# Physical mechanisms for droplet size and effective viscosity asymmetries in
turbulent emulsions

Lei Yi  1 Cheng Wang  1 Thomas van Vuren  2 Detlef Lohse  3,4 Frédéric Risso
5 Federico Toschi  2,6  Chao Sun1,7<EMAIL_ADDRESS>1Center for
Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of
Ministry of Education, Department of Energy and Power Engineering, Tsinghua
University, 100084 Beijing, China 2Department of Physics and Department of
Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB
Eindhoven, The Netherlands 3Physics of Fluids Group and Max Planck Center
Twente, MESA+ Institute and J.M. Burgers Center for Fluid Dynamics, University
of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands 4Max Planck
Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
5Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse,
CNRS, Toulouse, France 6CNR-IAC, Via dei Taurini 19, 00185 Roma, Italy
7Department of Engineering Mechanics, School of Aerospace Engineering,
Tsinghua University, Beijing 100084, China

###### Abstract

By varying the oil volume fraction, the microscopic droplet size and the
macroscopic rheology of emulsions are investigated in a Taylor-Couette (TC)
turbulent shear flow. Although here oil and water in the emulsions have almost
the same physical properties (density and viscosity), unexpectedly, we find
that oil-in-water (O/W) and water-in-oil (W/O) emulsions have very distinct
hydrodynamic behaviors, i.e., the system is clearly asymmetric. By looking at
the micro-scales, the average droplet diameter hardly changes with the oil
volume fraction neither for O/W nor for W/O. However, for W/O it is about
$50\%$ larger than that of O/W. At the macro-scales, the effective viscosity
of O/W is higher when compared to that of W/O. These asymmetric behaviors can
be traced back to the presence of surface-active contaminants in the system.
By introducing an oil-soluble surfactant at high concentration, remarkably, we
recover the symmetry (droplet size and effective viscosity) between O/W and
W/O emulsions. Based on this, we suggest a possible mechanism responsible for
the initial asymmetry. Next, we discuss what sets the droplet size in
turbulent emulsions. We uncover a $-6/5$ scaling dependence of the droplet
size on the Reynolds number of the flow. Combining the scaling dependence and
the droplet Weber number, we conclude that the droplet fragmentation, which
determines the droplet size, occurs within the boundary layer and is
controlled by the dynamic pressure caused by the gradient of the mean flow, as
proposed by Levich (1962), instead of the dynamic pressure due to turbulent
fluctuations, as proposed by Kolmogorov (1949). The present findings provide
an understanding of both the microscopic droplet formation and the macroscopic
rheological behaviors in dynamic emulsification, and connects them.

##  1 Introduction

Emulsions, such as mixtures of oil and water, have numerous industrial
applications, including enhanced oil recovery, liquid-liquid extraction, drug
delivery systems, and food processing (Mandal et al., 2010; Mcclements, 2007;
Maffi et al., 2021). We can distinguish two types of emulsions: oil droplets
in water and water droplets in oil, which we abbreviate with O/W and W/O,
respectively (Salager et al., 2000). What emulsion type is realized depends on
a number of variables, among which the dispersed phase volume fraction is
determinant (Zambrano et al., 2003). Typically, by increasing the dispersed
phase volume fraction, $\phi_{d}$, a point is reached where the system
experiences a so-called catastrophic phase inversion, by which the dispersed
phase suddenly becomes the continuous one and vice versa (Piela et al., 2009).
The evolution from O/W to W/O (or vice versa) can be accompanied by a dramatic
change of the emulsion properties, including its morphology, rheology, and
stability (Perazzo et al., 2015). Various studies show that asymmetric
behaviors between O/W and W/O emulsions can be found for both the phase
inversion characteristics and hydrodynamic behaviors, such as the critical
volume fraction for the phase inversion (Pacek et al., 1994), even when the
densities and the viscosities of the two phases in an oil-water system are
identical (Kumar, 1996). W/O emulsions in a gravity settler were found to
separate much more rapidly than their O/W counterparts (Kato et al., 1991).
The same holds for emulsion in a Taylor-Couette turbulent flow (Bakhuis et
al., 2021). Additionally, it is found that O/W and W/O emulsions have
different structures for a volume fraction $\phi_{d}>25\%$ of the dispersed
phase (Pacek et al., 1994). These experimental findings of the asymmetry in
emulsions cannot be easily explained within the scope of existing models
(Kumar, 1996). Although some models (e.g., charged droplet model) have been
proposed to account for the above mentioned observations (Tobin & Ramkrishna,
1999; Kumar, 1996), the understanding of asymmetric behaviors between O/W and
W/O emulsions is still very limited.

Turbulent emulsions are complex physical systems, characterized by a dynamical
coupling between small-scale droplets, large-scale flow and rheology. In the
low-volume-fraction regime, droplet fragmentation is generally caused by the
turbulent stress while the presence of droplets hardly affects the continuous
phase (Afshar Ghotli et al., 2013). The study of the droplet size in a
turbulent flow can be traced back to Kolmogorov (1949) and Hinze (1955), who
attributed the droplet break-up to turbulent fluctuations. Although the
Kolmogorov-Hinze (K-H) theory has been validated in a variety of experimental
and numerical studies on droplets or bubbles in a turbulent flow (Risso &
Fabre, 1998; Perlekar et al., 2012; Eskin et al., 2017; Rosti et al., 2019),
it was found to have limitations, for example, in non-homogeneous turbulent
flows (Hinze, 1955). In the high-volume-fraction regime (before phase
inversion), the microscopic droplet structure (droplet size and distribution),
generated by the turbulent stresses, has a strong feedback on the macroscopic
properties (viscosity) of the turbulent emulsion (De Vita et al., 2019; Yi et
al., 2021). It has been found that the effective viscosity of the emulsion
increases with increasing the volume fraction of the dispersed phase, which is
similar to what is found for the case of suspensions of solid particles (Rosti
& Takagi, 2021; Stickel & Powell, 2005; Guazzelli & Pouliquen, 2018). However,
when considering the statistics of deformation, coalescence, and breakup, the
dynamics of the droplets in emulsions is expected to be much richer than that
of solid particles in suspensions.

The problem becomes even more complicated when we consider turbulent emulsions
in practical environmental and industrial applications, where the appearance
of dirt and surfactant in liquids or on the interfaces has to be taken into
account (Soligo et al., 2020; Bazazi & Hejazi, 2020). The surfactant dynamics
can strongly modify the evolution of a flowing emulsion. On the one hand, the
surfactant directly changes the interfacial properties, affecting the
interface deformation and collision rate (Manikantan & Squires, 2020). On the
other hand, the presence of surfactant can alter the global properties of the
emulsion, such as its rheology (Kawaguchi, 2016). However, the current
understanding of the physics of turbulent emulsions with surfactant addition
is still limited.

In this work, we study the dynamics of the emulsion in a turbulent shear flow,
with an oil volume fraction ranging from $0\%$ to $100\%$. We focus on the
dispersed droplet size as a microscopic observer and the effective viscosity
of the emulsion as a macroscopic observer. By introducing a surfactant at a
controlled concentration into the system, we aim to reveal the physical
mechanism for the asymmetric behaviors between oil-in-water and water-in-oil
emulsions. Furthermore, we uncover the breakup mechanism of the droplet in
such turbulent emulsions, for which the droplet Weber number plays an crucial
role.

##  2 Experimental setup and procedure

Figure 1:  Sketch of the experimental set-up. The gap between the inner and
the outer cylinders is filled with two immiscible liquids: silicone oil and
ethanol-water. The emulsion is formed by rotating the inner cylinder at a
given angular velocity $\omega_{i}$, while the outer cylinder is kept fixed.
The torque of the inner cylinder is measured by a torque sensor. A circulating
water bath is used to maintain the working temperature at $\theta=22\pm
0.1^{\circ}$C. A high-speed camera equipped with a long-distance microscope is
used to capture the dispersed droplets in the flow.

In this study, the emulsion consists of two immiscible liquids: silicone oil
(density $\rho_{o}=866\rm~{}kg/m^{3}$ and viscosity $\nu_{o}=2.1\times
10^{-6}\rm~{}m^{2}/s$) and an aqueous ethanol-water mixture
($\rho_{w}=860\rm~{}kg/m^{3}$, $\nu_{w}=2.4\times 10^{-6}\rm~{}m^{2}/s$). The
experiments were carried out in a Taylor-Couette (TC) system (see figure
1(a)). The system has an inner cylinder radius $r_{i}=25\rm~{}mm$, an outer
cylinder radius $r_{o}=35\rm~{}mm$, and a gap $d=10\rm~{}mm$, giving a radius
ratio of $\eta=r_{i}/r_{o}=0.71$. The height of the inner cylinder is
$L=75\rm~{}mm$, so that the aspect ratio is $\Gamma=L/d=7.5$. The inner
cylinder is made of aluminum, while the outer one is made of glass to enable
optical measurements. Initially, the gap between the cylinders is filled by
the ethanol-water mixture and the oil. Then, the inner cylinder is set in
rotation at a constant angular velocity $\omega_{i}$, while the outer cylinder
is kept fixed (i.e., $\omega_{o}=0$). A strong turbulent shear flow is
generated which generates an emulsion. After a certain time, the emulsion
finally reaches a state where its statistical properties are steady. Note that
the density match of the two phases (oil and ethanol-water) eliminates the
effect of the centrifugal force on the liquid distribution in the system (see
appendix A). A circulating water bath is used to maintain the working
temperature at $\theta=22\pm 0.1^{\circ}$C. The temperature gradient in the
emulsion is negligible due to the efficient mixing induced by the turbulent
fluctuations (van Gils et al., 2011; Grossmann et al., 2016). The control
parameter of the Taylor-Couette flow is the Reynolds number defined as:
$Re=\omega_{i}r_{i}d/\nu_{w}$, where $\omega_{i}$ is the imposed angular
velocity of the inner cylinder, and $\nu_{w}$ is the viscosity of the ethanol-
water. Here we also define a modified Reynolds number,
$Re_{m}=\omega_{i}r_{i}d/\nu_{eff}$, where $\nu_{eff}$ is the effective
viscosity of the emulsion. We measured the total torque exerted on the inner
cylinder $T_{raw}$, which includes two parts: the torque contribution from the
cylindrical sidewall surfaces (the TC flow), $T$, and the torque contribution
from both the top and bottom end plates (the end flow), $T_{end}$. $T_{end}$
is measured using the same linearization method in previous studies (Hu et
al., 2017; Greidanus et al., 2011) (see appendix B). Thus, the torque
contribution of the TC flow can be determined by $T=T_{raw}-T_{end}$. Based on
this, the dynamic response of the emulsion to the imposed rotation is
characterized by the dimensionless torque: $G=T/2\pi L\rho\nu_{w}^{2}$, and a
modified one: $G_{m}=T/2\pi L\rho\nu_{eff}^{2}$.

