: Ejected mass $\Delta m$ as a function of the kinematic viscosity
$\nu$ for (a) silicone oils and (b) water-glycerol mixtures and different drop
heights. (c) Comparison of the dimensionless experimental data (symbols) to
the prediction of the scaling analysis (solid line). The shaded area shows the
range of $\mathcal{N}$ used by splat painting artists.

Experimentally, the acceleration is produced by dropping a weight (impactor)
from a given height $\Delta h$, onto the brush handle. The liquid mass $\Delta
m$ ejected from the bundle is measured as a function of the liquid kinematic
viscosity $\nu$ for various impactor drop heights. For both silicone oils (SO,
figure 2a) and water-glycerol mixtures (GLY, figure 2b), and regardless of
$\Delta h$, two regimes are observed depending on the viscosity. At low
viscosities, $\Delta m$ seems to plateau out toward a constant value while, at
larger viscosities, the mass decays with $\nu$ following a trend compatible
with a power law. In the following, we propose to rationalize those two
regimes by modelling the acceleration-driven flow in the brush bundle.

###  2.1 Brush kinematics

Figure 3: (a) Snapshots of the brush bundle loaded with
$40~{}\mathrm{c}\mathrm{St}$ silicone oil (SO-40), and subjected to an impact
from a drop height $\Delta h=3~{}\mathrm{c}\mathrm{m}$. (b) Sketch showing the
bundle deflection $\delta w_{B}$, vertical acceleration $\ddot{w}_{B}$ in the
reference frame of the laboratory, and its projection
$\ddot{w}_{B,\shortparallel}$ onto the longitudinal axis of the bundle. Time
variations of these magnitudes during impact are shown in panels (c), (d) and
(e), respectively. These data correspond to a position $X=0.85L$ along the
bundle of a brush loaded with $1000~{}\mathrm{c}\mathrm{St}$ silicone oil.

How the impact from a given height translates into the brush motion is a
complex solid mechanical problem, which we do not aim to address here.
Instead, we focus directly on the kinematics of the brush upon impact, as
revealed by high-speed imaging (see supplementary movie S2 and figure 3a). In
order to quantify the movement of the brush, we introduce in Figures 3b and 7
the $(X,Z)$ coordinate system tied to the (non-inertial) reference frame of
the accelerated brush handle. Digital image processing of sequences such as
Figure 3a allows us to measure vertical displacements in the reference frame
of the laboratory: $w_{H}(t)$ for the rigid handle, and $w_{B}(X,t)$ for the
deformable bundle. The bundle vertical displacement relative to the handle,
that is to say its deflection with respect to its equilibrium position, is
defined as $\delta w_{B}(X,t)=w_{B}(X,t)-w_{H}(t)$ (Figure 3c). Additionally,
the vertical displacement $w_{B}$ is time-differenciated twice to obtain the
instantaneous vertical acceleration $\ddot{w}_{B}(X,t)$ of the bundle (Figure
3d) in the reference frame of the laboratory.

###  2.2 Driving mechanism

In our experimental conditions, we observe that liquid is predominantly
ejected from the tip of the bundle, as shown in figure 3a. Assuming an
initially uniform liquid distribution in the bristles, this observation
implies that the vertical forcing $\ddot{w}_{B}$ mostly drives liquid _along_
the bundle to the tip, where it detaches and fragments. This is due to the
combination of two key ingredients.

First, at each point along the central line of the bundle, the local, vertical
forcing acceleration $\ddot{w}_{B}$ can be divided into a transversal and a
longitudinal component (Figure 3b). Because of the top-down symmetry of the
motion, the former averages out to zero over several oscillation cycles, while
the latter, denoted $\ddot{w}_{B,\shortparallel}$, imposes a forcing with non-
zero average to the fluid. Observing that the deflection angle $\alpha$ (see
figure 3b) is roughly uniform along the bundle, the projected acceleration can
be approximated as $\ddot{w}_{B,\shortparallel}\simeq\ddot{w}_{B}\times\delta
w_{B}/L$, under the assumption that $\alpha\ll 1$. As shown in Figure 3e, the
projected acceleration $\ddot{w}_{B,\shortparallel}$ is positive during most
of the motion, indeed pushing the liquid towards the bundle tip.

Second, as can be estimated from the anisotropic bundle geometry (see appendix
A), the transversal permeability (i.e. in the radial direction $r$) is about 5
times smaller than the longitudinal one [18]. Is is therefore easier for the
liquid to move in the longitudinal direction compared to the radial one.
Combining these two arguments, we arrive at the conclusion that the flow of
paint in the bundle mainly takes place in the direction parallel to the
bristles and that the main driving force is the longitudinal projection
$\ddot{w}_{B,\shortparallel}$ of the bundle acceleration.

###  2.3 Flow regimes in the bundle

As can be appreciated in Figure 3a, the wet bundle keeps a roughly cylindrical
shape of diameter $D_{\mathrm{wet}}$ during its oscillations. We therefore
model it as cylindrical slab of porous material of diameter
$D=D_{\mathrm{wet}}$ and length $L$ (Figure 4), in which all the pores (the
spaces between fibers) are assumed to be parallel. Measuring experimentally
the values of $D_{\mathrm{wet}}$, $L$, as well as the distribution of bristle
diameters $d$, we are able to determine the solid fraction $\phi_{s}$ and
typical pore diameter $\mathcal{D}_{h}$ to be assigned to the idealized bundle
(see appendix A).

Figure 4: Sketch of the idealized fiber bundle considered here, and
definitions of its relevant geometrical properties.

Let us now write an equation of motion for the incompressible and Newtonian
liquid flowing in this model bundle. Momentum conservation applied in the
(non-inertial) reference frame of the bundle reads

$\rho\frac{\partial\vec{v}}{\partial
t}+\rho\vec{v}\cdot\vec{\nabla}\vec{v}=-\vec{\nabla}P+\mu\vec{\nabla}^{2}\vec{v}-\rho\vec{a},$
(1)

where $\vec{v}$ is the (three-dimensional) fluid velocity, $P$ the pressure
and $\vec{a}=\ddot{w}_{B}\,\vec{e_{Z}}$ the linear (vertical) bundle
acceleration in the reference frame of the laboratory.

Equation (1) is then projected onto the longitudinal direction of the bundle,
which is expected to be the main flow direction. Additionally, the
characteristic lengthscale in the transversal direction is the hydraulic
diameter of the pores between the bristles, $\mathcal{D}_{h}\sim
0.14~{}\mathrm{m}\mathrm{m}$, while the one in the longitudinal direction is
the bundle length $L\sim 13~{}\mathrm{m}\mathrm{m}$. Under the approximation
of slender flow $\varepsilon=\mathcal{D}_{h}/L\ll 1$ and using the coordinate
system $(r,x)$ tied to the bundle (Figure 4), we arrive at

$\begin{split}\rho\frac{\partial v_{x}}{\partial
t}+&\rho\left(v_{x}\frac{\partial v_{x}}{\partial x}+v_{r}\frac{\partial
v_{x}}{\partial r}\right)=\\\ &-\frac{\partial P}{\partial
x}+\frac{\mu}{r}\frac{\partial}{\partial r}\left(r\frac{\partial
v_{x}}{\partial r}\right)-\rho\,\ddot{w}_{B,\shortparallel},\end{split}$ (2)

where $v_{x}$ and $v_{r}$ denote the components of the fluid velocity in the
longitudinal and radial directions, respectively.

