#  Membership Inference Attacks on Lottery Ticket Networks

Aadesh Bagmar  Shishira R Maiya  Shruti Bidwalkar  Amol Deshpande

###### Abstract

The vulnerability of the Lottery Ticket Hypothesis has not been studied from
the purview of Membership Inference Attacks. Through this work, we are the
first to empirically show that the lottery ticket networks are equally
vulnerable to membership inference attacks. A Membership Inference Attack
(MIA) is the process of determining whether a data sample belongs to a
training set of a trained model or not. Membership Inference Attacks could
leak critical information about the training data that can be used for
targeted attacks. Recent deep learning models often have very large memory
footprints and a high computational cost associated with training and drawing
inferences. Lottery Ticket Hypothesis is used to prune the networks to find
smaller sub-networks that at least match the performance of the original model
in terms of test accuracy in a similar number of iterations. We used CIFAR-10,
CIFAR-100, and ImageNet datasets to perform image classification tasks and
observe that the attack accuracies are similar. We also see that the attack
accuracy varies directly according to the number of classes in the dataset and
the sparsity of the network. We demonstrate that these attacks are
transferable across models with high accuracy.

Machine Learning, ICML

##  1 Introduction

Machine learning models are integrated into a wide range of applications
today, including ones involving critical, sensitive, or confidential data.
Despite the tremendous advantages of these models, they are vulnerable to
various kinds of attacks that compromise the efficacy [cite adv] and the
privacy (Shokri et al., 2017) of the system and the underlying data. In Shokri
et al. (2017), the authors designed a membership inference attack (MIA) which
made it possible to recover the individual records from training data, raising
major privacy concerns. They exploit the fact that most models tend to overfit
and have major differences in confidence values of samples belonging to
training and test splits to create a “shadow model” that distinguishes between
them. Lottery Ticket Hypothesis (Frankle & Carbin, 2018) is a breakthrough in
the area of neural network pruning. It shows the existence of sparse sub-
networks that at least match the test accuracy of the original dense network
and train in at most similar number of iterations. These sub-networks that are
composed of the “winning lottery ticket” have advantages of a smaller memory
footprint and lesser compute resources needed for inference while still
maintaining the test accuracy.

A natural question then is, are such “lottery ticket networks”, because of
their ability to generalize and not overfit to the same extent, less
susceptible to membership inference attacks? In this paper, we attempt to
answer that question empirically.

Our main contributions include performing experiments to evaluate the impact
of membership inference attacks on lottery ticket networks. We analyse the
performance of ResNet18 and ResNet50 classifiers trained on various datasets
with and without lottery tickets. We use a single shadow attack model approach
and employ a simple classifier (Multi-Layer Perceptron). We observe that the
accuracy and precision of our attacks increases with increase in the number of
classes and more importantly, we show that the lottery ticket networks are
equally vulnerable to membership inference attacks as the original dense
networks.

Rest of the paper is organized as follows. We first briefly describe the key
concepts of membership inference attacks and lottery tickets. We then describe
our experimental setup and report the results. Finally, we discuss the next
steps and provide some future directions for our work.

##  2 Background

In this section, we briefly describe the background required to understand our
work.

###  2.1 Membership Inference Attacks

Membership inference attacks (MIA) aim to identify whether a data sample was
used to train a machine learning model or not. These attacks have been
successfully carried out on centralized supervised learning and unsupervised
learning models and also distributed learning based Federated Learning models
(Hu et al., 2021).

These attacks work even if the attacker does not have access to the original
training data that was used to train the target model. Shokri et al. (2017)
describe a method wherein they train multiple “shadow models” that mimic the
behaviour of the target model. This is a type of a white-box attack where the
architecture of the targeted model and the training dataset membership of this
shadow model is known. Salem et al. (2018) showed that a single shadow network
is sufficient too.

Membership inference attacks have been studied extensively (Shokri et al.,
2017; Nasr et al., 2018; Li & Zhang, 2020) and across different domains
(Danhier et al., 2020; Salem et al., 2018; Liu et al., 2019; He et al., 2020).
Different types of attacks including neural network based and metric based
have been proposed and researchers have shown successful black box and white
box approaches. Defenses against such attacks have been studied as well and
mostly focus around reducing overfitting and reducing the influence of certain
data points. Nasr et al. (2018) suggest using adversarial regularization
training to defend against this. Shokri et al. (2017) suggested defense
techniques like restricting the prediction vector to top k classes; however,
highly accurate attacks are still possible even when the model reveals minimal
information (Li & Zhang, 2020).

###  2.2 Lottery Ticket Hypothesis

Lottery Ticket Hypothesis shows that it is possible to reduce the parameter
counts of trained neural networks by over $90\%$ while still maintaining the
accuracy by using efficient pruning techniques. This theory states that in
dense, randomly-initialized feed-forward networks there exist such sub-
networks which when trained in isolation reach test accuracy comparable to the
original network in a similar number of iterations (Frankle & Carbin, 2018).
To identify these sparse sub-networks, the authors aim to prune the weights
with the smallest-magnitude using Iterative Magnitude Pruning.

Since then there has been follow up work to further analyse the hypothesis
(Frankle et al., 2019; Malach et al., 2020; Frankle et al., 2020a; Morcos et
al., 2019) and apply this hypothesis to various domains such as object
detection (Girish et al., 2020), for graph neural networks (Chen et al.,
2020), for reinforcement learning and NLP (Yu et al., 2019), for pre-trained
BERT model (Chen et al., 2021), and so on.

##  3 Experimental Setup

Our goal is to understand if using lottery ticket networks has any impact on
privacy of the data used for training the model, given that lottery ticket
networks are smaller and more generalizable. We do this by empirically
comparing privacy leakages in regular neural networks against privacy leakages
in lottery ticket networks.

Figure 1: Performance of lottery tickets by varying sparsities for Resnet-18
and ResNet-50. Sparsity 0 indicates the original network. Algorithm 1 One Shot
Pruning

1: Randomly initialize network $f$ with initial weights $w_{0}$, mask
$m_{0}=1$, prune target percentage $p$

2: Train network for N iterations $f(x;w_{0})\rightarrow f(x;w_{n})$

3: Prune bottom $p\%$ of $w_{n}$ by magnitude to obtain $m_{n}$

4: Reset to initial weights $w_{0}$

5: Retrain pruned network for N iterations $f(x;m_{n}\odot w_{0})\rightarrow
f(x;m_{n}\odot w_{p})$

###  3.1 Datasets

We conducted our experiments on CIFAR-10, CIFAR-100 (CIF, ) and ImageNet
datasets. We use models trained on the ResNet18 architecture unless mentioned
otherwise.

