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End of Season Awards & Presentations 2015
Following our last home match of the 2014-15 season presentations were made to the London Irish Supporters Club Player of the Season as voted for by the LISC membership, the St Clare's Travel award to the Young Player of the Season presented by Tony and Maureen Byrne of St Clare's Travel, and the Robin Clive Memorial Trophy (Academy Player of the Season) as chosen by the Academy coaching staff.
Congratulations to our winners:
Tom Fowlie (Young Player of the Season)
Connor Murphy (Academy Player of the Season)
Alex Lewington (Player of the Season)
At the end of each season the London Irish Supporters Club make presentations to departing or retiring players & coaches who have given many years of service to London Irish.
For the 2014-15 season, a presentation was made to:
Guy Armitage.
We also have gifts for Kieran Low, Tom Homer and Brian Smith. We hope to catch up with Kieran over the next week or so, we plan to present Tom with his gift when London Irish play Bath next season and Brian's gift will be posted to him.
In addition we are also saying farewell to Ollie Adams, Can Cowell, Sean Cox, Myles Dorrian, Harry Lightfoot Brown, Dan Leo, Tomas O'Leary, Matt Parr and Eamonn Sheridan.
On behalf of all London Irish supporters, we would like to thank all those who are moving on for their service to London Irish and we wish them all good luck for their futures.

redpajama_set_name: RedPajamaCommonCrawl


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Download COVID-19 Mindset: The Collision of Issues
Hidden Depths – The Impact of COVID-19 on the Pharmaceutical, Biotech and Devices Industries
by Mark Senak
The COVID-19 pandemic is shifting gears and with it, the ways in which we cope and respond. We are moving from an initial state of surprise, fear, consternation and a general lack of knowledge to an evolving understanding of consequences of the pandemic in our lives – that we are going to have to adapt to several unpleasant realities both as a society and as individuals and that the stakes are long-term. From this developing understanding, there are some clear implications for the pharmaceutical, biotech and device sectors as the COVID-19 pandemic continues. In the most recent survey conducted by FleishmanHillard TRUE Global Intelligence – COVID-19 Mindset: The Collision of Issues – there are glimpses into the changes that people are experiencing in their consumption of healthcare services, and the impact that has on medicines and ultimately the pharmaceutical, biotech and device industries:
1. People Are Settling into a Lower Level of Access to Healthcare – In the beginning of the pandemic, FDA announced that it was "postponing" meetings. There was a clear implication of a "wait-and-see" approach to ascertain if circumstances might allow for in-person meetings later in the year. But under current circumstances and high transmission rates in the U.S., people are understanding they are in this for the long haul. That means that the initial downward spiral that was seen in the consumption of healthcare services will likely continue. In fact, the survey seems to confirm that. U.S. respondents are now estimating that a return to normal would take at least 29 weeks. Moreover, nearly half (48 percent) thought it would take from five months to more than one year. That perception will likely inform their behavior and healthcare access approach with nearly half (47 percent) saying that they will likely postpone routine doctor visits or elective procedures in the next 12 months. That may be good news for insurers, but not for healthcare systems of manufacturers of medicine.
2. Ongoing Financial Impacts Are Likely to Affect Care – As the July report from Families USA affirmed, over 5 million Americans have so far lost their insurance due to their loss of employment. With the current increase in cases, the economy will continue to be hindered, thereby making it likely that this number will not get better and may even worsen. In addition, there were impacts in specific regions and states where the rate of loss was much greater than others which would likely inform the future consumption of healthcare services and prescription medicines. The FleishmanHillard TRUE Global Intelligence Survey found that 58 percent of U.S. respondents stated that they either do not have enough money or have just enough money to pay monthly expenses and half stated that they have cut back on expenses – a behavior that they see continuing in a post-pandemic world. These financial realities inform pharmaceutical sales as well as the scope and presence of emerging need for patient assistance programs that may be needed to help patients get access to prescription medications. It is noteworthy that in the survey when asked why they were postponing doctor visits most people indicated concerns about safety, but nearly 1 in 5 said that they could not afford it and 14 percent said they no longer have insurance due to unemployment.
3. Expectations of Industry is High; Confidence Not So Much – The TRUE Global Intelligence Survey sought to ascertain how institutions are performing with respect to COVID-19 and perceptions about how committed these institutions are in doing the right thing. In the first round of the COVID-19 survey questions published in May 2020, 64 percent of U.S. respondents thought the pharmaceutical industry was doing excellent/great/good in their response to the pandemic, a number that increased to 69 percent in this second round survey and ahead of corporations more generally. It is possible the increase in regard for the industry is a result of the highly publicized efforts involved in vaccine development. However, when asked how committed the pharmaceutical industry is to do the right thing, only 29 percent of respondents gave high marks. How the industry executes on the vaccine promise may in turn influence the perception of commitment. Companies will be pressed to show they are committed to the right thing through the regulatory process and in issues related to access to any vaccine developed.
