]bibliography.bib

# Building an ID Card Repository with Progressive Web Application to Mitigate
Fraud

1st Kevin Akbar Adhiguna  Faculty of Mathematics and Natural Science
Padjadjaran University
Sumedang district, West Java, Indonesia
<EMAIL_ADDRESS>2nd Firhan Maulana Rusli  School of Computing
Telkom University
Bandung City, West Java, Indonesia
<EMAIL_ADDRESS>3rd Hendy Irawan  School of Computing
Telkom University
Bandung City, West Java, Indonesia
<EMAIL_ADDRESS>
###### Abstract

A lot of service requires identity of users to mitigate undesirable incidents,
such as fraud. To cut down probability of potential fraud, ID Card of users
are collected to be verified so people can verify users whenever an
undesirable activity happens. However, to verify a person’s identity through
his/her ID Card, a repository for the ID Card is required. To verify ID cards,
the ID Card repository will be connected to an automated ID Card Verification
API. The ID Card repository is meant to be used both on mobile phones and
desktop computers, so the concept of progressive web application is used. To
be able to upload images smoothly and build progressive web application, the
ID Card repository is built using ReactJS and Ant Design. Server side is
powered by Strapi and MongoDB. GraphQL API is utilized to connect client side
and server side. It involves queries to fetch data. To fetch data on the
client side, Apollo client is used to in ReactJS. Git is utilized for version
control system which gives contribution to Continuous Integration/Continuous
Delivery (CI/CD). In this paper, we will discuss why Strapi is best suited on
server side and how Ant Design, a library to style components in a web page,
can provide required components in this web application.

###### Index Terms:

ReactJS, Progressive Web Application, Ant Design, GraphQL, Strapi

##  I Introduction

Data and information have been a part of human life. They are mostly accessed
through the Internet and can be provided in various ways such as text, files,
and images. People upload and download data. However, when developing a web
application that enables its users to register, enterprises might want to
verify a user’s identity. One common way is to ask users to upload his/her
identity card. ID card repository will be used to save user’s ID card.

User Interface (UI) is important in creating an application, whether it is a
mobile application or a web application. However, creating application’s
functionality is a lot more important than UI and those things are often time-
consuming. Choosing an appropriate UI library becomes vital as necessary web
application components must be provided.

On the server side, there are notable things that must be considered
seriously. The first one is Application Programming Interface (API) creation.
Other than Simple Object Access Protocol (SOAP), Restful API (REST API) and
GraphQL API are available [1, 2]. Both REST API and GraphQL API have drawbacks
and advantages, so it is desirable to have a library or framework that
supports both. In order to focus on the web application’s functionality, API
creation time and effort need to be reduced.

Another notable thing is related to security which are authentication and
authorization. JSON Web Token (JWT) is used to control authentication and
authorization. Hence, public can not access pages that can only be seen by
authenticated users.

Connecting database and API on the server side should be as easy as possible.

To create API in no time and to configure database on server side, Strapi is
used.

To be able to create pages such as Login screen and Upload screen in this web
application, Ant Design is involved in building this ID card repository.
Nevertheless, placement of various web components can be a challenge in this
web application as importing a web component sometimes needs extra
configuration on style sheet.

Based on the introduction that has been described above, the following problem
formulations are obtained :

  1. 1.

Does Strapi provide adequate requirement on the server side such as API
creation, security, and database configuration?

  2. 2.

Are required components available in Ant Design?

##  II Literature Study

Here are technology stacks that power the ID Card Repository.

###  II-A ReactJS

ReactJS is a JavaScript library which is deployed to develop reusable user
interface (UI) components [3]. It is released by Facebook in 2013 and
maintained by Facebook and a community consisting of developers and companies.
ReactJS is optimized to provide speed, simplicity, and scalability [4]. The
fundamental behind the React is the concept of virtual Document Object
Manipulation (DOM) which React uses effectively. It renders sub-trees of nodes
based upon state changes. In order to keep components up to date, React does
the least amount of DOM manipulation possible. [5] It utilizes JSX to create
React elements. With power of JSX, there is no need to call
React.createElement() each time when building virtual DOM. JSX is an optional
HTML-like syntax that allows its users to create a virtual DOM tree without
using the React.createElement() method. [6] Other than JSX and virtual DOM,
React has components which are categorized as stateful and stateless. Stateful
components are used when data changes dynamically. State is a property of the
React components that includes any data that a component needs. [7] Before
React 16.8 was released, when data changes occur dynamically, a class-based
component must be used. However, with the use of React hooks, a functional
component can be used for it. React hooks is a mechanism for state management
in a functional style that avoids the use of classes [8].

###  II-B Progressive Web Application

Progressive Web Application (PWA) is a new set of standards advocated by the
Google Web Fundamentals group to bridge a gap by introducing features such as
offline support. PWA contributes toward the unification of the mobile
experience where web applications can be installed and distributed without
application marketplaces, work without Internet connectivity, receive push
notifications, and look like regular applications. [9]

###  II-C Ant Design

Ant Design is one of the most well-known ReactJS UI library that contains a
set of high quality components and demos for building rich and interactive UI
[10]. It is one of the most popular React UI library with more than 65000
stars on Github as well alongside Material UI, another famous React UI
library. Even though it is said that Ant Design is a ReactJS UI library, Ant
Design for VueJS and Angular are also available. Ant Design is applied on
designing UI components.

###  II-D Strapi

Strapi is a flexible and open-source headless content management system (CMS)
that gives developers the freedom to choose their favorite tools and
frameworks while also allowing editors to easily manage and distribute their
content [11]. Strapi can be used to create Application Programming Interface
(API) fast and easily. It provides a user-friendly interface and allows users
to create content-types which is similar with tables in databases without any
technical knowledge [12]. It has extensible plugin system including Admin
Panel, Authentication & Permission Management, Content Management and API
Generator. [13]. It offers Restful API (REST API) by default. However, GraphQL
API can be used alongside REST API by installing a GraphQL plugin. It is
possible to install a GraphQL plugin either through UI or terminal/command
prompt. With REST API, you might test your API using a third-party software,
such as Postman. However, if you install GraphQL plugin, Strapi provides
GraphQL Playground which helps a lot on testing your GraphQL API. It is
accessible at ’/graphql’ endpoint. Strapi cares about database configuration
as well. It can be used with various database management systems, such as
MySQL, MongoDB, SQLite, and PostgreSQL. To use one of them, installing locally
the one that will be used is a must. For instance, MongoDB is used to store
data of ID Card repository. Therefore, installing MongoDB in the machine that
will be used is a requirement.

###  II-E MongoDB

MongoDB is one of the most popular document based database [14]. It is a very
scalable NoSQL database management system that stores data in a BSON format, a
dynamic schema document structured like Javascript Object Notation (JSON)
[15].

###  II-F GraphQL

GraphQL is a query language for APIs and a runtime for fulfilling those
queries with your existing data. It provides clients the power to ask for
exactly what they need and makes it easier to evolve APIs over time. [16]
Unlike Restful API that changes its version when it needs to upgrade, GraphQL
API enables its users to add new fields and types to GraphQL API without
impacting existing queries. Unlike REST API that has several endpoints,
GraphQL API only utilizes a single endpoint which is ’/graphql’. Moreover,
GraphQL uses a schema to describe the organization of the data and declarative
query language to allow clients to access data [17].

###  II-G Git

Git is one of the most popular version control system. Distributed version
control systems and their hosting services such as Github and Gitlab has
revolutionized how developers collaborate by allowing them to freely exchange
and integrate code changes in a peer-to-peer fashion [18]. Git keeps track of
code changes, so if there is a change, what has been modified can be tracked
through commit messages and git status.

##  III Methodology

Creating API is one of the most vital things in building this ID card
repository on the server side. In this PWA, GraphQL API is utilized within
Strapi. However, before creating GraphQL API, Strapi requires its users to
choose a database management system then configure it easily.

Figure 1: MongoDB configuration in Strapi.

In setting up database management system through Strapi, develpoers are able
to choose default configuration provided by Strapi as shown in fig. 1 or
change it depends on what requirements of the application are.

Figure 2: Various fields that are provided by Strapi

Creating a content-type is a step that is similar to determining fields and
their data types in database management system. As shown in fig. 2, here are
fields with their data types that can be chosen in Strapi :

  * •

Text

    * –

Short Text

    * –

Long Text

  * •

Rich Text

  * •

Number

    * –

integer

    * –

big integer

    * –

decimal

    * –

float

  * •

Date

    * –

date

    * –

datetime

    * –

time

  * •

Boolean

  * •

Relation

  * •

Email

  * •

Password

  * •

Enumeration

  * •

Media

    * –

Multiple Media

    * –

Single Media

  * •

JSON

  * •

UID

On the client side, Ant Design which has a lot of components can give a hand
to style a web application. Not only components that usually can be seen in a
web application like an input field and typography, it provides components to
give spacing or margin as well. These are components of Ant Design that is
used or planned to be utilized in the ID card repository on each page :

Table I: Ant Design Components in each screen Screen Name | Ant Design Component
---|---
Login |  | \- Form
---
\- Input
\- Button
\- Space
Register |  | \- Form
---
\- Input
\- Button
\- Space
Dashboard |  | \- Layout
---
\- Table
\- Breadcrumb
\- Button
Figure 3: Dependencies of the PWA

fig. 3 illustrates dependency graphs in ID card repository PWA. Basically
there are three components which are ID Card PWA that acts as a client side,
Server side that generates GraphQL API and JWT provider within Strapi that is
connected with MongoDB, and the last one is ID Card Verification API whose
function is to verify ID Cards which are uploaded to ID card repository. On
the client side, everything is dependent on yarn as a package manager. On the
server side, JWT provider gives a JWT as one of the authentication response
alongside username and password that are fetched from MongoDB.

Figure 4: Authentication Flow of the ID Card Repository

fig. 4 describes data flows on authentication process. First, React Apollo
client will send an authentication request which contains username or email
and password. Then Strapi GraphQL server will send back JWT, username, and
email of the account as an authentication response.