###  3.2 System Setup

####  3.2.1 Obtaining Lottery Tickets

The first requirement of our experiments is lottery tickets at varying levels
of sparsity across different datasets. For this, we utilize a variant of
Iterative Magnitude Pruning (IMP) algorithm used in (Frankle & Carbin, 2018)
called ”One shot pruning” where the number of pruning rounds to obtain desired
sparsity is reduced to one. The detailed algorithm is presented in 1.

####  3.2.2 Training shadow models

We generated a dataset to train an attack model using the following steps. We
follow these steps for evaluating both, a normal Neural Network and a Lottery
Ticket Network created in a similar way as discussed in Frankle et al.
(2020b).

  * •

We pass a subset of images which have been used to train our initial network
and generate a set of confidence vectors $P(y|x)$. We choose equal number of
samples which the model has seen (coming from the original model’s training
set) and those that the model hasn’t seen (from original model’s testing set).
This is the dataset we use to train our attack model.

  * •

Once we have the confidence vectors, we train a classifier to identify whether
a sample belongs to the train set or the test set. This is our shadow model
for membership inference.

We chose the MLP (Multi-layer Perceptron) classifier because it gave the
highest accuracy (0.94) among all the classifiers we tried [SVM(0.89), Random
Forests (0.88)]. The MLP classifier optimizes the log-loss function using the
stochastic gradient descent. We use the standard implementation from scikit-
learn.

##  4 Results and Discussions

In this section, we discuss the results of our experiments.

| Accuracy Score of MIA (Using MLP) | Precision Score
---|---|---
Dataset | Baseline Model | Lottery Ticket Network | Baseline Model | Lottery ticket Network
CIFAR-10 | 0.503 | 0.504 | 0.57 | 0.59
CIFAR-100 | 0.744 | 0.744 | 0.97 | 0.94
| ImageNet
---
20k samples
0.944 | 0.940 | 0.945 | 0.931
| ImageNet
---
60k sample
0.9499 | 0.942 | 0.96 | 0.956
| ImageNet
---
100k samples
0.952 | 0.944 | 0.96 | 0.957
Table 1: Accuracy and Precision of MIA for different datasets

###  4.1 Primary Analysis

We observe that the accuracy and precision for membership inference attacks on
both our baseline model and its corresponding lottery ticket network are
almost similar. We show our results across various datasets and models in
Table 1. We observe minute differences after 20k samples in the accuracy of
our attack model. The attack model trained on Image Net showed high recall as
well.

####  4.1.1 Attack efficiency and number of classes

Our results are similar to those reported by Shokri et al. (2017) where they
observe an increase in accuracy with an increase in the number of classes.
This is because the attack model receives sparser data as the number of
classes increase. We observe that the trend translates to lottery ticket
networks as well where we observe an accuracy of 0.5, 0.74 and 0.94 for
datasets having 10, 100 and 1000 classes.

####  4.1.2 Attack accuracy and model sparsity

We evaluate how attack accuracy varies based on network sparsity (Frankle &
Carbin, 2018). We observe similar levels of accuracy for different sparsities
when trained on ImageNet ranging between 0.94 and 0.96 for ResNet18 and
between 0.95 and 0.97 for ResNet50. We show our results in Figure 1. We
observe a trend of decrease in attack accuracy at very high levels of
sparsity.

###  4.2 Comparing different architectures

Figure 2: The above figure shows the accuracy of our membership inference
attack and how it varies if we change the lottery ticket sparsity. We fix the
number of training samples to 50,000. We compare results across ResNet18 and
ResNet50 Figure 3: The above figure shows performance of our membership
inference attack and how it varies if we change the sample size used for
training. We fix the network sparsity to 80%. We compare results across
ResNet18 and ResNet50

We wanted to understand if changing the architecture leads to a change in
accuracy. We observe a higher accuracy in our attacks in ResNet50 compared to
ResNet18 implying that ResNet50 may be more vulnerable. We also observe that
the attack accuracy first increases with increase in sparsity and then drops.
This is coherent with the results observed by Girish et al. (2020) where they
show an increase in accuracy with increase in sparsity followed by a steep
drop at higher sparsity levels. We show our results in Figures 2 and 3.

###  4.3 Do membership attacks transfer?

In this section evaluate the efficacy of membership inference attack when the
shadow models are trained using the outputs of one model and then used to
attack an unknown model. We consider the case where shadow models are obtained
by training on confidence outputs from Resnet-18 and Resnet-50 architectures
on Imagenet. These are the observations from these set of experiments:

  * •

We can clearly see that the highest attack success are when we attack the same
model, as shown in the diagonal of 4.

  * •

By observing the columns of the heatmap across the two architectures, there
appears to be slight decrease in the efficacy of shadow model as we increase
sparsity

  * •

We can also see that the attack is more successful on Resnet-50 models
compared to Resnet-18.

Figure 4: The above heat-map shows the accuracy with which attacks are
transferable to other models. Source denotes the model whose outputs were used
to train an attack model and the target is the model which is attacked. Thus,
the diagonal shows our base case where the source and the target are the same.

##  5 Conclusion and Future Work

Our results seem promising and open future research directions. We observed
that the lottery ticket models are as vulnerable as their original
counterparts and we observe minute differences in accuracies. We also see that
the attacks vary based on number of classes in the dataset, where the accuracy
increases with increase in the number of classes. We also observed that the
attacks improve in accuracy as sparsity of the lottery ticket increases and
then drops. Our results demonstrate that these attacks are transferable across
models with a high accuracy. We see the following research directions for the
future:

  * •

We would like to explore other pruning methods like Grasp, SNIP, in the
context of membership inference attacks.

  * •

We feel more empirical study is required into examining the relationship
between attack susceptibility and generalizing capabilities of the model.

  * •

We wish to study if the differences in the accuracy affect classes (based on
their density in the training data) differently.

