[1]Ehsan Ataie

1]Department of Computer Engineering, University of Mazandaran, Babolsar, Iran

# Kubernetes in Action: Exploring the Performance of Kubernetes Distributions
in the Cloud

Hossein Aqasizade<EMAIL_ADDRESS><EMAIL_ADDRESS>Mostafa Bastam
<EMAIL_ADDRESS>[

###### Abstract

Kubernetes has emerged as a leading open-source platform for container
orchestration, allowing organizations to efficiently manage and deploy
containerized applications at scale. This paper investigates the performance
of four Kubernetes distributions, namely Kubeadm, K3s, MicroK8s, and K0s when
running OpenFaaS as a containerized service on a cluster of computing nodes on
CloudLab. For this purpose, experiments are conducted to examine the
performance of two virtualization modes, namely HVM and PV, supported by Xen
as the underlying hypervisor. Moreover, two container runtimes that are
integrated with Kubernetes, namely Docker, and Containerd, are examined to
assess their performance on both disk-intensive and CPU-intensive workloads.
After determining the appropriate underlying Xen mode and container runtime,
the Kubernetes distributions are set up and their performance is measured
using various metrics, such as request rate, CPU utilization, and scaling
behavior.

###### keywords:

Kubernetes, Serverless Computing, Performance Evaluation, Benchmarking

##  1 Introduction

Cloud computing is transforming a large part of the IT industry by improving
the attractiveness of software as a service, as well as changing how hardware
is designed and purchased [1]. It offers unlimited storage and processing
power, as well as unlimited access to resources [2]. Virtualization is a
crucial technology allowing for efficient hardware resource sharing,
encompassing two main types: virtual machines (VMs), which emulate complete
computer systems [3], and containers, lightweight units that share the host
operation system kernel for more streamlined deployment and scalability [4,
5].

CloudLab [6], a cloud computing testbed for research and education, provides
an infrastructure for running other platforms, including clusters of virtual
machines and containers. Xen’s default virtualization mode is hardware virtual
machine (HVM), which is also supported by CloudLab. HVM virtualization allows
virtual machines to operate as virtual hardware devices, allowing a wide range
of operating systems to be executed concurrently as guest operating systems.
To run paravirtualization (PV) mode on CloudLab, one needs to provide a script
to define a new profile. With PV, the guest operating system is modified to be
aware of its virtualized environment. This mode often results in better
performance as it reduces the overhead of virtualization. As a result of these
two approaches, users can tailor the benefits to meet their specific
requirements.

Container orchestration can be provided by Kubernetes [7], Docker Swarm [8],
or Apache Mesos [9, 10]. Kubernetes is widely recognized as a clustering
platform to run and manage containerized services such as OpenFaaS. Figure 1
illustrates the Kubernetes architecture. Kubernetes comes in several
distributions, including Kubeadm [7], Kubespray [11], K3s [12], Rancher [13],
MicroK8s [14], and K0s [15]. It includes a container runtime that manages
containers’ life cycles, including launching, stopping, and monitoring them.
The operation of containers can be affected when container runtime use user-
space kernels to enforce isolation [16]. Therefore, container runtime
performance is important as it ensures the proper and efficient execution of
containers, guarantees system performance, and enables a balanced allocation
of computing, storage, and network resources. Figure 2 shows the architectures
of Docker and Containerd [17], the two most popular container runtimes.

Figure 1: Architecture of Kubernetes Figure 2: Containerd vs. Docker runtimes

Function-as-a-Service (FaaS) is a form of serverless computing that abstracts
away the underlying infrastructure management, enabling developers to focus on
writing and deploying code [18]. Such platforms are usually integrated with
containerization technologies, which let developers encapsulate their
dependencies in containers. A FaaS platform manages resource provisioning and
function execution automatically based on a developer’s function code [19].
There are many popular FaaS platforms, such as Knative [20], Apache OpenWhisk
[21], Fission [22], Nuclio [23], but OpenFaaS [24] is one of the most popular
due to its github stars (24.3k) [25] and its popularity among researchers [26,
27, 28, 29, 30, 31, 32]. The architecture of OpenFaaS is shown in Figure 3.

Figure 3: Architecture of OpenFaaS

This study evaluates the performance of four Kubernetes distributions, namely
Kubeadm, K3s, MicroK8s, and K0s, from various aspects. Since related works
have shown that Xen is an efficient hypervisor for virtualization [33, 34, 35,
36], this hypervisor is adopted to provide the platform for running
Kubernetes. At first, we evaluate the performance of both HVM and PV modes to
check which mode provides better performance. For this purpose, a new profile
is added to CloudLab to provide Xen-PV mode on this testbed. After specifying
the appropriate mode, Docker and Containerd, two well-known container runtimes
that can be integrated with Kubernetes, are investigated. To this end,
Kubernetes has been modified to also use Docker as an alternative to
Containerd, so that we can evaluate both runtimes based on disk and CPU
performance. To do this, we run Sysbench to compare the runtimes in terms of
disk performance when MySQL, as a disk-intensive application, is deployed on
Kubernetes. Then, OpenFaaS is run as a compute-intensive application to
compare runtime environments in terms of CPU performance. When the underlying
Xen mode and the container runtime of Kubernetes are adopted, we investigate
the performance of the Kubernetes distributions using various metrics.

The remainder of this article is organized as follows: Section 2 provides an
overview of the research conducted in this area. Section 3 provides the
research methodology and the steps of the experiments. Section 4 outlines the
experimental setup, including hardware and software configurations and the
benchmarks. Section 5 summarizes the experimental results, including
comparisons between Xen virtualization modes, container runtimes, and
performance behavior of Kubernetes distributions. Finally, Section 6 draws the
final remarks and outlines some future research directions.

##  2 Related Work

Recent research indicates that virtualization technologies, containerization,
container orchestration, and serverless computing have been topics of
considerable interest to many researchers. In this section, an investigation
of studies on Kubernetes as well as serverless computing platforms is
conducted. The literature review aims to be concise, with a focus on research
studies relevant to this article.

Koziolek et al. [37] have compared MicroK8s, K3s, K0s, and MicroShift as
lightweight Kubernetes distributions. The authors found that K3s and K0s
marginally showed the highest control plane throughput in stress scenarios,
while MicroShift achieved the highest data plane throughput. Kjorveziroski et
al. [38] have assessed the performance of three Kubernetes distributions:
Kubespray, K3s, and MicroK8s using serverless benchmarks on the OpenFaaS
platform. They found that lightweight distributions like K3s and MicroK8s
offered better performance than Kubespray, particularly in reducing deployment
time and complexity. Despite some performance variations in extreme load
tests, K3s and MicroK8s generally performed comparably. Costa et al. [39] have
investigated software aging issues within Kubernetes, examining its behavior
in a digital twin cloud infrastructure for UAM-ODT systems. Through
accelerated lifespan experiments, utilizing Nginx and K3S tools, the authors
monitored resource usage and application performance. Notably, they detected
memory leaks persisting beyond cluster halts, shedding light on software aging
within Kubernetes-based infrastructures.

Mohanty et al. [40] have examined open-source serverless platforms, including
Kubeless, OpenFaaS, Fission, and OpenWhisk. All these platforms, except
OpenWhisk, were deployed in a Kubernetes cluster. The authors assessed the
features and architectures of each platform and compared them based on
response time and success rate under different load conditions. They found
that the Kubeless framework exhibited more stable performance than Fission and
OpenFaaS under various conditions. Palade et al. [41] have evaluated four
open-source serverless platforms, namely Kubeless, OpenWhisk, OpenFaaS, and
Knative in an edge computing environment for Internet of Things (IoT)
applications. The authors utilized JMeter to assess response time, operational
capacity, and success rates of deployed functions under various workloads. As
a result, Kubeless demonstrated superior performance in terms of response time
and operational capacity to other platforms.