##  IV Evaluation Methodology

Strapi provides GraphQL playground which is used to test GraphQL API. To
evaluate GraphQL API, both registration and login queries were tested. Below
is a query that register a user with GraphQL API :

⬇

mutation createUser(

$input: createUserInput

) {

createUser(

input: $input

) {

user {

username

email

}

}

}

To login successfully, one must retrieve a JWT that will be stored in local
storage. Below is a GraphQL query to obtain a JWT :

⬇

mutation Login(

$input: UsersPermissionsLoginInput!

) {

login(

input: $input

) {

jwt

user {

username

email

}

}

}

##  V Evaluation Result

Strapi can be utilized to create API fast. Soon after selecting fields and
saving the content type, the API will be created. Table 1 below is the fields
in a content type of Strapi GraphQL API with their data types :

Table II: Fields and Data Types in ID Card Repository PWA Field | Data Type
---|---
kind | enumeration
identifier | short text
name | short text
birthPlace | short text
birthDate | date
gender | enumeration
bloodType | enumeration
address | long text
religion | short text
marriageStatus | enumeration
occupation | short text
nationalityCode | short text
expiryDate | date
facePhoto | single media
cardImage | single media
personWithCardPhoto | single media
issuerCountryCode | short text
issuedDate | date
faceTop | integer
faceLeft | integer
faceWidth | integer
faceHeight | integer
statusCode | enumeration
uploadedAt | datetime
extractedAt | datetime
verifiedAt | datetime
issuerProvince | short text
issuerCity | short text
uploaderId | short text

In fig. 5, Strapi GraphQL server responds to createUser mutation in 755ms.

Figure 5: Strapi GraphQL server Registration response

fig. 6 shows Strapi GraphQL server gives response to Login mutation in 223ms.

Figure 6: Strapi GraphQL server Authentication Response

Having Strapi run on the server side, on the client side, The styling was done
with Ant Design, including its web components and icons as shown in fig. 7.

Figure 7: Login screen build with Ant Design

##  VI Conclusion

Strapi comes up with required fields and data types to create ID card
repository. It also helps developers on API creation quickly and easily.
GraphQL API and mutation queries that were tested with GraphQL playground in
Strapi worked well as well. Ant Design that has a lot of web components
enables styling to be done easily alongside ReactJS and Apollo Client on the
client side.

For the future development, connecting to ID card verification API, creating
registration, upload, and dashboard screens, making UI responsive, and
implementing a JWT to control authentication and authorization flow are
planned.

## Acknowledgment

First, we would like to thank Lovia (https://about.lovia.life), who supported
us to finish this research paper. Next, we would like to thank to our parents,
who supported us with love and understanding. Without them, we could never
have reached this current level of success.

## References

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# On a discrete composition of the fractional integral and Caputo
derivative111This is an accepted version of the manuscript published in
Communications in Nonlinear Science and Numerical Simulations 108 (2022),
106234 with DOI: https://doi.org/10.1016/j.cnsns.2021.106234

Łukasz Płociniczak$\;{}^{,}$222Email<EMAIL_ADDRESS>Faculty of
Pure and Applied Mathematics, Wrocław University of Science and Technology,
Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

###### Abstract

We prove a discrete analogue for the composition of the fractional integral
and Caputo derivative. This result is relevant in numerical analysis of
fractional PDEs when one discretizes the Caputo derivative with the so-called
L1 scheme. The proof is based on asymptotic evaluation of the discrete sums
with the use of the Euler-Maclaurin summation formula.

Keywords: fractional integral, Caputo derivative, Euler-Maclaurin formula

AMS Classification: 26A33, 34A08, 65R20

##  1 Introduction

Let $I^{\alpha}$ be the fractional integral operator of order
$\alpha\in(0,1)$, i.e. for any locally integrable function
$y:(0,T)\mapsto\mathbb{R}$ we define

$I^{\alpha}y(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}y(s)ds.$
(1)

Further, the Caputo derivative is defined on smooth functions by

$D^{\alpha}y(t)=I^{1-\alpha}y^{\prime}(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}y^{\prime}(s)ds,\quad
0<\alpha<1.$ (2)

An elementary result (see for ex. [5]) states that the composition
$I^{\alpha}D^{\alpha}$ is given by

$I^{\alpha}D^{\alpha}y(t)=I^{\alpha}I^{1-\alpha}y^{\prime}(t)=I^{1}y^{\prime}(t)=y(t)-y(0),$
(3)

which is a straightforward generalization of the fundamental theorem of
calculus. In what follows we are interested in finding an analogue of the
above relation when we allow for the time variable to take only a discrete
number of possible values. This is especially relevant in numerical analysis
where one constructs various schemes for approximately solving differential
equations with fractional derivatives. For example, finite difference or
finite element methods lead to a nonlocal recurrence relations that may be
inverted by the use of the fractional integral or the fractional version of
the discrete Grönwall’s lemma [7, 2, 11, 9]. Some results related to the same
family as ours can be found in [3] where authors consider the stability of the
L1 scheme on graded meshes. In particular, they give estimates for the case
with power function on the right-hand side of (3) in the discrete context. For
more information concerning fractional calculus and numerical methods we refer
the reader to [1, 5, 6].

##  2 Main result

In numerical analysis, some very common discretizations of the above operators
are constructed by simple quadratures. Fix a time step $h$ and define the mesh

$t_{n}=nh,$ (4)

where $h>0$ is the time step. For brevity we denote $y_{n}:=y(t_{n})$. Some
very useful discretizations of $I^{\alpha}$ and $D^{\alpha}$ can be
constructed by a simple rectangle quadrature applied to defining integrals. In
particular, we have

$I^{\alpha}y_{n}=J^{\alpha}y_{n}+Q_{n},\quad
D^{\alpha}=\delta^{\alpha}y_{n}+R_{n},$ (5)

where the discretizations $J^{\alpha}$ and $\delta^{\alpha}$ are defined by

$J^{\alpha}y_{n}=\frac{h^{\alpha}}{\Gamma(1+\alpha)}\sum_{i=0}^{n-1}b_{n-i}(\alpha)y_{i+1},\quad\delta^{\alpha}y_{n}=\frac{h^{-\alpha}}{\Gamma(2-\alpha)}\sum_{i=0}^{n-1}b_{n-i}(1-\alpha)(y_{i+1}-y_{i}),$
(6)

with weights

$b_{j}(\beta)=j^{\beta}-(j-1)^{\beta}.$ (7)

Furthermore, the remainders satisfy

$Q_{n}\sim-\frac{h}{\Gamma(1+\alpha)}\left(\frac{1}{2}t^{\alpha}+\frac{\zeta(-\alpha)}{n^{\alpha}}+\frac{\alpha}{12n}\right)t^{\alpha}y^{\prime}(\tau),\quad
R_{n}\sim-\frac{h^{2-\alpha}}{\Gamma(2-\alpha)}\zeta(\alpha-1)y^{\prime\prime}(\tau),\quad
n\rightarrow\infty,\,nh\rightarrow t,$ (8)

where $\zeta$ is Riemann-Zeta function and $\tau\in(0,t)$ is some number. The
bounds above are sharp (see [10]). In the literature this discretization of
the fractional derivative is called the L1 scheme (see [8]).

We are interested in discrete version of the composition formula (3), that is
we expect that $J^{\alpha}\delta^{\alpha}y_{n}=y_{n}-y_{0}+r_{n}$, with some
remainder $r_{n}$. In order to prove this result and find the form of $r_{n}$
we need to recall the Euler-Maclaurin formula written in the form that we need
(for a proof see for ex. [4]).

######  Theorem 1 (Euler-Maclaurin).

For $f\in C([0,m])$ we have

$\sum_{k=1}^{m}f(k)=\int_{1}^{m}f(x)dx+\frac{1}{2}\left(f(1)+f(m)\right)+\int_{1}^{m}f^{\prime}(x)P_{1}(x)dx,$
(9)

where $P_{1}(x)=B_{1}(x-\lfloor x\rfloor)$ is the periodized Bernoulli
polynomial $B_{1}(x)=x-\frac{1}{2}$.

We can now proceed to our main result. Notice that in the below the remainder
is of order $h^{\min(\alpha,1-\alpha)}$, that is, the exponent is always not
larger than $1/2$. This interesting fact comes from the singularity of
kernels. When composed, the fractional integral and Caputo derivative produce
a kernel that has two types of singularities at each endpoint of the
integration interval.

######  Theorem 2.

Let $\alpha\in(0,1)$ and $y\in C^{1}(0,T)$. For any fixed $t\in(0,T)$ with
$n\rightarrow\infty$ and $h=t/n$ we have

$J^{\alpha}\delta^{\alpha}y_{n}=y_{n}-y_{0}+r_{n}$ (10)

where the remainder $r_{n}$ satisfies

$|r_{n}|\leq
Ch^{\beta}\int_{0}^{t}(t-s)^{-\beta}|y^{\prime}(s)|ds,\quad\beta:=\min(\alpha,1-\alpha).$
(11)

and the constant $C$ depends only on $\alpha$ and $y$.