## Acknowledgements

We would like to thank Sharath Girish from University of Maryland for helping
us with the lottery ticket code. We would also like to thank Prof. Jim Purtilo
for his support during this work.

## References

  * (1) Cifar-100 dataset from university of toronto.  https://www.cs.toronto.edu/~kriz/cifar.html.  Accessed: 2021-03-07.
  * Chen et al. (2020) Chen, T., Frankle, J., Chang, S., Liu, S., Zhang, Y., Wang, Z., and Carbin, M.  The lottery ticket hypothesis for pre-trained bert networks.  _arXiv preprint arXiv:2007.12223_ , 2020.
  * Chen et al. (2021) Chen, T., Sui, Y., Chen, X., Zhang, A., and Wang, Z.  A unified lottery ticket hypothesis for graph neural networks.  _arXiv preprint arXiv:2102.06790_ , 2021.
  * Danhier et al. (2020) Danhier, P., Massart, C., and Standaert, F.-X.  Fidelity leakages: Applying membership inference attacks to preference data.  In _IEEE INFOCOM 2020-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_ , pp. 728–733. IEEE, 2020.
  * Frankle & Carbin (2018) Frankle, J. and Carbin, M.  The lottery ticket hypothesis: Finding sparse, trainable neural networks.  _arXiv preprint arXiv:1803.03635_ , 2018.
  * Frankle et al. (2019) Frankle, J., Dziugaite, G. K., Roy, D. M., and Carbin, M.  Stabilizing the lottery ticket hypothesis.  _arXiv preprint arXiv:1903.01611_ , 2019.
  * Frankle et al. (2020a) Frankle, J., Dziugaite, G. K., Roy, D., and Carbin, M.  Linear mode connectivity and the lottery ticket hypothesis.  In _International Conference on Machine Learning_ , pp. 3259–3269. PMLR, 2020a.
  * Frankle et al. (2020b) Frankle, J., Dziugaite, G. K., Roy, D. M., and Carbin, M.  Stabilizing the lottery ticket hypothesis, 2020b.
  * Girish et al. (2020) Girish, S., Maiya, S. R., Gupta, K., Chen, H., Davis, L., and Shrivastava, A.  The lottery ticket hypothesis for object recognition.  _arXiv preprint arXiv:2012.04643_ , 2020.
  * He et al. (2020) He, Y., Rahimian, S., Schiele, B., and Fritz, M.  Segmentations-leak: Membership inference attacks and defenses in semantic image segmentation.  In _European Conference on Computer Vision_ , pp. 519–535. Springer, 2020.
  * Hu et al. (2021) Hu, H., Salcic, Z., Dobbie, G., and Zhang, X.  Membership inference attacks on machine learning: A survey.  _arXiv preprint arXiv:2103.07853_ , 2021.
  * Li & Zhang (2020) Li, Z. and Zhang, Y.  Label-leaks: Membership inference attack with label.  _arXiv preprint arXiv:2007.15528_ , 2020.
  * Liu et al. (2019) Liu, G., Wang, C., Peng, K., Huang, H., Li, Y., and Cheng, W.  Socinf: Membership inference attacks on social media health data with machine learning.  _IEEE Transactions on Computational Social Systems_ , 6(5):907–921, 2019.
  * Malach et al. (2020) Malach, E., Yehudai, G., Shalev-Schwartz, S., and Shamir, O.  Proving the lottery ticket hypothesis: Pruning is all you need.  In _International Conference on Machine Learning_ , pp. 6682–6691. PMLR, 2020.
  * Morcos et al. (2019) Morcos, A. S., Yu, H., Paganini, M., and Tian, Y.  One ticket to win them all: generalizing lottery ticket initializations across datasets and optimizers.  _arXiv preprint arXiv:1906.02773_ , 2019.
  * Nasr et al. (2018) Nasr, M., Shokri, R., and Houmansadr, A.  Comprehensive privacy analysis of deep learning: Stand-alone and federated learning under passive and active white-box inference attacks.  2018\.
  * Salem et al. (2018) Salem, A., Zhang, Y., Humbert, M., Berrang, P., Fritz, M., and Backes, M.  Ml-leaks: Model and data independent membership inference attacks and defenses on machine learning models.  _arXiv preprint arXiv:1806.01246_ , 2018.
  * Shokri et al. (2017) Shokri, R., Stronati, M., Song, C., and Shmatikov, V.  Membership inference attacks against machine learning models.  In _2017 IEEE Symposium on Security and Privacy (SP)_ , pp. 3–18. IEEE, 2017.
  * Yu et al. (2019) Yu, H., Edunov, S., Tian, Y., and Morcos, A. S.  Playing the lottery with rewards and multiple languages: lottery tickets in rl and nlp.  _arXiv preprint arXiv:1906.02768_ , 2019.


# Circular hydraulic jumps: where does surface tension matter?

A. Duchesne 1<EMAIL_ADDRESS>L. Limat 2  1 Univ. Lille, CNRS,
Centrale Lille, Univ. Polytechnique Hauts-de-France, UMR 8520 - IEMN, F-59000
Lille, France 2 Université de Paris, CNRS, Laboratoire Matière et Systèmes
Complexes (MSC), UMR 7057 - Bâtiment Condorcet, 10 rue Alice Domon et Léonie
Duquet, 75013 Paris, France

###### Abstract

Recently, an unusual scaling law has been observed in circular hydraulic jumps
and has been attributed to a supposed missing term in the local energy balance
of the flow [Bhagat et al. (2018)]. In this paper, we show that - though the
experimental observation is valuable and interesting - this interpretation is
presumably not the good one. When transposed to the case of a axial sheet
formed by two impinging liquid jets, the assumed principle leads in fact to a
velocity distribution in contradiction with the present knowledge for this
kind of flows. We show here how to correct this approach by keeping
consistency with surface tension thermodynamics: for Savart-Taylor sheets,
when adequately corrected, we recover the well known $1/r$ liquid thickness
with a constant and uniform velocity dictated by Bernoulli’s principle.