The dispersed oil (or ethanol-water) droplets in the emulsion were captured
using the high-speed camera equipped with a long-distance microscope. Videos
and images from experiments were analyzed for the drop size determination
using ImageJ software and Matlab codes. The numerical average of the droplet
diameter $\left<D\right>$ is used as the indicator of the droplet size in this
study. To ensure achieving enough statistics, the average droplet diameter is
calculated based on $\mathcal{O}(10^{3})$ droplet samples. Experiments were
performed for various oil volume fractions, $\phi_{o}$, and Reynolds numbers,
$Re$.

Figure 2:  (a) Average droplet diameter as a function of the oil volume
fraction. The droplet diameter hardly changes with the oil volume fraction for
both O/W emulsions ($\phi_{o}\leq 50\%$, hollow points) and W/O emulsions
($\phi_{o}\geq 60\%$, solid points). The ethanol-water droplets in W/O are
found to be about $50\%$ larger than the oil droplets in O/W. The bluish band
spans one standard deviation around the average diameter. The dashed lines
denote the average values of the droplet diameters for O/W (or W/O). A typical
O/W emulsion ($\phi_{o}=5\%$) and a typical W/O emulsion ($\phi_{o}=95\%$) are
shown in the insets. The scalar bar represents $200\rm\mu m$. (b) The
normalized effective viscosity of the emulsion as a function of the oil volume
fraction, $\phi_{o}$, for various Reynolds numbers, $Re$. The inset shows the
asymmetry factor $S$ as a function of $\phi_{o}$ for various $\omega_{i}$. The
asymmetric trend of the effective viscosity between O/W (left branch) and W/O
(right branch) emulsions is more pronounced at low Reynolds numbers.

##  3 Results and discussion

###  3.1 Asymmetric behaviors of the droplet size and the effective viscosity

The size of the dispersed droplets in a turbulent emulsion characterizes the
microscopic structure of the emulsion, which affects the macroscopic stability
and rheology of the emulsion. Also, the volume fraction of the dispersed
phase, $\phi_{d}$, determines both the micro-scale structure (droplet size)
and, consequently, the macro-scale rheological behaviors (effective
viscosity).

Firstly, we focus on the effect of the oil volume fraction, $\phi_{o}$, on the
droplet size, for a given Reynolds number of $Re=5.2\times 10^{3}$. The volume
fraction of oil, $\phi_{o}$, is varied from 0 (ethanol-water mixture) to
$100\%$ (pure oil) by fixing the volume of each phase initially put into the
TC system. After emulsification, the final state of the emulsion is observed
to be O/W for $\phi_{o}\leq 50\%$ and W/O for $\phi_{o}\geq 60\%$. A phase
inversion process, in which the continuous phase and the dispersed phase are
exchanged (Salager et al., 2000), is thus expected to occur in between these
two volume fractions. In this range, the behavior of the system is too complex
to allow for an accurate determination of the inversion point. In the
following, we will only consider values of $\phi_{o}$ either below $50\%$ or
larger than $60\%$, for which the nature of the dispersed and continuous
phases are unambiguously determined. In this study, we focus on the global and
local properties of O/W and W/O emulsions in a turbulent flow. Note that all
emulsions we obtained are of a simple type, and we did not observe multiple
emulsions, such as O/W/O or W/O/W (Perazzo et al., 2015). Typical O/W emulsion
for $\phi_{o}=5\%$ and W/O emulsion for $\phi_{o}=95\%$ are shown in inset
images in figure 2(a). Under steady stirring conditions, the droplet size in
the turbulent emulsion eventually shows a statistically stationary
distribution, giving an average of the droplet diameter, $\left<D\right>$, as
an indicator for the droplet size. The average droplet diameters for various
oil fractions are shown in figure 2(a). For both O/W emulsions (left branch,
$\phi_{o}\leq 50\%$) and W/O (right branch, $\phi_{o}\geq 60\%$) emulsions, it
is found that the droplet size is almost independent of the oil fraction, at
fixed $Re$. Remarkably, we find that the ethanol-water droplets in the right
branch are about $50\%$ larger than the oil droplets in the left branch,
indicating an obvious asymmetry of the droplet size between O/W and W/O
emulsions. One may think that this asymmetric behavior is due to the slight
difference in physical properties of the two liquids used in experiments.
However, we note that the densities of these two liquids are too close to
account for the observed asymmetry. The interfacial tension between the two
immiscible liquids is also identical for O/W and W/O emulsions. What about the
viscosity? The viscosity of the silicone oil, $\nu_{o}=2.1\times
10^{-6}\rm~{}m^{2}/s$, is slightly lower than that of the ethanol-water,
$\nu_{w}=2.4\times 10^{-6}\rm~{}m^{2}/s$, at the experimental temperature of
$\theta=22^{\circ}$C, while it has been found that at least an order of
magnitude difference between the viscosities of the two phases could change
the droplet size by a measurable amount (Pacek et al., 1994). Moreover,
additional experiments we performed show that the asymmetry of the droplet
size remains, even when we eliminate viscosity difference by adjusting the
working temperature (see appendix D for more details). Thus, the small
viscosity difference cannot account for the observed asymmetry of the droplet
size in these experiments. The origin of the asymmetry of the droplet size
must therefore have another origin.

Apart from the droplet size, also the effective viscosity of the emulsion
shows an asymmetric behavior. For various oil volume fractions
($0\%\leq\phi_{o}\leq 100\%$), we measured the effective viscosity of the
emulsion, $\nu_{eff}$, which is calculated using a method that has been
recently proposed for viscosity measurements in a turbulent Taylor-Couette
flow (Bakhuis et al., 2021; Yi et al., 2021). As well known, an effective
power-law dependence $G\propto Re^{\alpha}$ holds in the single-phase
Taylor–Couette turbulent flow, where the power-law exponent $\alpha$ depends
on the regime of the Reynolds number (Grossmann et al., 2016). Indeed, if the
modified Reynolds number and dimensionless torque are used, the current two-
phase emulsion flow still follows the effective power-law dependence:
$G_{m}\propto Re_{m}^{\alpha}$, where $\alpha=1.58$ in this study (see
appendix B). Based on this relation, we can calculate the effective viscosity
of the emulsion, $\nu_{eff}$. The detailed calculation of the effective
viscosity is documented in appendix B. The results of the effective viscosity
are shown in figure 2(b), which can also be divided into two parts: O/W
emulsions for $\phi_{o}\leq 50\%$ (left branch) and W/O emulsions
$\phi_{o}\geq 60\%$ (right branch). Here the effective viscosity is
characterized by its normalized value, $\nu_{eff}/\nu_{n}$, where
$\nu_{n}=\nu_{w}$ for O/W ($\phi_{o}\leq 50\%$) and $\nu_{n}=\nu_{o}$ for W/O
($\phi_{o}\geq 60\%$). For each branch, the effective viscosity increases with
the increasing dispersed phase volume fraction $\phi_{d}$ for all Reynolds
numbers. Note that the dispersed phase refers to oil for O/W or ethanol-water
for W/O. The effective viscosity has only a weak dependence on the dispersed
phase volume fraction in the dilute regime (i.e., for $\phi_{d}<5\%$), while
it displays a stronger dependence at larger dispersed phase volume fractions.
The increase of the effective viscosity with increasing $\phi_{d}$ originates
from the hydrodynamic or contact interactions between dispersed droplets, as
it is observed in similar turbulent droplet dispersions (Pouplin et al., 2011)
and in solid particle suspensions (Guazzelli & Pouliquen, 2018). Furthermore,
the effective viscosity is found to decrease with the increasing Reynolds
number for a given $\phi_{d}$, indicating that the turbulent emulsion somehow
shows a shear shinning behavior (Yi et al., 2021). Though the qualitative
trend of the effective viscosity versus the dispersed phase volume fraction is
similar for both the left and the right branches, an asymmetry of the
effective viscosity between O/W and W/O emulsions is measured. The effective
viscosity of O/W (left branch) is found to be higher than that of W/O (right
branch) for a given Reynolds number, particularly for the case of the high
dispersed phase volume fraction (see figure 2(b)). To quantitatively represent
the degree of asymmetry, we define an asymmetry factor as:

$S=\frac{\Big{(}\frac{\nu_{eff}}{\nu_{n}}\Big{)}_{\phi_{o}}-\Big{(}\frac{\nu_{eff}}{\nu_{n}}\Big{)}_{1-\phi_{o}}}{\Big{(}\frac{\nu_{eff}}{\nu_{n}}\Big{)}_{\phi_{o}}+\Big{(}\frac{\nu_{eff}}{\nu_{n}}\Big{)}_{1-\phi_{o}}},$
(1)

where the subscripts $\phi_{o}$ and $1-\phi_{o}$ denote the emulsion at
$\phi_{o}$ and at $1-\phi_{o}$, respectively. An asymmetry factor $S$
deviating from $0$ indicates asymmetry. The asymmetry factor, as a function of
the oil volume fraction, is shown as the inset of figure 2(b). It is found
that the asymmetry decreases with the increasing Reynolds number. In addition,
the asymmetric trend between O/W and W/O is more pronounced for high dispersed
phase volume fractions. Since it was already found that the droplet size has a
dramatic influence on the emulsion rheology (Pal, 1996), the macroscopic
asymmetry of the effective viscosity between O/W and W/O could be connected to
the microscopic asymmetric behavior of the droplet size.