The characteristic timescale $t_{c}$, defined as the average half-period of
the bundle oscillations, and characteristic longitudinal acceleration,
$a_{\shortparallel}$, can be estimated from the brush kinematics (see figure 3
and appendix B). Typical values are $t_{c}\sim 3~{}\mathrm{m}\mathrm{s}$ and
$a_{\shortparallel}\sim 0.2-2\times
10^{3}~{}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$, depending on the drop height
$\Delta h$. The velocity scale $V$ of the flow in the bundle remains a priori
unknown and will be set by the dominant force balance in equation (2).

The fate of the momentum injected by the impact (last term in equation (2)) is
mostly to accelerate the liquid and to be dissipated by viscous friction.
Which of these mechanisms is dominant depends on the ratio between the viscous
term (second on right-hand side) and the transient inertial term (first in
left-hand side) in equation (2), quantified by the dimensionless number

$\mathcal{N}=\frac{\nu t_{c}}{\mathcal{D}_{h}^{2}}.$ (3)

Note that for a periodic flow, this parameter would be equivalent to the
Womersley number $Wo=\mathcal{N}^{-1/2}$. In our setting, we simply refer to
$\mathcal{N}$ as the dimensionless viscosity. Experimentally,
$\mathcal{N}=0.2-50$, therefore requiring to discuss both the inertia-and
viscosity-dominated regimes.

For $\mathcal{N}\ll 1$, the forcing acceleration is mostly used to accelerate
the liquid through the transient term of equation (2). This leads to a typical
velocity scale

$V_{i}=a_{\shortparallel}t_{c},$ (4)

of the order of a few $\mathrm{m}\mathrm{/}\mathrm{s}$. Note that, in this
regime, the value of the Strouhal number $Sr=L/t_{c}V_{i}$, comparing the
transient to convective inertial terms, is of the order of a few units.
Convective effects are therefore expected to be subdominant.

For $\mathcal{N}\gg 1$, the forcing acceleration is mostly dissipated by
viscous friction, leading to a typical velocity scale

$V_{v}=\frac{a_{\shortparallel}\mathcal{D}_{h}^{2}}{\nu}=V_{i}\,\mathcal{N}^{-1}.$
(5)

###  2.4 Modelling the ejected mass

Knowing the characteristic flow velocity $V$ in the different regimes, the
total mass $\Delta m$ expelled from the brush can be estimated as

$\Delta m\sim\rho Q\Delta t,$ (6)

where $Q$ is the liquid flow rate going through the bundle and $\Delta t$ the
total duration of impact, typically in the range $6-10~{}\mathrm{m}\mathrm{s}$
(see appendix B). The flow rate is simply expressed as
$Q=\mathcal{A}^{\prime}V$, where $\mathcal{A}^{\prime}$ is the total cross-
sectional area available for the fluid to flow in the bundle, computed from
its geometry (see appendix A). Combining equation (6) with the characteristic
velocities in the different regimes (equations (5) and (4)), we arrive at

$\displaystyle\Delta m_{i}$ $\displaystyle\sim\rho\mathcal{A}^{\prime}\Delta
t\,a_{\shortparallel}t_{c},$ (7) $\displaystyle\Delta m_{v}$
$\displaystyle\sim\rho\mathcal{A}^{\prime}\Delta
t\,\frac{a_{\shortparallel}\mathcal{D}_{h}^{2}}{\nu}=\Delta
m_{i}\,\mathcal{N}^{-1},$ (8)

in the inertial and viscous limits, respectively.

The predictions of equations (7) (for $\mathcal{N}<1$) and (8) (for
$\mathcal{N}>1$) are shown in figure 2c (solid line). Experimental masses are
made dimensionless with the predicted detached mass in the inertial limit
$\Delta m_{i}$ (equation (7)) and plotted as a function of the dimensionless
viscosity $\mathcal{N}$ (equation (3)). Comparison between the non-
dimensionalized data (symbols) and the predictions of the scaling analysis
shows an overall good agreement, both in trend and orders of magnitude. For a
given liquid family (SO or GLY), the dimensionless data corresponding
different drop heights do collapse onto a single mastercurve. Interestingly,
the mastercurves for SO and GLY overlap in the viscous regime
($\mathcal{N}\gtrsim 1$) but depart slightly in the inertial regime
($\mathcal{N}\lesssim 1$). This discrepancy is likely due to capillary
effects, as the typical surface tension of water-glycerol is more than 3 times
larger than the one of silicone oils (see appendices C and D). The shaded area
in figure 2c shows the range of dimensionless viscosities $\mathcal{N}$
corresponding to artist-chosen splat painting conditions with acrylic paint
(see appendix E). Strikingly, those values are located around $\mathcal{N}\sim
1$, suggesting that artists dilute their paint and/or adjust their tapping
strength to work close to the inertia-dominated regime.

##  3 How is liquid distributed on the substrate?

Figure 5: Example of patterns obtained by splat painting silicone oils of
various viscosities $\nu$ with different brush accelerations (quantified by
the drop height $\Delta h$). The substrate size is A4 ($21\times
29.7~{}\mathrm{c}\mathrm{m}$) and the distance between the brush and the
substrate is $H=15.5~{}\mathrm{c}\mathrm{m}$.

Of paramount importance in the context of artistic painting is the final
spotted pattern obtained on the substrate (see figure 1 and videos S1). Figure
5 shows examples of the patterns generated in our experimental conditions for
a selection of drop heights $\Delta h$ and silicone oil kinematic viscosities
$\nu$. Given an initial mass of liquid expelled from the brush, those patterns
are the result of a complex interplay between (i) liquid fragmentation, (ii)
drop flight dynamics and (iii) droplet impact dynamics on the substrate. Each
of these processes is the subject of a whole body of literature on its own. We
will therefore simply focus on key ideas allowing us to draw a connection
between the total painted area $A_{p}$ (sum of all the black areas for a given
pattern in figure 5) and the ejected mass $\Delta m$, which was rationalized
in the previous section.

###  3.1 Liquid fragmentation

Once the fluid is expelled from the brush, the elongation of fluid filaments
leads to fragmentation and droplet formation. The droplet size distribution
resulting from this complex process has been abundantly studied in the
literature [13], including under radial (centrifugal) acceleration conditions
[19, 12]. Although this distribution is not quantified in our experiments, a
typical drop diameter can be estimated, based on the idea that the expelled
liquid filaments leave the brush with a diameter of the order of the pore size
$\mathcal{D}_{h}$ in the bundle. Subsequent filament destabilization under the
action of the Rayleigh-Plateau instability does not modify this length scale
substantially [20]. For the sake of simplicity, we will henceforth assume that
the droplet population resulting from splat painting consists of $N$ identical
drops of diameter $D_{0}$, with $D_{0}\sim\mathcal{D}_{h}$.