In the case of circular hydraulic jumps, we propose here a simple approach
based on Watson description of the flow in the central region [Watson (1964)],
combined with appropriate boundary conditions on the formed circular front.
Depending on the specific condition, we find in turn the new scaling by Bhagat
et al. (2018) and the more conventional scaling law found long ago by Bohr et
al. (1993). We clarify here a few situations in which one should hold rather
than the other, hoping to reconcile Bhagat et al. observations with the
present knowledge of circular hydraulic jump modeling. However, the question
of a possible critical Froude number imposed at the jump exit and dictating
logarithmic corrections to scaling remains an opened and unsolved question.

##  1 Introduction

Stationary axisymmetrical liquid structures formed by jet impacts, have
motivated an enormous amount of literature. Three examples that will be
important here are sketched on Fig. 1. First of all, the well-known circular
hydraulic jump [Rayleigh (1914); Tani (1949); Watson (1964); Craik et al.
(1981); Bohr et al. (1993); Bush & Aristoff (2003); Duchesne et al. (2014);
Mohajer & Li (2015); Salah et al. (2018); Bhagat et al. (2018); Wang & Khayat
(2019, 2021)], sketched on Fig.1-a, with a well developed liquid film
extending all around. Its equivalent on a ”dry” surface, possibly
superhydrophobic [Jameson et al. (2010); Button et al. (2010); Maynes et al.
(2011)], the ”rim atomization” is sketched on Fig.1-b. Finally the well-known
radial liquid sheet [Savart (1833); Huang (1970); Clanet & Villermaux (2002);
Villermaux et al. (2013)], formed either by impinging two opposite symmetrical
liquid jets, having the same central axis or by impinging a liquid jet on a
solid surface with a diameter similar to the jet diameter is depicted on
Fig.1-c.

Figure 1: Three axisymmetric film flows are discussed in the present article.
(a) The classical circular hydraulic jump formed by a jet impacting a solid
disk at its center, (b) atomization ring formed by a jet impacting a dry
surface possibly superhydrophobic, (c) liquid sheet formed by impact of two
liquid jets of opposite direction.

These three geometries are of course linked together by the same general
equation for the energy balance. In this article, we will therefore show that
apparent paradoxes raised by the modeling of the surface tension on the
circular hydraulic jump by Bhagat et al. (2018) may be solved or at least
clarified by considering the geometry depicted in Fig.1-c.

The selection of jump radius $R_{J}$ in the circular hydraulic jump case
(Fig.1-a) has motivated many studies. The two most well known approaches are
the one from Watson and Bush [Watson (1964); Bush & Aristoff (2003)], in which
the height of the outer film remains a control parameter, and the one from
Bohr et al. (1993), rather devised when a liquid film extends all around at
large distance, and inspired from boundary layer theories. As well known, this
second approach leads to a scaling law dependence of $R_{J}$ upon flow rate
$Q$ and the physical parameters ($\nu$ the kinematic viscosity of the fluid,
$g$ gravity), that reads:

$R_{J}\sim Q^{\frac{5}{8}}\nu^{-\frac{3}{8}}g^{-\frac{1}{8}}.$ (1)

Later, Duchesne et al. (2014) emphasized the importance of logarithmic
corrections to scaling, due to viscous dissipation in the outer film, yet
observed numerically by Bohr, and also showed that the prefactor was
experimentally linked to the value of the Froude number at the exit of the
jump, that seemed to be locked to a critical value. This phenomenon was
recovered by Mohajer & Li (2015) and by Argentina et al. (2017) with a non-
linear modeling of film flow equations including the first finite slope terms.

Very recently, an attempt of revision of this picture has been published by
Bhagat et al. (2018), who performed new experiments, and reported the
observation of a different scaling in which surface tension $\gamma$ was
involved, but not gravity:

$R_{J}\sim
Q^{\frac{3}{4}}\rho^{\frac{1}{4}}\nu^{-\frac{1}{4}}\gamma^{-\frac{1}{4}},$ (2)

in which $\rho$ is the liquid density. To rationalize this finding, these
authors claimed that most available approaches of surface tension influence
lead to only small corrections [Bush & Aristoff (2003)] and that the
description of the circular hydraulic jump had thus to be completely
reconsidered. They introduced an energy balance, between two radii $r$ and
$r+\delta r$, that reads:

$\left[\rho\frac{\bar{u}^{2}}{2}\bar{u}rh\right]^{r+\delta r}_{r}=\left[\gamma
r\bar{u}\right]^{r+\delta r}_{r}-\left[p\bar{u}h\right]^{r+\delta
r}_{r}-\left[\rho g\frac{h^{2}}{2}r\bar{u}\right]^{r+\delta
r}_{r}-r\tau_{W}\bar{u}\delta r,$ (3)

with the notation $\left[A\right]^{r+\delta r}=A(r+\delta r)-A(r)$, and in
which $\bar{u}$ designates the flux-average radial velocity, $r$ the distance
to the axis, $h(r)$ the thickness of the liquid layer, $p(r)$ the pressure at
$z=0$ and $\tau_{W}$ the wall shear stress. The last term on the right
designates the viscous dissipation by friction on the substrate, while the
first one is an additional term compared to previous approaches, that is
presumed to be “at the origin” of the new scaling (2). This conjecture has
been contested [Duchesne et al. (2019); Bohr & Scheichl (2021)] (see
alsoBhagat & Linden (2020) answer), and it is also known that a scaling like
(2) can also appear without such an assumption, as shown for instance byButton
et al. (2010) for liquid bells formed below a ceiling.

We find here useful to have a look on what would happen in the simplified
geometry of Fig.1-c, when applying this principle. As we shall show in section
2, this modeling leads to a velocity distribution in complete contradiction
with the present knowledge of liquid sheets (and with Bernoulli’s principle),
which suggests that Bhagat et al argument is flawed. In fact the obtained flow
field is not new, and has been proposed in the past by Bouasse (1923) who
attributed the calculation to Hagen (see Hagen (1849)). It will be instructive
here to remind the argument followed by Hagen and Bouasse, in section 3, in a
Lagrangian frame, analyzing a circular expanding piece of film. We will then
show, in the same section, how one can correct the argument to get the more
classical and now admitted result deduced from Bernoulli’s principle of a
uniform radial velocity around the impact point, and how, missing some terms
in the balance, one can get the flawed result of Bouasse and Hagen. Finally,
coming back to a Eulerian description, we will explain how these
considerations impact the principle proposed in eq.(3). We will show that an
extra term exactly equal and opposite to the capillary contribution should
cancel this one, in a way consistent with classical thermodynamics, leading to
the expression usually written from the balance of momentum.