###  3.2 Recovering the asymmetry between O/W and W/O emulsions using
surfactant

Figure 3:  (a-b) Schematic diagrams and microscopic images of the effects of
surfactants (or surface-active contaminants) on the emulsions. Part (a)
depicts the emulsions without adding surfactant. The asymmetric behaviors of
the droplet size between O/W emulsions and W/O emulsions are observed due to
the difference in the concentration of contaminants on the interfaces. The
deformation of the droplet is observed clearly in W/O emulsions. Part (b)
represents emulsions with adding DBE at $250~{}\mu\text{L/L}$. The surfactant
is adsorbed into the interface at saturation for both O/W and W/O emulsions.
The symmetry of droplet size is almost recovered. Snapshots of the emulsion at
an oil volume fraction of $\phi_{o}=5\%$ are used to represent O/W emulsions,
while W/O emulsions are denoted by snapshots of the emulsion at
$\phi_{o}=95\%$. For all snapshots here, $Re=5.2\times 10^{3}$. The scalar bar
represents $200~{}\rm\mu m$. (c) The droplets size in emulsions for various
oil volume fractions, with and without adding the surfactant, for
$Re=5.2\times 10^{3}$. The droplet size almost does not change with the oil
volume fraction for both O/W ($\phi_{o}\leq 50\%$) and W/O ($\phi_{o}\geq
60\%$) emulsions. The asymmetric trends of the droplet size between O/W and
W/O are found to disappear when adding surfactant DBE into the emulsion
systems. Note that $12.5~{}\mu\text{L/L}$ and $250~{}\mu\text{L/L}$ here refer
to the concentrations of DBE in oil. The dashed lines represent the average
values of the droplet diameters for O/W (or W/O). The colored bands span the
average diameter ± one standard deviation. For the sake of clarity, the
colored band for the case using $12.5~{}\mu\text{L/L}$ DBE is not present
here. (d) The normalized effective viscosity of the emulsion for the oil
volume fraction from $\phi_{o}=0\%$ to $\phi_{o}=100\%$, with adding
surfactant DBE at $250~{}\mu\text{L/L}$, at various Reynolds numbers. The
inset shows the asymmetry factor $S$ as a function of the oil volume fraction.
The asymmetric behavior of the effective viscosity between O/W emulsions and
W/O emulsions becomes considerably smaller due to the presence of surfactant
DBE at saturation.

We hypothesize that the possible reason for the asymmetry of the droplet size
between O/W and W/O emulsions is the presence of surface-active contaminants.
These surface-active contaminants are widely found on the liquid-liquid
interface in practical environments, which is focused on by various studies
related to interfacial phenomena (Calvo et al., 2019; De Malmazet et al.,
2015; de Gennes, 2001). On the one hand, these surface-active contaminants can
modify the interfacial properties, yielding the change of the droplet size in
the emulsion (Bazazi & Hejazi, 2020; Manikantan & Squires, 2020). On the other
hand, the solubility of these contaminants is usually different in the oil
phase and the aqueous phase (Kawaguchi, 2016). The preferential solubility can
induce a different distribution of contaminants and different interfacial
properties in O/W and W/O, which could be the source of the asymmetric
behaviors.

To investigate the effect of surfactants on the asymmetric behavior of
turbulent emulsions, an effective way is to add a controlled amount of a
selected surfactant into the system. Firstly, we perform experiments using a
kind of oil-soluble nonionic surfactant: dimethylsiloxane block copolymer
(30-35% Ethylene Oxide). For convenience, we use its abbreviation (DBE) from
the manufacturer. For the purpose of the present study, two contrasted
concentrations of DBE in oil are selected. One is $12.5~{}\mu\text{L/L}$,
which is comparable to the critical micelle concentration (CMC) of DBE in
water (around $13~{}\mu\text{L/L}$) (Rheingans et al., 2000), and the second,
20 times larger, is $250~{}\mu\text{L/L}$. The DBE is well mixed with the oil
before each experiment. Two microscopic images of O/W and W/O emulsions with
adding $250~{}\mu\text{L/L}$ DBE are shown in figure 3(b). Figure 3(c) shows
the results of the droplet size in emulsions using DBE, for various oil volume
fractions. Here, the Reynolds number is fixed at $Re=5.2\times 10^{3}$. The
measured droplet sizes for emulsions using $12.5~{}\mu\text{L/L}$ and
$250~{}\mu\text{L/L}$ DBE are denoted by yellow marks and red marks in figure
3(c), respectively. It is found that DBE only slightly reduces the droplet
size in O/W emulsions (left branch) when compared to what has been found in
emulsions without adding surfactant (blue marks). But for W/O emulsions (right
branch), the droplet size decreases with the increasing concentration of DBE.
When $250~{}\mu\text{L/L}$ DBE is used, remarkably, the droplet size
difference between O/W and W/O emulsions is eliminated (see red marks in
figure 3(c)). Consequently, we nearly recover the symmetry of the droplet size
between O/W and W/O by adding oil-soluble surfactant DBE at high concentration
into the emulsion system.

The above results can be explained using the schematic diagrams in figure
3(a-b). We first focus on the cases without adding surfactant. In the
practical environment, even without adding any surfactant, the emulsion
inevitably contains some surface-active contaminants (Soligo et al., 2020;
Bazazi & Hejazi, 2020) (black indicators in figure 3(a)), which measurably
originate from the wall of the container in this study. These surface-active
contaminants are measurably preferentially soluble in aqueous ethanol-water
and act as a surfactant. As illustrated in the left part of figure 3(a) (O/W),
the surface-active contaminants from the wall dissolve into the continuous
phase of ethanol-water. These contaminants are then adsorbed into the liquid-
liquid interface and modify the oil droplet size for two reasons. On the one
hand, contaminants on the surface suppress droplet coalescence, which is known
to be a common effect of surfactants on emulsion systems (Baret, 2012; Dai &
Leal, 2008; Ha et al., 2003; Cristini et al., 1998). On the other hand, these
surface-active contaminants fully cover the surface of oil droplets, inducing
a reduction of the interfacial tension (Manikantan & Squires, 2020). Thus, the
breakup of droplets could be promoted. The effect of contaminants on the
breakup and coalescence of the droplets finally reflects on the smaller size
of oil droplets in O/W. As the interfacial tension is found to only slightly
decrease with the concentration of DBE (see appendix A), it is reasonable to
assume that the inhibition of the droplet coalescence is the dominant factor
affecting the droplet size here. However, when oil is the continuous phase
(W/O) as shown in the right part of figure 3(a), ethanol-water droplets
embedded within the oil are not in contact with walls. Therefore, only a few
surface-active contaminants are adsorbed into the interface. The cleaner
liquid-liquid interface brings less inhibition to the coalescence, yielding
the larger droplet size for the given turbulent strength (see figure 3(a)).

The effective viscosity difference between O/W and W/O could be related to the
deformability of the dispersed phase, which is closely connected to the
droplet size (Pal, 1996; van Gils et al., 2013; Verschoof et al., 2016; Saiki
et al., 2007). When compared to the larger ethanol-water droplets, the small
and non-deformable oil droplets could yield a larger resistance to the
straining component of the shearing flow, thereby increasing the effective
viscosity of the emulsion (Stickel & Powell, 2005; Guazzelli & Pouliquen,
2018; Bakhuis et al., 2018). In addition, it was already found that surface-
active contaminants give rise to surface-excess rheology as compared to a
clean surface (Brenner, 2013; Elfring et al., 2016; Langevin, 2014), which is
also called interfacial rigidification effect (Erni, 2011). Thus, oil droplets
covered by contaminants are less deformable, which brings an extra
contribution to the effective viscosity of O/W (Guazzelli & Pouliquen, 2018).
Furthermore, surface-active contaminants could modify the dynamics of the
emulsion through the hydrodynamic coupling interaction between the oil-droplet
surface and the surrounding flow (Baret, 2012). As a result, the surface-
active contaminants account for the measured higher effective viscosity of O/W
(left branch) than that of W/O (right branch) (see figure 3(b)).

Next, we consider the experimental results using the oil-soluble surfactant
DBE. For the case of O/W emulsion shown in the left part of figure 3(b), some
DBE added in the system is competitively adsorbed into the interface. As the
interface is already saturated due to the contaminants, the interfacial
properties show no significant change when adding DBE, which is consistent
with the previous result that the droplet sizes for O/W emulsions only
slightly decrease when DBE is added. When the oil is the continuous phase
(W/O) as shown in the right part of figure 3(b), the surface of ethanol-water
droplets is expected to be mostly covered by DBE adsorbed from oil, i.e, the
surface is at saturation ($250~{}\mu\text{L/L}$ case). Therefore, the
coalescence of droplets is now inhibited. Consequently, the asymmetric trend
of the droplet size between O/W and W/O is eliminated using DBE (figure 3(c)).

In this part, we focus on the effect of surfactant on another feature of the
emulsion system: the effective viscosity. As shown in figure 3(d), it is found
that the symmetry of the effective viscosity between O/W (left branch) and W/O
(right branch) emulsions is partially recovered using $250~{}\mu\text{L/L}$
DBE. This is clearly indicated by the asymmetry factor $S$ close to $0$ (see
the inset of figure 3(d)), which is expected to be mainly attributed to the
recovery of the symmetry of the droplet size using DBE (see figure 3(c)). It
should be noted that the symmetry of the effective viscosity is not fully
recovered. In general, there are always some differences between O/W and W/O
emulsions, such as the distribution of the surfactant in the flow. The
effective viscosity for each case is found to be slightly larger than that for
its corresponding case without adding surfactant (compare figure 3(d) to
figure 3(b)). The reason could be that the polymeric surfactant (DBE in this
study) with high molecular weight enhances the interfacial rigidification
effect of the droplet surface when compared to the case with only contaminants
(Erni, 2011). Indeed, the copolymer molecules of surfactant could form shell-
like structures around the drops (Sundararaj & Macosko, 1995). This increases
the resistance of the droplet to the surrounding flow, yielding the extra
contribution of the viscous dissipation of the flow.

###  3.3 The dependence of the droplet size on the Reynolds number

Figure 4:  (a) The average droplet diameter normalized by the gap width as a
function of the Reynolds number for oil volume fractions of $\phi_{o}=1\%$ and
of $99\%$. The black and red symbols denote droplet sizes in emulsions using
no surfactant and using $250~{}\mu\text{L/L}$ DBE, respectively. The error
bars are based on the errors of the edge detection. The black lines denote the
scaling prediction of the Kolmogorov-Hinze theory, $\left<D\right>/d\sim
Re^{-6/5}$. The asymmetry of the droplet size in O/W ($\phi_{o}=1\%$) and W/O
($\phi_{o}=99\%$) emulsions is removed using DBE at saturation. (b) The Weber
number as a function of the Reynolds number in a log-log plot for the two
theoretical approaches. The yellow and blue diamonds denote the results using
K-H theory based on the energy dissipation rate in the bulk area and that
using the Levich’s theory, respectively. O/W emulsions at $\phi_{o}=1\%$
without surfactant are considered here.