###  3.2 Drop flight dynamics

Considering a droplet of diameter $D_{0}$ leaving the brush with an initial
velocity $U_{0}$ at a height $H$ above the substrate, we wish to estimate its
velocity $U_{\ast}$ at the moment of its impact on the substrate.
Approximating the initial drop velocity by the typical (vertical) velocity of
the tip of the bundle in the reference frame of the laboratory, $U_{0}$ is
found experimentally to be of the order of
$2-5~{}\mathrm{m}\mathrm{/}\mathrm{s}$, depending on $\Delta h$. The impact
velocity $U_{\ast}$ of the droplet then depends on how much it was slowed down
by air friction during its flight. This can be quantified comparing the flight
time $t_{f}$ of the droplet and the characteristic time $\tau$ of viscous
friction in air (see appendix F). In our experimental conditions, we can show
that $t_{f}\sim\tau$ for droplets leaving the brush with a vertical velocity
$U_{0}$ pointing downwards, which is the situation leading to the shortest
flight time. The impact velocity $U_{\ast}$ on the substrate may therefore be
approximated terminal velocity $U_{t}$ of the droplet in air.

###  3.3 Drop impact

The impact dynamics of a drop on a solid surface is essentially governed by
the Weber and Reynolds numbers [14, 15], defined respectively as

$We_{\ast}=\frac{\rho D_{0}U_{\ast}^{2}}{\sigma}\quad\text{and}\quad
Re_{\ast}=\frac{D_{0}U_{\ast}}{\nu}.$ (9)

In particular, those parameters control the maximum lateral extension of the
impacting drop (or maximum spreading diameter $D_{m}$) [21] and the occurrence
of splashing (i.e. fragmentation of the drop upon impact) [15], that happens
for $K=We_{\ast}\sqrt{Re_{\ast}}\gtrsim 3000$. Across the whole $(\nu,\Delta
h)$ parameter space covered in our experiments, we estimate $We_{\ast}\sim 1$,
$Re_{\ast}\sim 0.5-30$, and $K\sim 0.65-6.5$ in the limit of a drop impacting
at its terminal velocity ($U_{\ast}=U_{t}$, see appendix F). We therefore do
not expect the impacting drops to undergo splashing, as confirmed by visual
inspection of the patterns.

For a drop impacting on a rough surface like paper, no rebound or recoil can
happen as the contact line remains pinned to the substrate [22]. Consequently,
the maximum spreading diameter $D_{m}$ can be considered a good approximation
for the diameter $D_{\mathrm{sp}}$ of the spot left on the paper.

###  3.4 Modelling the total painted area

Figure 6: Dimensionless plot of the total painted area $A_{p}$ as a function
of the mass $\Delta m$ detached from the brush for various silicone oil
dimensionless viscosities $\mathcal{N}$ and brush accelerations (quantified by
the impactor drop height $\Delta h$).

Taking advantage of the abovementionned simplifications, we now set out to
find a relationship between the detached mass $\Delta m$ and the total painted
area $A_{p}$. The volume corresponding to $N_{d}$ identical drops of diameter
$D_{0}$ is $\mathcal{V}=\pi N_{d}D_{0}^{3}/6$, which is experimentally
measured as $\Delta m/\rho$. Making the additional assumption that the spots
left by the droplets on the substrate do no overlap, the total painted area
can be written as $A_{p}=\pi N_{d}D_{\mathrm{sp}}^{2}/4$, where
$D_{\mathrm{sp}}$ denotes the diameter of a spot. Combing those expressions
and recalling that $D_{\mathrm{sp}}\sim D_{m}$, we obtain

$A_{p}\sim\frac{3}{2}\frac{\Delta m}{\rho
D_{0}}\left(\frac{D_{m}}{D_{0}}\right)^{2}=\frac{3}{2}\frac{\Delta m}{\rho
D_{0}}\mathcal{S}^{2},$ (10)

where we have introduced the drop spreading factor $\mathcal{S}=D_{m}/D_{0}$.
Scaling $\Delta m$ by the prediction in the inertial limit $\Delta m_{i}$
(equation (7)) leads to

$\frac{A_{p}}{A_{p,i}}\sim\frac{3}{2}\mathcal{S}^{2}\,\frac{\Delta m}{\Delta
m_{i}},$ (11)

where the total painted area is made dimensionless with $A_{p,i}=\Delta
m_{i}/\rho D_{0}\sim\mathcal{A}^{\prime}\Delta
t\,a_{\shortparallel}t_{c}/D_{h}$.

Figure 6 reveals that the dimensionless painted areas and detached masses,
obtained experimentally for silicone oils of various viscosities splatted onto
paper, are proportional. The drop spreading factor $\mathcal{S}$ is therefore
roughly constant in our experiments. This is compatible with predictions [15]
showing that $\mathcal{S}$ is only a weak function of the impact parameters
(Weber number $We_{\ast}$ and Reynolds number $Re_{\ast}$). More
quantitatively, fitting the data of figure 6 with equation (11) yields a
typical drop spreading factor of $\mathcal{S}\sim 1.5$. This value is
consistent with the one expected for $We_{\ast}\sim 1$ [21].

##  4 Conclusion

In summary, we investigated the artistic technique of splat painting from a
fluid mechanical perspective. Inspired by the empirical know-how developed by
artists, we explored the influence of two key parameters – liquid viscosity
and brush acceleration – on the quantity of liquid ejected from the brush and
on the final spotted pattern laid down on the substrate.

We showed that the amount of liquid leaving the brush exhibits two regimes,
depending on the value of the dimensionless parameter $\mathcal{N}$ that
compares viscous dissipation to liquid inertia in the hair bundle. For
$\mathcal{N}\gg 1$, most of the brush’s momentum is dissipated by viscosity
and the detached mass is proportional to the Stokes number. For
$\mathcal{N}\ll 1$, the brush’s momentum is mostly used to accelerate the
paint and the expelled mass tends to its upper limit, which is independent of
$\mathcal{N}$. The viscosities of acrylic paint solutions used for splat
painting correspond to this inertia-dominated regime, while native acrylic
paint would fall within the viscous-dominated regime. We thus argue that
artists dilute their paint to perform splat painting (i.e. decrease
$\mathcal{N}$) until they reach the inertial regime $\mathcal{N}\sim 1$, which
allows them to maximize the expelled mass. Additionally, we evidenced that the
total paint-covered area on the substrate is essentially proportional to the
detached mass: drop spreading upon impact does not contribute significantly to
the spot size. The total area covered is therefore essentially governed by the
flow in the bundle, which sets the amount of liquid expelled, and not by the
drop impact conditions on the substrate (at least in the absence of slippage).

We believe that the inertia-dominated regime we evidenced may also be
encountered in other strongly accelerated flows in porous media. Those include
for example sudden expiratory events (sneezing, coughing) [17] or violent
geophysical flows (volcanic eruptions, geysers) [23].