Balla et al. [42] have conducted a comprehensive evaluation of the language
runtime performances of Python, Node.js, and Golang on platforms such as
OpenFaaS, Kubeless, Fission, and Knative. The study explored the impact of
supported auto-scaling algorithms on the runtime of the examined functions and
proposed solutions to enhance Python runtime performance in OpenFaaS and
Kubeless. The research revealed an issue related to session timeout in
OpenFaaS, directing all requests to a single function instance and negatively
affecting function performance at scale. Additionally, in high concurrent
loads, Python execution in Kubeless faced issues due to unresponsiveness to
health check messages, which was resolved by adjusting runtimes. Li et al.
[43] have analyzed key characteristics of open-source serverless platforms,
including Nuclio, OpenFaaS, Knative, and Kubeless, and performance gaps in
service exporting and auto-scaling are addressed. Nuclio stood out as having
the lowest latency with 99 percentile latency. This is achieved by queuing
exclusively within function pods. On the other hand, OpenFaaS and Knative
suffer from delays due to multiple components queuing at the same time. Decker
et al. [44] have revealed an assessment of open-source serverless platforms,
including OpenFaaS and Nuclio. They found that Nuclio had 1.5x higher data
throughput than OpenFaaS. A key factor contributing to enhanced performance
was the integration of user functions with function instances. The study found
that existing platforms need improvement for optimal high performance
computing (HPC) usage, underscoring the need for broader benchmarking efforts,
as well as the limitations of Fission. Lee et al. [45] have evaluated
serverless applications for use in research and commercial cloud scenarios,
demonstrating their cost-effective deployment in Big Data map-reduce and
concurrent image processing. In the study, serverless was demonstrated to be
effective in specific applications, such as IoT-driven live migrations in data
centers.

Wen et al. [46] have characterized four mainstream commodity serverless
computing platforms (AWS Lambda, Google Cloud Functions, Azure Functions, and
Alibaba Cloud Function Compute) through qualitative and quantitative analysis.
The researchers have found that too many function instances can lead to poor
performance in a single virtual machine. Mavridis et al. [47] have evaluated
security-oriented container runtimes for improved isolation in multi-tenant
and serverless cloud scenarios. Additionally, OSv Unikernels and Kubernetes
integration was introduced for enhanced isolation. Nabla demonstrated superior
performance, while Kata and gVisor showed competitive results. The Kubeless-
based serverless framework exhibited longer function start times and reduced
performance. Bensalem et al. [48] have addressed the problem of efficiently
scaling and allocating resources for serverless functions in edge networks,
especially for delay-sensitive applications, using a reinforcement learning
(RL) approach. RL and Deep RL algorithms were compared with monitoring-based
heuristics for practical solutions. Simulation results demonstrated that the
RL algorithm outperformed monitoring-based algorithms, reducing the total
delay of function requests by up to 50%. Basic RL performed as well as DRL,
making it a fast and efficient solution.

Unlike previous studies, this paper specifically focuses on running Kubernetes
on the Xen hypervisor, as an efficient underlying virtualization platform, and
harnessing its benefits. Although Kubernetes has replaced Docker with
Containerd since version 1.24, another objective of this research is to
consider whether Containerd is actually an effective replacement for Docker.
Most importantly, this study uses K6 and breakpoint testing to highlight
failure points and performance behaviors in different Kubernetes
distributions, specifically when OpenFaaS is deployed, which have not been
covered in previous studies.

##  3 Research Methodology

In this study, the experiments are divided into three steps. The
specifications and methodologies of each step are discussed in the following
sections.

###  3.1 Virtualization in CloudLab

As several studies have found virtualization to be more secure and functional
[36, 49, 50, 51], virtual machines are considered to be the Kubernetes
infrastructure in this study. Further, in our previous work[33], we have found
that Xen is an efficient hypervisor for VMs as well as containers running on
top of VMs. Since CloudLab’s default Xen virtualization mode is HVM and lacks
PV mode, we have written a Python script to add PV mode, which creates a
profile called “h_aqasiz"[52]. After the PV mode is manually added to
CloudLab, IOzone benchmarks are used to compare the performance of the PV and
HVM modes. The work is then continued in the virtualization mode, which is
empirically proved to be most satisfactory.

###  3.2 Runtimes

In Kubernetes, container runtimes are responsible for load balancing, service
creation, and optimum container execution. By default, Containerd is used as
the container runtime in recent Kubernetes versions. In this paper, Kubernetes
has been modified to also use Docker as an alternative to Containerd, so that
we can measure and evaluate the performance of these two runtimes in terms of
disk and CPU operation. To do this, we run Sysbench to compare the runtimes in
terms of disk performance when MySQL, as a disk-intensive application, is
deployed on Kubernetes. Then, OpenFaaS is run as a compute-intensive
application to compare runtime environments in terms of processor performance.
After determining the winner container runtime, we will continues on that one.

###  3.3 Kubernetes distributions

When the underlying Xen mode and the container runtime of Kubernetes are
specified, we investigate four Kubernetes distributions, namely Kubeadm, K3s,
MicroK8s, and K0s using various metrics. Kubeadm is a command-line tool for
installing and initializing Kubernetes clusters, and it is designed to
simplify the boot-up of new clusters on Linux. In contrast, as a lightweight
distribution, K3s includes only the necessary components to reduce memory and
CPU consumption. It is mainly designed for use in constrained environments
such as edge computers and IoT devices. MicroK8s, like K3s, is a lightweight
Kubernetes distribution developed by Canonical. It provides a straightforward
Kubernetes distribution for development and testing environments. As the
newest Kubernetes distribution, Mirantis’ K0s can be deployed on both local
and cloud networks, simplifying and reducing complexity through Kubernetes’
standard design. In order to ensure fairness, all Kubernetes distributions
were considered version 1.27.2.

##  4 Experimental Setup

The purpose of this section is to describe the hardware and software,
benchmarks that are used, as well as the execution method and configuration of
the applications.

###  4.1 Software and Hardware

The experiment is conducted on C6620 machines in CloudLab, which are equipped
with an Intel Xeon E5-2650v2 processor, 64 GB of memory, a 1 TB hard drive,
and an X520 PCIe Dual-port 10Gb Ethernet NIC. Ubuntu Server 20.04 [53] is used
for all experiments, and the Xen hypervisor is used to virtualize the
environment in PV and HVM modes. As mentioned earlier, a script was written to
create a PV profile in CloudLab for public access. With this profile, one can
set the number of nodes and cores, and the value of memory and other
parameters.

Furthermore, we evaluated two popular container runtimes, Containerd and
Docker, using the MySQL [54] service and the OpenFaaS platform in Kubernetes
for throughput, CPU usage, and scalability. A serverless service can place
considerable load on the processor, so choosing OpenFaaS as an execution
service may be a wise choice. Lastly, four Kubernetes distributions were
implemented, namely Kubeadm, K3s, MicroK8s, and K0s. All distributions, except
Kubeadm, were automatically installed with Calico, but we manually installed
it in every distribution to ensure accuracy. The Container Network Interface
(CNI) plugin was used by Calico [55]. A master and three worker nodes were
used in the study. A separate node was assigned to generate workloads and
measure them in order to ensure that the cluster did not interfere with
workload generation. Figure 4 illustrates the CloudLab implementation in more
detail.

Figure 4: Cluster implementation architecture

Additionally, we used Prometheus [56] and Grafana [57] to visualize the
results, which were installed on the measurement node. In addition, a query
was written to obtain values specific to the OpenFaaS and OpenFaaS-Fn
namespaces that can be found on our GitHub [58]. As a final note, OpenFaaS
platform version 0.26.4 has been installed on all Kubernetes distributions
using Helm [59] version 3.11.0, and the maximum number of pods was 5 due to
OpenFaaS Community Edition. Details of the installation are provided in Table
1.

Table 1: Hardware and software specifications in CloudLab Criterion | Values
---|---
Operating System | Ubuntu Server 20.04
Number of nodes | 4 (1 master, 3 workers)
Processor | Intel® Xeon® Silver 4114
Memory | 64 GB
Disk | 1 TB
Network Card | Intel X520 PCIe Dual port 10Gb Ethernet NIC
Kubernetes Version | 1.27.2
CNI Plugin | Calico v3.16
OpenFaaS Version | 0.26.4
Helm Version | 3.11.0

###  4.2 Benchmarks

There are several tools available for creating workloads on a serverless
platform and clustering services, and in this study, the Grafana K6 load
testing tool[60] and Sysbench [61] benchmark were used. The breakpoint test
from K6 was used to generate workloads for the serverless platform. Breakpoint
experiments identify system limitations by progressively increasing the load
to an unrealistic level. Understanding how and when a system begins to fail
can therefore help to prepare for it. The generated load also allows us to see
the function rate as well as other performance behaviors of each Kubernetes
distribution. An illustration of the breakpoint test model can be found in
Figure 5. Using Sysbench, a script-based multi-threaded benchmark tool, we
simulate a heavy load on the MySQL service in Kubernetes by running up to 150
threads simultaneously.