As the droplet size and the associated asymmetry have been discussed for
various oil fractions, the next question is what sets the droplet size in
turbulent emulsions. In this part, we study the dependence of the droplet size
on the Reynolds number, at the low volume fraction of the dispersed phase
(i.e., $\phi_{o}=1\%$ and $\phi_{o}=99\%$). Since the volume fraction of the
dispersed phase is very low in these cases, the viscosity of the emulsion is
approximately equal to that of the continuous phase, giving $Re_{m}\approx Re$
and $G_{m}\approx G$. We firstly consider the cases at $\phi_{o}=1\%$ without
using surfactant. Note that the coalescence of droplets is inhibited due to
the surface-active contaminants on the droplet surface. Therefore, the droplet
size is mainly determined by the turbulent breakup mechanism. As shown in
figure 4(a) by the black circles, the droplet size normalized by the gap,
$\left<D\right>/d$, is found to have a scaling dependence on the Reynolds
number, $Re$, with an effective exponent of $-1.18\pm 0.05$ obtained by a
direct fit for $\phi_{o}=1\%$.

We now explore the physical mechanism behind the scaling dependence of the
droplet size on the Reynolds number. According to the Kolmogorov-Hinze theory,
the droplet formation in a turbulent flow is determined by the competition
between the deforming external dynamic pressure force (turbulent fluctuations)
and the resisting interfacial tension over the droplet surface (Kolmogorov,
1949; Hinze, 1955), of which the ratio is usually indicated by the droplet
turbulent Weber number: $We=\rho\overline{\delta u^{2}}D/\gamma$, where $\rho$
is the density of the continuous phase, $\overline{\delta u^{2}}$ is the mean-
square velocity difference over a distance equal to the droplet diameter $D$,
and $\gamma$ is the interfacial tension between the two phases (Risso & Fabre,
1998). If the droplet diameter $D$ belongs to the inertial turbulent sub-
range, $\overline{\delta u^{2}}$ could be expressed as a function of the local
energy dissipation rate: $\overline{\delta u^{2}}=C_{1}(\varepsilon D)^{2/3}$,
where the constant $C_{1}\approx 2$ according to Batchelor (Batchelor, 1953).
This yields the Weber number as: $We=2\rho\varepsilon^{2/3}D^{5/3}/\gamma$.
The force balance implies the existence of a critical value of the Weber
number beyond which breakup occurs (Hinze, 1955), and this value is found to
be of order of unity (i.e. $We\sim\mathcal{O}(1)$) in various studies (Hesketh
et al., 1991a; Risso & Fabre, 1998; Lemenand et al., 2017). Thus, the
prediction of the maximum stable droplet size in a homogeneous and isotropic
turbulent flow can be given by:
$D_{max}=C(\rho/\gamma)^{-3/5}\varepsilon^{-2/5}$ ($C$ is a constant
coefficient), which is the main result of the work by Hinze (1955). Moreover,
various studies have shown that the average droplet diameter,
$\left<D\right>$, can be used as the indicator of the droplet size in the
Kolmogorov-Hinze prediction (Boxall et al., 2012; Perlekar et al., 2012;
Lemenand et al., 2003).

Firstly, we speculate that the droplet size could be dominated by the
turbulent fluctuations in the bulk flow of the system, where most droplets
distribute. The local energy dissipation rate in the bulk can be estimated as
$\varepsilon_{b}\sim u_{{}_{T}}^{3}/\ell$, where $u_{{}_{T}}$ and $\ell$ are
the typical velocity fluctuation and the characteristic length scale of the
flow (Ezeta et al., 2018). As the typical velocity fluctuation can be
expressed as $u_{{}_{T}}\sim\omega_{i}r_{i}\sim Re\cdot\nu/d$ (Van Gils et
al., 2012), we obtain $\varepsilon_{b}\sim Re^{3}\nu^{3}/d^{4}$ by assuming
$\ell\sim d$. Inserting $\varepsilon_{b}$ into the K-H prediction, the scaling
dependence of the droplet size on the Reynolds number is obtained as
$\left<D\right>/d\sim Re^{-6/5}$, which agrees well with the experimental
results for $\phi_{o}=1\%$ without surfactant (see black circles in figure
4(a)). However, the discussion above is only a simple analysis based on the
scaling law. A further quantitative study on the droplet formation in a
turbulent flow needs to consider the Weber number, which can be calculated as
$We=2\rho_{w}\varepsilon_{b}^{2/3}D^{5/3}/\gamma$, where $\varepsilon_{b}$ can
be estimated as $\varepsilon_{b}\approx
0.1T\omega_{i}/[\pi(r_{o}^{2}-r_{i}^{2})L\rho_{w}]$ in a TC turbulent flow
(Ezeta et al., 2018). As shown in figure 4(b) (yellow diamonds, approach 1),
the Weber number ranges between $0.013$ to $0.018$, two orders of magnitude
smaller than the critical value obtained in previous studies, suggesting that
the bulk of the system, where most droplets flow around, is not the place
where the droplet size is determined.

Indeed, the droplet breakup is most often observed close to the area where the
most intense stress participates in the deformation (Hesketh et al., 1991b;
Afshar Ghotli et al., 2013). Considering that the coalescence is almost
inhibited in the current system, the droplet size is mainly dominated by the
place where small droplets are generated. Thus, the droplet size is expected
to be dominated by the boundary layer area close to the wall, where the K-H
theory has some limitations. The resulting droplet Weber number is too small
as well when using the local energy dissipation rate near the wall, suggesting
that the K-H theory is not appropriate for modeling the droplet size in the
present system (see appendix C for details).

A prediction of the droplet size in the non-homogeneous turbulent flow past a
solid wall was proposed by Levich (1962), who gave the dynamic pressure force
exerted on the two sides of the droplet using the logarithmic distribution of
the mean velocity in the boundary layer. Note that the Reynolds number in the
current study is in the interval where the logarithmic mean velocity
distribution in the boundary layer can exist (Huisman et al., 2013). According
to Levich (1962), the droplet diameter can be written as
$\left<D\right>=2\sqrt{\gamma\nu/(25\rho_{w}u_{\ast}^{3})}$, where we use the
shear velocity: $u_{\ast}=\sqrt{\tau_{w}/\rho_{w}}=\sqrt{T/(2\pi\rho
r_{i}^{2}L)}$. Using the effective scaling of $G\sim Re^{1.58}$ obtained in
the current system, the scaling dependence of the droplet diameter on the
Reynolds number is derived as $\left<D\right>/d\sim Re^{-1.19}$, where the
exponent $-1.19$ is very close to the $-6/5$ in the K-H prediction and agrees
again with the scaling dependence observed in experiments for $\phi_{o}=1\%$.
Based on the Levich theory, we also calculate the Weber number as the ratio of
the dynamic pressure force induced by the mean flow to the interfacial
tension: $We=25\rho_{w}u_{\ast}^{3}\left<D\right>^{2}/(2\nu_{w}\gamma)$. As
shown in figure 4(b) (blue diamonds, approach 2), the Weber number for the
Levich prediction is about $5$, which is consistent with the critical value
for the droplet breakup in a turbulent flow (Risso & Fabre, 1998; Lemenand et
al., 2017). The comparison of the Weber numbers based on the energy
dissipation rate and that based on Levich’s theory lead to the conclusion that
the droplet fragmentation, which determines the droplet size, occurs within
the boundary layer and is controlled by the dynamic pressure caused by the
gradient of mean flow, in agreement with the mechanism originally proposed by
Levich. Note that this conclusion also requires that the boundary layer
thickness is larger than the droplets diameter, which is supported by the fact
that boundary layer thickness is estimated as $5$ times the droplet size in
this study (see appendix C).

The discussion of the two approaches above is based on the droplet size at
$\phi_{o}=1\%$, whereas the droplet size at $\phi_{o}=99\%$ is found to follow
the same $-6/5$ scaling dependence (see black triangles in figure 4(a)),
indicating the robustness of the scaling law. Furthermore, this figure also
shows the existence of the asymmetry of the droplet size between O/W and W/O
at high $Re$, at least for low dispersed phase volume fractions (i.e.,
$\phi_{o}=1\%$ and $\phi_{o}=99\%$). Considering that the droplet size for W/O
is about $50\%$ larger than that for O/W, only a slight variation of the Weber
number is expected at $\phi_{o}=99\%$ as compared to the case at
$\phi_{o}=1\%$. Therefore, all the qualitative conclusions given above are
valid for emulsions at $\phi_{o}=99\%$ as well. In this case, the droplet
coalescence needs to be considered. The fact that the breakup theory, without
accounting for coalescence, describes well the experimental result is
particularly interesting. Indeed, for a steady state can be finally reached,
the coalescence rate of droplets has to be equal to the breakup rate of
coalesced droplets. We therefore observed an average size that is larger than
that predicted by the breakup theory, but which remains proportional to it.
Note that the interpretation given here is still not complete, and the results
remain open for discussion. Additionally, the finding that the droplet
formation is controlled by the boundary layer also provides a reasonable
explanation for the observations that the droplet size hardly depends on the
dispersed phase volume fraction $\phi_{d}$ (see figure 4(a)). The droplets
that are generated close to the wall, where similar mean velocity gradient
could distribute for various $\phi_{d}$, are expected to have similar size.
Therefore, the similar droplet size is observed at various $\phi_{d}$ in the
entire system for O/W (or W/O) emulsions.

Next, we turn to the results using surfactant. Experiments using
$250~{}\mu\text{L/L}$ DBE are performed, and the results are shown in figure
4(a). We note that the scaling dependence of the droplet size on the Reynolds
number remains, suggesting the robustness of the scaling law for turbulent
emulsions containing DBE. For the case of $\phi_{o}=1\%$, it is found that the
droplet size only slightly decreases due to DBE. However, the droplet size at
$\phi_{o}=99\%$ shows a dramatic reduction to a value close to that at
$\phi_{o}=1\%$, yielding the elimination of the asymmetry of the droplet size,
for various Reynolds numbers. Since we have found that the recovery of the
symmetry using DBE is due to the inhibition of droplet coalescence for the
lowest $Re$ case (i.e., $Re=5.2\times 10^{3}$), it is reasonable to conclude
that the similar behaviors of the droplet size observed here at high $Re$ have
the same physical interpretations. Moreover, similar results have been
observed in additional experiments using a lower concentration of DBE
($12.5~{}\mu\text{L/L}$) (see appendix D).