## Materials and methods

### Materials

Figure 7: Sketch of the experimental setup. The brush is clamped in a
horizontal position at a distance $H$ from the substrate. An impact is
produced by dropping a weight from a height $\Delta h$ onto the brush handle.
As a result of this acceleration, the bristles at tip of the brush release
liquid. Scale: $D_{\mathrm{dry}}=4.7~{}\mathrm{m}\mathrm{m}$.

#### Brush

In our experiments, we use a set of identical commercial round brushes,
purchased from a local crafts shop. Each brush is composed of a wooden handle
connected to a bundle of natural hairs (also called bristles) by a metallic
ferrule. The only modification we make to this brush is to trim the tip of the
bundle, cutting all the bristles to a uniform length $L=13.0\pm
0.5~{}\mathrm{m}\mathrm{m}$ (see close-up in figure 7). At the exit of the
ferrule, the dry bundle has a diameter $D_{\mathrm{dry}}=4.7\pm
0.2~{}\mathrm{m}\mathrm{m}$. Examination under the microscope reveals that the
hairs are quite polydisperse, with an average diameter $\langle
d\rangle=169~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\pm
53~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (std), see appendix A. We will
henceforth assume that all our brush copies share these same properties.

#### Liquids

Most paints are non-Newtonian fluids, exhibiting shear-thinning, yield-stress
or even viscoelastic properties [24, 25]. However, to be able to vary the
fluid shear viscosity independently from the brush acceleration (and therefore
shear rate), we choose to focus our study on Newtonian liquids. Two families
of Newtonian liquids are used: silicone oils, denoted by prefix SO, and water-
glycerol mixtures, denoted by prefix GLY. Both these families allow us to vary
viscosity over three orders of magnitudes while keeping density and surface
tension roughly constant (see appendix C).

#### Substrate

The substrate on which splatted liquid droplets are collected is A4 white
paper ($80~{}\mathrm{g}\mathrm{/}\mathrm{m}^{2}$, Navigator). The distance
between the brush and the substrate is set to a constant value
$H=15.5~{}\mathrm{c}\mathrm{m}$ throughout all the experiments.

### Experimental setup and protocol

#### Impact setup

We developed a custom-made experimental setup to mimic the conditions in which
an artist accelerates the paint-loaded brush to produce splat patterns. As
shown in figure 7, the brush is held horizontally at a fixed height $H$ above
the substrate. The wooden handle is clamped to simulate how the painter would
hold the brush. Brush acceleration is generated by dropping onto the wooden
handle a ring-shaped brass weight guided along a fixed vertical brass tube.
The impact generates a transverse wave that propagates in the handle all the
way to the tip of the brush, where it is transmitted to the paint-loaded
bristles. The magnitude of the maximum acceleration can be adjusted by
changing the height $\Delta h$ from which the weight is dropped (see appendix
B). Three drop heights are reported in this study: $\Delta h=1.5$, $3$ and
$6~{}\mathrm{c}\mathrm{m}$.

#### Measurement setup and protocol

The brush is initially loaded with a fixed mass of liquid $m_{0}=155\pm
5~{}\mathrm{m}\mathrm{g}$, which is kept constant throughout the experiments.
We also make sure that the clamping of the handle and impact location are the
same for all experiments.

During impact, the vertical displacements of the extremity of the handle and
of the bristles, as well as the resulting liquid atomization, are visualized
from the side and recorded by a high-speed camera (NAC MEMRECAM HX 3) at a
framerate of 30,000 fps. A typical image sequence is presented in Fig. 3a. In-
house image processing is subsequently used to extract displacements and
accelerations.

The total mass of liquid ejected $\Delta m$ is measured by weighting the brush
before and after impact with a high-precision scale (Sartorius, resolution
$\pm 0.1~{}\mathrm{m}\mathrm{g}$). In addition, for a selection of silicone
oils, the ejected droplets resulting from the tapping are collected on the
substrate laid down horizontally at a fixed distance $H$ below the brush.
Immediately after splatting ($<10~{}\mathrm{s}$ after impact), a top view of
the final drop pattern is photographed using a Samsung Galaxy S20 camera
(resolution of 12 megapixels) and backlighting with a homogeneous light panel
(dimensions $41\times 23.5~{}\mathrm{c}\mathrm{m}$).

#### Image processing of patterns

Photographs of the patterns are subsequently processed using an in-house
Matlab code. Under backlighting conditions, the silicone oil drops on paper
appear as bright spots on a darker background. Spot detection is therefore
performed through image binarization, using an adaptive threshold. However,
due to the heterogeneity of the paper substrate, the background itself
features some granularity. The resulting spurious spots are filtered out using
as a reference the photograph of the blank paper under the same imaging
conditions. Examples of the final processed patterns are shown in Figure 5.

## Acknowledgments

J.R.R. acknowledges funding from the Spanish MCIN/AEI/10.13039/501100011033
through Grant No. PID2020-114945RB-C21. This project has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 882429 (L.C.). We are very grateful
to splat paiting artists Octavio Moctezuma and Caroline Champougny for sharing
their expertise on this technique, as well as their paint recipes. Many thanks
as well to Mithun Ravisankar for his help with the rheological measurements of
paints.

##  Appendix A Bundle geometry

In this section, we characterize experimentally the geometry of brush’s hair
bundle. We then deduce and discuss geometrical properties of the anisotropic
porous medium used as an idealized representation of the bundle. The values of
the most important properties are summarized in table 1.

Figure 8: Diameter probability distribution of the bristles in the bundle, measured on a reduced sample of about $110$ fibers. Red crosses mark the average diameter within each bin of the histogram. Property | Value | Eq.
---|---|---
Wet diameter $D_{\mathrm{wet}}$ | $4.7\pm 0.2~{}\mathrm{m}\mathrm{m}$ | -
Solid fraction $\phi_{s}$ | $0.57\pm 0.03$ | (14)
Hydraulic diameter $\mathcal{D}_{h}$ | $138\pm 16~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ | (16)
Permeability ratio $\kappa$ (squ.) | $6.2-7.9$ | (19)
Permeability ratio $\kappa$ (hex.) | $4.6$ | (19)
Table 1: Geometrical properties of the model wet bundle for a loading
$m=m_{0}$ of the reference liquid SO-1000. The permeability ratio $\kappa$ is
computed in both square and hexagonal packing geometries for a monodisperse
bundle with the same solid fraction $\phi_{s}\pm\delta\phi_{s}$ as our
polydisperse bundle.

###  A.1 Bristle diameter distribution

To determine geometrical properties such as bristle number and bristle
diameter distribution, the bundle of one of the brush copies is severed at its
base to be characterized. The total number of hairs in the bundle is
determined by weighting several known amount of hairs on a high precision
scale (Sartorius, with a resolution of $\pm 0.1~{}\mathrm{m}\mathrm{g}$).
Measuring the total mass of all the bristles then allows us to deduce their
number, which is found to be $N=404\pm 6$. A reduced sample of hairs from the
brush (about $25~{}\%$ of the total) is examined under the microscope to
measure their diameter distribution.