Figure 5: Breakpoint test

Furthermore, the IOzone [62] benchmark is also used in this research to
compare different modes of virtualization. IOzone is a benchmarking tool for
file systems that analyzes the performance of input/output operations by
reading and writing to the disk. This study examined a 512 MB file with 2 MB
records in the IOzone benchmark to analyze read and write performance. As part
of OpenFaaS, we implemented a factorial function written in Python and invoked
it in order to calculate the factorial of 500, placing pressure on the
processor. To find the breaking point, concurrency was increased up to 20,000
over a 30-minute period. According to our observations, it takes almost two
minutes for the input queue to become empty after generating a heavy load. All
measurements were repeated independently and separately 10 times to ensure
greater accuracy. To prevent the experiments from influencing one another, we
introduced a 5-minute delay between runs.

##  5 Evaluation Results

Herein, the results of the experiments conducted for the three mentioned steps
of Section 3 are presented. The first section examines disk performance in
CloudLab and compares two types of Xen virtualization modes. Next, two
different types of container runtimes are examined using MySQL and OpenFaaS.
As a final section, the OpenFaaS serverless platform is evaluated from a
variety of perspectives when it is run on different Kubernetes distributions.

###  5.1 Virtualization in CloudLab

This section describes how the underlying virtualization environment is
configured on CloudLab servers and how it affects performance. Xen virtual
machines are supported by CloudLab, but the default mode is HVM. As mentioned
earlier, we have written and executed a Python script in CloudLab to provide
Xen-PV mode in that testbed. Then, we evaluate the performance of both HVM and
PV modes using the IOzone benchmark. The goal is to check which mode provides
better disk performance. Figures 7 and 7 show disk read and write performance,
respectively, for the two virtualization modes. As can be seen in the figures,
PV mode performs significantly better than HVM mode both for reading and
writing data. To provide additional confidence, we repeated these measurements
with a variety of CloudLab settings, including different server locations and
machine types. However, the results remain the same. The results are in line
with related work showing that PV offers higher disk performance than HVM.
This is because PV drivers allow direct communication between the guest
operating system and the hypervisor, thus reducing the overhead associated
with emulating hardware in HVM.

Figure 6: Read performance

Figure 7: Write performance

###  5.2 Runtimes

This section examines the performance of two popular Kubernetes runtimes,
i.e., Docker and Containerd. To this end, the MySQL service is initially
executed by both runtimes, and measurements are taken. Then, the OpenFaaS
serverless platform is executed by these two runtimes, and the results are
examined. For further assurance, the results reported are averages of 10
repetitions.

####  5.2.1 MySQL Experiment

At first, an experiment is conducted for the MySQL service. The graph in
Figure 9 illustrates the trend in Transactions Per Second (TPS). TPS is not
significantly different between the two runtimes at lower thread counts,
although Docker has slightly better performance. With increasing thread
counts, the difference between the two runtimes becomes more apparent, with
Docker proving to be the winner under massive loads. The most significant
difference between Docker and Containerd performance occurs at 100 threads,
where Docker shows almost 37% more TPS than Containerd. The latency of the two
runtimes when MySQL is run is shown in Figure 9. As can be seen, the trend is
almost the same as Figure 9 and Docker provides up to 27% less latency than
Containerd. It happens when the number of threads is 100.

Figure 8: MySQL TPS

Figure 9: MySQL latency

####  5.2.2 OpenFaaS Experiment

After examining the Kubernetes runtimes for disk operations using the MySQL
service, here, the OpenFaaS serverless platform is used to examine them
regarding CPU performance measures. Figure 11 shows the result of assessment
in terms of requests per second (RPS) when the concurrency level is changed.
At concurrency 1, Containerd shows better performance than Docker. When the
concurrency level is 25 and 50, both runtimes perform more or less the same.
As concurrency increases, Docker gradually outperforms Containerd. The maximum
difference is at concurrency 200 where Docker can handle almost 70% more
requests per second than Containerd. Figure 11 illustrates the latency of the
two runtimes under different workloads in a similar manner. It can be seen
that Docker shows less latency as concurrency increases. At concurrency 200,
OpenFaaS latency, when Docker is adopted, is around 46% less than when
Containerd is chosen.

As a result of the two experiments, it can be concluded that Containerd cannot
demonstrate good performance and Docker runtime is a better choice than
Containerd for MySQL and OpenFaaS services, particularly under heavy loads.
Docker outperforms Containerd in handling increased concurrencies probably due
to its optimized network stack and efficient management of container
processes. Docker’s architecture, which includes built-in load balancing
capabilities, can more effectively distribute incoming requests across
containers, thus maintaining higher throughput and lower latency as
concurrency levels rise. This could be particularly noticeable in complex
applications like MySQL and OpenFaaS, where the overhead of container
management and network communication significantly impacts performance.
Therefore, the continuation of this study will be conducted using Docker as
the container runtime of Kubernetes.

Figure 10: OpenFaaS RPS

Figure 11: OpenFaaS latency

###  5.3 Clustering and OpenFaaS

After determining the appropriate underlying Xen mode and container runtime,
in this section, various Kubernetes distributions are examined. All of the
considered distributions, including Kubeadm, K3s, MicroK8s, and K0s, are set
up and executed over Xen-PV virtual machines when Docker is set as the
Kubernetes container runtime. For the experimentation, we continue with the
OpenFaaS deployment and then invoke the factorial function using breakpoint
testing.

####  5.3.1 RPS

This section examines the request rate for various Kubernetes distributions.
Figure 12 illustrates the performance of distributions in a single frame.

Figure 12: Kubernetes distributions function rate

By applying the workload to various distributions, it appears that all of them
grow rapidly. They reach their peak at around minute 4 where K3s and MicroK8s
are almost at the same point, taking the lead. These two distributions are
followed by a significant difference with Kubeadm and then, K0s with a slight
difference. As time passes and the rate of input requests increases according
to the breakpoint test, queues build up and the function rate gradually
decreases. However, it fluctuates within a limited range. In the final
minutes, all distributions except K0s grow again.

Regarding K0 behavior, there is a noteworthy point. It turns out that around
minute 4, the system experiences a failure, indicated by 503 errors, and the
rate of successful responses declines sharply. The system cannot resume normal
operation afterwards, so it breaks down. As a result, K0s might perform well
under low loads, but under pressure, it fails unlike other distributions.
Performance issues could be related to various factors such as settings,
resource constraints, program bottlenecks, or deployment-related issues, and
most importantly, to K0s’ inherent nature. Overall, K0s does not perform well
in the RPS section, and K3s offers the highest performance among the
distributions. MicroK8s and Kubeadm are ranked next.

####  5.3.2 CPU Consumption

The CPU usage for each of the four different distributions is shown in Figure
13. Starting from zero, CPU usage gradually increases over time for all
distributions. In the 4th minute, when the queue builds up, utilization
reaches its local maximum. This is in agreement with the function rate
results. Afterward, CPU usage remains practically constant with some
fluctuations as time passes except for K0s. Between 22nd and 28th minutes the
utilization grows up and then, after the load generation stops at 30th minute,
it decreases suddenly and becomes zero at minute 32. Kubeadm, unlike K3s and
MicroK8s, maintains relatively low CPU usage throughout the benchmark, which
indicates efficient resource utilization or scalability.

Figure 13: Kubernetes distributions CPU usage

K0s utilization is very low since it drops all requests after minute 6 as
explained in Section 5.3.1. MicroK8s has interesting behavior regarding CPU
utilization exceeding 100%, which is the result of hyper-threading performed
by an Intel Xeon Silver 4114 processor with 10 cores and 20 threads.
Consequently, it can perform up to 20 tasks at the same time, and each core
can increase its frequency up to 3.00 GHz if necessary. Very high CPU
utilization for MicroK8s may lead to server failure if the CPUs are not
powerful enough. The average CPU usage during the 6th and 22nd minutes, for
Kubeadm, K3s, and MicroK8s is around 45%, 60%, and 105%, respectively.

####  5.3.3 Scaling

Figure 14 shows a chart illustrating scale changes for different Kubernetes
distributions in terms of pod number. All distributions except K0s follow a
similar trend, gradually increasing the number of pods to 5, which represents
the final value for the community edition of OpenFaaS. However, K0s fails to
meet expectations and ultimately decreases its scale from four to just one
pod. This result is consistent with previous results explained in Sections
5.3.1 and 5.3.2.