##  4 Conclusions

In summary, we investigated the hydrodynamic behaviors of emulsions in a
turbulent shear flow by varying the oil volume fraction from $0\%$ to $100\%$.
Firstly, it is found that the average droplet diameter hardly changes with the
oil volume fraction for O/W (or W/O) emulsions, while the ethanol-water
droplets in W/O are $50\%$ larger than the oil droplets in O/W. Secondly, the
increasing trend of the effective viscosity versus the dispersed phase volume
fraction is similar for both O/W and W/O emulsions, whereas the effective
viscosity of O/W is found to be higher than that of W/O for the same Reynolds
number, particularly for the case of high dispersed phase volume fractions.
The asymmetric behaviors of the droplet size and the effective viscosity
between O/W and W/O emulsions can be traced back to the presence of
unavoidable surface-active contaminants, mainly from the wall, which probably
preferentially dissolve in ethanol-water. In the presence of the contaminants,
the coalescence of the oil droplets in O/W is suppressed when compared to the
ethanol-water droplets with cleaner surface in W/O, yielding the smaller
droplet size for O/W than that for its W/O counterpart. Moreover, the higher
effective viscosity of O/W than that of W/O can be connected to the smaller
and non-deformable oil droplets due to the contaminants. By introducing the
oil-soluble surfactant DBE at a controlled concentration, we recover the
symmetries of both the droplet size and the effective viscosity between O/W
and W/O emulsions. This is consistent with the explanation of the mechanism
responsible for the initial asymmetry.

Next, we discuss what sets the droplets size in turbulent emulsions. Firstly,
the normalized droplet size is found to be close to a $-6/5$ scaling
dependence on the Reynolds number for the oil volume fraction of $1\%$ and of
$99\%$, which is robust for both emulsions with and without surfactant.
Theoretically, the $-6/5$ scaling dependence can be obtained using either the
K-H theory with the energy dissipation rate or the theory by Levich. However,
the Weber numbers being much less than 1 for the K-H theory indicates that the
energy dissipation rate in the bulk flow is not enough to cause the breakup of
such small droplets in this study. According to the Weber number based on
Levich’s theory, we conclude that the droplet fragmentation, which determines
the droplet size, occurs within the boundary layer and is controlled by the
dynamic pressure caused by the gradient of mean flow.

The present findings provide a better understanding of the hydrodynamic
behaviors for both O/W emulsions and their W/O counterparts. The results on
the effective viscosity open the possibility for active drag reduction during
the oil recovery and transport through controlling the dispersed phase. Our
finding of Levich’s droplet fragmentation mechanism also has some potential
implications for the modulation of droplet size in chemical processing related
to the dynamic emulsification. In future studies, more effects that may affect
the effective viscosity and droplet size will be studied, aiming at attaining
a complete understanding on the hydrodynamic behaviors of the turbulent
emulsion at various conditions, and in particular near the phase inversion,
where the phenomena are most striking.

## Acknowledgements

We acknowledge Mingbo Li, Sijia Lyu, and Stijn van Aartsen for their
insightful suggestions and discussions. We thank Oliver Masbernat for his help
in selecting the surfactants. This work is financially supported by the
Natural Science Foundation of China under grant nos. 11988102 and 91852202,
and the Tencent Foundation through the XPLORER PRIZE.

## Declaration of interests

The authors report no conflict of interest.

##  Appendix A Liquids and surfactants

Figure 5:  (a) The kinematic viscosity $\nu$ as a function of the temperature
$\theta$. The red circles denote measured viscosity of ethanol-water, and the
blue circles depict that of silicone oil. At the temperature for experiments
$\theta=22^{\circ}$C, the viscosity of ethanol-water is $\nu_{w}=2.4\times
10^{-6}\rm~{}m^{2}/s$, while that of silicone oil is $\nu_{o}=2.1\times
10^{-6}\rm~{}m^{2}/s$. (b) The interfacial tension as a function of the
concentration of DBE in oil.

The silicone oil (KF-96L-2cSt), purchased from Shin-Etsu, has a viscosity of
$\nu_{o}=2.1\times 10^{-6}\rm~{}m^{2}/s$ and a density of
$\rho_{o}=866\rm~{}kg/m^{3}$. The ethanol-water mixture ($\nu_{w}=2.4\times
10^{-6}\rm~{}m^{2}/s$, $\rho_{w}=860\rm~{}kg/m^{3}$) is prepared with $25\%$
deionized water and $75\%$ ethanol in volume. Note that the viscosity of this
mixture is very close to that of silicone oil. The viscosity values of both
these two liquid phases are measured using a hybrid rheometer type of TA DHR-1
at a temperature of $\theta=22^{\circ}$C (see figure 5(a)).

The density match of these two liquid phases eliminate the effect of
centrifugal force on the liquid distribution. Furthermore, the dispersed
droplets are expected to experience pressure fluctuations due to the strong
turbulent liquid velocity fluctuations that develop in the current system.
Here, we can compare the force induced by the velocity fluctuations to the
centrifugal force by introducing a centrifugal Froude number (van Gils et al.,
2013);

$Fr_{\text{cent }}(r)=\frac{\rho u^{\prime
2}_{\theta}/\left<D\right>}{\Delta\rho U_{\theta}^{2}/r},$ (2)

where $u^{\prime}_{\theta}$ denotes the standard deviation of the azimuthal
liquid velocity fluctuations in the bulk, $U_{\theta}$ the mean azimuthal
liquid velocity, and $r$ the radial position of the droplet to be considered.
Based on the measurements in previous studies (Grossmann et al., 2016), we
take the estimation of $u^{\prime}_{\theta}\sim 0.01\omega r_{i}$ and
$U_{\theta}\sim 0.1\omega r_{i}$. Consequently, we find that $Fr_{\text{cent
}}(r)$ is in the order of $10^{2}$, indicating that the centrifugal force is
negligible compared to the force induced by the velocity fluctuations, which
leads the droplets to get spread in the entire system.

A nonionic surfactants is used in experiments, which is a kind of
dimethylsiloxane block copolymer (30-35% Ethylene Oxide) purchased from
Gelest. For convenience, we use its abbreviation (DBE) from the manufacturer.
The density and molecular weight reported by the manufacturer are
$970\rm~{}kg/m^{3}$ and around $10^{3}\rm~{}g/mol$, respectively. This
surfactant is non-soluble in water but highly soluble in silicone oil. The
interfacial tension between the two liquids (oil and ethanol-water) was
measured using the pendent drop technique on a goniometer instrument (SCA20).
Without using surfactant, the interfacial tension between oil and ethanol-
water is $\gamma=5.7~{}\text{mN/m}$. We performed measurements for emulsions
containing DBE at various concentrations. As shown in figure 5(b), the
interfacial tension between the two liquids only slightly decreases with the
increasing concentration of DBE. Considering that the Levich prediction
($\left<D\right>=2\sqrt{\gamma\nu/(25\rho u_{\ast}^{3})}$) gives a scaling
between the droplet size and the interfacial tension as
$\left<D\right>\sim\gamma^{1/2}$, we can estimate the droplet size reduction
due to the interfacial tension reduction is around only $5\%$, which is much
less than the $50\%$ jump of droplet size measured in experiments with adding
250 $\mu\text{L/L}$ DBE. The effect of the interfacial tension reduction due
to the DBE on the droplet size is unimportant in the current work. Thus, it is
reasonable to conclude that the inhibition of the droplet coalescence is the
dominating factor in affecting the droplet size when using DBE in emulsions.

##  Appendix B The torque and the effective viscosity

Figure 6:  (a) The calibration of the torque measurements. The torque
contribution of the end effect can be determined as the longitudinal intercept
of the fitting line. Here, the oil volume fraction is $\phi_{o}=0\%$. (b) The
dependence between the modified dimensionless torque, $G_{m}$, and the
modified Reynolds number, $Re_{m}$, by using the effective viscosity. All
these sets of data at the various oil volume fractions collapse in a master
curve, and the error is less than $1\%$. The inset shows the dimensionless
torque compensated with $Re_{m}^{-1.58}$. Here, the results are from emulsions
without using surfactant.

The torque is a response parameter of the emulsion system in the current
study. The torque is directly measured by the rheometer through the shaft
connected to the inner rotating cylinder with high accuracy up to
$0.1\rm~{}nN\cdot m$. For each experiment, we set the angular velocity
$\omega_{i}$ of the inner cylinder at a constant value. After the system
reaches a statistically stable state, the measurement of the torque is
performed.

Note that the total torque ($T_{raw}$) directly measured by the torque sensor
can be split up two parts: (1) the torque $T$ due to the sidewall of the inner
cylinder (Taylor-Couette flow); (2) the torque $T_{end}$ related to the top
and bottom end plates (von Kámán flow). In this study, what we need is $T$,
which can be determined by using a linearisation method (Hu et al., 2017;
Greidanus et al., 2011). We performed torque measurements in three TC devices
with different heights of, $L$, $2L$, and $3L$. Since the contribution of the
cylindrical sidewall increases linearly with the height of the cylinder
(Greidanus et al., 2011; Hu et al., 2017), we can obtain the $T_{end}$ as the
longitudinal intercept of the linear fit (see figure 6(a)). The ratio of the
torque caused by the TC flow to the total torque can be given as
$\beta=1-T_{end}/T_{raw}$, which is determined by performing experiments for
two cases of single-phase flow (i.e., $\phi_{o}=0\%$ and $\phi_{o}=100\%$).
Then, $\beta$ obtained can be applied to the flow with internal dispersed
phase (i.e., emulsions). Consequently, the value of the torque caused by the
TC emulsion flow can be calculated for various oil volume fractions.

For a TC turbulent emulsion, the control parameter can be defined using the
modified Reynolds number,

$Re_{m}=\omega_{i}r_{i}d/\nu_{eff},$ (3)

where $\nu_{eff}$ is the effective viscosity of the emulsion. The response
parameter is the modified dimensionless torque given by:

$G_{m}=T/2\pi L\rho\nu_{eff}^{2},$ (4)

where $T$ denotes the torque that is required to maintain the inner cylinder
rotating at a constant angular velocity $\omega_{i}$.