The experimental probability distribution of bristles’ diameters $d$ is
presented in figure 8. This distribution is characterized by its first and
second moments, defined respectively as

$\displaystyle\mathcal{M}_{1}$
$\displaystyle=\sum_{i=1}^{N_{b}}p(d_{i})\,d_{i}\approx(169\pm 3)\times
10^{-6}~{}\mathrm{m},$ (12) $\displaystyle\mathcal{M}_{2}$
$\displaystyle=\sum_{i=1}^{N_{b}}p(d_{i})\,d_{i}^{2}\approx(31\pm 1)\times
10^{-9}~{}\mathrm{m}^{2},$ (13)

where $N_{b}=10$ is the number of bins used to compute the probability
distribution, and $d_{i}$ the average diameter in bin $i$ (red crosses in
figure 8). Assuming that the bristles’ diameters in the idealised bundle
follow the same distribution (i.e. have the same $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$), we are able to compute the following properties of the
model porous medium.

###  A.2 Solid fraction

The solid volume fraction $\phi_{s}$ is defined as the total volume occupied
by the fibers, normalized by the total volume of the bundle. Denoting
$\mathcal{A}$ the total cross-sectional area of the bundle, we have

$\phi_{s}=\frac{\pi}{4}\frac{N\mathcal{M}_{2}}{\mathcal{A}}=\frac{N\mathcal{M}_{2}}{D^{2}},$
(14)

where we recall that $N$ is the total number of fibers in the bundle and
$D=D_{\mathrm{wet}}$ the diameter of the liquid-loaded bundle. The
corresponding estimation of the bundle’s solid fraction is provided in table
1.

###  A.3 Hydraulic diameter

The hydraulic diameter $\mathcal{D}_{h}$ of the bundle is defined as

$\mathcal{D}_{h}=\frac{4\mathcal{A}^{\prime}}{\mathcal{P}},$ (15)

where $\mathcal{A}^{\prime}=\mathcal{A}(1-\phi_{s})$ is the total cross-
sectional area available for the liquid to flow and $\mathcal{P}=\pi
N\mathcal{M}_{1}$ is the total wet perimeter in the bundle. Eventually, the
hydraulic diameter can be expressed as

$\mathcal{D}_{h}=\frac{4\mathcal{A}(1-\phi_{s})}{\mathcal{P}}=\frac{(1-\phi_{s})}{\phi_{s}}\frac{\mathcal{M}_{2}}{\mathcal{M}_{1}},$
(16)

where we took advantage of equation (14) to replace $\mathcal{A}$. Note that,
in the case of a bundle of identical fibers of radius $r$, the quantity
$\mathcal{M}_{2}/\mathcal{M}_{1}$ reduces to $2r$ and we find the expression
proposed by Gebart [26]. For the diameter distribution presented in figure 8,
we estimate the hydraulic diameter to be of the order of a hundred microns
(see table 1).

###  A.4 Permeabilities

An anisotropic porous medium (such as the one depicted in figure 4 of the main
text) has different permeabilities, depending if the flow is parallel or
perpendicular to the fiber direction. For a bundle of identical fibers of
radius $r$, Gebart [26] derived analytical expressions for the longitudinal
and transversal permeabilities, denoted $K_{L}$ and $K_{T}$, respectively:

$\displaystyle K_{L}$
$\displaystyle=\frac{8r^{2}}{c}\frac{(1-\phi_{s})^{3}}{\phi_{s}^{2}},$ (17)
$\displaystyle K_{T}$
$\displaystyle=C_{1}r^{2}\left(\sqrt{\frac{\phi_{s,\mathrm{max}}}{\phi_{s}}}-1\right)^{5/2},$
(18)

where $c$, $C_{1}$ and $\phi_{s,\mathrm{max}}$ are constants that depend on
the geometry of the fiber packing. For a square packing, $c=57$,
$C_{1}=16/(9\pi\sqrt{2})$ and $\phi_{s,\mathrm{max}}=\pi/4$, while for a
hexagonal packing $c=53$, $C_{1}=16/(9\pi\sqrt{6})$ and
$\phi_{s,\mathrm{max}}=\pi/(2\sqrt{3})$. Interestingly, the ratio $\kappa$
between the longitudinal and transversal permeabilities is independent of the
fiber radius $r$:

$\kappa=\frac{K_{L}}{K_{T}}=\frac{8}{cC_{1}}\frac{(1-\phi_{s})^{3}}{\phi_{s}^{2}\left(\sqrt{\frac{\phi_{s,\mathrm{max}}}{\phi_{s}}}-1\right)^{5/2}}.$
(19)

To the best of our knowledge, there are no expressions equivalent to equations
(17), and (18) in the literature for a bundle of polydisperse fibers. However,
since equation (19) does not depend on the fiber radius, we may compute the
permeability ratio $\kappa$ for a monodisperse bundle with the same solid
fraction $\phi_{s}\pm\delta\phi_{s}$ as our polydisperse bundle. The
corresponding values are reported in table 1 for two possible ordered fiber
packing geometries (square and hexagonal).

Assuming that the permability ratios $\kappa$ estimated in the monodisperse
case provide a reasonable order of magnitude for the polydisperse case (for a
given solid fraction), we expect the longitudinal permeability to be about 5
times larger than the transversal one for our brush’s bundle.

###  A.5 Effect of liquid loading

Figure 9: Illustration of the clumping effect in a wet bundle of fibres
(cross-sectional view), depending on the volume of liquid loaded. The same
mass $m$ of liquid 1 (density $\rho_{1}$) and liquid 2 (density
$\rho_{2}<\rho_{1}$) is loaded on a given brush. Because the corresponding
volumes are such that $V_{2}>V_{1}$, and liquids are incompressible, the
diameter of the wet bundles satisfy $D_{\mathrm{wet},2}>D_{\mathrm{wet},1}$.

When a hairy structure is wet, capillary forces tend to bring the hairs closer
together, an effect that is referred to as clumping [27, 9, 8, 28, 29]. This
is the reason why the brush bundle is tighter when it is wet as compared to
when it is dry (Figure 10). Because liquids are incompressible, we expect that
the diameter of the wet bundle depends on the volume of liquid loaded on it
(as long as there is no dripping). However, our protocol involves loading a
fixed _mass_ of liquid in the bundle. Since density can vary up to $30~{}\%$
between our liquids (see table 3), the loaded volume and therefore the wet
diameter $D=D_{\mathrm{wet}}$ may vary slightly between experiments, as
illustrated in Figure 9.

In the following, we show how the bundle geometrical properties can be
corrected for this effect. Volume conservation of the bundle imposes that

$\frac{\pi D^{2}}{4}L=\frac{m}{\rho}+N\mathcal{M}_{2}L,$ (20)

where we recall that $N\mathcal{M}_{2}$ is the cross-sectional occupied by the
bristles. The cross-sectional area available for the liquid to flow is then
simply be expressed as

$\mathcal{A}^{\prime}=\frac{m}{\rho L},$ (21)

and the hydraulic diameter $\mathcal{D}_{h}$ is deduced from equation (15).