Figure 14: Kubernetes distributions replication

##  6 Conclusion

In this study, the performances of four popular and widely-used Kubernetes
distributions were examined when OpenFaaS was run as a FaaS application. At
first, The Xen virtualization modes (PV and HVM) in CloudLab environment were
assessed, and Xen-PV was selected for use in later phases due to its better
performance. After that, the performance of two container runtimes for
Kubernetes was evaluated, where Docker was adopted consequently because of up
to 37% higher TPS and 27% lower latency in MySQL experiments and also up to
70% higher RPS and 46% lower latency in OpenFaaS experiments, compared to
Containerd. Finally, Kubeadm, K3s, MicroK8s, and K0s were investigated when
Xen-PV and Docker were set up as the underlying virtualization and container
runtime environments, respectively. It turned out that K03 did not work under
high input loads, while the other distributions were able to withstand our
breakpoint tests. To sum up, K3s is the winner because it offered a high RPS
while maintaining an acceptable level of CPU utilization. Following K3s,
MicroK8s demonstrated the highest RPS, but put a heavy load on CPUs. Kubeadm
ranks third in terms of RPS, but consumes CPU moderately.

Machine Learning and Deep Learning techniques are gaining relevance in
contemporary computing landscapes, prompting us to explore their potential
applications in enhancing Kubernetes and serverless platforms as future work.
Several ways can emerge for further investigation and development, such as
optimizing resource allocation, identifying bottlenecks and inefficiencies,
and integrating anomaly detection. Our goal is to unlock machine learning and
deep learning’s full potential in Kubernetes and serverless platforms through
continued research and experimentation.

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# Compressible and immiscible fluids with arbitrary density ratio

Fei Wang<EMAIL_ADDRESS>Institute for Applied Materials, Karlsruhe
Institute of Technology (KIT), Strasse am Forum 7, 76131 Karlsruhe, Germany
Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT),
Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany  Britta
Nestler  Institute for Applied Materials, Karlsruhe Institute of Technology
(KIT), Strasse am Forum 7, 76131 Karlsruhe, Germany  Institute of
Nanotechnology (INT), Karlsruhe Institute of Technology (KIT),
Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
Institute of Digital Materials Science, Karlsruhe University of Applied
Sciences, Moltkestraße 30, 76133 Karlsruhe, Germany

###### Abstract

For the water-air system, the density ratio is as high as about 1000; no model
can fully tackle such a high density ratio system. Here, we present an
alternative theory for the density evolution equations in computational fluid
dynamics, differing from the concept of Navier-Stokes and Euler equations. Our
derivation is built upon the physical principle of energy minimization from
the aspect of thermodynamics. The present results are consistent with Landau’s
theory of sound speed, and, most importantly, provide a generalization of
Bernoulli’s principle for energy conservation. The present model can be
applied for immiscible fluids with arbitrary density ratios, thereby, opening
a new window for computational fluid dynamics both for compressible and
incompressible fluids.

Introduction. Immiscible fluids with a high density ratio exist everywhere in
our daily lives Lohse  _et al._ (2015). A typical example is the water-air
system, where the density ratio is about 1000. In the classic fluid dynamics,
for the water-air system, the density is either assumed to be a constant
according to the Boussinesq approximation Lohse and Xia (2010); Xia (2013) or
interpolated over the composition Wu  _et al._ (2024); Reder  _et al._ (2021);
Wörner (2012). The latter consideration leads to the paradox of density
interpolation. In these considerations, the variation of the density at the
water-air interface has not yet been resolved, preventing us from a
quantitative understanding on the fundamentals of interfacial fluid dynamics
for immiscible fluids. Some recent attempts Wöhrwag  _et al._ (2018); Abels
_et al._ (2012); Ding  _et al._ (2007) have been made to shed light on the
density ratio problem.

Another daily example for the density variation is evaporation of water and
condensation of vapour. The effect of diffusion and convection in the
evaporation and condensation processes cannot be coped by the Navier-Stokes
equations Abels  _et al._ (2012). Even for pure fluids, like gas phase, when
the fluid velocity is close to the speed of sound, the density varies in space
with time (see Landau’s theory of sound wave Landau and Lifshitz (2013)). The
density evolution equation for high speed fluids is beyond the scope of Euler
and Navier-Stokes equations.

In this Letter, we will propose a generalized theory for the evolution
equations of fluids. Our concept is based on the physical principle of energy
minimization. We will derive a novel kinetic equation for the density at the
water-air interface. In special cases, the present results are shown to be
well consistent with Landau’s theory of sound speed for inviscid fluids. The
derived density evolution equation also provides a generalization of
Bernoulli’s law for the conservation of potential energy plus kinetic energy.
We show that for inviscid fluids, the classic Bernoulli’s equation loses its
validity when there is a change in the density at the droplet-air interface.

Figure 1: Origin of the pressure energy for uniform and non-uniform systems.
For a uniform system (left), the pressure at any position is the same, which
is taken as a reference value. In a non-uniform system with density variation
(right), the pressure is generally non-uniform due to the asymmetric
collisions of different sized molecules. Green and red spheres depict two
molecules or atoms with distinct sizes, leading to a variation of the density
in space. The arrows sketch the thermal motion of the molecules/atoms.

Setup. We consider an isothermal closed system consisting with two immiscible
phases, say $\alpha$ and $\beta$. These two phases consist of $K\in\mathbb{Z}$
components in a domain $\Omega$. The volume concentration of component $i$
($i=1,\ 2,\ \cdots,\ K$) is denoted by $\phi_{i}(\bm{r})$,
$\forall\bm{r}\in\Omega$, leading to a concentration vector
$\bm{\phi}=(\phi_{1},\phi_{2},\cdots,\phi_{K})$. The boundary of the domain
$\Omega$ is denoted by $\Gamma$. For a binary system, we use the notation
$\phi_{1}=\phi$, $\phi_{2}=1-\phi$. The density of the mixture at any position
$\bm{r}\in\Omega$ is denoted by $\rho(\bm{r})$.

Paradox of the density interpolation. For a binary system, the density of a
mixture is often considered to be a function of the composition $\phi$, namely
$\rho=\rho(\phi)$. In such a consideration, it implies that the time evolution
of the density is defined when the time evolution of the composition is given;
the system is over-defined if both $\rho$ and $\phi$ are considered to be
unknowns. As an example, we discuss the linear interpolation of the density
over the composition for a binary system as

$\rho(\phi)=\rho_{1}\phi+\rho_{2}(1-\phi),$ (1)

where $\rho_{1}$ and $\rho_{2}$ are respectively the density of the species 1
and 2, which are constants but generally $\rho_{1}\neq\rho_{2}$. According to
the definition of density, we have the following expression for the mixture
density

$\rho=\frac{\rho_{1}v_{1}+\rho_{2}v_{2}}{V}=\rho_{1}\frac{v_{1}}{V}+\rho_{2}\frac{v_{2}}{V},$
(2)

where $V\subset\Omega$ is a volume element and expressed as
$V=v_{1}+v_{2}+v_{e}$; $v_{e}$ depicts the excess volume and $v_{i}$ ($i=1,\
2$) is the volume of the $i$ species. The volume element includes sufficient
amount of fluid particles for statistic thermodynamics. By assuming a zero
excess volume, i.e., $v_{e}=0$, we have $V=v_{1}+v_{2}$. Substituting this
expression into Eq. (2) and using the definition of volume concentration
$\phi_{i}=v_{i}/V$, Eq. (1) is replicated. Suffice to say, the linear
interpolation in Eq. (1) indicates a zero excess volume of mixing throughout
the system. Any other kinds of interpolations define a certain excess volume,
which is uniform everywhere in the domain $\Omega$. The assumption of uniform
excess volume (either zero or non-zero) deviates from the physical fact that
the local excess volume $v_{e}$ is affected by the local pressure, which
should be space dependent; the excess volume at each position in the domain
should have its own characteristics. An additional paradox in the
interpolation concept is the conflict of incompressibility with mass
diffusion. The incompressibility requires $\dot{\rho}=0$. When diffusion takes
place, i.e., $\dot{\phi}\neq 0$, the density evolution based on the linear
interpolation is $\dot{\rho}=(\rho_{1}-\rho_{2})\dot{\phi}\neq 0$. In the
following, we will shed light on this density paradox from the physical
principle of energy minimization.

System energy. We propose that the total energy of the system consists of
potential free energy $\mathcal{F}$, potential pressure energy $\mathcal{P}$,
macroscopic kinetic energy $\mathcal{K}$, and wall free energy $\mathcal{W}$,
reading

$\displaystyle\mathcal{E}=\mathcal{F}+\mathcal{P}+\mathcal{K}+\mathcal{W}.$
(3)

As derived by Cahn Cahn and Hilliard (1958) and van der Waals Rowlinson
(1979), the potential free energy $\mathcal{F}(\bm{\phi},\nabla\bm{\phi})$ is
written as the integration over the non-local potential energy density
$e(\bm{\phi},\nabla\bm{\phi})$ as Chen and Doolen (1998)

$\displaystyle\mathcal{F}(\bm{\phi},\nabla\bm{\phi})=\int_{\Omega}e(\bm{\phi},\nabla\bm{\phi})d\Omega,$
(4) $\displaystyle
e(\bm{\phi},\nabla\bm{\phi})=f(\bm{\phi})+\sum_{i=1}^{K}\frac{1}{2}\kappa_{i}(\nabla\phi_{i})^{2},$
(5)

where $f(\bm{\phi})$ depicts the bulk free energy density and $\kappa_{i}$
represents the gradient energy coefficient which is related to the interfacial
tension of the $\alpha$-$\beta$ interface, $\sigma_{\alpha\beta}$.