This does not mean, however, that Bhagat et al. scaling discovery is of none
interest. In section 4 and 5, we will try to precise to which capillary
structures – different from the stationary hydraulic jump observed by Bohr –
it could apply, and a possible way to justify its occurrence.

##  2 A look to a simple situation: the axisymmetrical liquid sheet.

Let us try to apply the principle suggested in eq.(3) to the case suggested on
Fig.1-c, i.e. to a axisymmetrical sheet formed by the coaxial impact of two
jets in a situation of negligible gravity. The viscous shear on the substrate
having disappeared, eq.(3) reduces to a very simple balance that reads:

$\left[\rho rh\frac{u^{3}}{2}\right]^{r+\delta r}_{r}=\left[\gamma
ru\right]^{r+\delta r}_{r},$ (4)

where the horizontal velocity $u$ has no dependence upon the transverse
direction, and coincides with any of its average values. This implies that the
following quantity is constant all over the sheet:

$\rho rh\frac{u^{3}}{2}-\gamma ru=\mathrm{Cte}.$ (5)

Combined with the mass balance $Q=2\pi rhu$, this leads to the following
expression for $u$:

$u=2\pi\frac{\gamma}{\rho Q}r+\sqrt{u_{0}^{2}-4\pi\frac{\gamma}{\rho
Q}r_{0}u_{0}+4\pi^{2}\frac{\gamma^{2}}{\rho^{2}Q^{2}}r^{2}},$ (6)

in which $r_{0}$ designates the jet radius at impact and $u_{0}$ the
asymptotic value for $u$, reached when $r=r_{0}$, which satisfies the equality
$Q=\pi r_{0}^{2}u_{0}$ in a quasi-elastic shock approximation [Villermaux et
al. (2013)]. In the limit of large jet velocity, i.e. $u_{0}^{2}\gg
2\gamma\textfractionsolidus(\rho r_{0})$, this expression reduces to the
slowly varying upon $r$ approximate:

$u\approx u_{0}+2\pi\frac{\gamma}{\rho Q}(r-r_{0}),$ (7)

which is known to be false, as it has been checked experimentally that the
velocity is constant all over the sheet, recovering the Bernoulli’s principle
(see in particular Fig. 3 in [Villermaux et al. (2013)]). It is however
amazing to remind that a similar expression is proposed by Bouasse (1923) who
attributed this result to Hagen (1849), but with a slight sign change, that is
in fact due to a mistake on his own:

$u\approx u_{0}-2\pi\frac{\gamma}{\rho Q}(r-r_{0}).$ (8)

Though obtained erroneously, this expression is very seductive and Bouasse
used it to calculate the radius of the liquid sheet $R_{LS}$ assuming that the
sheet border should stay at the place in which $u$ vanishes which leads to
$R_{LS}=(\rho Qu_{0})\textfractionsolidus 2\pi\gamma(=\rho
r_{0}^{2}u_{0}^{2})\textfractionsolidus 2\gamma$ . Surprisingly this result
coincides with the right one that is in fact obtained, now, by assuming a
constant velocity, dictated by Bernoulli’s principle, and the balance of
momentum at the sheet perimeter, i.e. $\rho hu^{2}=\gamma$ [Villermaux et al.
(2013)]. But on the other hand, we would like to stress out that the radial
velocity is uniform in the sheet of Fig.1-c, which means that the principle
proposed in eq.(3), and therefore the basis of the theory developed by Bhagat
et al. (2018) is flawed.

##  3 Reconsidering Hagen argument, and its implications for hydraulic jump.

We try to understand the fault underlying Bouasse and Hagen principle. Their
line of thought is easier to explain considering a Lagrangian frame, and more
precisely the balance of energy on a annular piece of fluid, convected by the
radial flow, and it is in fact the method proposed by Bouasse himself in his
treatise of fluid mechanics [Bouasse (1923)].

Let us consider a piece of annular piece of film as on Fig.2-a, convected and
distorted by the flow. Mass conservation implies that, at any time $hr\delta
r=\mathrm{Cte}$, while the balance of energy for the whole annulus reads, in
the limit of $\delta r$ small enough to satisfies the condition $\delta
r\frac{\partial u}{\partial r}\ll u$ of a slowly varying velocity field :

$\frac{\partial}{\partial t}\left[2\pi\left(\frac{1}{2}\rho u^{2}rh\delta
r+\gamma r\delta r\right)\right]\approx 2\pi\gamma\delta ru.$ (9)

Figure 2:  Lagrangien (a) and Eulerian (b) frame for discussing energy balance
in a annular portion of a liquid film used in the text.

The first term in the left hand side of this equation stands for kinetic
energy, and the second for the surface energy enclosed between $r$ and
$r+\delta r$. The right hand term comes from the work of surface forces, and
does not vanish. Indeed, the same surface tension force is pulling on a
different arc length, as the external boundary has a larger perimeter than the
other (note that this is the intuitive argument underlying Bhagat’s analysis).
Still in the limit of a slowly varying velocity field at the scale $\delta r$,
after noting that $\frac{\partial}{\partial t}=u\frac{\partial}{\partial r}$,
eq.(9) reads:

$rh\delta ru\frac{\partial}{\partial r}\left(\frac{\rho
u^{2}}{2}\right)+\gamma u\delta r\approx\gamma u\delta r.$ (10)

In fact, the two surface tension terms are canceling each other, which means
that the work provided to the annulus by surface tension of the outer
interfaces is completely transformed into the surface energy stored at the
free surface of the annulus, in agreement with simple thermodynamic
considerations. As a result, the fluid velocity is unaffected by surface
tension balance and remains constant as one would deduce from a more classical
argument in terms of Bernoulli’s principle, i.e. $u(r)$ is in fact independent
of $r$:

$u(r)=u_{0}=\mathrm{Cte}.$ (11)

Note here that skipping from eq.(9) to eq.(10) is not completely trivial as
there is an extra $\gamma$ term remaining, but this one vanishes for the
constant and uniform $u_{0}$ solution.