Firstly, we calculate the $Re_{m}$ and $G_{m}$ at various angular velocities
$\omega_{i}$ for pure ethanol-water mixture ($\phi_{o}=0\%$) with a known
viscosity. When we plot these data in a $G_{m}-Re_{m}$ plot, we find a scaling
law as $G_{m}\sim Re_{m}^{1.58}$ (see figure 6(b)). Further, we can write this
relation as $G_{m}=KRe_{m}^{1.58}$, where $K$ denotes a constant prefactor. If
we insert the definitions of $G_{m}$ and $Re_{m}$ to this dependence, we
obtain a dependence of torque $T$ and viscosity $\nu_{eff}$ as

$T=AK\nu_{eff}^{0.42},$ (5)

where $A$ equals to $2\pi L\rho(\omega_{i}r_{i}d)^{1.58}$. This relation is
expected to be valid for emulsion systems with various oil volume fractions
and Reynolds numbers as well, which is supported by previous studies (Ravelet
et al., 2007; Bakhuis et al., 2021). The torque and the effective viscosity of
the emulsion system can be denoted as $T$ and $\nu_{eff}$ for a constant
angular velocity $\omega_{i}$ at the oil volume fraction of $\phi_{o}$. For
the pure ethanol-water mixture ($\phi_{o}=0\%$) system at the same angular
velocity, we obtain the measured torque value $T_{w}$ and the viscosity
$\nu_{w}$. Both these two systems follow the relation given above. Since the
angular velocities of these two systems are the same, the prefactor $A$ is the
same too. Then, we can derive the following relation:

$\frac{\nu_{eff}}{\nu_{w}}=\Big{(}\frac{T}{T_{w}}\Big{)}^{2.38}.$ (6)

The effective viscosity of emulsion systems $\nu_{eff}$ can be obtained based
on this relation. By using the effective viscosity obtained for each cases, we
calculate $G_{m}$ and $Re_{m}$ for various volume fractions and angular
velocities. When we plot together all data in a $G_{m}$-$Re_{m}$ plot, we find
that all data sets of $G_{m}$ versus $Re_{m}$ collapse in a master curve, for
various oil fractions (see figure 6(b)).

##  Appendix C The dependence of the droplet size on the Reynolds number

In a TC turbulent flow, the boundary layer thickness can be estimated as:

$\lambda_{u}=d/(2Nu_{\omega}).$ (7)

Here, we use another typical response parameter in a TC turbulent flow, the
angular velocity Nusselt number:

$Nu_{\omega}=\frac{T}{2\pi L\rho J_{\omega}^{lam}},$ (8)

where $J_{\omega}^{lam}=2\nu
r_{i}^{2}r_{o}^{2}\omega_{i}/(r_{o}^{2}-r_{i}^{2})$ is the angular velocity
transport for the laminar TC flow. Based on this, the boundary layer
thickness, $\lambda_{u}$, is found to be around 5 times larger than droplet
size, $\left<D\right>$, for various Reynolds numbers, which supports the
Levich theory in the main paper.

It is found that the droplet breakup is most often observed close to the area
where the most intense stress participates in the deformation (Hesketh et al.,
1991b; Afshar Ghotli et al., 2013). Thus, the droplet size is expected to be
dominated by the boundary layer region close to the wall, where the energy
dissipation rate ($\varepsilon_{bl}$) is largest (Ezeta et al., 2018). The
$\varepsilon_{bl}$ can be estimated as
$\varepsilon_{bl}=u_{\ast}^{3}/\lambda_{u}$, where we use the shear velocity:
$u_{\ast}=\sqrt{\tau_{w}/\rho}=\sqrt{T/(2\pi\rho r_{i}^{2}L)}$ and the
boundary layer thickness: $\lambda_{u}=d/(2Nu_{\omega})$ in a TC turbulent
flow (Eckhardt et al., 2007). Thus, the normalized energy dissipation rate can
be estimated as $\varepsilon_{bl}/(\nu^{3}d^{-4})\sim G^{3/2}Nu_{\omega}$.
Since the volume fraction of the dispersed phase is very low here
($\phi_{o}=1\%$ or $\phi_{o}=99\%$), the viscosity of the emulsion is
approximately equal to that of the continuous phase, giving $Re_{m}\approx Re$
and $G_{m}\approx G$. Using the effective scaling of $G=T/2\pi
L\rho\nu^{2}\sim Re^{1.58}$ obtained in the current system (see figure 6(b)),
we can get $Nu_{\omega}\sim Re^{0.58}$. Consequently, the energy dissipation
rate is found to scale as $\varepsilon_{bl}/(\nu^{3}d^{-4})\sim Re^{2.95}$.
Inserting this expression of $\varepsilon_{bl}$ into the K-H prediction (i.e.,
$\left<D\right>=C(\rho/\gamma)^{-3/5}\varepsilon_{bl}^{-2/5}$), one obtains
$\left<D\right>/d\sim Re^{-1.18}$, which is also in good agreement with the
experimental data. The scaling exponent of $-1.18$, which is close to the
$-6/5$ given by the K-H theory using dissipation rate in the bulk, suggests
that the energy dissipation at the boundary layer ($\varepsilon_{bl}$) is just
proportional to the local energy dissipation in the bulk of the system
($\varepsilon_{b}$). Note that this is similar to what observed for the case
of the liquid-liquid dispersion in an agitated vessel (Wichterle, 1995).

However, the Weber numbers calculated using $\varepsilon_{bl}$ are about
$0.08$, which is an order magnitude smaller than the critical value
($\mathcal{O}(1)$). This indicates that the energy dissipation rate
$\varepsilon_{bl}$ near the wall is not large enough to cause the breakup of
such small droplets. These results again suggest that the K-H theory is not
appropriate for modeling the droplet size in the present system.

##  Appendix D Additional experiments

Figure 7:  (a) The droplet size for various Reynolds numbers at
$\theta=22^{\circ}$C and at $\theta=30^{\circ}$C. (b) The average droplet
diameter normalized by the gap width as a function of the Reynolds number for
oil volume fractions of $\phi_{o}=1\%$ and of $99\%$. The black and magenta
symbols denote droplet sizes in emulsions using no surfactant and using
$12.5~{}\mu\text{L/L}$ DBE, respectively. The black lines denote the scaling
prediction of the Kolmogorov-Hinze theory. The asymmetry of the droplet size
in O/W ($\phi_{o}=1\%$) and W/O ($\phi_{o}=99\%$) emulsions is partially
removed using $12.5~{}\mu\text{L/L}$ DBE.

One may think that the difference of the viscosity between oil
($\nu_{o}=2.1\times 10^{-6}\rm~{}m^{2}/s$) and ethanol-water
($\nu_{w}=2.4\times 10^{-6}\rm~{}m^{2}/s$) could be the source of the
asymmetry of the droplet size. By adjusting the temperature of the emulsion
from $\theta=22^{\circ}$C to $\theta=30^{\circ}$C, we eliminate the viscosity
difference between the two liquids (see figure 5(a)) and measure the droplet
size at oil volume fraction of $\phi_{o}=1\%$ and of $\phi_{o}=99\%$. As shown
in figure 7(a), the droplet sizes only have slight change when compared to the
results obtained at $\theta=22^{\circ}$C. Clearly, the ethanol-water droplets
in W/O are larger than the oil droplets in O/W as well, for all Reynolds
numbers. Thus, this small viscosity difference between the two liquids used in
experiments cannot account for the obvious asymmetry of the droplet size.

Here, we provide experimental results for emulsions using
$12.5~{}\mu\text{L/L}$ DBE. As shown in figure 7(b), the -6/5 scaling
dependence of the droplet size on the Reynolds number remains, suggesting the
robustness of the scaling law for turbulent emulsions containing
$12.5~{}\mu\text{L/L}$ DBE. For the case of $\phi_{o}=1\%$, it is found that
the droplet size only slightly decreases due to DBE. However, the droplet size
at $\phi_{o}=99\%$ shows an obvious reduction, yielding the partial
elimination of the asymmetry of droplet size, for various Reynolds numbers.

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Pa$\alpha$ spectral region, spline fitting for continuum estimation is
applied. The vertical dashed lines indicate the location of the nonsystemic
corrected emission line.

##  Appendix E CL and Black Hole Mass

Further comparisons between the black hole mass and the CL emission strength
(in this case [Si vi] and [Si x]) are shown in

Figure E.1.

Figure E.1: Correlation between the CL luminosities and the mass of the
central black hole. Only moderate correlations are found. $R_{\rm pear}$ is
the Pearson correlation and $p$ the Pearson $p$-value.

##  Appendix F Hidden Broad Lines

Figure F.1 presents an example, of LEDA 157443 (BAT ID 597), showing hidden
broad lines. Parts of the spectrum are shown. The H$\beta$ and H$\alpha$
emission lines show no clear sign of broad components. The Pa$\alpha$ emission
line, on the other hand, shows a clear broad component.

Figure F.1: Example of hidden broad lines: the Pa$\alpha$ emission line in the
NIR regime clearly shows a broad component, while the H$\beta$ and H$\alpha$
emission lines in the optical regime do not show clear broad components. The
source shown is LEDA 157443 (BAT ID 597).

##  Appendix G Continuum Fitting

For most of the spectral ranges, we fit the continuum using a fourth-order
polynomial. In certain ranges, due to an irregular shape, we use a spline fit
to correct the continuum. In total, we have applied a spline fit in 123
regions for 88/168 sources (as a reminder, we have separated each spectrum
into seven spectral regions; see Table 1). Table G.1 lists the spectral
regions where a spline fit is used. Figure G.1 shows a comparison of the two
continuum fit methods in the Pa$\alpha$ region in source BAT 1138.

Table G.1: List of spectral regions where spline fit is applied (indicated by "x"). BAT ID | Pa$\epsilon$ |  $[$S viii$]$ | Pa$\gamma$ | Pa$\beta$ |  $[$Si x$]$ |  $[$Fe xiii$]$ | Pa$\alpha$
---|---|---|---|---|---|---|---
10 |  |  |  |  |  |  |
17 |  |  |  |  |  |  | x
20 |  |  |  |  |  |  |
31 |  |  |  |  |  |  | x
32 |  |  |  |  |  |  |
37 |  |  |  |  |  |  | x
50 |  |  |  |  |  |  |
52 |  |  |  |  | x |  |
53 |  |  | x |  |  |  | x
57 |  |  |  | x |  |  | x
… |  |  |  |  |  |  |
Figure G.1: Comparison of the two continuum fitting methods. The black
spectrum in the upper part shows the raw spectrum. The spline fit is indicated
in blue and the fourth-order polynomial fit in brown. The fourth-order
polynomial fit corrected spectrum is shown in the center (in brown) and the
spline fit corrected spectrum is shown at the bottom (in blue). The red shaded
regions are excluded from the continuum fit due to intrinsic emission.