###### Proof.

Fix $n=1,...,N$ and start with writing the composition as

$\begin{split}J^{\alpha}\delta^{\alpha}y_{n}&=\frac{h^{\alpha}}{\Gamma(1+\alpha)}\sum_{i=0}^{n-1}b_{n-i}(\alpha)\delta^{\alpha}y_{i+1}\\\
&=\frac{1}{\Gamma(1+\alpha)\Gamma(2-\alpha)}\sum_{i=0}^{n-1}b_{n-i}(\alpha)\sum_{j=0}^{i}b_{i-j+1}(1-\alpha)(y_{j+1}-y_{j}).\end{split}$
(12)

Now, interchanging the order of summation we can write

$\begin{split}J^{\alpha}\delta^{\alpha}y_{n}&=\frac{1}{\Gamma(1+\alpha)\Gamma(2-\alpha)}\sum_{j=0}^{n-1}\left(\sum_{i=j}^{n-1}b_{n-i}(\alpha)b_{i-j+1}(1-\alpha)\right)(y_{j+1}-y_{j})\\\
&=\frac{1}{\Gamma(1+\alpha)\Gamma(2-\alpha)}\sum_{j=0}^{n-1}\left(\sum_{k=1}^{n-j}b_{n-j-k+1}(\alpha)b_{k}(1-\alpha)\right)(y_{j+1}-y_{j})\end{split}$
(13)

where in the second equality we have introduced a new summation variable
$k=i-j+1$ for the sum in the parenthesis. Put $m=n-j$ and denote the sum in
the parenthesis above

$S_{m}=\sum_{k=1}^{m}b_{m-k+1}(\alpha)b_{k}(1-\alpha).$ (14)

We would like to find the approximation of $S_{m}$ and to this end we use
Euler-Maclaurin formula (9) by using (7) and writing

$\begin{split}S_{m}&=\sum_{k=1}^{m}\left((m-k+1)^{\alpha}-(m-k)^{\alpha}\right)\left(k^{1-\alpha}-(k-1)^{1-\alpha}\right)\\\
&=m\sum_{k=1}^{m}\left(\left(1-\frac{k}{m}+\frac{1}{m}\right)^{\alpha}-\left(1-\frac{k}{m}\right)^{\alpha}\right)\left(\left(\frac{k}{m}\right)^{1-\alpha}-\left(\frac{k}{m}-\frac{1}{m}\right)^{1-\alpha}\right)\\\
&=m\sum_{k=1}^{m}f(k),\end{split}$ (15)

where we defined the function $f$. Now, using (9) we can write

$S_{m}=m\int_{1}^{m}f(x)dx+\frac{m}{2}\left(f(1)+f(m)\right)+m\int_{1}^{m}f^{\prime}(x)P_{1}(x)dx.$
(16)

We will estimate the above three components when $m$ is large. We quickly can
see that the easiest part is

$\begin{split}\frac{m}{2}\left(f(1)+f(m)\right)&=\frac{m}{2}\left(\left(\frac{1}{m}\right)^{1-\alpha}\left(1-\left(1-\frac{1}{m}\right)^{\alpha}\right)+\left(\frac{1}{m}\right)^{\alpha}\left(1-\left(1-\frac{1}{m}\right)^{1-\alpha}\right)\right)\\\
&=O\left(\frac{1}{m^{1-\alpha}}+\frac{1}{m^{\alpha}}\right),\quad
m\rightarrow\infty,\end{split}$ (17)

where the asymptotic behaviour follows from the Taylor expansion. Further, we
turn to the analysis of the first term in (16) for which we change the
integration variable $y=mx$

$m\int_{1}^{m}f(x)dx=m^{2}\int_{\frac{1}{m}}^{1}\left(\left(1-y+\frac{1}{m}\right)^{\alpha}-\left(1-y\right)^{\alpha}\right)\left(y^{1-\alpha}-\left(y-\frac{1}{m}\right)^{1-\alpha}\right)dy.$
(18)

Note that the above is invariant under the transformation $\alpha\mapsto
1-\alpha$ what can be seen by a substitution $y=1-x+1/m$. By inspection we can
see that the integrand converges to
$\alpha(1-\alpha)(1-y)^{\alpha-1}y^{-\alpha}$ as $m\rightarrow\infty$ (each
expression in parenthesis converges to its derivative), hence, by the Lebesgue
Monotone Convergence Theorem and the definition of Euler beta function we
conclude that

$m\int_{1}^{m}f(x)dx\rightarrow\alpha(1-\alpha)\int_{0}^{1}(1-y)^{\alpha-1}y^{-\alpha}dy=\Gamma(1+\alpha)\Gamma(2-\alpha)\quad\text{as}\quad
m\rightarrow\infty.$ (19)

This, together with (13) proves that in this limit, the discrete composition
verifies (3). However, we would like to investigate the rate of this
convergence to have a more useful formula. Due to singularity of the
integrand, we cannot expand it into Taylor series for large $m$ and then
integrate since such expansion would diverge. Since the behaviour of the
integrand is different for each terminal: $t=0$, $t=1$ we split it into two
terms

$m\int_{1}^{m}f(x)dx=m^{2}\left(\int_{\frac{1}{m}}^{\frac{1}{2}}+\int_{\frac{1}{2}}^{1}\right)=K_{0}+K_{1}.$
(20)

Because $1/m>0$, the integrand in $K_{0}$ does not have any singularities we
can safely expand for $1/m\rightarrow 0$

$K_{0}=\alpha(1-\alpha)\int_{\frac{1}{m}}^{\frac{1}{2}}(1-y)^{\alpha-1}y^{-\alpha}dy+\frac{\alpha(1-\alpha)}{2m}\int_{\frac{1}{m}}^{\frac{1}{2}}(1-y)^{\alpha-2}y^{-\alpha-1}(\alpha-y)dy+O\left(\frac{1}{m^{2}}\right).$
(21)

And the first integral above is

$\begin{split}\int_{\frac{1}{m}}^{\frac{1}{2}}(1-y)^{\alpha-1}y^{-\alpha}dy&=\int_{0}^{\frac{1}{2}}(1-y)^{\alpha-1}y^{-\alpha}dy-\int_{0}^{\frac{1}{m}}(1-y)^{\alpha-1}y^{-\alpha}dy\\\
&=\int_{0}^{\frac{1}{2}}(1-y)^{\alpha-1}y^{-\alpha}dy-\frac{1}{1-\alpha}\frac{1}{m^{1-\alpha}}+O\left(\frac{1}{m^{2-\alpha}}\right),\end{split}$
(22)

since the term $(1-y)^{\alpha-1}=O(1)$ as $m\rightarrow\infty$. By the same
argument, the second integral in $K_{0}$ can be expanded as follows

$\int_{\frac{1}{m}}^{\frac{1}{2}}(1-y)^{\alpha-2}y^{-\alpha-1}(\alpha-y)dy=m^{\alpha}+O(1),\quad
m\rightarrow\infty.$ (23)

Therefore,

$K_{0}=\alpha(1-\alpha)\int_{0}^{\frac{1}{2}}(1-y)^{\alpha-1}y^{-\alpha}dy+O\left(\frac{1}{m^{1-\alpha}}\right),\quad
m\rightarrow\infty.$ (24)

A similar analysis cannot be conducted for $K_{1}$ since we would arrive at a
divergence. A roundabout can be constructed by observing that

$\left(1-y+\frac{1}{m}\right)^{\alpha}-(1-y)^{\alpha}=\alpha\int_{0}^{\frac{1}{m}}(1-y+z)^{\alpha-1}dz.$
(25)

Then, by Tonelli’s theorem and noting that
$m(y^{1-\alpha}-(y-1/m)^{1-\alpha})=(1-\alpha)y^{-\alpha}+O(1/m)$ we have

$\begin{split}K_{1}&=\alpha
m\int_{0}^{\frac{1}{m}}\left(\int_{\frac{1}{2}}^{1}(1+z-y)^{\alpha-1}y^{-\alpha}dy\right)\left(1+O\left(\frac{1}{m}\right)\right)dz\\\
&=\alpha
m\int_{0}^{\frac{1}{m}}\left(\int_{\frac{1}{2(1+z)}}^{\frac{1}{1+z}}(1-u)^{\alpha-1}u^{-\alpha}du\right)\left(1+O\left(\frac{1}{m}\right)\right)dz,\end{split}$
(26)

where we have put $y=(1+z)u$ what removes the singularity from the integrand.
The integral in parenthesis can now be expanded for $z\rightarrow 0^{+}$
(since $z\in(0,1/m))$ yielding the leading order

$K_{1}=\alpha\int_{\frac{1}{2}}^{1}(1-u)^{\alpha-1}u^{-\alpha}du+O\left(\frac{1}{m^{\alpha}}\right),\quad
m\rightarrow\infty.$ (27)

Finally, we can go back to (20) to obtain

$m\int_{1}^{m}f(x)dx=\Gamma(1+\alpha)\Gamma(2-\alpha)+O\left(\frac{1}{m^{\alpha}}+\frac{1}{m^{1-\alpha}}\right),\quad
m\rightarrow\infty.$ (28)

The next step is to proceed with the remainder in (16). Its analysis is
similar to the above and we sketch only the most important details. By
calculating derivatives we see that the remainder has almost the same form as
before

$\begin{split}m\int_{1}^{m}&f^{\prime}(x)P_{1}(x)dx\\\ &=-\alpha
m\int_{\frac{1}{m}}^{1}\left(\left(1-y+\frac{1}{m}\right)^{\alpha-1}-\left(1-y\right)^{\alpha-1}\right)\left(y^{1-\alpha}-\left(y-\frac{1}{m}\right)^{1-\alpha}\right)P_{1}(my)dy\\\
&-(1-\alpha)m\int_{\frac{1}{m}}^{1}\left(\left(1-y+\frac{1}{m}\right)^{\alpha}-\left(1-y\right)^{\alpha}\right)\left(y^{-\alpha}-\left(y-\frac{1}{m}\right)^{-\alpha}\right)P_{1}(my)dy.\end{split}$
(29)

By counting powers and utilizing the fact that $P_{1}$ is bounded we can
specify the correct convergence order of the above. For example, the first
integral has two singularities for large $m$: $y=0$ and $y=1$. In the former
case we can expand in the Taylor series for $m\rightarrow\infty$ which will
consume two powers of $m$ leaving $y^{-\alpha}$ singularity. After integration
we obtain a term proportional to $m$ with an exponent $1-2+1-\alpha=-\alpha$.
On the other hand, to deal with the singularity at $y=1$ we use the trick with
Tonelli’s theorem to obtain a $O(m^{-\alpha})$ term. The second integral above
can be analysed in the same way with the difference that the singularity at
$y=1$ yields a $O(m^{-1})$ term (because the integrand after expansion in
$m\rightarrow\infty$ is integrable there), while the one at $y=0$ produces
$O(m^{\alpha-1})$. We see that the remainder introduces terms of the same
order, hence putting everything together, recalling that $m=n-j$, and
returning to (13) brings us to