To reconnect with Bouasse, instead of this, if one would forget the internal
surface energy contribution in the left hand member of eq.(10), one would get
the following equation for $u$, that reduces to:

$rh\delta r\frac{\partial}{\partial r}\left(\frac{\rho
u^{2}}{2}\right)\approx\gamma\delta r.$ (12)

After simplifying $\delta r$, and using the fact that $Q=2\pi ruh$, this
equation leads to:

$\frac{\partial u}{\partial r}\approx 2\pi\frac{\gamma}{\rho Q},$ (13)

which leads finally to eq.(7). Alternatively, eq.(8) is obtained when one
forgets the work provided to the annulus by the outer parts of the liquid
sheet, i.e. by neglecting the right hand member of eq.(10), following the
intuitive but erroneous idea of Hagen (1849) that surface tension could slow
down the flow. Historically Bouasse followed the first argument, but committed
a sign mistake, obtaining eq.(8), that was physically more natural,
considering presumably what Hagen said long ago.

To summarize, a correct treatment of the expansion of liquid annula in the
flow leads to the classical result of a uniform velocity, while the
approximates defended by Hagen and Bouasse would follow from neglecting a part
of capillary terms. We believe that a similar problem is involved in eq.(3).
If we consider now a Eulerian description of the flow, as suggested on
Fig.2-b, the balance of energy should rather read:

$\left[\rho\frac{\bar{u}^{2}}{2}\bar{u}rh+\gamma r\bar{u}\right]^{r+\delta
r}_{r}=\left[\gamma r\bar{u}\right]^{r+\delta
r}_{r}-\left[p\bar{u}h\right]^{r+\delta r}_{r}-\left[\rho
g\frac{h^{2}}{2}r\bar{u}\right]^{r+\delta r}_{r}-r\tau_{W}\bar{u}\delta r,$
(14)

in which we have added in the left hand side the surface energy convected by
the film. It is true that one can consider a capillary force, as in Bhagat et
al, in the right hand member, but in this case, one should not miss the flux
of surface crossing the two circles displayed on Fig.2-b in the left hand side
of the equation. And just as what happens in a Lagrangian frame, the physics
being the same in both frame, the capillary effects should exactly compensate
each other in this equation, that should then reduce to the more conventional
form:

$\left[\rho\frac{\bar{u}^{2}}{2}\bar{u}rh\right]^{r+\delta
r}_{r}=-\left[p\bar{u}h\right]^{r+\delta r}_{r}-\left[\rho
g\frac{h^{2}}{2}r\bar{u}\right]^{r+\delta r}_{r}-r\tau_{W}\bar{u}\delta r,$
(15)

that apart some coefficients that will depend on the detailed structure of the
flow profile is consistent with what people are used to write starting rather
from the balance of momentum [Bohr et al. (1993)]. Therefore, we do not
consider, in the interpretation of eq.(2), that one should add a new capillary
force distributed all over space as proposed by Bhagat et al. (2018). To our
opinion, this would imply to redo the initial mistake of Hagen and Bouasse.
Just as for the calculation of the size of radial liquid sheets, the solution
should rather lies inside the boundary conditions written at the circle which
radius is under question. We are now developing more this idea.

##  4 Alternative explanation of unusual scaling: the boundary condition at
the ”jump” radius. Comparison with atomization rings.

To interpret the occurrence of Bhagat et al. (2018) scaling, we propose an
alternative approach. We just treat the two ideal situations of Fig.1-a and
Fig.1-b with the same method, and see what happens. We will then see that the
situation obtained in Fig.1-b may be compared to the one suggested by Bhagat
et al. (2018).

To simplify the analysis, the “internal” flow for $r_{0}<r<R_{J}$ is
assimilated to the one discussed long ago by Watson (1964), in which fluid
inertia is progressively dissipated by viscous friction, i. e. for $r<R_{J}$:

$u(r,z)=\frac{27c^{3}}{8\pi^{4}}\frac{Q^{2}}{\nu(r^{3}+l^{3})}f\left(\frac{z}{h}\right),$
(16)

in which $c\approx 1.402$, $l=0.567r_{0}R$ (with $R$ the Reynolds number of
the jet) and $f$ is the function:
$f(\eta)=\sqrt{3}+1-\frac{2\sqrt{3}}{1+cn(3^{\frac{1}{4}}c(1-\eta))}$. Mass
conservation implies that the film thickness and the flux of momentum are
given by:

$\rho h<u^{2}>=\frac{27\sqrt{3}c^{3}}{16\pi^{6}}\frac{\rho
Q^{3}}{R_{J}\nu(R_{J}^{3}+l^{3})},$ (17)

where $<u^{2}>=\int_{0}^{h}u^{2}\mathrm{d}z$. In the case of Fig.1-a, this
flow must be matched for $r>R_{J}$ to a film flow under the action of gravity,
that, according to lubrication [Duchesne et al. (2014)], has a thickness
distribution $H(r)$ given by:

$H(r)^{4}=H_{\infty}^{4}+\frac{6}{\pi}\frac{\nu
Q}{g}ln\left(\frac{R_{\infty}}{r}\right),$ (18)

where $R_{\infty}$ designates the outer radius of the substrate, where the
thickness $H$ reaches a value called $H_{\infty}$ that will depend on the
specific geometrical conditions of the flow there (see Fig.1-a for the
graphical definition). At $r=R_{J}$, one has to write some matching condition,
that is consistent with the approximations made on each side of $r=R_{J}$, and
stands for a shock [Bélanger (1841); Rayleigh (1914)]. If we assume $h\ll H$
and neglect the surface tension at the shock (i.e. for circular hydraulic
jumps large enough such as the ones considered by Bhagat et al. (2018)), this
shock condition reads:

$\rho h(r)<u(R_{J})^{2}>\approx\rho gH(R_{J})^{2},$ (19)

In the limit of negligible values for $H_{\infty}$ and $r_{0}$, compared to
the other scales, it is easy to check that these equations lead to the
following scaling law for $R_{J}$:

$R_{J}ln\left(\frac{R_{\infty}}{R_{J}}\right)^{\frac{1}{8}}=\frac{(3c)^{\frac{3}{4}}}{2^{\frac{9}{8}}\pi^{\frac{1}{1}}8}\frac{Q^{\frac{5}{8}}}{\nu^{\frac{3}{8}}g^{\frac{1}{8}}},$
(20)

i.e. the scaling obtained by Bohr et al. (1993) and modified by logarithmic
corrections.