The $\mathcal{P}=\int_{\Omega}pd\Omega$ term, often overlooked previously and
introduced here, describes the potential pressure energy. We formulate the
pressure energy density as

$\displaystyle-p=e-\sum_{i=1}^{K}\mu_{i}\phi_{i},$ (6)

where $\mu_{i}$ is the chemical potential of species $i$, the formulation of
which will be derived in the following via the energy dissipation principle.
Herein, an essential consideration is that the pressure $p$ is always
associated with the potential energy $\mathcal{F}$. A physical interpretation
of the pressure is shown in Fig. 1. A homogenous mixture of two distinct
species (Fig. 1a) leads to a uniform pressure $p_{0}(\bm{r})=\text{const},\
\forall\bm{r}\in\Omega$, which can be taken as a reference value. However,
when the two species are not homogeneously mixed (Fig. 1b), the pressure is
non-uniform in space; the non-uniform pressure is caused, for example, by the
collisions of different molecules with distinct sizes due to the asymmetry in
space. As considered in Dalton’s law, the partial pressure from different
phases is additive, namely
$p=p_{\alpha}\omega_{\alpha}+p_{\beta}\omega_{\beta}$, where $\omega_{\theta}$
($\theta=\alpha,\ \beta$) depicts the volume fraction of phase $\theta$. In
the special case of a homogenous mixture, the pressure defined in Eq. (6) is
consistent with the Landau potential, $-p=f(\bm{\phi})-\sum_{i=1}^{K}(\partial
f/\partial\phi_{i})\phi_{i}$. The difference in the Landau potential of
different phases is the driving force for the movement of the $\alpha$-$\beta$
interface, e.g., phase transformation. A mathematical derivation of Eq. (6) is
similar to that for Landau potential; the derivation is based on the fact that
the internal energy of the system is an extensive variable.

The $\mathcal{K}$ term denotes the macroscopic kinetic energy, which is
formulated as

$\mathcal{K}=\int_{\Omega}\bm{\zeta}\cdot\mathbf{u}d\Omega,$ (7)

where $\bm{\zeta}$ is the linear momentum density expressed in the classic
physics as $\bm{\zeta}=\rho\mathbf{u}$. We present three remarks for the
kinetic energy: (I) One should not write the kinetic energy density as
$\frac{1}{2}\rho\mathbf{u}^{2}$; the reason will be elucidated in the
following. (II) The macroscopic kinetic energy should not be confused with the
microscopic kinetic energy, which has already been considered in the potential
energies, $\mathcal{F}$ and $\mathcal{P}$. (III) One should not interpolate
the velocity via $\mathbf{u}=\mathbf{u}_{1}\phi_{1}+\mathbf{u}_{2}\phi_{2}$;
this interpolation leads to an artificial contribution to the kinetic energy
$\rho\mathbf{u}_{1}\cdot\mathbf{u}_{2}$ Wang  _et al._ (2023). In other words,
the fluid velocity is not a thermodynamically extensive variable.

The $\mathcal{W}$ term in Eq. (3) represents the wall free energy, which is
integrated over the wall free energy density $f_{w}$ on the boundary $\Gamma$
as $\mathcal{W}=\int_{\Gamma}f_{w}d\Gamma$. The formulation of the wall free
energy density $f_{w}$ has been comprehensively discussed elsewhere Wang and
Nestler (2024); Wang  _et al._ (2023); Wang and Nestler (2021) and is not the
scope of this work.

Energy dissipation. We derive the equilibrium condition and the kinetic
equations based on the principle of energy minimization, namely

$\frac{d\mathcal{E}}{dt}=\dot{\mathcal{E}}\leq 0.$

Next, we evaluate the total time derivative of the free energy $\mathcal{F}$,
the pressure energy $\mathcal{P}$, and the kinetic energy $\mathcal{K}$. By
using the chain rule and noting that the functional $\mathcal{F}$ has two
arguments, $\bm{\phi}$ and $\nabla\bm{\phi}$, the total time derivative of the
potential free energy $\mathcal{F}$ reads

$\dot{\mathcal{F}}(\bm{\phi},\nabla\bm{\phi})=\int_{\Omega}\sum_{i=1}^{K}\frac{\partial
e}{\partial\phi_{i}}\frac{d\phi_{i}}{dt}+\frac{\partial
e}{\partial\nabla\phi_{i}}\cdot\frac{d\nabla\phi_{i}}{dt}d\Omega.$ (8)

By using the vector calculus
$\nabla\cdot(\nabla\phi_{i}\otimes\nabla\phi_{i})=\nabla^{2}\phi_{i}\nabla\phi_{i}+(\nabla\phi_{i}\cdot\nabla)\nabla\phi_{i}$,
the definition of materials derivative
$\dot{\nabla\phi_{i}}=\partial_{t}\nabla\phi_{i}+\mathbf{u}\cdot\nabla\nabla\phi_{i}$,
and integration by parts with no-flux boundary condition for a closed system
(more details can be found in Refs. Wang  _et al._ (2023); Zhang  _et al._
(2024)), we obtain the following equations

$\displaystyle\dot{\mathcal{F}}=\int_{\Omega}\sum_{i=1}^{K}\mu_{i}\dot{\phi}_{i}+(\nabla\cdot\underline{\underline{\bm{\Theta}}})\cdot\mathbf{u}d\Omega;$
(9) $\displaystyle\mu_{i}=\frac{\partial
e}{\partial\phi_{i}}-\nabla\cdot\frac{\partial e}{\partial\nabla\phi_{i}};$
(10) $\displaystyle\underline{\underline{\bm{\Theta}}}=\sum_{i}\frac{\partial
e}{\partial\nabla\phi_{i}}\otimes\nabla\phi_{i}.$ (11)

The first part of Eq. (9) defines a generalized formulation for the chemical
potential, as stated in Eq. (10). At thermodynamic equilibrium, the chemical
potential is a constant value throughout the system. In a special case of a
zero chemical potential, i.e., $\mu_{i}=0$, Eq. (10) is equivalent to the
Euler-Lagrange equation; Eq. (10) also provides a physical interpretation for
the variational derivative in mathematics. The second part of Eq. (9) defines
a stress tensor $\underline{\underline{\bm{\Theta}}}$, which is responsible
for the transformation of the potential free energy into the macroscopic
kinetic energy Jacqmin (1999). This stress tensor is consistent with the
Korteweg stress by evaluating the derivative via considering Eq. (5), namely
$\partial_{\nabla\phi_{i}}e=\kappa_{i}\nabla\phi_{i}$.

Noteworthily, a uniform chemical potential throughout the system cannot well
define the thermodynamic equilibrium; the pressure evolution should be
considered as well. An evaluation of the time derivative for the pressure
energy and the kinetic energy leads to

$\displaystyle\dot{\mathcal{P}}=\int_{\Omega}\frac{dp(\mathbf{r})}{dt}d\Omega=\int_{\Omega}\nabla
p\cdot\dot{\mathbf{r}}d\Omega=\int_{\Omega}\nabla\cdot(p\mathbf{I})\cdot\mathbf{u}d\Omega,$
(12)
$\displaystyle\dot{\mathcal{K}}=\int_{\Omega}(\dot{\bm{\zeta}}\cdot\mathbf{u}+\bm{\zeta}\cdot\dot{\mathbf{u}})d\Omega,$
(13)

where $\mathbf{I}$ is an identity tensor. For the derivation of Eq. (12), we
have used the chain rule and the definition of the velocity,
$\dot{\mathbf{r}}=\mathbf{u}$; for the derivation of Eq. (13), the chain rule
of total derivative has been applied. Summarizing Eq. (9), Eq. (10), Eq. (12),
and Eq. (13) and collecting all the $\cdot\mathbf{u}$ terms, we obtain the
total time derivative of the system energy $\mathcal{E}$ as

$\displaystyle\dot{\mathcal{E}}=\int_{\Omega}\sum_{i=1}^{K}\underbrace{\mu_{i}\dot{\phi_{i}}}_{\text{I}}+\underbrace{[\nabla\cdot(\underline{\underline{\bm{\Theta}}}+p\mathbf{I})+\dot{\bm{\zeta}}]}_{\text{II}}\cdot\mathbf{u}+\underbrace{\bm{\zeta}\cdot\dot{\mathbf{u}}}_{\text{III}}d\Omega.$
(14)

The physical interpretation for the three parts of the energy evolution in Eq.
(14) is sketched in Fig. 2.