Let us assume we know the wet diameter $D_{\mathrm{ref}}$ of the bundle loaded
with a given mass $m$ of a reference liquid of density $\rho_{\mathrm{ref}}$.
Since the bundle is always loaded with the same mass $m$, then its geometry
(i.e. available cross-sectional area and hydraulic diameter) can be
straightforwardly deduced for all liquids from the reference measurement:

$\displaystyle\mathcal{A}^{\prime}$
$\displaystyle=\mathcal{A}_{\mathrm{ref}}^{\prime}\frac{\rho_{\mathrm{ref}}}{\rho},$
(22) $\displaystyle\mathcal{D}_{h}$
$\displaystyle=\mathcal{D}_{h,\mathrm{ref}}\frac{\rho_{\mathrm{ref}}}{\rho}.$
(23)

Table 1 shows the geometrical of the bundle loaded with a mass $m=m_{0}$ of
the reference liquid, chosen to be $1000~{}\mathrm{c}\mathrm{St}$ silicone oil
(SO-1000). The available cross-sectional area $\mathcal{A}^{\prime}$ and
hydraulic diameter $\mathcal{D}_{h}$ were deduced for all the other liquids in
table 3 using equations (22) and (23), and used to non-dimensionalize the data
in presented in Figure 2c of the main text.

##  Appendix B Brush impact characteristics

Liquid family |  $\Delta h$ ($\mathrm{c}\mathrm{m}$) |  $t_{c}$ ($\mathrm{m}\mathrm{s}$) |  $\Delta t$ ($\mathrm{m}\mathrm{s}$) |  $a_{\shortparallel}$ ($\times 10^{3}~{}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$) |  $a_{0}$ ($\times 10^{3}~{}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$) |  $U_{0}$ ($\mathrm{m}\mathrm{/}\mathrm{s}$)
---|---|---|---|---|---|---
(SO) | $1.5$ | $2.96\pm 0.07$ | $6.44\pm 0.05$ | $0.23\pm 0.02$ | $2.4\pm 0.2$ | $2.0\pm 0.1$
$3$ | $2.29\pm 0.02$ | $6.41\pm 0.02$ | $0.69\pm 0.02$ | $4.3\pm 0.1$ | $3.33\pm 0.08$
$6$ | $2.22\pm 0.02$ | $7.0\pm 0.6$ | $1.5\pm 0.1$ | $6.1\pm 0.3$ | $4.7\pm 0.3$
(GLY) | $1.5$ | $2.72\pm 0.01$ | $9.8\pm 0.2$ | $0.23\pm 0.02$ | $2.5\pm 0.1$ | $2.05\pm 0.09$
$3$ | $2.6\pm 0.2$ | $9.81\pm 0.08$ | $0.56\pm 0.03$ | $4.0\pm 0.2$ | $3.11\pm 0.07$
$6$ | $2.70\pm 0.01$ | $9.76\pm 0.06$ | $1.6\pm 0.1$ | $5.8\pm 0.1$ | $4.6\pm 0.1$
Table 2:  Observables characterizing the bundle kinematics for two different
copies of the same brush (one for the SO family and one for the GLY family)
and various drop heights $\Delta h$ of the impactor.

In splat painting, the acceleration of the liquid-loaded bristles is driving
the expulsion of liquid from the bundle. However, unlike other geometries
where the acceleration can be imposed (for example by setting the angular
frequency of a rotating device [12]), our configuration does not allow us have
a direct control over the acceleration of the brush. In our experimental set-
up, the control parameter is the drop height of the weight, which determines
the energy of the impact on the brush’s handle. How this translates into the
accelerated motion of the bundle depends on the geometry and mechanical
properties of the different parts of the brush (handle, ferrule, bundle). Our
approach here is not to solve this solid mechanics problem, but rather to
encapsulate it using a few key observables extracted from the high-speed
imaging of the bundle motion.

Liquid detachment impedes the automatic detection of the bundle motion on the
video footages (see snapshots in figure 3a of the main text). This is the
reason why we use here only the experimental videos for the most viscous
liquids (SO-1000 and GLY-100), for which all the liquid remains trapped
between the bristles. We checked that the bundle kinematics is essentially
independent of the viscosity of liquid loaded in the bristles, thus allowing
us to extrapolate the results obtained on SO-1000 and GLY-100 to lower
viscosities. In the following, we explain how the observables characterizing
the bundle kinematics are defined and computed. Table 2 summarizes their
values for the two different copies of the same brush, used for the (SO) and
(GLY) families respectively.

###  B.1 Oscillation timescale of the bundle

The characteristic timescale $t_{c}$ of the flow is set by the oscillations of
the bundle during the impact. Since both upwards and downwards flinging may
cause paint detachment, we define $t_{c}$ as the average half-period of
oscillation of the bundle deflection $\delta w_{B}(t)$ (see figure 3c in the
main text).

###  B.2 Total duration of impact

The impact duration $\Delta t$ is defined as the total amount of time during
which the longitudinal projection of the bundle acceleration
$\ddot{w}_{B,\shortparallel}(X,t)$ takes non-zero values. In practice, we use
as a threshold $thr$ the average of $\ddot{w}_{B,\shortparallel}$ after impact
(i.e. for $t>10~{}\mathrm{m}\mathrm{s}$ in the example of figure 3e in the
main text). The duration $\Delta t$ is then computed as the total time during
which $\ddot{w}_{B,\shortparallel}>thr$.

###  B.3 Characteristic accelerations

The typical longitudinal acceleration $a_{\shortparallel}$ driving the flow
along the bundle is obtained from the time integral of
$\ddot{w}_{B,\shortparallel}(X,t)$ (see figure 3e in the main text). It is
defined as

$a_{\shortparallel}=\frac{1}{\Delta
t}\int\ddot{w}_{B,\shortparallel}(X,t)\,\mathrm{d}t,$ (24)

and evaluated at the bundle tip ($X=0.85L$), where most of the liquid
expulsion occurs.

Similarly, for the purpose of estimating the effective Bond number (see
section D), we also define the typical _vertical_ acceleration $a_{0}$ of the
bundle as

$a_{0}=\frac{1}{\Delta t}\int|\ddot{w}_{B}(X,t)|\,\mathrm{d}t,$ (25)

where the vertical bundle acceleration $\ddot{w}_{B}(X,t)$ is taken at
$X=0.85L$.

###  B.4 Characteristic vertical velocity

Liquid drops and filaments leave the brush with a velocity that is equal to
the one of the tip of the bundle (in the reference frame of the lab). The
typical vertical velocity $U_{0}$ of a drop right after detachment can
therefore be estimated as

$U_{0}=\frac{1}{\Delta t}\int|\dot{w}_{B}(X,t)|\,\mathrm{d}t,$ (26)

where $X=0.85L$ is again a proxy for the location of the bundle tip.