We now consider the regime described in Fig.1-b that may be obtained in
stationary regime with particular superhydrophobic treatment [Maynes et al.
(2011)] or with inverse gravity [Jameson et al. (2010); Button et al. (2010)].
In this regime the force opposed to fluid inertia at the boundaries is
dictated only by surface tension and not by gravity, there is no developed
shock, no liquid ”wall”. In other words, the flux of momentum is only balanced
by surface tension, which means that equations (18) and (19) are simply
replaced now by:

$\rho h(r)<u(R_{J})^{2}>\approx\gamma(1-\cos\theta),$ (21)

with $\theta$ the static contact angle

Using eq.(17) in the limit $r=R_{J}\gg r_{0}$, this condition yields a new
scaling that reads:

$R_{J}=\left(\frac{27\sqrt{3}c^{3}}{16\pi^{6}}\right)^{\frac{1}{4}}(1-cos\theta)^{\frac{1}{4}}Q^{\frac{3}{4}}\nu^{-\frac{1}{4}}\rho^{\frac{1}{4}}\gamma^{-\frac{1}{4}}.$
(22)

This scaling is the same than the one suggested by Bhagat et al. (2018) and
previously by Button et al. (2010). It explains why the scaling obtained by
Bhagat et al. (2018) applies to the experimental data of Jameson et al. (2010)
even if the theory leading to this scaling is not the right one.

We thus do not believe that there is a “universal” scaling that should hold
for any circular ”print” formed around an impacting jet. Sometimes one can
find Bohr’s scaling and sometimes Baghat and Button one, it is the analysis of
the conditions around the impact that will matter.

##  5 Another possible occurrence of Bhagat and Button scaling.

We now show that Bahgat’s scaling may also be observed in classical circular
hydraulic jumps. In Bhagat et al. (2018) paper, the authors consider an
intermediate regime where the liquid has not yet reached the edge of the plate
(see Fig.3). In their experimental evidence the authors consider partial
wetting conditions (they use Perspex, glass and Teflon) and aqueous solutions.
Given that the front propagation speed is rather small, we can consider that
the liquid front height is approximately given by

$h_{cap}\approx\left(\frac{\gamma}{\rho
g}\right)^{\frac{1}{2}}(1-\cos\theta)^{\frac{1}{2}}.$ (23)

Figure 3: sketch of the intermediate regime for a low viscosity liquid in
partial wetting.

Considering the eq.(18) for low viscosity liquid and moderate flow rate one
can conclude that:

$H(r)\approx H_{\infty}\approx h_{cap}.$ (24)

Therefore the (simplified) shock condition (19) previously obtained leads to:

$\rho h(r)<u(R_{J})^{2}>\approx\frac{1}{2}\rho gh_{cap}^{2}.$ (25)

Surprisingly, this argument leads again exactly to the ”surface tension
dominated” scaling :

$R_{J}=\left(\frac{27\sqrt{3}c^{3}}{16\pi^{6}}\right)^{\frac{1}{4}}(1-cos\theta)^{\frac{1}{4}}Q^{\frac{3}{4}}\nu^{-\frac{1}{4}}\rho^{\frac{1}{4}}\gamma^{-\frac{1}{4}}.$
(26)

Following now a remark from Bhagat et al. (2018), one can also denote that by
defining the Weber number as

$\mathrm{We}\approx\frac{\rho h(r)<u(R_{J})^{2}>}{\gamma}\approx cste,$ (27)

i.e., a constant Weber number that replaces the constant Froude number
encountered in fully established hydraulic jump with a complete, flowing outer
film.

Considering $\theta=\frac{\pi}{2}$, we obtain that:

$\mathrm{We}\approx\frac{1}{2},$ (28)

which is the order of magnitude of the Weber number reported in Bhagat et al.
(2018).

##  6 Conclusion

In summary, we have reconsidered the problem of scaling law selection of the
”radius of influence” in the problem of vertical jet impact on a horizontal
solid surface. In our opinion, the ideal law (1) proposed by Bohr and
coworkers (to which, one should not forget to add logarithmic corrections as
in Duchesne et al. (2014)) corresponds to the ideal situation of a stationary
hydraulic jump formed inside a liquid film extending on the whole solid
surface. On the opposite, the scaling (2) suggested in ref. [Bhagat et al.
(2018)] rather holds in different situations, some of these ones being:

\- stationary impact of a jet on a dry surface, possibly superhydrophobic,
without formation of the outer film (atomization ring),

\- stationary impact of a jet on a dry surface in inverse gravity (impact of a
jet on a ceiling),

\- transient regime of circular hydraulic jump formation for low viscosity
liquids in partial wetting.

It would be interesting to explore in more details these three situations, and
to identify possible other ones. In our opinion, there is no need to imagine
some universal extra capillary term imposing the scaling (2) as imagined in
ref [Bhagat et al. (2018)]. Though this extra term really exists, when the
control volume contains the free surface of the film instead of excluding it,
it is in practice compensated by another one in a way consistent with
classical thermodynamics. As usual in free surface flows there is no increase
or decrease of velocity that could be due alone to the action of surface
tension, except when Marangoni effects are involved [Marmottant et al.
(2000)]. Going on in this direction would be just reproducing for thin film
flows on a solid, the initial mistake of Hagen and Bouasse.

If we come back to the question of Bohr scaling we have left a bit aside the
questions of the logarithmic corrections and the possible existence of a
critical Froude number at the jump exit, suggested in [Duchesne et al.
(2014)]. The possible existence of this critical Froude number leads to a
different exponent for the Logarithmic corrections ($3/8$ instead of $1/8$)
and this question is still not solved. As told in the introduction, recent
non-linear analytical treatment of the film flow suggests that such a critical
Froude number could exist, but this remains to be established and convincingly
explained.

A specific problem of great interest where these considerations should mater
is the question of jet impacts on inclined plates. It is not obvious in this
kind of problem that a perfect hydraulic jump can exist, or not, and the two
scaling should compete against each other in a way that merits to be
investigated. The influence of a external fields, here the tangent component
of gravity on a circular shock is a fundamental question of great interest. A
specific effort should be done in this direction [Wilson et al. (2012);
Duchesne et al. (2013)].