4. A Vaccine Will Help, But Not Solve the COVID-19 Issues – As companies race to develop, test and get regulatory blessing for a vaccine, consumers do not perceive it as a singular solution. While just over half (57 percent) of U.S. respondents said that a vaccine will end the threat, over one-third of respondents said that the end comes when there are either no more infections or there are medications that can effectively address symptoms. In short, while vitally important, it may be equally important to put the spotlight on what companies are doing to shore up testing and treatments for COVID-19 in addition to vaccine development. This may be particularly true given recent reporting from The New York Times citing a growing number of polls that report a high level of distrust respecting a vaccine casting doubt on a willingness to accept any new vaccine for COVID-19.
In short, there is a lot being demanded of industry both in terms of action (finding a safe and effective vaccine) – and expectation (that once found, access will not be an issue). This is happening at a time when it is likely that sales of existing medicines and the launch of new ones will be less financially rewarding in the past as people put off care either because of safety or financial concerns. Last year, prior to the emergence of the pandemic, a Gallup poll found the pharmaceutical industry held in record low regard at the bottom of U.S. industry rankings. In the midst of the pandemic and in between the demands and expectations of industry, there is opportunity for change.
With Development of the COVID-19 Vaccine, Pharma Gets an Opportunity to Address Reputational Standing
Roche: Glamorizing Breast Health Puts Breast Cancer Awareness On Display.
Is It Too Risky to Visit Your Doctor?


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The Storyline Slam: Skool'd
#ArtLitPhx: The Storyline SLAM: Skool'd
September 7, 2018 September 2, 2018 Superstition Review
Date: September 14th
Time: 7 pm-9 pm
10 STORYTELLERS. 6 MINUTES. 1 WINNER
Ten tellers will have 6 minutes each to share a story based on the theme Skool'd.
Sign up on TheStoryline.org August 11th through September 8th to tell a story. Eight names will be drawn September 9th and posted on the TheStoryline.org. Two more names will be drawn live at the beginning of the show.
Five members of the audience will be the judges and the story with the most points at the end of the show receives a $30 cash prize.
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HomeAll PostsProduct ReviewsThe 15 Best Astrology Apps for 2022 &#8211...
The 15 Best Astrology Apps for 2022 – The Complete List
by omotayoaola
The best astrology apps can teach you more about the stars, provide a detailed birth chart, and even spice up your dating life. While Co-Star and The Pattern are the most popular among general audiences, if you want to learn more about astrology, there are a variety of applications available.
It's difficult for an astrology app to describe what is happening collectively and on the tiniest level of personal emotions and wishes. If you discover one that accomplishes both, that's an excellent app.
Some applications are data-driven, providing an in-depth look at the planets' positions, while others connect users with specialists for readings and insights into their horoscopes.
Whether you're searching for a daily horoscope app or software that tracks every planet's retrograde, a mindfulness tool that interacts with the moon, check what shadow work is, or an astrological dating app, heck or even tells you what to eat on your periods. You've come to the correct spot.
On the App Store, the best astrology apps on the list have received at least four out of five stars, and the majority of them are free to download.
1 Methodology for Selecting the Best Astrology App
2 Best Overall Astrology App
2.1 Astrology Zone
3 The Best Astrology Apps For Beginners
3.1 Chani
3.2 Sanctuary
3.3 Time Nomad
4 Best Astrology Apps for Relationships
4.1 The Pattern
4.2 Astrolis Horoscopes and Tarot
4.3 Daily Horoscope AstrologyZone
5 Best Astrology Apps for Marriage Prediction
5.1 Time Passages
5.2 Nebula
6 Best Astrology Apps for Birth Charts
6.1 Co-Star
6.2 AstroFuture
7 Best Astrology Apps for iPhone
7.1 Zodiac Touch – Psychic Reading
7.2 Why it stands out
8 Best Astrology Apps for Android
8.1 TimePassages
8.3 The Daily horoscope
10 FAQs About Best Astrology Apps
10.1 omotayoaola
Methodology for Selecting the Best Astrology App
To choose the best astrology apps, we looked at each feature and thought about how each one may be fascinating, helpful, or just plain fun to use.
Because these applications have distinct strengths and drawbacks, they appeal to different users according to how they utilize astrology in their everyday life. That's why we didn't rule out the simple, no-frills applications and looked at a variety of solutions with many bells and whistles.
I also considered pricing. Even an excellent software might be a headache if the price of its top features isn't clear. The user ratings and app popularity are also checked to see which apps were the best.
Best Overall Astrology App
You've found the best daily horoscope app if you're seeking accurate daily and monthly horoscopes, compatibility reports, essays, and more. Astrology Zone is an award-winning program that provides you with all the information you need to plan your day, month, or year ahead. Phunware, Inc.'s Astrology Zone Daily Horoscope is a Lifestyle app.