##  Appendix C Liquid preparation and properties

Experiments are carried out at ambient temperatures ranging from $19.4$ to
$25.7~{}\degreecelsius$, with an average $\langle
T\rangle=21.6~{}\degreecelsius$. Table 3 summarizes the physical properties of
all the liquids investigated at the average temperature $\langle T\rangle$.
However, note that the values of density and viscosity are corrected to
account for the actual temperature at the moment of each experiment.

###  C.1 Silicone oils (SO)

Silicone oils SO-2, 5, 10, 20, 50, 100, 200, 350, 1000 were used off the
shelf. In addition, silicone oils SO-40, 70, 150 were obtained by blending
commercially available oils using the Grunberg-Nissan mixing rule [30]. The
exact kinematic viscosities of the blends were subsequently measured with an
Ostwald viscometer.

###  C.2 Water-glycerol mixtures (GLY)

Aqueous glycerol solutions are prepared mixing pure glycerol ($\geq 99~{}\%$)
and ultrapure (Milli-Q) water in various proportions. Solutions are labelled
GLY-$\xi$, where $\xi$ stands for the mass fraction of glycerol in $\%$, while
GLY-100 simply denotes pure glycerol. The density and viscosity of these
solutions were computed based on $\xi$ and the ambient temperature using the
correlation-based calculator developed by Andreas Volk and Chris Westbrook
[31, 32].

###  C.3 Precautions and cleaning protocol

Silicone oils and aqueous liquids (water-glycerol and acrylic paint solutions)
are tested on separate copies of the same brush. When changing the liquid on a
given brush, the bristles are thoroughly cleaned, either by wiping with
absorbing paper (silicone oils) or rinsing in ultrapure water and drying
(aqueous liquids) to ensure the brush returns to its initial state.

Contrarily to silicone oils, water-glycerol mixtures are observed to evaporate
in our experimental conditions (ambient relative humidity in the range
$20-40~{}\%$ and timescales of $\sim 1~{}\min$). For each liquid in the GLY
family, the mass lost to evaporation during the experiment, denoted $\Delta
m_{\mathrm{evap}}$, is therefore characterized experimentally. Glycerol being
hygroscopic, the evaporated mass decreases when increasing the glycerol
concentration, ranging from $\Delta m_{\mathrm{evap}}\sim
3~{}\mathrm{m}\mathrm{g}$ for GLY20 to values below the resolution of our
scale ($\Delta m_{\mathrm{evap}}<0.1~{}\mathrm{m}\mathrm{g}$) for GLY80. The
values of the ejected mass $\Delta m$ plotted in figure 2 b of the main text
are corrected from the effect of evaporation.

For each liquid family and each set of control parameters (liquid viscosity
and brush acceleration), a minimum of three repetitions of the same
experiments are conducted, both for mass and pattern measurements.

Liquid code | $\nu$ | $\rho$ | $\sigma$
---|---|---|---
| $(\mathrm{c}\mathrm{St})$ | $(\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{{}^{3}}\mathrm{m})$ | $(\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m})$
SO-2 | $2.1$ | $876$ | $18.7$
SO-5 | $5.3$ | $921$ | $19.7$
SO-10 | $11$ | $938$ | $20.1$
SO-20 | $21$ | $953$ | $20.6$
SO-40 | $43$ | $958$ | $20.7$
SO-50 | $53$ | $963$ | $20.8$
SO-70 | $63$ | $967$ | $20.9$
SO-100 | $107$ | $969$ | $20.9$
SO-150 | $147$ | $970$ | $21.0$
SO-200 | $213$ | $971$ | $21.0$
SO-350 | $373$ | $973$ | $21.1$
SO-1000 | $1070$ | $974$ | $21.2$
GLY-20 | $1.6$ | $1045$ | $71.7$
GLY-40 | $3.2$ | $1098$ | $70.0$
GLY-60 | $8.8$ | $1152$ | $68.5$
GLY-70 | $17$ | $1178$ | $67.0$
GLY-78 | $35$ | $1200$ | $67.4$
GLY-80 | $48$ | $1208$ | $67.1$
GLY-87 | $100$ | $1225$ | $66.7$
GLY-100 | $970$ | $1260$ | $63.4$
Table 3: Properties at $\langle T\rangle=21.6~{}\degreecelsius$ of the model
liquids used in this study: silicone oils (prefix SO) and water-glycerol
solutions (GLY).

##  Appendix D Effects of surface tension on the flow in the bundle

Figure 10: Snapshots of bundle motion during an impact from a drop height
$\Delta h=3~{}\mathrm{c}\mathrm{m}$ (a) in the absence of liquid, (b) when
loaded with SO-2 ($\nu=2.1~{}\mathrm{c}\mathrm{St}$,
$\sigma=18.7~{}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$), (c) when loaded
with GLY-20 ($\nu=1.6~{}\mathrm{c}\mathrm{St}$,
$\sigma=71.7~{}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$). The timestamps are
identical in the three sequences.

In this section, we identify and discuss three ways in which surface tension
may affect (directly or indirectly) the amount of liquid ejected from the
brush. The surface tension of water-glycerol solutions (GLY) is typically more
than three times larger than the one of silicone oils (SO), while other
properties remain similar (see table 3). These two families of liquids
therefore allow us to test our hypotheses against experimental observations.

###  D.1 Bundle cohesion

The snapshots in Figure 10a show that the wet bundle says cohesive during
impact. This observation is crucial, as it allows us to define the bundle
diameter $D_{\mathrm{wet}}$ and the associated bundle geometrical properties
(see Appendix A).

The bundle’s cohesion is due to a combination of surface tension and viscous
forces. Capillary bridges resist hair separation, as it would result in an
increased interfacial area. Additionally, for two wet bristles to separate,
the liquid between them must flow away, which becomes more and more difficult
as viscosity increases.

Figure 10 shows that, even for the smallest viscosities tested (SO-2 on panel
b, GLY-20 on panel c), the wet hair bundle remains more cohesive during impact
than in the dry case (panel a). This suggests that, for both SO and GLY,
capillary forces are strong enough to overcome the bristle inertia.

###  D.2 Transverse flow

Figure 10 also reveals differences in the phenomenology of paint detachment
for low viscosity silicone oils and water/glycerol solutions. In the case of
SO-2 (figure 10b), many liquid filaments are observed to detach from various
locations along the “body” of the bundle, although most of the liquid still
leaves from the tip. In contrast, for GLY-20 (figure 10c), which has similar
viscosity and density as SO-2 but a surface tension more than 3 times larger,
liquid is observed to detach exclusively from the tip of the bundle.

The effective Bond number compares the vertical inertia of the liquid to
capillary forces. It can be defined as

$Bo=\frac{\mathcal{D}_{h}^{2}\rho a_{0}}{\sigma},$ (27)

where $a_{0}$ is the typical _vertical_ bundle acceleration at its tip. For
silicone oils, this number lies in the range $Bo=2.2-6.8$, while for
water/glycerol solutions, $Bo=0.56-1.4$. Since $Bo>1$ for the SO family, the
vertical acceleration is therefore strong enough to overcome surface tension
and break capillary bridges at the periphery of the bundle. This results in an
additional mass of liquid detached from the “body” of the bundle. On the
contrary, $Bo\lesssim 1$ for the GLY family, and surface tension is able to
keep liquid inside the bundle along its length. Liquid thus only leaves from
the bundle tip.