Before further analyzing the energy dissipation, we state the following
dissipation-conservation theorem which connects the first law and the second
law of thermodynamics.

Figure 2: The energy dissipation mechanisms in Eq. (14). I: The diffusion via
the exchange of the molecules and vacancies leads to the dissipation of the
free energy $\mathcal{F}$. II: The gradient of the free energy $\mathcal{F}$
and the associated pressure $\mathcal{P}$ result in an increase in the
momentum $\bm{\zeta}=\rho\mathbf{u}$. III: The difference in the velocity,
density, and viscosity in distinct subdomains (dashed rectangle) gives rise to
the dissipation of the velocity $\mathbf{u}$ caused by the “spring” drag
effect; the spring symbol denote the interaction between the atoms/molecules
in different subdomains.

Dissipation-conservation theorem. In a closed system, for a conserved scalar
field $\phi$ and the conjugate potential $\mu$, the energy dissipation
$\int_{\Omega}\mu\dot{\phi}d\Omega=-\int_{\Omega}M(\nabla\mu)^{\scriptscriptstyle
2}\leq 0$ is equivalent to the conservation equation $\dot{\phi}=\nabla\cdot
M\nabla\mu$. Here, $M$ is a positive mobility. The proof is straightforward by
using integration by parts
$\int_{\Omega}\mu\dot{\phi}d\Omega=\int_{\Omega}\mu\nabla\cdot M\nabla\mu
d\Omega=\int_{\Gamma}\mu
M\nabla\mu\cdot\mathbf{n}d\Gamma-\int_{\Omega}M(\nabla\mu)^{\scriptscriptstyle
2}d\Omega$ and the no-flux boundary condition, $\nabla\mu\cdot\mathbf{n}=0$.
The dissipation-conservation theorem shows that the conservation and
dissipation are not independent, but always associated.

According to the dissipation-conservation theorem, the first part in Eq. (14),
I, defines the generalized diffusion-convection equation as

$\displaystyle\dot{\phi_{i}}$
$\displaystyle=\nabla\cdot\varsigma_{i}\nabla\mu_{i}.$ (15)

The coefficient $\varsigma_{i}$ is the mobility subjected to the Gibbs-Duhem
constraint; see Ref. Wang  _et al._ (2019) for the derivation of the mobility.
For a binary case, this equation returns to the Cahn-Hilliard equation.

The second part in Eq. (14), II, reveals the momentum conservation

$\dot{\bm{\zeta}}=\frac{d(\rho\mathbf{u})}{dt}=-\nabla\cdot(\underline{\underline{\bm{\Theta}}}+p\mathbf{I}).$
(16)

The conservative force at the right hand side is responsible for the exchange
of the microscopic potential energy $\mathcal{F}+\mathcal{P}$ with the
macroscopic kinetic energy $\mathcal{K}$. This result is nothing but Newton’s
law of momentum balance. Here, a more clear physical meaning of the force is
presented; the force term consists of the pressure $p$ and the stress tensor
$\underline{\underline{\bm{\Theta}}}$. The stress tensor is due to the
gradient term in the potential energy $\mathcal{F}$, while the pressure is
contributed by the associated free energy.

The only term left in the time evolution of system energy, III, is
$\int_{\Omega}\bm{\zeta}\cdot\dot{\mathbf{u}}d\Omega$. By applying the same
dissipation principle as done for the scalar field in Eq. (15), we obtain the
dissipation equation for the velocity as

$\dot{\mathbf{u}}=\nabla\cdot\tau_{1}(\nabla\bm{\zeta}+\nabla\bm{\zeta}^{T})+\nabla\cdot\tau_{2}(\nabla\cdot\bm{\zeta})\mathbf{I}.$
(17)

Unlike the dissipation rule for the scalar variable $\phi_{i}$ with the
conjugate potential $\nabla\mu_{i}$ (see Eq. (15)), three terms
$\nabla\bm{\zeta}$, $\nabla\bm{\zeta}^{T}$, and
$(\nabla\cdot\bm{\zeta})\mathbf{I}$ have to be taken into account for the
dissipation of a vector $\mathbf{u}$ Landau and Lifshitz (2013) (for the
dissipation of a tensor field, like strain tensor, one has to consider Oldroyd
derivative Oldroyd (1950)).

The right hand side of Eq. (17) is equivalent to Newton viscous stress tensor
if we define the mobilities $\tau_{1}$ and $\tau_{2}$ as

$\tau_{1}=\eta/\rho^{2},\ \tau_{2}=\lambda/\rho^{2},$ (18)

where $\eta$ and $\lambda$ are the shear and bulk viscosities Tanaka (2000),
respectively.

Another formulation for the kinetic energy is
${\mathcal{K}}^{\prime}=\int_{\Omega}\frac{1}{2}\rho\mathbf{u}^{2}d\Omega$. In
this case, the time derivative of the kinetic energy reads

$\frac{d{\mathcal{K}}^{\prime}}{dt}=\int_{\Omega}\frac{1}{2}\frac{d\rho}{dt}\mathbf{u}^{2}+\rho\mathbf{u}\cdot\frac{d\mathbf{u}}{dt}d\Omega.$
(19)

The second part leads to the classic Navier-Stokes equation by applying the
dissipation-conservation theorem to the momentum vector $\rho\mathbf{u}$. A
noteworthy shortcoming for the formulation of Eq. (19) is the missing
information of the momentum balance as well as the transformation between the
potential and kinetic energies. Moreover, the first part in Eq. (19) requires
a prior knowledge about the time evolution of the density, which is unknown
for two immiscible phases. The density evolution can be avoided if the kinetic
energy is written as
$\rho\mathbf{u}\cdot\mathbf{u}=\bm{\zeta}\cdot\mathbf{u}$; in this case, the
time evolution of the momentum $\bm{\zeta}$ as a whole term is sufficient,
subjected to the momentum conservation. Noteworthily, the so-called continuity
equation $\partial_{t}\rho+\nabla\cdot(\rho\mathbf{u})=0$ cannot be applied
here for density evolution. The reason is as follows: (I) The classic
continuity equation is derived for pure fluid, where there is no density
variation in space. (II) When considering two immiscible fluids, the classic
continuity equation cannot be straightforwardly adopted, since the density
varies across the interface. In this case, the mass conservation and energy
dissipation are not independent. Suffice to say, one cannot directly apply a
mass conservation without considering the energy dissipation across the
interface. (III) At the interface of two immiscible fluids, one cannot use the
condition $\nabla\cdot\mathbf{u}=0$ which is used for pure fluids. The
divergence free condition $\nabla\cdot\mathbf{u}=0$ implies a uniform excess
volume everywhere in the whole domain $\Omega$ and that the mass is uniformly
distributed in the speed of sound. In other words, when the fluid velocity is
much less than the speed of sound, the density almost has no variation in
space; only in this case, the condition $\nabla\cdot\mathbf{u}=0$ can be
applied. For pure homogenous fluids, the condition $\nabla\cdot\mathbf{u}=0$
is used to solve the unknown pressure $p_{0}$, which is homogenous and can be
considered as a reference value (Fig. 1a). However, for two immiscible fluids,
the pressure is inhomogeneous across the interface, which cannot be solved by
$\nabla\cdot\mathbf{u}=0$ due to the density variation.

Density evolution equation and generalization of Bernoulli’s law. Eq. (16)
subtracting Eq. (17) with the relation
$\dot{\bm{\zeta}}=\dot{\rho}\mathbf{u}+\rho\dot{\mathbf{u}}$, we obtain the
temporal equation for the density as

$\mathbf{u}^{2}\dot{\rho}=\\{\nabla\cdot[(p\mathbf{I}+\underline{\underline{\bm{\Theta}}})+\tau_{1}(\nabla\bm{\zeta}+\nabla\bm{\zeta}^{T})+\tau_{2}(\nabla\cdot\bm{\zeta})\mathbf{I}]\\}\cdot\mathbf{u}.$
(20)

As demonstrated in the following, Eq. (20) is consisent with Landau’s theory
of sound speed and provides a generalization of Bernoulli’s law.