$J^{\alpha}\delta^{\alpha}y_{n}=y_{n}-y_{0}+\sum_{j=0}^{n-1}\frac{c_{n-j}}{(n-j)^{\beta}}(y_{j+1}-y_{j}),\quad\beta=\min(\alpha,1-\alpha),$
(30)

with $c_{n-j}$ bounded by, say, $C$. Hence, for $n\rightarrow\infty$ with
$nh\rightarrow t\in(0,T)$ by the definition of Riemann integral we have

$\begin{split}\left|\sum_{j=0}^{n-1}\frac{c_{n-j}}{(n-j)^{\beta}}(y_{j+1}-y_{j})\right|&\leq
Ch\sum_{j=0}^{n-1}\frac{|y^{\prime}(\xi_{j})|}{(n-j)^{\beta}}\\\
&=C\frac{nh}{n^{1+\beta}}\sum_{j=0}^{n-1}\left(1-\frac{j}{n}\right)^{-\beta}|y^{\prime}(\xi_{j})|\sim
C\frac{t}{n^{\beta}}\int_{0}^{1}(1-x)^{-\beta}|y^{\prime}(xt)|dx\\\ &\sim
Ch^{\beta}\int_{0}^{t}(t-s)^{-\beta}|y^{\prime}(s)|ds.\end{split}$ (31)

with $\xi_{j}$ being a intermediate point and the new integration variable
$s=xt$. The proof is complete. ∎

We close the paper with several numerical verifications of the above theorem.
First, it is interesting to see how the asymptotic relation for the Euler-
Maclaurin’s integral (28) behaves. In Fig. 1 a loglog plot of

$\left|m\int_{1}^{m}f(x)dx-\Gamma(1+\alpha)\Gamma(2-\alpha)\right|,$ (32)

is depicted for increasing $m$ with $f$ defined in (15). Recall that the
integrand is invariant under the transformation $\alpha\mapsto 1-\alpha$ and
thus we consider only one exemplary case of $\alpha=0.75$. As can be seen, the
integral approaches its limit with the correct rate. Numerical simulations
with other values of $\alpha$ give very similar results.

Figure 1: The loglog plot of (32) for increasing $m$ and $\alpha=0.75$ (solid,
circles) and the reference line (dashed, triangles).

We can also verify the main relation of this paper, that is the discrete
composition (10). As an example we choose three test functions: a polynomial
$t^{3}$, $\sin t$, and a non-smooth function $|t-1/2|$. We compute the residue

$\rho=\left|J^{\alpha}\delta^{\alpha}y_{n}-y_{n}+y_{0}\right|,$ (33)

with the final time $t=1$, $\alpha=0.5$, and a decreasing sequence of steps
$h=1/n$. Results of calculations are presented in Fig. 2. Immediately we can
see that now, the convergence to zero is not monotone. Rather, for the
majority of chosen steps $h$ the results of computations cluster along or
parallel to the reference line $h^{0.5}$. This confirms the predicted order of
the remainder $r_{n}$ in (11). As can also be seen, the results are not
sensitive to the chosen test function even if it does not have a continuous
derivative. Interestingly, the residue for $h=10^{-i}$ for $i=1,2,3,...$ is
smaller than in the other cases. In each case, however, our estimate of the
remainder (11) is confirmed.

Figure 2: The loglog plot of (33) as a function of $h=1/n$ with a fixed value
of $\alpha=0.5$ for different test functions.

## Acknowledgement

Ł.P. has been supported by the National Science Centre, Poland (NCN) under the
grant Sonata Bis with a number NCN 2020/38/E/ST1/00153.

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# Vertical impact of a water jet on a hot plate: from a growing drop to spray
formation.

A. Goerlinger<EMAIL_ADDRESS>A. Germa  F. Zoueshtiagh
A.Duchesne  Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-
France, UMR 8520 - IEMN - Institut d’Electronique de Microélectronique et de
Nanotechnologie, F-59000 Lille, France

###### Abstract

In this article, we experimentally investigate the impact of a submillimetric
water jet on a horizontal surface heated well above the "static" Leidenfrost
temperature of water. We observe the transition from a regime where a single
drop grows at the impingement point to a regime of spray formation. The main
control parameter appears to be the jet Weber number ($\mathrm{We}$). The
first regime persists until $\mathrm{We}\lesssim 30$ whereas the spray
formation occurs for $\mathrm{We}\gtrsim 40$. Surprisingly, we found no
influence of the hot plate’s temperature on the reported phenomena. We
particularly focus on the second regime, where the liquid jet spreads on the
plate, forming a liquid sheet that eventually lifts off and breaks into
droplets. We characterized this regime by the radius $r_{c}$ of the liquid
sheet when it is still in contact with the plate and the angle of ejection
$\theta$ of the droplets. We further examine the ejected droplets by
characterizing their speed and sizes. Simple models are proposed to predict
the dependencies and order of magnitudes of $r_{c}$ and $\theta$. We also aim
to predict the critical Weber number at which the transition between the two
regimes occurs. Our models show reasonable agreement with our experimental
data. Finally, we compare the energy transferred from the jet to the droplets
with results reported in the literature for impacts on unheated surfaces,
finding a difference of nearly a factor 2.

##  I Introduction

When a drop of liquid is placed on a surface heated well above the boiling
point of the liquid, the drop’s lifetime becomes significantly greater than
what can be expected. This phenomenon is due to the formation of a vapor layer
between the droplet and the hot plate, hindering efficient heat transfer. This
phenomenon was initially observed by Leidenfrost in 1756 and has been named
after him ever since [1]. Because of its importance in many industrial
applications such as cooling systems [2, 3] or sprays [4], the Leidenfrost
effect has been the topic of extensive research in the case of a sessile drop
on a hot plate [5]. Other research delved into the dynamics of a drop with an
initial vertical velocity colliding with a hot plate [6, 7, 8]. For instance,
it was observed that the drop can spread upon impact [9, 10], elastically
bounce off the heated surface, [11], or generate a spray [12, 10, 13]. The
existence of different boiling regimes as well as impact patterns have also
been investigated [12, 14, 15]. However, unlike the impact of a drop, little
is known about jet impingement on a heated surface beyond the classical
industrial applications of quenching. Studies in this area are primarily
engineering-focused, centering on the heat transfer between the liquid jet and
the plate, and distinguish two main areas: a wet area near the jet’s impact
point, where the liquid is in contact with the hot plate, and a dry area far
from the impact point [16]. These investigations cover planar [17, 18, 19, 20,
21] or circular [22, 23, 24, 25] jets, considering horizontal as well as
inclined surfaces [26]. They mostly propose empirical modelings of the heat
transfer in a transient regime where the temperature of the surface is not
held constant, and very few of these works include a hydrodynamic model [23,
25]. To remedy this lack of hydrodynamic descriptions, we explore in this
study the scenario of a circular jet impacting a hot surface with a
temperature significantly exceeding the boiling point of the liquid. We will
demonstrate that upon impact, the jet can atomize, and under the tested
experimental conditions of a sub-millimetric jet and a constant temperature
maintained on the heated surface, the system exhibits quasi-steady behavior.
Special attention will be given to the hydrodynamic characterization of the
liquid as the jet impacts the hot plate and the conditions that lead to
droplet ejections. We explore the system behaviour for varying surface
temperatures, jet radii, and different flow rates. We employ simplified models
to derive scaling laws consistent with experimental observations and
characterize the transition that leads to this regime. Finally, we examine the
velocity and radius of the ejected droplets.

##  II Experimental apparatus

Our experimental apparatus is shown Fig. 1. All experiments were conducted
with deionized water. A syringe pump (Harvard Apparatus Pump 11 Elite) mounted
with a 50 mL glass syringe provided flows of desired flowrates, $Q$, through a
needle of radius $a$ ranging from 85 $\mathrm{\SIUnitSymbolMicro m}$ to
250$\mathrm{\SIUnitSymbolMicro m}$. Flow rates ranging from 0.075 mL/s to 1.55
mL/s were employed depending on the needle diameter. The Reynolds number in
the injector, defined as $\mathrm{Re}=Q/\nu a$ with $\nu$ the kinematic
viscosity of water, is estimated to range between 900 and 6700. The water jet
impacts a Duralumin disk with a thickness of 1.1 cm and a diameter of 16 cm,
positioned 2 cm below the needle. The Duralumin disk is heated using a
Fisherbrand Isotemp hot plate, enabling us to raise the plate’s temperature
from 300$\mathrm{\SIUnitSymbolCelsius}$ up to
500$\mathrm{\SIUnitSymbolCelsius}$ with a 600 W heating power. The surface
temperature $T$ is monitored via a K-type thermocouple with a 0.1 K accuracy.
Experiments feature a side-view via a Mikrotron Motionblitz Cube 4 camera
mounted with a Tamron SP AF28-75mm camera lens, and a top-view via a Phantom
Miro C211 camera mounted with a Sigma 105 mm 1:2.8 DG Macro camera lens.

Figure 1: Experimental apparatus. The thermocouple indicates the temperature
$T$ of the surface of the Duralumin disk. The syringe pump enables us to
control the flowrate $Q$ of the jet.

Side-view acquisitions are made at 150 FPS whereas top-view acquisitions are
made at much higher FPS, between 3300 FPS and 6800 FPS. Lighting is provided
by 110 W LED flood lights. Data were analyzed via Python, Matlab and ImageJ
softwares. We performed initial experiments to verify that the outcomes were
unaffected by the precise location of the impinging jet, thereby ensuring
uniform temperature distribution across the heated disk. Following each
experiment, we noted a small temporary decrease in the temperature of the disk
caused by the impact of the water jet. To ensure the plate’s temperature
returned to its initial value before subsequent jet impacts, various waiting
times between each run were tested, yielding consistent results. Finally, our
observations remain consistent throughout the entire experiment duration.
Moreover, the recorded temperature change on the disk never surpasses 2% of
the disk temperature setting. This indicates that our experiments are in a
steady-state condition.