# Flow-based SVDD for anomaly detection

Marcin Sendera  Marek Śmieja  Łukasz Maziarka  Łukasz Struski  Przemysław
Spurek  Jacek Tabor

###### Abstract

We propose FlowSVDD – a flow-based one-class classifier for anomaly/outliers
detection that realizes a well-known SVDD principle using deep learning tools.
Contrary to other approaches to deep SVDD, the proposed model is instantiated
using flow-based models, which naturally prevents from collapsing of bounding
hypersphere into a single point. Experiments show that FlowSVDD achieves
comparable results to the current state-of-the-art methods and significantly
outperforms related deep SVDD methods on benchmark datasets.

Machine Learning, ICML

##  1 Introduction

Anomaly (novelty/outlier) detection refers to the identification of novel or
abnormal patterns embedded in a large amount of typical (normal) data
(Miljković, 2010). Anomaly detection algorithms find application in fraud
detection systems, discovering failures in the industrial domain, detection of
adversarial examples, etc..

In contrast to typical binary classification problems, where every class
follows some probability distribution, an anomaly is a pattern that does not
conform to the expected behavior. In consequence, a completely novel type of
anomalies can occur at test time, which is not similar to any known anomalies.
Moreover, in most cases, we do not have access to any anomalies at training
time. In consequence, novelty detection is usually solved using unsupervised
approaches, such as one-class classifiers, which focus on describing the
behavior of available data (inliers). Any observation, which deviates from
this behavior, is labeled as an outlier.

Our research is motivated by the idea of Support Vector Data Description
(SVDD) (Tax & Duin, 2004), which obtains a spherically shaped boundary around
a dataset by usage of soft margin and penalization of data points from outside
the bounding region. We propose FlowSVDD – a one-class classifier based on
flow-based models (Dinh et al., 2014), which finds a hypersphere with a
minimal volume that encloses data. Since flow-based models are commonly used
in the context of generative models, we redefine their cost function to
minimize the volume of the bounding hypersphere instead of maximizing the log-
likelihood function. On one hand, flow-based models allow us to calculate a
Jacobian of a neural network at every point. In consequence, minimizing the
volume of the hypersphere in the feature space leads to the minimization of
the volume of the corresponding bounding region in the input space. On the
other hand, since flow-based models give an explicit formula for the inverse
mapping, we automatically get a parametric form for the corresponding bounding
region in the input space. In contrast to deep SVDD models, our approach
eliminates the problem of hypersphere collapse, which makes it easy to use.

Extensive experiments performed on typical benchmark datasets show that our
method significantly outperforms the deep SVDD model while being comparative
to state-of-the-art models for anomaly detection.

Our contribution is summarized as follows:

  1. 1.

We propose an adaptation of the SVDD method to deep neural networks with the
use of flow models.

  2. 2.

We show that the realization of the SVDD loss function on flow-based models
prevents from hypersphere collapse.

  3. 3.

We experimentally compare FlowSVDD with Deep SVDD and current state-of-the-art
methods.

##  2 Proposed model

### Preliminaries: SVDD.

Our approach is motivated by a classical Support Vector Data Description
(SVDD) (Tax & Duin, 2004), which tries to find a minimal hypersphere to
enclose the data. To allow the possibility of outliers in the training set,
SVDD uses a soft margin and penalizes data points that lie outside the
bounding hypersphere. If $f$ maps input data to the output kernel space, then
SVDD loss equals:

$F(R,c;f)=R^{2}+\frac{1}{\nu n}\sum_{i}\max(0,\|f(x_{i})-c\|^{2}-R^{2})$ (1)

where $c\in\mathbb{R}^{D},R\in\mathbb{R}$ is the center and the radius of the
hypersphere, respectively, and $\nu$ is the trade-off between the volume and
boundary violations of the hypersphere, i.e. fraction of outliers.

The realization of SVDD using deep neural networks was presented in (Ruff et
al., 2018) (it was termed DSVDD). However, direct minimization of the SVDD
loss may lead to a trivial solution, i.e. the hypersphere collapses to a
single point $c$. To avoid this negative behavior, it has been recommended
that the center $c$ must be something other than the all-zero-weights
solution, and the network should use only unbounded activations and omit bias
terms. While the two first conditions can be accepted, omitting bias terms in
a network may lead to a sub-optimal feature representation due to the role of
bias in shifting activation values.

To eliminate the above restrictions a recent work (Chong et al., 2020)
proposes two regularizers, which prevent hypersphere collapse, and uses an
adaptive weighting scheme to control the amount of penalization between the
SVDD loss and the respective regularizer.

### Flow-based SVDD.

As an alternative to DSVDD, we realize the SVDD objective using flow models.
Let us recall that a neural network $f:\mathbb{R}^{D}\to\mathbb{R}^{D}$ is a
flow model if the inverse mapping $f^{-1}$ is given explicitly and the
Jacobian determinant $w(x)=\det df(x)$ can be easily calculated. In our
approach, we use a special class of flow models, in which Jacobian determinant
is constant at every point, i.e. $w(x)=w$, such as NICE (Dinh et al., 2014).
In this case, we get a natural correspondence between the volume of the
bounding hypersphere in the output space and the volume of a bounding region
in the input space, see below.

Let us first consider the simplest situation when $w=1$. In such a scenario,
the volume of any shape in the input space equals the volume of its image in
the output feature space. In consequence, a direct minimization of the SVDD
objective does not lead to the hypersphere collapse.

In a more general scenario, when $w\neq 1$, we need to include $w$ in the SVDD
objective. Observe that the Jacobian determinant of the mapping $f/w^{1/D}$
equals 1. Thus to get the equality of the volume in the input and output
space, we redefine the SVDD loss (1) as follows:

$F(R,c;f)=R^{2}+\frac{1}{\nu
n}\sum_{i=1}^{n}\max(0,\|f(x_{i})/w^{1/D}-c\|^{2}-R^{2}).$ (2)

In a test phase, a given example $x$ is deemed as an outlier if:

$\|f(x)/w^{1/D}-c\|>R,$

which is equivalent to

$\|f(x)-w^{1/D}c\|>Rw^{1/D}.$

In other words, inliers lie inside the ball $B(w^{1/D}c;Rw^{1/D})$.

##  3 Experiments

In this section, we experimentally examine FlowSVDD and compare it with
several state-of-the-art approaches. FlowSVDD is implemented using the
architecture of the NICE flow model (4 coupling layers – each consisted of 4
layers and 256 hidden dimensions) with constant Jacobian determinant and
$\nu=0.05$.

### Illustrative example.

To get the intuition behind FlowSVDD, we first consider 2-dimensional
examples, which are easy to visualize. The results presented in Figure 1 show
the resulting hyperspheres in the latent space and the corresponding bounding
regions in the input space. At first glance, we can observe that the bounding
region in the original space is close to the structure of inliers. In the
latent space, FlowSVDD finds the center point $c$ and radius $R$ to enclose
$(1-\nu)$ percentage of data inside the ball $B(c;R)$. Observe that, unlike
the density-based flow models, FlowSVDD does not transform data into Gaussian
distribution in a latent space.

Figure 1: Enclosing hyperspheres in the latent space of FlowSVDD (right) and
the corresponding bounding regions in the input space (left).

### Benchmark data for anomaly detection.

To provide quantitative assessment, we take into account
Thyroid111http://odds.cs.stonybrook.edu/thyroid-disease-dataset/ and
KDDCUP222http://kdd.ics.uci.edu/databases/kddcup99/kddcup.testdata.unlabeled_10_percent.gz
datasets, which are typically used for anomaly detection. We use the standard
training and test splits and follow exactly the same evaluation protocol as in
(Wang et al., 2019). In particular, we use the F1 score and the Area Under
Receiver Operating Characteristic curve (AUC).

Our model is compared with the following algorithms: (1) One-class SVM (OC-
SVM) (Schölkopf et al., 2001), (2) Deep structured energy-based models (DSEBM)
(Zhai et al., 2016), (3) Deep autoencoding Gaussian mixture model (DAGMM)
(Zong et al., 2018), (4) variants of MQT – multivariate quantile map (NLL,
TQM1, TQM2, TQM∞) (Wang et al., 2019) and (5) Deep Support Vector Data
Description (DSVDD) (Ruff et al., 2018) \- another implementation of SVDD cost
function in deep neural networks.

Table 1: Performance on two anomaly detection datasets. Thyroid
---
| OC-SVM | DSEBM | DAGMM | NLL | TQM1 | TQM2 | TQM∞ | DSVDD | FlowSVDD
F1 | .3887 | .0403 | .4782 | .7312 | .5269 | .5806 | .7527 | - | .7097
AUC | - | - | - | - | - | - | - | 0.749 | .9797
KDDCUP
| OC-SVM | DSEBM | DAGMM | NLL | TQM1 | TQM2 | TQM∞ | DSVDD | FlowSVDD
F1 | .7954 | .7423 | .9369 | .9622 | .9621 | .9622 | .9622 | - | .9030
AUC | - | - | - | - | - | - | - | - | .9384

The results presented in Table 1 show that FlowSVDD model performs better than
most methods on the Thyroid dataset and is significantly better than DSVDD in
terms of the AUC metric. In the case of KDDCUP, FlowSVDD achieves a score in
between the classical methods and current state-of-the-art.

### Image datasets.

To provide further experimental verification, we use two image datasets: MNIST
and Fashion-MNIST. In contrast to the previous comparison, these two datasets
are usually used for multiclass classification and thus need to be adapted to
the problem of anomaly detection. For this purpose, each of the ten classes is
deemed as the nominal class while the rest of the nine classes are deemed as
the anomaly class, which results in 10 scenarios for each dataset.

We additionally compare FlowSVDD with the following models: (1) Geometric
transformation (GT) (Golan & El-Yaniv, 2018), Variational autoencoder (VAE)
(Kingma & Welling, 2013), Denoising autoencoder (DAE) (Vincent et al., 2008),
Generative probabilistic novelty detection (GPND) (Pidhorskyi et al., 2018),
Latent space autoregression (LSA) (Abati et al., 2019). In contrast to
previous experiment, we only use TQM2 and NLL as the only implementations of
MTQ, because they output the highest value of AUC (Wang et al., 2019).

Figure 2: Box plots for rankings calculated on MNIST (left) and Fashion-MNIST
(right). The median ranking is marked by a line, while the average ranking is
marked with a number.