Declaration of Interests. The authors report no conflict of interest.

## References

  * Argentina et al. (2017) Argentina, Médéric, Cerda, Enrique, Duchesne, Alexis & Limat, Laurent 2017 Scaling the viscous circular hydraulic jump. APS Division of Fluid Dynamics Meeting Abstracts .
  * Bélanger (1841) Bélanger, J. B. 1841 Notes sur l’hydraulique. Ecole Royale des Ponts et Chaussées, Paris, France, session 1842, 223.
  * Bhagat et al. (2018) Bhagat, Rajesh K, Jha, NK, Linden, PF & Wilson, D Ian 2018 On the origin of the circular hydraulic jump in a thin liquid film. Journal of Fluid Mechanics 851.
  * Bhagat & Linden (2020) Bhagat, Rajesh K & Linden, Paul F 2020 The circular capillary jump. Journal of Fluid Mechanics 896.
  * Bohr et al. (1993) Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–648.
  * Bohr & Scheichl (2021) Bohr, Tomas & Scheichl, Bernhard 2021 Surface tension and energy conservation in a moving fluid. Phys. Rev. Fluids 6, L052001.
  * Bouasse (1923) Bouasse, Henri 1923 Jets, tubes et canaux. Librairie Delagrave, Paris.
  * Bush & Aristoff (2003) Bush, J. W. M. & Aristoff, J. M. 2003 The influence of surface tension on the circular hydraulic jump. J. Fluid Mech. 489, 229–238.
  * Button et al. (2010) Button, E. C., Davidson, J. F., Jameson, G. J. & Sader, J. E. 2010 Water bells formed on the underside of a horizontal plate. part 2. theory. Journal of Fluid Mechanics 649, 45–68.
  * Clanet & Villermaux (2002) Clanet, C. & Villermaux, E. 2002 Life of a smooth liquid sheet. Journal of fluid mechanics 462, 307–340.
  * Craik et al. (1981) Craik, A. D. D., Latham, R. C., Fawkes, M. J. & Gribbon, P. W. F. 1981 The circular hydraulic jump. J. Fluid Mech. 112, 347–362.
  * Duchesne et al. (2019) Duchesne, Alexis, Andersen, Anders & Bohr, Tomas 2019 Surface tension and the origin of the circular hydraulic jump in a thin liquid film. Physical Review Fluids 4 (8), 084001\.
  * Duchesne et al. (2013) Duchesne, Alexis, Lebon, Luc & Limat, Laurent 2013 Jet impact on an inclined plate: contact line versus hydraulic jump. In European Coating Symposium, ECS 2013, pp. 48–51. UMONS, Université de Mons.
  * Duchesne et al. (2014) Duchesne, Alexis, Lebon, Luc & Limat, Laurent 2014 Constant froude number in a circular hydraulic jump and its implication on the jump radius selection. EPL (Europhysics Letters) 107 (5), 54002.
  * Hagen (1849) Hagen, G 1849 Ueber die scheiben, welche sich beim zusammenstossen von zwei wasserstrahlen bilden und über die auflösung einzelner wasserstrahlen in tropfen. Annalen der Physik 154 (12), 451–476.
  * Huang (1970) Huang, JCP 1970 The break-up of axisymmetric liquid sheets. Journal of Fluid Mechanics 43 (2), 305–319.
  * Jameson et al. (2010) Jameson, G. J., Jenkins, C. E., Button, E. C. & Sader, J. E 2010 Water bells formed on the underside of a horizontal plate. part 1. experimental investigation. Journal of Fluid Mechanics 649, 19–43.
  * Marmottant et al. (2000) Marmottant, Philippe, Villermaux, Emmanuel & Clanet, Christophe 2000 Transient surface tension of an expanding liquid sheet. Journal of colloid and interface science 230 (1), 29–40.
  * Maynes et al. (2011) Maynes, D., Johnson, M. & Webb, B. W. 2011 Free-surface liquid jet impingement on rib patterned superhydrophobic surfaces. Phys. Fluids 23 (052104).
  * Mohajer & Li (2015) Mohajer, Behzad & Li, Ri 2015 Circular hydraulic jump on finite surfaces with capillary limit. Physics of Fluids 27 (11), 117102.
  * Rayleigh (1914) Rayleigh, L. 1914 On the theory of long waves and bores. Proc. R. Soc. London A 90 (619), 324–328.
  * Salah et al. (2018) Salah, Sofiene Ouled Taleb, Duchesne, Alexis, De Cock, Nicolas, Massinon, Mathieu, Sassi, Khaled, Abrougui, Khaoula, Lebeau, Frédéric & Dorbolo, Stéphane 2018 Experimental investigation of a round jet impacting a disk engraved with radial grooves. European Journal of Mechanics-B/Fluids 72, 302–310.
  * Savart (1833) Savart, F. 1833 Mémoire sur le choc d’une veine liquide lancée contre un plan circulaire. Ann. chim 54, 56–87.
  * Tani (1949) Tani, I. 1949 Water jump in the boundary layer. J. Phys. Soc. Japan 4, 212–215.
  * Villermaux et al. (2013) Villermaux, Emmanuel, Pistre, Violaine & Lhuissier, Henri 2013 The viscous savart sheet. Journal of Fluid Mechanics 730, 607–625.
  * Wang & Khayat (2019) Wang, Yunpeng & Khayat, Roger E 2019 The role of gravity in the prediction of the circular hydraulic jump radius for high-viscosity liquids. Journal of Fluid Mechanics 862, 128–161.
  * Wang & Khayat (2021) Wang, Yunpeng & Khayat, Roger E 2021 The effects of gravity and surface tension on the circular hydraulic jump for low-and high-viscosity liquids: A numerical investigation. Physics of Fluids 33 (1), 012105.
  * Watson (1964) Watson, E. J. 1964 The radial spread of a liquid over a horizontal plane. J. Fluid Mech. 20, 481–499.
  * Wilson et al. (2012) Wilson, D. I., Le, B. L., Dao, H. D. A., Lai, K. Y., Morison, K. R. & Davidson, J. F. 2012 Surface flow and drainage films created by horizontal impinging liquid jets. Chemical Engineering Science 68 (1), 449–460.