##  III Results

We explore the behavior of the system by analyzing the impact of temperature
in relation to the Weber number, $\mathrm{We}$. This parameter, which compares
the inertia to surface tension effects, is defined as:

$\mathrm{We}=\frac{2\rho u_{\mathrm{jet}}^{2}a}{\gamma}=\frac{2\rho
Q^{2}}{\pi^{2}\gamma a^{3}}$ (1)

where $u_{\mathrm{jet}}$ is the jet’s flow velocity, $a$ is the radius of the
jet, $\rho=1000$ $\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$ is the density of
water at 20$\mathrm{\SIUnitSymbolCelsius}$ and $\gamma=73$
$\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$ is the surface tension of water at
20$\mathrm{\SIUnitSymbolCelsius}$. In the present experiments, $\mathrm{We}$
ranges from 30 to 660 with the impinging jet remaining continuous. Two primary
regimes, namely (i) and (ii) were identified in the experiments. Fig. 2 shows
the characteristic shapes of the jet upon its impact with the hot plate for
regimes (i) and (ii), respectively. Regime (i) is associated with
$\mathrm{We}\lesssim 30$ where a single drop stays under the jet and grows
until detaches from the jet and a new drop starts forming (see Fig. 2(a) and
Supplemental Material [27] M1). Here, the drop’s behavior under the jet
recalls the classical Leidenfrost effect. When $\mathrm{We}$ exceeds
approximately 40, the regime (ii) occurs, i.e. the jet striking the hot plate
results in the formation of a roughly circular liquid sheet that rapidly
breaks up into small droplets, which are ejected at a characteristic angle
$\theta$ (see Fig. 2(b) and Supplemental Material [27] M2). In this regime the
downward pressure from the impinging jet results in direct contact between the
water and the heated surface of the disk, thus inhibiting any Leidenfrost
effect near the point of jet impact. This was corroborated by the presence of
water vapor emission, accompanied by the distinctive sizzling noise associated
with boiling water. Furthermore, Karwa et al.’s formulation [25] for the
minimum temperature at which film boiling can occur beneath a jet indicates
840$\mathrm{\SIUnitSymbolCelsius}$, a value significantly higher than the
temperatures investigated in this study. For intermediary Weber numbers
($30\lesssim\mathrm{We}\lesssim 40$), we observe a transition state in which
features of both regimes can be observed. The emergence of these regimes in
relation to the Weber number ($\mathrm{We}$) remains mostly unaffected by
changes in plate temperature or jet radius.

(a) $\mathrm{We}\simeq 24$

(b) $\mathrm{We}\simeq 98$

Figure 2: Impact of a jet of water on a hot Duralumin disk for two different
Weber numbers of the jet. The radius of the jet is $a=195$
$\mathrm{\SIUnitSymbolMicro m}$, the surface temperature of the Duralumin disk
is $T=350$$\mathrm{\SIUnitSymbolCelsius}$. (a) Growing drop regime. A single
drop remains beneath the jet, enlarges, and eventually separates from the jet,
making room for a new drop to form. (b) Droplet ejection regime. A circular
liquid pad with a radius $r_{c}$ forms beneath the jet and fractures at its
periphery into small droplets that are expelled at an angle $\theta$ relative
to the horizontal.

In regime (ii) two characteristics, the ejection angle $\theta$ of the
droplets with respect to the horizontal and the radius $r_{c}$ of the liquid
sheet while it is still in contact with the hot plate, were investigated.
Figure 3 displays images of experiments conducted at a fixed jet radius but
with varying Weber numbers. It can be observed that the ejection angle
$\theta$ decreases as the Weber number increases. This seems somewhat
counterintuitive, as one might naturally expect that a stronger jet would lead
to a higher angle, as is typically observed when a drop impacts an unheated
surface [28]. Furthermore, the radius $r_{c}$ is observed to increase with the
Weber number.  Figure 3: Droplet ejection regime observed when a jet of radius
$a=195$ $\mathrm{\SIUnitSymbolMicro m}$ impact a Duralumin disk heated at
$T=350$$\mathrm{\SIUnitSymbolCelsius}$ for different Weber numbers
$\mathrm{We}$. (a) $\mathrm{We}=47$. (b) $\mathrm{We}=100$. (c)
$\mathrm{We}=190$. (d) $\mathrm{We}=560$. As $\mathrm{We}$ increases, one can
observe the ejection angle $\theta$ flattening and the radius $r_{c}$ of the
contact area widening. One can also note that the jet starts to become
unstable for Weber numbers above 200. Figure 4: Droplet ejection regime
observed when a jet of Weber number $\mathrm{We}\simeq 185$ impacts a
Duralumin disk heated at $T=350$$\mathrm{\SIUnitSymbolCelsius}$ for different
jet radii $a$. (a) $a=85$ $\mathrm{\SIUnitSymbolMicro m}$. (b) $a=125$
$\mathrm{\SIUnitSymbolMicro m}$. (c) $a=195$ $\mathrm{\SIUnitSymbolMicro m}$.
(d) $a=250$ $\mathrm{\SIUnitSymbolMicro m}$. As $a$ increases, one can observe
the radius $r_{c}$ of the contact area widening, whereas the ejection angle
$\theta$ does not seem to depend on the jet radius $a$. Additionally, $r_{c}$
also increases with the jet radius $a$, as demonstrated in Fig. 4, unlike the
ejection angle $\theta$, which remains constant with respect to $a$. It is
noteworthy that for Weber numbers exceeding approximately 200, instabilities
begin to emerge in the jet. We attribute these instabilities to corrugation
generated by the turbulent nature of the jet at high Reynolds number
($\mathrm{Re}\gtrsim 3000$). Experimental results show that both $\theta$ and
$r_{c}$ remain constant over time, provided that the jet flow rate remains
unchanged. This indicates that our experiments are indeed conducted in a
stationary regime. Regarding the variations in $\theta$, the results are
illustrated in Fig. 5, confirming a decrease in the ejection angle as the
Weber number increases. All experimental points collapse on the same curve
irrespective of the surface temperature $T$ and jet radius $a$, strongly
suggesting that the ejection angle $\theta$ is not influenced by these two
parameters. As for the radius $r_{c}$ of the contact area, we noticed its
increase with $\mathrm{We}$, a trend that aligns with intuition given the rise
in inertia with $\mathrm{We}$. We observed a power law dependency for $r_{c}$
with respect to We, and, similarly to $\theta$, the temperature $T$ of the
disk also has no impact on $r_{c}$ (Fig. 6(a)). However, this similarity ends
with the clear influence of the jet radius $a$ over $r_{c}$ (Fig. 6(b)) as
$r_{c}$ increases with $a$. Measurement uncertainties are estimated to be at
most 10% for $\theta$ and $r_{c}$. In the following section, we will discuss
these results.

Figure 5: Ejection angle $\theta$ plotted against the Weber number
$\mathrm{We}$ for various surface temperatures $T$ and jet radii $a$. The data
illustrates that $\theta$ solely varies with $\mathrm{We}$, without being
influenced by either $T$ or $a$.

(a) Fixed jet radius $a=195\ $\mathrm{\SIUnitSymbolMicro m}$$

(b) Fixed surface temperature $T=400$\mathrm{\SIUnitSymbolCelsius}$$

Figure 6: The radius $r_{c}$ of the contact area between the water and the
heated surface plotted against the Weber number for either a fixed (a) jet
radius $a=195\ $\mathrm{\SIUnitSymbolMicro m}$$ or a fixed (b) surface
temperature $T=400$\mathrm{\SIUnitSymbolCelsius}$$. The data indicates a
power-law relationship between $r_{c}$ and $\mathrm{We}$, with a clear
dependency on the jet radius.

##  IV Discussion

The upcoming sections will introduce simple models founded on three
distinctive regions that emerge experimentally and theoretically [16] with the
distance from the point of impact of the jet on the hot plate. We refer to
them as labelled in Fig. 7 as zone (I), (II) and (III). Hereafter, we first
describe their characteristics before providing models for each of them in the
subsequent sections. Zone (I) corresponds to the area close to the impact
point and is called wet area. The pressure imposed by the jet prevents the
vapor layer of the Leidenfrost effect from forming. Therefore, the water in
this area is in direct contact with the heated surface. The radius of this
area is $r_{c}$. Zone (II), the launching area, delineates the area where the
liquid layer separates from the heated surface, indicating the formation of an
underlying vapor layer. This region also influences the angle of droplet
ejection, which is one of the distinguishing features of zone (III). This last
zone, called the fragmentation area, witnesses the fragmentation of the layer
into tiny droplets similar to a Savart liquid sheet. The droplets are ejected
in all directions at an angle $\theta$ relative to the horizontal.

Figure 7: Division of the study into 3 areas: the wet area (zone I) where the
water is in contact with the heated surface, the launching area (zone II)
where a vapor layer lifts the water off the surface and the fragmentation area
(zone III) where the liquid sheet breaks into small droplets.