For inviscid flow, there is no energy dissipation for kinetic energy, i.e.,
$\tau_{1}=\tau_{2}\approx 0$. In this case, Eq. (20) is rewritten as

$\mathbf{u}^{2}\frac{d\rho}{dt}=\nabla\cdot(p\mathbf{I}+\underline{\underline{\bm{\Theta}}})\cdot\mathbf{u}.$
(21)

By using the definition of the velocity and pressure gradient in one
dimension, $u=dr/dt$, and $\nabla p=dp/dr$, we obtain the relation between
velocity, pressure, and density as

$u^{2}=\bigg{(}\frac{dp}{d\rho}\bigg{)}_{s},$ (22)

where the stress tensor has been overlooked. The result stated by Eq. (22) is
fully consistent with the definition of sound speed. The condition for this
definition is that there is no energy dissipation, namely, isentropic, as
indicated by the subscription $s$ denoting the isentropic condition. This
result coincides with Landau’s theory of sound speed Landau and Lifshitz
(2013).

Considering inviscid fluids via setting $\tau_{1}=\tau_{2}\approx 0$, and the
steady state, $\dot{\rho}=\mathbf{u}\cdot\nabla\rho$ where we set
$\partial_{t}\rho=0$, we rewrite Eq. (20) as

$\mathbf{u}^{2}(\mathbf{u}\cdot\nabla\rho)=[\nabla\cdot(p\mathbf{I}+\underline{\underline{\bm{\Theta}}})]\cdot\mathbf{u}.$
(23)

By considering an one-dimensional setup and using the relation $d(\rho
u^{2})=u^{2}d\rho+2\rho udu$, Eq. (23) is further simplified as

$d(p+\rho u^{2})=2ud\rho.$ (24)

This result is nothing but a generalized Bernoulli’s equation. When there is
no density variation, namely, $d(p+\rho u^{2})=0$, we have $p+\rho
u^{2}=\text{constant}$, which is Bernoulli’s principle Lohse (2003); Anderson
(2016). However, when density varies, $d\rho\neq 0$, the associated energy is
changed as well; in this case, the classic Bernoulli’s equation loses its
validity.

Discussion. The key point of the present work is the energy minimization based
on an alternative formulation for the macroscopic kinetic energy,
$\rho\mathbf{u}\cdot\mathbf{u}$, rather than $\frac{1}{2}\rho\mathbf{u}^{2}$.
In our derivation, the dissipation of the kinetic energy is divided into two
parts. The first part is the momentum conservation, $d(\rho\mathbf{u})/dt$
which changes not only the fluid velocity $\mathbf{u}$ but also the density
$\rho$. This derivation differs from Euler and Navier-Stokes equations where
the momentum variation is only caused by the evolution of the fluid velocity.
The second part of the kinetic energy dissipation is the velocity dissipation
due to the difference in the density, viscosity, and velocity, as explained in
Fig. 2III.

Another vital point of the current work is the pressure energy, the
formulation of which is consisent with many pioneers Chen and Doolen (1998).
The physical meaning of the pressure has been elucidated in Fig. 1. The
minimization of the pressure energy leads to the so-called grand potential
model Wang  _et al._ (2012). In the lattice model, the free energy and the
pressure can be explicitly formulated according to the theory of
thermodynamics, such as Landau potential. However, for non-lattice models, the
pressure cannot be explicitly identified. This case is the consideration in
Euler and Navier-Stokes equations, where the unknown pressure is solved by the
divergence free of the velocity for incompressible flow. For liquid and solid
phases, the lattice model can be applied because of the regularized order of
the structure within the short range. For gas phases, the lattice model loses
its validity due to the disorder of the structure; one has to use the equation
of state to formulate the partial pressure of the gas phase Wöhrwag  _et al._
(2018); Ledesma-Aguilar  _et al._ (2014).

Additional remark is the dissipation of the velocity. Following the
dissipation-conservation principle, the dissipation of the velocity equals to
$\nabla\cdot\tau_{1}\nabla\bm{\zeta}=\nabla\cdot\tau_{1}\nabla(\partial\mathcal{K}/\partial\mathbf{u})$,
where $\bm{\zeta}=\partial\mathcal{K}/\partial\mathbf{u}$ is analogous to the
chemical potential $\mu=\partial\mathcal{F}/\partial\phi$ with the associated
dissipation term
$\nabla\cdot\varsigma\nabla\mu=\nabla\cdot\varsigma\nabla(\partial\mathcal{F}/\partial\mathbf{\phi})$.
Within the context of the dissipation-conservation theorem, the Newton viscous
stress tensor is obtained when defining $\tau_{1}=\eta/\rho^{2}$. The question
is that if the velocity is a conserved variable or not. If not, the
dissipation has to be modified by the non-conserved form as

$\dot{\mathbf{u}}=-\tau\bm{\zeta},$ (25)

where $\tau$ is a positive mobility. The non-conserved form is known as the
Allen-Cahn type dissipation following the gradient descendent path, being
consisent with the Langevin equation. In the current work, we consider the
linear momentum $\bm{\zeta}=\rho\mathbf{u}$. Non-linear terms in terms of
fluid velocity may be accounted for the dissipation if we consider non-linear
momentum, for example, $\bm{\zeta}\propto\mathbf{u}^{n}$ or
$\bm{\zeta}\propto(\nabla\mathbf{u})^{n}$. The non-linear momentum not only
modifies the velocity dissipation but also changes the momentum balance, which
may be applied for understanding highly nonlinear effects, such as turbulence.
These specialized cases are out of the scope of the present discussion.

In conclusion, we have proposed an alternative theory differing from Navier-
Stokes and Euler equations to address the high density ratio problem in fluid
dynamics. Our consideration is based on the basic principle of energy
minimization. A noteworthy outcome of the present work is a novel evolution
equation for the high density ratio system, such as water-air. For inviscid
fluids, our derivation of the density evolution is fully consistent with
Landau’s theory of sound speed. Our derivation also leads to a generalization
of Bernoulli’s law for steady-state flow when there is a density variation in
space. The current work reshapes the understanding of momentum conservation in
Newton’s law. We expect that the present model can be used for immiscible
fluids with arbitrary density ratios as well as for pure fluids when the fluid
velocity is close to the speed of sound.

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# Isolated photon-jet correlations in Pb-Pb collisions at
$\sqrt{s_{\text{NN}}}=5.02$ TeV in ALICE††thanks: Presented at Quark Matter
2022

Alwina Liu
On behalf of the ALICE Collaboration  University of California, Berkeley

###### Abstract

Jets correlated with isolated photons are a promising channel to study jet
quenching in heavy-ion collisions, as photons do not interact strongly and
therefore constrain the $Q^{2}$ of the initial hard scattering. We present the
isolated photon-jet correlations measured in Pb–Pb collisions at
$\sqrt{s_{\text{NN}}}$ = 5.02 TeV by the ALICE collaboration. We study
correlations of isolated photons above 20 GeV/$c$ with charged-particle jets
above 10 GeV/$c$, reconstructed with the anti-$k_{\text{T}}$ algorithm. The
correlations probe the lowest jet $p_{\text{T}}$ range ever measured at LHC
energies, and larger modifications due to the QGP are expected in the lower
$p_{\text{T}}$ regime.

##  1 Introduction

In an ultra-relativistic collision between two nuclei, occasionally two
partons will interact to produce particles with large transverse momentum.
Sometimes, a hard scattering will produce a photon, known as a prompt photon,
which can then traverse the quark-gluon plasma (QGP) produced by the rest of
the collision without strongly interacting with the QGP. Therefore, when a jet
is produced along with a prompt photon in a hard scattering, while the jet
loses energy to the QGP, the photon tags the initial energy of the scattered
parton that produces the jet. By studying these photon-jet events, one can
study partonic energy loss in the QGP, probe bound nucleon structure, and do
detailed studies of jet fragmentation and the potential modification thereof
by the QGP. Photon-jet correlations are not a new concept; they have been
measured in $\sqrt{s_{\text{NN}}}=5.02$ TeV Pb-Pb collisions at the LHC by
both CMS [1] and ATLAS [2]. However, this measurement considers
lower-$p_{\text{T}}$ photons (down to 20 GeV/$c$) and charged-particle jets
(down to 10 GeV/$c$), which probes a low-$x$, low-$Q^{2}$ regime.

The data for this measurement was collected in 2018 with the ALICE detector
[3, 4] at the LHC from Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV.
The Inner Tracking System and Time Projection Chamber are used together to
reconstruct charged particles (tracks). The VZERO system is used for the
minimum-bias trigger and to determine the centrality of the collisions. The
Electromagnetic Calorimeter system (EMCal) is used to measure the photons by
grouping neighboring cells into clusters; it is also used to trigger on
collisions with a large deposit of energy in the EMCal.