This mechanism may explain why, in the inertial regime ($\mathcal{N}<1$) the
expelled mass is systematically larger for SO as compared to GLY (see figure
2c in the main text). In the viscous regime ($N>1$), liquid viscosity impedes
any transverse flow in the bundle, regardless of the value of surface tension.
However, in the inertial regime, the liquid can flow more easily in the
transverse direction, and only a strong enough surface tension (i.e. such that
$Bo<1$) may prevent liquid from leaving the body of the bundle.

###  D.3 Longitudinal capillary pressure gradient

In the framework of our scaling approach, the magnitude of capillary effects
relative to the forcing longitudinal acceleration is quantified by the
dimensionless surface tension

$\Sigma=\frac{\sigma}{\rho a_{\shortparallel}\mathcal{D}_{h}L},$ (28)

which is of the order of $0.008-0.06$ for SO and $0.03-0.2$ for GLY. For both
liquid families, the capillary pressure gradient therefore remains only a
small correction to the total pressure gradient driving the liquid in the
longitudinal direction in the bundle.

##  Appendix E Artist-chosen splat painting conditions

In this section, we determine the range of dimensionless viscosities
$\mathcal{N}$ used in practice by splat painting artists.

Table 4: Concentrations and properties of acrylic paint aqueous solutions elaborated by splat painting artist Caroline Champougny. Reference | Color Name | Concentration | Kinematic viscosity $\nu$ | Density $\rho$ | Surface tension $\sigma$
---|---|---|---|---|---
|  |  | $(\mathrm{c}\mathrm{St})$ | $(\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{{}^{3}}\mathrm{m})$ | $(\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m})$
104 | Zinc White | $19~{}\%$ | $5.6$ | $1010$ | $57$
269 | Azo Yellow Medium | $7.9~{}\%$ | $1.9$ | $1001$ | $75$
504 | Ultramarine | $8.3~{}\%$ | $3.8$ | $1004$ | $78$
507 | Ultramarine violet | $15~{}\%$ | $5.4$ | $1014$ | $57$
735 | Oxide Black | $7.7~{}\%$ | $2.3$ | $1003$ | $66$

###  E.1 Acrylic paint solutions

We asked artist Caroline Champougny to prepare acrylic paint solutions
suitable for splat painting. The solutions were prepared using commercial
acrylic paint (Amsterdam - Standard Series, see references in table 4) of
various colors diluted in ultrapure (Milli-Q) water. The procedure was the
following: first, a given mass of native acrylic paint was weighted in a test
tube. Then, the artist was adding water little by little, trying the mixture
along the way, until the desired consistency for splat painting was obtained.
The test tube was weighted again, allowing to deduce the paint concentrations
(in gram of paint per gram of water) presented in table 4. The kinematic
viscosities $\nu$, densities $\rho$ and surface tensions $\sigma$ of those
solutions were subsequently measured in the lab.

###  E.2 Brush kinematics

We also asked artist Caroline Champougny to perform the splat painting
technique with a brush similar to the one used in our experiments, and the
acrylic paint solutions she had prepared. We recorded her in the process with
a color high-speed camera at a frame rate of 1000 fps (see supplementary video
S3). The video allowed us to estimate a typical half-period of the bundle
oscillations $t_{c}\sim 7~{}\mathrm{m}\mathrm{s}$. Assuming a similar brush
hydraulic diameter as in our experiments $\mathcal{D}_{h}\sim
0.14~{}\mathrm{m}\mathrm{m}$, we arrived at the values of $\mathcal{N}=\nu
t_{c}/\mathcal{D}_{h}^{2}$ highlighted in figure 2c of the main text.

##  Appendix F Effect of air friction on drop flight

###  F.1 Terminal velocity

In this section, we aim at evaluating the effect of air friction on the impact
velocity of droplets on the substrate, denoted $U_{\ast}$. We consider
spherical droplets of diameter $D_{0}\sim\mathcal{D}_{h}$ and density $\rho$,
initially launched from a height $H$ above the substrate with an initial
velocity $\pm U_{0}$. The modulus of the velocity $U_{0}$ is assumed to be
equal to the one at the tip of the bundle, measured in the reference frame of
the laboratory (see table 2).

The initial Reynolds number $Re_{a,0}=\rho_{a}U_{0}\mathcal{D}_{h}/\mu_{a}$,
based on the density $\rho_{a}$ and dynamic viscosity $\mu_{a}$ of air, is of
the order of a few tens. The drops are thus a priori submitted to non-linear
drag, and their terminal velocity $U_{t}$ (i.e. in steady state) is given by
the implicit equation

$U_{t}=\sqrt{\frac{4}{3}\frac{\rho}{\rho_{a}}\frac{gD_{0}}{C_{d}}},$ (29)

where $g$ is the acceleration of gravity and $C_{d}$ is the drag coefficient,
which itself depends on the terminal velocity through the Reynolds number

$Re_{a}=\frac{\rho_{a}U_{t}D_{0}}{\mu_{a}}.$ (30)

For the drag coefficient, we use the Torobin-Gauvin correlation [33] which is
valid for a sphere in the range $1<Re_{a}<100$. Solving equation (29) for
$U_{t}$ through an iterative procedure, we find $U_{t}\sim
0.4~{}\mathrm{m}\mathrm{/}\mathrm{s}$ and $Re_{a}\sim 4$. The time $\tau$
required for the droplet to reach this terminal velocity is then

$\tau=\frac{U_{t}}{g}=\sqrt{\frac{4}{3}\frac{\rho}{\rho_{a}}\frac{D_{0}}{gC_{d}}}\sim
40~{}\mathrm{m}\mathrm{s}.$ (31)

###  F.2 Flight times

To compute the flight time of droplets, we consider two limiting cases. The
maximum flight time $t_{f,+}$ is reached for a vertical initial velocity
pointing upwards (i.e. away from the substrate), while the minimum is obtained
for a vertical initial velocity pointing downwards (i.e. towards the
substrate). A lower bound for those flight times is given by

$t_{f,\pm}=\frac{U_{0}}{g}\left(\pm 1+\sqrt{1+\frac{2gH}{U_{0}^{2}}}\right),$
(32)

in the absence of air friction. Comparing to the time required for the
droplets to reach their terminal velocity, we find: $t_{f,+}\sim
0.48-0.98~{}\mathrm{s}\gg\tau$, and $t_{f,-}\sim
32-66~{}\mathrm{m}\mathrm{s}\sim\tau$. We therefore conclude that the terminal
velocity $U_{t}$ can be used as a reasonable estimation of the typical drop
velocity $U_{\ast}$ upon impact on the substrate.

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