###  IV.1 Zone I: the wet area

As previously mentioned, in this region, the water maintains contact with the
disk even at plate temperatures reaching up to
500$\mathrm{\SIUnitSymbolCelsius}$, which is below the minimum film boiling
temperature of 840°C predicted by Karwa et al. [25]. The contact area being
characterized by $r_{c}$, we shall therefore endeavour to obtain an associated
scaling law with respect to experimental parameters. To this end, we consider
$r_{c}$ to be the distance the water must travel away from the jet to be
heated from its initial temperature of 20°C up to 100°C. We define the time
$\tau$ needed for the water to travel from the impact point of the jet to the
edge of the wet area, i.e. $r_{c}$. The characteristic velocity in this
simplified model is defined as $v=Q/\pi a^{2}$, assuming a fully inertial
regime, leading to the expression for the characteristic time $\tau$ as:

$\tau=\frac{r_{c}}{v}\sim\frac{\pi a^{2}}{Q}r_{c}.$ (2)

During this period, the water receives a heat flux $\varphi_{\mathrm{plate}}$
from the hot plate which can be expressed as

$\varphi_{\mathrm{plate}}=h_{\mathrm{heat}}\Delta T_{\mathrm{plate}}$ (3)

where $h_{\mathrm{heat}}$ is the heat transfer coefficient, expressed in
$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}\text{\,}{\mathrm{K}}^{-1}$, and $\Delta
T_{\mathrm{plate}}=T_{\mathrm{plate}}-T_{\mathrm{amb}}$ denotes the
temperature difference between the plate ($T_{\mathrm{plate}}$) and the water
($T_{\mathrm{amb}}=20$\mathrm{\SIUnitSymbolCelsius}$$), which corresponds to
the ambient temperature. Assuming that the heat transfer occurs only through
convection, we can estimate $h_{\mathrm{heat}}$ using a conductive model,
yielding

$h_{\mathrm{heat}}\sim\frac{\lambda}{e}$ (4)

where $\lambda=0.6$
$\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$ is the
thermal conductivity of liquid water and $e$ is the average thickness of the
water layer. Assuming $e$ remains constant across the entire wet area, the
volume $V$ of fluid that has flowed during the time $\tau$ is given by

$V=Q\tau\quad\text{and}\quad V=\pi r_{c}^{2}e.$ (5)

Using equation (2), these two expressions yield

$e\sim\frac{a^{2}}{r_{c}}.$ (6)

Finally, we calculate the energy absorbed by the water from the plate as it
travels from the point of impact of the jet to the end of the wetted region,
given by

$\mathcal{E}_{\mathrm{rec}}=\varphi_{\mathrm{plate}}\tau S$ (7)

where $S$ is the surface of the wetted area. Using equations (2), (3), (4) and
(6), one obtains

$\mathcal{E}_{\mathrm{rec}}\sim\frac{\pi\lambda r_{c}^{2}}{Q}S\Delta
T_{\mathrm{plate}}.$ (8)

To establish a scaling law for $r_{c}$, we presume that the energy
$\mathcal{E}_{\mathrm{rec}}$ transferred to the water is offset by the energy
required to elevate its temperature from its initial ambient temperature
$T_{\mathrm{amb}}=20$\mathrm{\SIUnitSymbolCelsius}$$ to its boiling
temperature $T_{\mathrm{boil}}=100$\mathrm{\SIUnitSymbolCelsius}$$,
facilitating the boiling of water and the formation of a vapor layer in the
launch area. An estimate of this energy can be obtained by:

$\mathcal{E}_{\mathrm{boil}}=\rho C_{p}V\Delta T_{\mathrm{water}}$ (9)

where $C_{p}=4187$
$\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$ is the
specific heat of water, $V$ is the volume of water that is boiled and $\Delta
T_{\mathrm{water}}=T_{\mathrm{boil}}-T_{\mathrm{amb}}$. The heated volume of
water $V$ can be estimated by the expression $V=Se$. Recalling the expression
for $e$ given by equation (6), equation (9) reads

$\mathcal{E}_{\mathrm{boil}}=\rho C_{p}\frac{a^{2}}{r_{c}}S\Delta
T_{\mathrm{water}}.$ (10)

We can now derive an expression for $r_{c}$ by equating equations (8) and
(10), resulting in:

$\mathcal{E}_{\mathrm{boil}}=\mathcal{E}_{\mathrm{rec}}\Rightarrow r_{c}\sim
Q^{1/3}a^{2/3}\left(\frac{\rho C_{p}\Delta
T_{\mathrm{water}}}{\pi\lambda\Delta T_{\mathrm{plate}}}\right)^{1/3}.$ (11)

We make one final assumption in this simplified model, regarding $\Delta
T_{\mathrm{plate}}$. Data presented in Fig. 6(a) indicates that $r_{c}$ does
not depend on the temperature of the heated surface. Therefore, we hypothesize
that the localized surface beneath the wet area (i.e., directly beneath the
jet) is sufficiently cooled to maintain a temperature equal to that of boiling
water, i.e., $100$\mathrm{\SIUnitSymbolCelsius}$$, while every other point on
the heated surface exceeds the prescribed temperature. One can note that this
assumption is consistent with the expression of the heat transfer coefficient
$h_{\mathrm{heat}}$ between the plate and the water as we use the thermal
conductivity $\lambda$ of liquid water. Therefore, we write $\Delta
T_{\mathrm{plate}}=\Delta T_{\mathrm{water}}=80^{\circ}$ $\mathrm{K}$.
Finally, equation (11) is re-expressed in terms of $\mathrm{We}$ by
substituting $Q$ using equation (1):

$r_{c}\sim\mathrm{We}^{1/6}a^{7/6}\left(\frac{C_{p}}{\lambda}\sqrt{\frac{\gamma\rho}{2}}\right)^{1/3}.$
(12)

As a result, we can establish the following scaling law:

$\frac{r_{c}}{Ka^{7/6}}\propto\mathrm{We}^{1/6},$ (13)

where $K=\left(\frac{C_{p}}{\lambda}\sqrt{\frac{\gamma\rho}{2}}\right)^{1/3}$.
This scaling law is compared to our experimental data in Figure 8. Our
measurements collapsed relatively well and the 1/6 exponent corresponds
closely with our data. Furthermore, equation (12) allows us to estimate
$r_{c}$. For instance, with $a=125$ $\mathrm{\SIUnitSymbolMicro m}$ and
$\mathrm{We}=60$, we obtain $r_{c}\sim 2$ $\mathrm{mm}$, which is within the
expected range compared to Fig. 8. Hence, our simple model yields reasonably
accurate results.

Figure 8: Scaling of $r_{c}$ with respect to $\mathrm{We}$ obtained via
equation (13), where $K$ is a constant derived from equation (13).

###  IV.2 Zone II: the launching area

In this area, the water layer detaches from the heated surface as a vapor
layer forms beneath it. The objective of this section is to characterize the
transition from the growing drop regime to the droplet ejection regime.
Furthermore, we aim to elucidate the observed value of the Weber number of
approximately 35 for the jet at this transition. To achieve this, we assume
the water temperature in the launching area is around
100$\mathrm{\SIUnitSymbolCelsius}$, and we introduce a local Weber number
$\mathrm{We}_{\mathrm{loc}}$, defined at the distance $r_{c}$ from the jet:

$\mathrm{We}_{\mathrm{loc}}=\frac{\rho_{100}u_{\mathrm{loc}}^{2}h_{\mathrm{loc}}}{\gamma_{100}}=\frac{\rho_{100}u^{2}(r_{c})h(r_{c})}{\gamma_{100}},$
(14)

where $\rho_{100}=960$ $\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$ is the density
of water at 100$\mathrm{\SIUnitSymbolCelsius}$, $\gamma_{100}=59$
$\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$ is the surface tension of water at
100$\mathrm{\SIUnitSymbolCelsius}$ and $u(r_{c})$ and $h(r_{c})$ are the local
speed and the depth, respectively, of the flow in the launching area. As a
Weber number compares the effect of inertia over the effect of surface
tension, the two observed regimes can be rationalized as follows. For
$\mathrm{We}_{\mathrm{loc}}<1$ the droplet lacks sufficient inertia to
overcome surface tension effects, causing the drop to grow as it is supplied
by the jet. Conversely, for $\mathrm{We}_{\mathrm{loc}}>1$ surface tension
effects are insufficient to counteract inertia, resulting in fragmentation of
the water-air interface into small droplets. Previous studies in the
literature have documented instances of instabilities in radially expanding
liquid sheets [29, 30]. These investigations revealed that the local Weber
number approached unity at the onset of instability. Consequently, we aim to
derive an expression for $\mathrm{We}_{\mathrm{loc}}$ and determine the Weber
number of the jet at the transition point, namely when
$\mathrm{We}_{\mathrm{loc}}=1$. To achieve this, we employ the expressions for
$u(r_{c})$ and $h(r_{c})$ as given by Watson’s equations [31], which describe
the radial spread of a liquid jet over a horizontal surface. Consequently, the
height of the water layer at $r_{c}$ can be expressed as [31]:

$h(r_{c})=\frac{2\pi^{2}\nu_{100}}{3\sqrt{3}}\frac{r_{c}^{3}+\ell^{3}}{Qr_{c}},$
(15)

where $\ell=0.567a\mathrm{Re}^{1/3}$ represents a length derived from
computation, $\mathrm{Re}=Q/a\nu_{100}$ denotes the Reynolds number of the
flow, and $\nu_{100}=0.3\cdot 10^{-6}$
${\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$ stands for the kinematic
viscosity of liquid water at 100$\mathrm{\SIUnitSymbolCelsius}$. One can note
that in our case, $\ell\sim 2\cdot 10^{-3}$ $\mathrm{m}$ and therefore is not
negligible at all compared to $r_{c}\sim 10^{-3}$ $\mathrm{m}$. The local flow
velocity is then determined by the conservation of mass:

$u(r_{c})=\frac{Q}{2\pi
r_{c}h(r_{c})}=\frac{3\sqrt{3}}{4\pi^{3}\nu_{100}}\frac{Q^{2}}{r_{c}^{3}+\ell^{3}}.$
(16)

Equation (14) can then be rewritten as

$\mathrm{We}_{\mathrm{loc}}=\frac{3\sqrt{3}}{8\pi^{4}}\frac{\rho_{100}}{\gamma_{100}\nu_{100}}\frac{Q^{3}}{r_{c}(r_{c}^{3}+\ell^{3})}.$
(17)

By replacing $r_{c}$ by its expression in eq. (11), one obtains

$\mathrm{We}_{\mathrm{loc}}=\frac{3\sqrt{3}}{8\pi^{4}}\frac{\rho_{100}}{\gamma_{100}\nu_{100}}\frac{\left(\dfrac{\pi\lambda}{\rho
C_{p}}\right)^{1/3}}{\dfrac{\rho
C_{p}}{\pi\lambda}+\dfrac{0.567^{3}}{\nu_{100}}}Q^{5/3}a^{-8/3}.$ (18)