##  2 Analysis

The goal of this analysis is to measure the correlation between isolated
prompt photons and the charged-particle jets produced in the same hard
scattering. We make an isolation cut on the photon of
$p_{\text{T}}^{\text{iso, ch}}<1.5$ GeV/$c$ within $R=0.2$. To reconstruct the
jets from tracks with $p_{\text{T}}>0.15$ GeV/$c$, we use FastJet [5] with the
anti-$k_{\text{T}}$ algorithm with $R=0.2$. The angular correlations
($\Delta\varphi$) and momentum balance
($p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\gamma}$) of photon-jet pairs are
measured for photons with $20<p_{\text{T}}^{\gamma}<40$ GeV/$c$ within
$\absolutevalue{\eta}<0.67$ and for charged-particle jets with
$p_{\text{T}}^{\text{ch jet}}>10$ GeV/$c$ within $\absolutevalue{\eta}<0.7$.

At leading order, prompt photons are produced in isolation, whereas neutral
mesons (which decay into photons) and fragmentation photons tend to be
produced within jets. The isolation energy of a photon candidate can therefore
be used to reduce the contribution of these non-prompt photons. It is defined
as the sum of the $p_{\text{T}}$ of the charged tracks within $R=0.2$ of the
EMCal cluster after subtracting the underlying event density $\rho$:

$p_{\text{T}}^{\text{iso,
ch}}=\Sigma_{\text{tracks}}p_{\text{T}}^{\text{track}}-\rho(\pi R^{2})$ (1)

The underlying event from the Pb-Pb collisions causes both the reconstructed
jet $p_{\text{T}}$ and the photon $p_{\text{T}}^{\text{iso, ch}}$ to be too
high. The underlying event density $\rho$ is estimated with the jet-
area/median method [6] and used to correct both the photon isolation (Eq. 1)
and the jet $p_{\text{T}}$.

The underlying event also causes “fake” jets to be reconstructed from
particles not associated with any hard scattering. Along with these “fake”
jets, there are also jets produced from a different process than the one that
produced the photon; collectively, we call this the combinatorial background.
To estimate its size, we use an event mixing technique in which photons from
an EMCal-triggered event are paired with charged-particle jets from a
different minimum-bias event. Events are matched to each other based on
centrality, $z$-vertex position, and event-plane angle. By construction, the
photon and jets are not produced in the same hard scattering, so this mixed-
event correlation can be subtracted from the same-event correlation to remove
the combinatorial background.

The other large source of background is photons from neutral meson decays, and
we use the shower profile in the EMCal to distinguish between signal and
background photons and measure the purity with a template fit. As in [7], the
shower profile of a photon in the EMCal, i.e. the shower shape, is encoded
geometrically and used to define the signal region (SR), which contains most
of the prompt photons, and the background region (BR), which is dominated by
decay photons. Unlike in [7], a variation of the standard shower shape
variable is used; it considers the cells surrounding the cell with the most
energy in the cluster and is called $\sigma^{2}_{\text{long(5x5)}}$.

The prompt photon candidates are selected to be the EMCal clusters that pass
the isolation cut and have a signal-like shower shape
($0.1<\sigma^{2}_{\text{long(5x5)}}<0.3$). However, in addition to the
isolated prompt photons we wish to measure, this sample also contains isolated
decay and fragmentation photons with a signal-like shower shape. In order to
estimate the purity of the photon candidate sample, we fit the shower shape
distribution of isolated clusters to a linear combination of templates. This
purity is then extracted in bins of centrality and photon candidate
$p_{\text{T}}$, and does not change very much with either, as seen in Figure
1.

Figure 1: The photon purity is measured as a function of photon candidate
$p_{\text{T}}$ across various centrality ranges for photon candidates with
$p_{\text{T}}^{\text{iso, ch}}<1.5$ GeV/$c$.

With the photon candidate purity, we can estimate and subtract the non-prompt
photon background from the photon candidate sample. First, we pair each prompt
photon candidate with all charged-particle jets in the event with
$p_{\text{T}}>10$ GeV/$c$, leading to a per-trigger (i.e. per-photon-
candidate) yield of photon-jet pairs, which we call $C_{\text{SR}}$. This is a
linear combination of the per-trigger yield for prompt photons and non-prompt
photons, with the relative weight given by the purity. The non-prompt photon
correlation can be estimated by the clusters in the shower shape background
region ($0.6<\sigma^{2}_{\text{long(5x5)}}<1.5$), as this region is dominated
by photons from neutral meson decays; we call this correlation
$C_{\text{BR}}$. To get the prompt photon correlation $C_{\text{S}}$, we do
the following weighted subtraction with purity $P$:

$C_{\text{S}}=\frac{C_{\text{SR}}-(1-P)C_{\text{BR}}}{P}$ (2)

Four correlations are measured with the purity weight described in Eq. 2 and
combined to get the fully-subtracted correlation signal: the same-event
signal-region (SESR), the same-event background-region (SEBR), the mixed-event
signal-region (MESR), and the mixed-event background-region (MEBR). As
mentioned above, the background-region (BR) correlations account for the non-
prompt photon background, while the mixed-event (ME) correlations account for
the combinatorial background. To avoid double-counting the combinatorial
background associated with non-prompt photons, we add MEBR back to get the
final signal yield:

$\gamma_{\text{prompt}}+\text{correlated
jets}=\text{SESR}-\text{SEBR}-\text{MESR}+\text{MEBR}$ (3)

The dominant sources of systematic uncertainty are the uncertainty associated
with the shape of the mixed-event correlation and the uncertainty associated
with the selection of the background-region used to estimate the non-prompt
photon background. To evaluate the mixed-event shape uncertainty, we consider
the difference in the shape of the $\Delta\varphi$ correlations between the
same-event and mixed-event correlations in the region with minimal signal. To
evaluate the background-region uncertainty, we vary the
$\sigma^{2}_{\text{long(5x5)}}$ range used to measure $C_{\text{BR}}$.

##  3 Results

The fully-subtracted angular correlations $\Delta\varphi$ are shown in Figure
2. They are compared to PYTHIA embedded into minimum-bias Pb-Pb data, to
account for the effects of the underlying event present in the Pb-Pb
environment, as the data are not unfolded. They are also compared to a non-
embedded PYTHIA simulation as an approximation of a pp reference. We see a
trend of a higher back-to-back yield in more central collisions in both the
data and the embedded PYTHIA, suggesting that this arises from underlying
event fluctuations. We also see an apparent suppression in the most peripheral
bin, but we note that 50% is not very peripheral and there is much less
“extra” yield from underlying event fluctuations, so this is not as surprising
as it might seem.

Figure 2: The angular correlations between isolated photons and jets are
compared with embedded gamma-jet and dijet PYTHIA treated like data and also
non-embedded gamma-jet and dijet PYTHIA treated like data

The fully-subtracted momentum imbalance correlations
$p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\gamma}$ are shown in Figure 3, and,
as with the $\Delta\varphi$ observable, are compared to both embedded and non-
embedded PYTHIA. A shape difference as a function of centrality is seen in the
embedded PYTHIA due to detector effects.

Figure 3: The momentum imbalance between isolated photons and jets is compared
with embedded gamma-jet and dijet PYTHIA treated like data and also non-
embedded gamma-jet and dijet PYTHIA treated like data

In order to study the centrality dependence of jet energy loss, we calculate
the mean of the $p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\gamma}$ distribution
within $0<p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\gamma}<1$. This is shown in
Figure 4. Within our uncertainties, we do not observe a centrality dependence.
For higher-energy photons and jets, CMS and ATLAS reported modifications for
more central Pb-Pb collisions compared to a pp reference.

Figure 4: The truncated mean (between 0 and 1) of momentum imbalance between
isolated photons and jets is compared with a combined embedded gamma-jet and
dijet PYTHIA sample which has been treated like data.

##  4 Summary

We present the first measurement of isolated photon-jet correlations in Pb-Pb
collisions in ALICE and the first such measurement at the LHC down to photon
$p_{\text{T}}=20$ GeV/$c$. This extends the LHC analyses down to a lower
$Q^{2}$ and $x$ regime. No centrality-dependent medium modification is seen
for the photon-jet angular correlation or $p_{\text{T}}$ imbalance within the
uncertainties. We plan to compare to model predictions such as CoLBT and are
looking forward to more precise measurements with the upcoming LHC Run 3 data.

## References

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