To present the results, we compute the ranking on each of 10 scenarios and
summarize it using a box plot, see Figure 2. The results show that FlowSVDD
significantly outperforms DSVDD in both datasets. In the case of the MNIST
dataset, we observe that FlowSVDD is almost as good as the current state-of-
the-art methods, like GT and NLL.

Finally, we analyze, which samples are localized close to or furthest from the
center of bounding hypersphere. Results in Figure 3 shows that FlowSVDD maps
regular images in the hypersphere center. Contrary, examples localized far
from the center, could be hard to identify. It means that FlowSVDD gives
results consistent with our intuition.

Figure 3: Best nominal (left) and worst nominal (right) examples determined by
FlowSVDD for MNIST (top) and Fashion-MNIST (bottom).

##  4 Conclusion

The paper introduced FlowSVDD, which realizes the SVDD paradigm in the case of
neural networks. Making use of flow-based models and an appropriate SVDD-like
cost function, we find a minimal bounding region for a majority of data.
Unlike other deep SVDD realizations, FlowSVDD does not change the determinant
of a Jacobian matrix, which means that the resulting hypersphere cannot
collapse in a latent space. The experimental results demonstrate that FlowSVDD
presents a very good performance in the case of both artificial and real-world
one-class settings.

## Acknowledgements

The work of M. Śmieja was supported by the National Centre of Science (Poland)
Grant No. 2018/31/B/ST6/00993. The work of P. Spurek was supported by the
National Centre of Science (Poland) Grant No. 2019/33/B/ST6/00894. The work of
J. Tabor was supported by the National Centre of Science (Poland) Grant No.
2017/25/B/ST6/01271.

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a relatively small number of interacting spins can be simulated, outliers are
wastefully problematic as each of them reduces the effective interacting
Hilbert space by a factor of $2$.

The pseudo-random graphs are generated iteratively: first, we randomly propose
a new spin position $j$ and check if (i) the inter-spin vectors with all other
spins $k$ on the graph satisfy $|\vec{r}_{jk}|>r_{\mathrm{min}}$, while (ii)
at least one inter-spin vector satisfies $|\vec{r}_{jk}|<r_{\mathrm{max}}$. If
(i) and (ii) are satisfied, we accept the proposal and update the random graph
with the spins at position $j$. Otherwise, we discard the proposed spin. We
repeat this procedure until a graph of $L$ spins is constructed. The
simulations are performed with $L{=}14$, $r_{\mathrm{min}}{=}0.9$ and
$r_{\mathrm{max}}{=}1.1$; here $r_{\mathrm{min}}$ and $r_{\mathrm{max}}$ are
given in units of $\sqrt[3]{\mu_{0}\hbar\gamma_{n}^{2}}$.

For numerical simulations, unless stated explicitly otherwise, we initialize
the system in a $\hat{\textbf{x}}$-polarized pure product state
$|\psi_{0}\rangle\\!=\\!\bigotimes_{j=1}^{L}\frac{1}{\sqrt{2}}\left(|\\!\uparrow_{j}\rangle\\!+\\!|\downarrow_{j}\rangle\right)$
and perform numerically exact time evolution using the protocol of Eq. (3).
Due to the critical long-range interaction strength, the characteristic energy
scales of the model are not immediately obvious (the energy density integral
diverges in three spacial dimensions, which indicates the necessity to impose
a lattice cutoff scale). However, they can be estimated from the dynamics of a
free-induction decay: to this end, the initial state $|\psi_{0}\rangle$ is
evolved under $\mathcal{H}_{\mathrm{dd}}$ which induces a decay of the initial
$\hat{\textbf{x}}$-magnetization. Eventually, this provides an energy scale
$J=1/\tau_{d}$ where $\tau_{d}$ is the timescale on which the
$\hat{\textbf{x}}$-magnetization decays to $1/e$ of its initial value as the
system approaches equilibrium. We note that the value of $J$ obtained in this
way can only serve as a rough estimate of energy scales, as different initial
states might lead to different values of $J$.

To minimize detrimental finite size effects and increase the ergodic
properties of the drive we add a small uniformly distributed random noise
$\delta t_{\textrm{acq}}\in[-0.05t_{\textrm{acq}},0.05t_{\textrm{acq}}]$ to
the time duration $t_{\textrm{acq}}$, i.e. $t_{\textrm{acq}}\rightarrow
t_{\textrm{acq}}+\delta t_{\textrm{acq}}$ [76, 77] which results in a slightly
different unitary for each driving cycle. Such a ”noisy” driving protocol can
remove left-over synchronization effects emerging from the discreteness of the
many-body spectrum. However, since the noise term $\delta t_{\textrm{acq}}$
breaks the periodicity of the drive, one needs to carefully examine that no
additional physics, such as an extra contribution to heating, is induced due
to finite $\delta t_{\textrm{acq}}$. For the system under investigation, this
has been demonstrated in Ref. [14].

Finally, let us present some additions to the data presented in the main text,
i.e. in Fig. 2 and Fig. 4. In Fig. S3, we present the full numerical data for
the simulated time evolution partially depicted in Fig. 2B. In particular, the
simulated data also leads to the square shapes when projected to the
$\hat{\textbf{x}}{\text{-}}\hat{\textbf{y}}$ or
$\hat{\textbf{y}}{\text{-}}\hat{\textbf{z}}$ plane. In addition to the
excursion angle $\Phi$ and linear scale $\ell$ shown in Fig. 4C-D, we show the
corresponding raw data and decay times as functions of the AC amplitude in
Fig. S4. Let us in particular point out the increase in the lifetime with
increasing AC Amplitude by a factor of up to $2$, suggested by the numerical
simulation. This is not observed in the experiment and obtaining analytical
expressions for the decay time are beyond reach for our non-integrable long-
range interacting systems. Notice however that the lifetime is only affected
by a factor of $2$ over a large range of AC amplitudes and that the impact of
finite size effects on the various quantities is not well-understood.

##  App. F  Role of the Phase in the Transformation to the Rotating Frame

In App. G we introduced the rotation to the dressed rotating frame,
$W(t,\chi){=}\exp\left[-i\left(\omega_{\mathrm{SL}}t+\chi\right)I_{z}\right]$
which features the phase $\chi$. A finite phase, $\chi{\neq}0$, corresponds to
a constant shift in the experimentally observed dressed frame phase
$\varphi{=}\arctan\left(I_{x}/I_{y}\right)-\chi$. While the derivations in
App. G and following are done for $\chi{=}0$, one can simply account for the
additional phase $\chi$ by performing a subsequent static unitary
transformation: $W_{\mathrm{static}}=\exp(-i\chi I_{z})$ on the final results.

Note that the phase $\chi$ naturally enters the expression for the
experimentally measured phase $\varphi$. However, due to the subtraction of
the ramp-like profile of the phase $\varphi$ from the raw experimental data,
see Fig. 11 and App. B, the actual observable becomes
$\varphi_{0}{=}\varphi(B_{\mathrm{AC}})-\varphi(B_{\mathrm{AC}}\\!=\\!0)$.
Therefore, this subtraction procedure rather can be naturally viewed as fixing
the gauge freedom associated with the phase $\chi$, so that $\varphi{=}0$ at
$B_{\mathrm{AC}}{=}0$.

##  App. G Role of detuning and imperfect pulses on observed shapes

In the main text we discussed two imperfections of the SL pulse drive: the
detuning $\delta$ from the Lamor frequency $\omega_{L}$, and the deviation in
the pulse angle $\delta\vartheta{=}\vartheta-2\pi k/N$. Here, we show that
they can lead to an enhancement of the signal. In fact, they are essential for
reproducing the experimental observations. Eventually, we also investigate the
related question of having noise in the kick sequence.

We start with the detuning $\delta$. Let us stress again that, due to the
large scale separation $\omega_{L}/J\leq 10^{-5}$, even a small deviation
($0.01\%$) can lead to a comparably large detuning $\delta\leq J$. Note that
the detuning affects the direction of the kicks
$\hat{\textbf{x}}\to\hat{\textbf{n}}=\cos(\alpha)\hat{\textbf{x}}+\sin(\alpha)\hat{\textbf{z}}$,
cf. Eq. (10). Therefore, the obtained shapes lie on circles which are centered
around the $\hat{\textbf{n}}$ axis. Note that $\hat{\textbf{n}}$ has a finite
$\hat{\textbf{z}}$ contribution, $\hat{\textbf{n}}_{z}\propto\delta$, which
leads to a finite $\hat{\textbf{z}}$-tilt, as can also be observed in the
simulations in Fig. S5. Therefore, the detuning is crucial for explaining the
experimentally observed $\hat{\textbf{z}}$-tilt.

Moreover, the detuning, together with the imperfection in the pulse strength
$\delta\vartheta$, leads to a finite $\hat{\textbf{x}}$-magnetization,
$M_{x}\propto\sin(\alpha)+\delta\vartheta/\tau$, in the leading-order Floquet
Hamiltonian $\mathcal{H}_{F;\,k}^{(0)}$, cf. Eq. (5). This leads to an
increase in the $\hat{\textbf{x}}$-polarization strength, as can also be
observed in Fig. S5A(i) and Fig. S6A(i). In addition, as mentioned in App. I
is it vital that $\delta$ or $\delta\vartheta$ are finite, in order for the
initial state to have a finite energy density $\epsilon$, and thus also a
finite temperature $|\beta|\propto|\sin(\alpha)+\delta\vartheta/\tau|>0$. Such
an increase in temperature is correlated with the increase in
$\hat{\textbf{y}}$\- and $\hat{\textbf{z}}$-magnetizations in Fig. S5 and Fig.
S6, cf. Eq. (24).

Note that the robustness of the shapes with respect to imperfection in the
kicks, i.e. for $\delta\vartheta\neq 0$, also demonstrates that the shapes are
not simply a single-particle effect but are stabilized by the many-body
interactions.

Finally, let us emphasize that we numerically do not observe a significant
change in the lifetime when varying the detuning $\delta$ or the angle
$\delta\theta$.

To explore this behavior, we simulate noisy pulses
$\vartheta{=}{(1+\zeta)\times 2\pi/4}$ for the special case of $N{=}4$, where
$\zeta$ are random numbers that differ for every pulse uniformly distributed
in the interval $[-\eta,\,\eta]$ (with $\eta$ being the noise strength). We
observe a strong decrease in the lifetime of the prethermal plateau with
increasing $\eta$, see Fig. S7. This result is not surprising as the periodic
nature of the drive is lost with increasing randomness.