We can compute $Q$ with respect to $\mathrm{We}_{\mathrm{loc}}$ and $a$ using
equation (18), and then use equation (1) to express the Weber number of the
jet in relation to $\mathrm{We}_{\mathrm{loc}}$ as:

$\mathrm{We}=\left[\frac{8\pi^{4}}{3\sqrt{3}}\frac{\gamma_{100}\nu_{100}}{\rho_{100}}\left(\frac{\rho
C_{p}}{\pi\lambda}\right)^{1/3}\left(\frac{\rho
C_{p}}{\pi\lambda}+\frac{0.567^{3}}{\nu_{100}}\right)\right]^{6/5}\frac{2\rho}{\pi\gamma}a^{1/5}\mathrm{We}_{\mathrm{loc}}^{6/5}.$
(19)

Given that $\mathrm{We}_{\mathrm{loc,trans}}=1$ at the transition point, one
can infer from this equation the corresponding value of
$\mathrm{We}_{\mathrm{trans}}$. One can note a very slight dependency on $a$,
which is not experimentally observable given the range of jet radii we
explored. The absence of the hot plate temperature on
$\mathrm{We}_{\mathrm{trans}}$ is also consistent with our observations. For a
needle of radius $a=195$ $\mathrm{\SIUnitSymbolMicro m}$, equation (19) yields

$\mathrm{We}_{\mathrm{trans,model}}\sim 515.$ (20)

This value far exceeds our experimental observations. However, if we use
experimental values for $r_{c}$ ($\sim 0.8$ mm) instead of relation (11), we
recover

$\mathrm{We}_{\mathrm{trans,meas}}\sim 20,$ (21)

which is actually quite close to the expected value of 35.

###  IV.3 Zone III: the fragmentation area

In this zone, the liquid sheet, upon detachment from the heated surface in the
launching region, disintegrates into small droplets. This region is
characterized by an angle of ejection, denoted as $\theta$, which specifies
the trajectory of the droplets relative to the horizontal plane. To elucidate
the ejection angle, one may revisit the launching area and examine the
influence of the following two forces acting on a droplet (refer to Fig. 7).
The first force, stemming from the vaporization of water into the vapor layer
in the launching area, induces an overpressure, lifting the water layer from
the heated surface and resulting in an upward force denoted as
$F_{\mathrm{up}}$. As the thickness of the liquid sheet is a few dozen
micrometres, this force is likely to have a substantive impact on the sheet.
The second force, attributed to the inertia of the water, creates a radial
pressure that propels the water outward from the jet, thereby generating a
radial outward force denoted as $F_{\mathrm{rad}}$. The interplay of these two
forces leads to a diagonal trajectory and an ejection angle $\theta$ for the
droplets, which can be formulated as

$\tan(\theta)=\frac{F_{\mathrm{up}}}{F_{\mathrm{rad}}}$ (22)

We can estimate $F_{\mathrm{rad}}$ by considering that an outward radial
pressure driven by inertia is applied on the inner vertical section of the
water layer in the launching area. Therefore,

$F_{\mathrm{rad}}=\frac{1}{2}\rho_{100}u(r_{c})^{2}\times 2\pi r_{c}h(r_{c}).$
(23)

By applying the conservation of mass, $Q=2\pi r_{c}h(r_{c})u(r_{c})$, and
substituting the expression of $u(r_{c})$ from Eq. (16) into equation (23), we
derive:

$F_{\mathrm{rad}}=\frac{3\sqrt{3}}{8\pi^{3}}\frac{\rho_{100}}{\nu_{100}}\frac{Q^{3}}{r_{c}^{3}+\ell^{3}}$
(24)

For the upward force $F_{\mathrm{up}}$, we can reasonably assume that an
overpressure $\Delta P$ acts on the bottom surface of the water layer in the
launching area. By defining the width of the launching area as $\delta_{\ell}$
(see Fig. 7), the force $F_{\mathrm{up}}$ can be expressed as:

$F_{\mathrm{up}}=\Delta P\pi\left[(r_{c}+\delta_{\ell})^{2}-r_{c}^{2}\right]$
(25)

One can approximate $\Delta P$ using an analogy with Celestini’s study [32],
which investigates the take-off of Leidenfrost drops and provides the
following expression for the overpressure within the vapor layer:

$\Delta P\sim L^{2}\frac{\nu_{v}\lambda_{v}\Delta
T_{v}}{h_{v}^{4}\mathcal{L}},$ (26)

where $L$ is a characteristic length of the vapor layer, $\nu_{v}=2\cdot
10^{-5}$ ${\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$ is the viscosity of
water vapor at 100$\mathrm{\SIUnitSymbolCelsius}$ and ${\lambda_{v}=0.03}$
$\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$ its thermal
conductivity at 100$\mathrm{\SIUnitSymbolCelsius}$, $\mathcal{L}=2.26\cdot
10^{6}$ $\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}$ is the latent heat of
vaporization of water, $\Delta T_{v}=T-T_{\mathrm{boil}}$ is the difference
between the temperature of the hot plate in the launching area and the
temperature at the top of the water layer (100$\mathrm{\SIUnitSymbolCelsius}$)
and $h_{v}$ is the thickness of the vapor layer in the launching area. As
$L^{2}$ represents a characteristic surface area for the vapor layer, we
assume it can be substituted by the surface area of the vapor layer in the
launching area, given by
$\pi\left[(r_{c}+\delta_{\ell})^{2}-r_{c}^{2}\right]$. The upward force can
thus be expressed as

$F_{\mathrm{up}}\sim\pi^{2}\left[(r_{c}+\delta_{\ell})^{2}-r_{c}^{2}\right]^{2}\frac{\nu_{v}\lambda_{v}\Delta
T_{v}}{h_{v}^{4}\mathcal{L}}$ (27)

Combining equations (22), (24) and (27) yields

$\tan(\theta)\sim\frac{8\pi^{5}}{3\sqrt{3}}\frac{\nu_{100}\nu_{v}\lambda_{v}\Delta
T_{v}}{\rho_{100}h_{v}^{4}\mathcal{L}}\frac{\delta_{\ell}^{2}\left(2r_{c}+\delta_{\ell}\right)^{2}(r_{c}^{3}+\ell^{3})}{Q^{3}}$
(28)

Experimental observations indicate that $\delta_{\ell}\ll r_{c}$.
Additionally, we can estimate the height of the vapor layer by writing
$\tan(\theta)\sim h_{v}/\delta_{\ell}$. Consequently, equation (28) becomes

$\tan^{5}(\theta)\sim\frac{32\pi^{5}}{3\sqrt{3}}\frac{\nu_{100}\nu_{v}\lambda_{v}\Delta
T_{v}}{\rho_{100}\mathcal{L}}\frac{r_{c}^{2}(r_{c}^{3}+\ell^{3})}{\delta_{\ell}^{2}Q^{3}}$
(29)

We can derive from this equation an approximate value of $\tan(\theta)$. We
estimate $\delta_{\ell}$ to be around 0.1 mm, a reasonable approximation based
on our experimental findings. Considering $a=195$ $\mathrm{\SIUnitSymbolMicro
m}$, $\mathrm{We}\sim 50$, and utilizing equations (1) and (12) as well as the
fact that $\ell=0.567\nu_{100}^{-1/3}a^{2/3}Q^{1/3}$, we obtain
$\tan(\theta)\approx 0.52$, which aligns with the expected order of magnitude
when compared with our measured values presented in Fig. 9. However, as
$\delta_{\ell}$ is not accessible experimentally, we need to obtain a relation
for $\delta_{\ell}$. To do so, we assume that only inertia has an impact on
the water layer in the launching area. The water layer is therefore equivalent
to a Savart sheet in this area, and as $\mathrm{We}<1000$ in our experiments,
we can write $\delta_{\ell}\propto\mathrm{We}$ [33]. Using this estimation
along with equations (1) and (12), one obtains:

$\tan(\theta)\propto a^{4/15}\mathrm{We}^{-8/15}\Delta T_{v}^{1/5}$ (30)

This scaling law is compared to experimental data in Figure 9. The
experimental results suggest a power dependency of $\mathrm{We}$ with an
exponent of $-8/15$, akin to the model, particularly when examining data for a
constant jet diameter. Nevertheless, the collective dataset appears to exhibit
a slight deviation from the anticipated power of $-8/15$. This discrepancy can
be attributed to a nuanced reliance on both $a$ and $T$ (via $\Delta T_{v}$)
in the model, which is not accounted for in the plotted data. Our proposed
explanation for the observed weak dependency is that the plate is locally
cooled to such an extent by the jet that the initial temperature near the jet
impact point becomes irrelevant.

Figure 9: Scaling of $\tan(\theta)$ with respect to $\mathrm{We}$ obtained via
equation (30). The predicted $-8/15$ exponent for $\mathrm{We}$ is in a
relatively good agreement with our experimental measurements.

Having established the scaling laws for different zones, we now turn our
attention to the energy contained in droplets during atomization. This will
enable us to estimate the efficiency of our system in breaking up a liquid
sheet into droplets and to compare our results with similar data in the
literature that does not involve heating, such as the Savart sheet reported by
Clanet et al. [33]. To achieve this, we need to measure the size and velocity
of the ejected droplets, which is the focus of the next section.

##  V Energy of atomization

In this section, we aim to characterize the droplets emitted from the
fragmentation area. Specifically, we explore how experimental parameters
impact their properties, including velocities and size. A Matlab code
employing a tracking algorithm [34] has been developed and implemented to
extract the trajectories and horizontal velocities of the droplets from the
recorded videos. Figure 10(a) shows a typical velocity distribution in the
experiments. Fitting these distributions with a Gaussian curve allows us to
determine the average horizontal velocity for the droplets. Given that we also
have information about the ejection angle of the droplets (see Fig. 5), we are
able to calculate the average magnitude of the velocity of the droplets,
denoted as $v_{\mathrm{drop}}$. Figure 10(b) illustrates $v_{\mathrm{drop}}$
plotted against $u(r_{c})-u(r_{c}^{*})$, where $u(r_{c}^{*})$ represents how
much speed the droplets lost in order to overcome surface tension. To compute
$u(r_{c}^{*})$, we set $\mathrm{We}_{\mathrm{loc}}=1$ in equation (14). We
then approximate the radius of contact $r_{c}^{*}$ and the flowrate $Q^{*}$ at
the transition between regimes (i) and (ii) by using our measurements for the
lowest Weber number at which regime (ii) is observed in figure 6(b). Our data
then show that $v_{\mathrm{drop}}$ matches quite well $u(r_{c})-u(r_{c}^{*})$,
indicating that the inertia of the droplets is indeed what remains of the
inertia of the liquid sheet after surface tension is overcome. One can notice
that neither $T$ nor $a$ affect $v_{\mathrm{drop}}$.

(a) Horizontal speed distribution.

(b) Average droplet speed.

Figure 10: Horizontal speed of ejected droplets for different jet radii $a$
and plate temperatures $T$. (a) Example of an observed distribution for the
measured horizontal speed of the droplets for $a=195$
$\mathrm{\SIUnitSymbolMicro m}$, $T=300$$\mathrm{\SIUnitSymbolCelsius}$ and
$\mathrm{We}=79$. (b) Average droplet speed $v_{\mathrm{drop}}$ with respect
to $u(r_{c})-u(r_{c}^{*})$, where $u(r_{c}^{*})$ is the speed of the flow in
the liquid sheet when $\mathrm{We}_{\mathrm{loc}}=1$. A linear dependence can
be observed between the two parameters. The black dotted line indicates a
slope of 1.

Our investigation also covered the distribution of the droplet radius. For
Weber numbers close to the transition, the radius distribution of the droplets
is found bidisperse (Fig. 11(a)) with two distinct mean radii. The release of
small satellite droplets in addition to the bigger main droplets when a liquid
bridge breaks [35, 36] is a likely explanation for this observation. However,
as the Weber number increases (Fig. 11(b)), the distribution of the radii of
the droplets becomes monodisperse and only one mean radius $r_{\mathrm{drop}}$
emerges. One possible explanation for this change in distribution is that the
conditions conducive to the formation of satellite droplets vanish or simply
they become too small to be reliably detected by our code. Henceforth, we will
disregard these satellite droplets and focus solely on the largest mean radius
$r_{\mathrm{drop}}$ in the case of bidisperse distributions. It is also
noteworthy that the droplets exhibit a radius on the order of
$\mathcal{O}(100)$ $\mu$m. This finding aligns with measurements reported in
the literature by Oulded Taled Salah et al. [37] when a Savart sheet breaks.

(a) $\mathrm{We}\simeq 38$

(b) $\mathrm{We}\simeq 68$

Figure 11: Radius distribution of the ejected droplets when a jet of radius
$a=250$ $\mathrm{\SIUnitSymbolMicro m}$ impacts the Plexiglas plate heated at
$T=400$$\mathrm{\SIUnitSymbolCelsius}$ for two different Weber numbers. (a)
For $\mathrm{We}\simeq 38$, we observe a bidisperse distribution in the radii
of the ejected droplets. (b) For $\mathrm{We}\simeq 68$, the distribution
becomes monodisperse. Insets show a top-view of the ejected droplets. White
scale bar is 1 cm.

To get some insights in the physics underlying the size of emerging droplets,
we derived the film thickness, $h(r_{c})$, at $r_{c}$ using Eq. 15, and
plotted the ratio $r_{\mathrm{drop}}/h(r_{c})$ as a function of $\mathrm{We}$
in Fig. 12. The results show that the ratio $r_{\mathrm{drop}}/h(r_{c})$
remains nearly constant with $\mathrm{We}$ even though we saw earlier that
both $r_{\mathrm{drop}}$ and $h(r_{c})$ vary with $\mathrm{We}$. Furthermore,
the plate temperature $T$ seems to have little influence on this ratio, while
the effect of the radius $a$ of the impacting jet appears unclear.

Figure 12: Ratio between $v_{\mathrm{drop}}$ and $h(r_{c})$ with respect to
the Weber number, $\mathrm{We}$.

While the previous scaling of $r_{\text{drop}}$ by $h(r_{c})$ appears to be
effective in providing insights into the system behavior, it is worth noting
another scaling reported in the literature. Clanet et al. [33]
nondimensionalized $r_{\text{drop}}$ by $a$ and found the following scaling:

$\left(\frac{r_{\mathrm{drop}}}{a}\right)^{3}\propto\mathrm{We}^{-1}.$ (31)

Our data show a comparable scaling behavior, albeit with a factor of 5 to 10
smaller than that reported by Clanet et al. This difference is anticipated
because in our system, the heat addition alters and diminishes the effect of
surface tension, leading to the production of smaller droplets for the same
Weber number, i.e., the same input energy. Moreover, in Clanet et al.’s
experiment, the liquid-solid contact area is at least five times bigger than
in our case. Therefore, the liquid flow experiences less friction in our
experiment and retain a considerable speed when the liquid sheet breaks down.
However, as the fragmentation must occur when $\mathrm{We}_{\mathrm{loc}}\sim
1$, the liquid sheet’s thickness must consequently be smaller in our
experiments compared to those of Clanet et al., which leads to the production
of smaller droplets. To assess the energy taken by the droplets, we followed a
similar approach to Clanet et al. [33] and considered the energy carried away
by the droplets per unit of time as:

$\mathcal{E}_{\mathrm{drop}}=n\left(\frac{2}{3}\pi\rho_{100}r_{\mathrm{drop}}^{3}v_{\mathrm{drop}}^{2}+4\pi\gamma_{100}r_{\mathrm{drop}}^{2}\right),$
(32)

where $n$ is the number of droplets emitted per unit of time. Using the
conservation of mass, one finds

$n=\frac{3}{4}\frac{a^{2}}{r_{\mathrm{drop}}^{3}}\sqrt{\frac{\gamma\mathrm{We}}{2\rho
a}}.$ (33)

Therefore, (32) becomes

$\mathcal{E}_{\mathrm{drop}}=\frac{3}{4}\frac{a^{2}}{r_{\mathrm{drop}}^{3}}\sqrt{\frac{\gamma\mathrm{We}}{2\rho
a}}\left(\frac{2}{3}\pi\rho_{100}r_{\mathrm{drop}}^{3}v_{\mathrm{drop}}^{2}+4\pi\gamma_{100}r_{\mathrm{drop}}^{2}\right).$
(34)

As for the initial energy per unit of time of the jet, one can write

$\mathcal{E}_{\mathrm{jet}}=\frac{\pi}{2}\rho
a^{2}\left(\frac{\gamma\mathrm{We}}{2\rho a}\right)^{3/2}.$ (35)

The two energies per unit of time are plotted against each other in Fig. 13.
In this figure, the black dashed line corresponds to the average experimental
ratio reported by Clanet et al. [33]. The slope of this line represents the
amount of energy transferred from the jet to the droplets or can alternatively
be viewed as the energy dissipation that has taken place, as stated by Clanet
et al. [33]. We prefer to discuss it as an energy transfer where our data
provide an indication of how much input energy per unit time
$\mathcal{E}_{\mathrm{jet}}$ is transferred to the energy carried by the
droplets $\mathcal{E}_{\mathrm{drop}}$. When comparing our data to the black
dashed line, one can observe that we measure an energy transfer exceeding the
10% reported by Clanet et al. [33]. This enhancement in energy transfer
efficiency is likely attributed to the heat absorbed by the liquid sheet,
which modifies the surface tension, thereby facilitating the detachment, and
hence, ejection of droplets from the sheet. Of course, this improvement comes
at the expense of heat addition to the system, which reduces its overall
efficiency. The energy lost in our system can be attributed to heat
dissipation, viscous effects, and losses due to evaporation and boiling.
Nevertheless, despite these losses, the present system remains of interest if
the goal is to dispense droplets on-demand with smaller sizes than what
classical systems can achieve, and all this with a fairly simple setup of a
liquid impacting a heated surface. One can also wonder what happens for high
values of $\mathcal{E}_{\mathrm{jet}}$, where our data seemingly matches the
predictions of Clanet et al. [33]. In particular, it may indicate that heat
addition to the system has no influence on fast droplets. A more thorough
study is however needed to fully address this question.

Figure 13: Energy taken away by the droplets per unit of time compared to the
initial energy of the jet per unit of time. The black dashed line correspond
to a 10% ratio between the two quantities as reported by Clanet et al. [33]
result.

##  VI Conclusion

We explored the effect of submillimetric liquid jets impacting a heated
surface above its Leidenfrost temperature. Two distinct regimes emerged based
on the Weber number ($\mathrm{We}$). For $\mathrm{We}\leq 30$ the experiments
showed the formation of a large centrimetric drops whereas for
$\mathrm{We}\geq 40$ the impact yielded to atomization of the jet, leading to
distinct physical characteristics such as ejection angle and droplet size. The
study mainly focused on characterising the regime showing atomization. While
the system displayed weak sensitivity to the temperature of the heated plate
and the jet size, simple models successfully explained various scaling
characteristics of the system behaviour. Moreover, these models offered
explanations for the transition between the regimes, attributed to the
interplay between liquid inertia and surface tension. We also conducted
detailed measurements of droplet size under various experimental conditions.
Our findings indicate an enhanced energy transfer from the jet to droplets
compared to systems without surface heating, as supported by existing models
and literature data. This system holds promise for practical applications,
enabling precise droplet dispensing with smaller sizes than traditional
methods, using a relatively straightforward setup of liquid impacting a heated
surface.

## Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la
Recherche (ANR), under Grant No. ANR-21-CE30-0014 (project IJET) and thank
Loïc Lam for his technical assistance.

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