It has a 5-star rating on the AppStore and has had over 16,637 ratings.
Horoscopes are detailed and to the point.
The majority of the information is available for free.
A somewhat upbeat view
What We Don't Like
It doesn't take the complete natal chart into consideration
The monthly horoscopes are frequently published late.
The Best Astrology Apps For Beginners
Chani
One of the best astrology apps was created by astrologer Chani Nicholas created the CHANI app for "self-discovery, mindfulness, and healing."
Nicholas and other professional astrologers have posted updates about the current sky, mindfulness and meditation techniques, personal horoscopes, and birth-chart readings, among other things. It's ideal for novices and those interested in using astrology as a tool for wellness and healing. It will also teach you how to have a scarcity mindset.
According to the CHANI website, the app was created in collaboration with AnnieCannons, Inc, a queer-led software agency, and 5% of all proceeds are donated to survivors of gender-based violence who are Black, Latinx, Native/Indigenous, queer, and trans.
Visually appealing and straightforward to use
The aspect of health and wellness
The accessible part of astrology
Various add-ons are available.
It might be pricey.
Some users have reported bugs.
The Sanctuary app not only offers free horoscopes, daily tarot card draws, and educational astrological materials, but it also connects you with psychics, tarot readers, and astrologers for on-demand paid readings.
This is an excellent tool for anybody interested in getting their horoscope, learning more about astrology, and connecting with various specialists for personal tasks, making it one of the best astrology apps.
Birth chart real-time in real-time
Options for tarot cards
The text is simple to comprehend.
Other applications are more thorough than this one.
It may be pricey.
Time Nomad
This app performs the role of a virtual astrologer. Time Nomad allows you to follow the positions of planets, minor bodies, and asteroids in real-time and analyze birth charts for yourself or your friends. It also provides you with updates about astrological events that are coming up, a great feature for one of the best astrology apps.
The program also sends weekly astrological lessons and forecasts to individuals who have problems interpreting the material.
Widget for your home screen
Horoscope for love and business daily
Option to keep an eye on the current graph
Calculate planetary hours using the built-in calculator.
The daily love horoscope may not always be accurate.
Advanced reporting is not available in the freemium version.
Best Astrology Apps for Relationships
The Pattern
This no-frills, free software can be pretty accurate: Just ask Channing Tatum, who expressed his surprise at its precision on Twitter in 2019. Users begin by revealing their name and gender and details such as their date, place, and precise moment of birth.
Yo anybody know of this shit and can explain it to me! Cause WTF! pic.twitter.com/4Qhp6gj7IU
— Channing Tatum (@channingtatum) July 12, 2019
The Pattern promotes itself as a social network, allowing users to connect with friends on the site to learn more about how the cosmos may affect their relationships. More than just the best astrology app, this is a great must-have app.
It's astonishingly accurate.
Allows you to form ties with your friends using the app.
Advice is more ambiguous; it does not provide solutions.
Astrolis Horoscopes and Tarot
One of the best astrology apps and a popular horoscope application is Astrolis Horoscopes and Tarot. It has a full-featured freemium version, unlike many other options on our list. As a result, you may keep track of your daily or weekly horoscopes without spending any money.
Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, and Pisces are among the 12 zodiac signs covered by the app. It displays a three-card tarot reading every day, which helps determine present circumstances and forecasts future occurrences.
Another element that makes Astrolis Horoscopes and Tarot so user-friendly is zodiac compatibility, which may be pretty helpful in assisting you in finding areal-life companion.
All 12 zodiac signs are represented.
Compatibility by Zodiac.
Chinese horoscopes are based on the lunar calendar, daily tarot horoscopes.
Have to buy credit to talk to psychic
Daily Horoscope AstrologyZone
If you are missing a horoscope app that can deliver a reliable daily horoscope, give a chance to Susan Miller's "AstrologyZone" to fill up the void.
By emphasizing simplicity, this best astrology app provides detailed horoscopes for all 12 signs. If you want to stay ahead of the game, you can also keep track of the monthly horoscopes for all 12 signs.
Daily and monthly horoscopes are available.
There are writings on a variety of themes in this collection.
Reports on compatibility
Make a list of the most auspicious days.
Some stories seem irrelevant
Best Astrology Apps for Marriage Prediction
Time Passages
Choose from add-ons such as a compatibility meter, which compares charts to determine the possibility of a reasonable conclusion, or forecasting for the future, which allows you to view forthcoming astrological events and assess how they could affect what lies ahead.
You don't even have to read the entire app to obtain the main points: Mercury retrograde periods and future moon phases are shown prominently on the app's homepage, so you'll never be caught off guard.
The best astrology app for marriage preparation, hands down!
Provides up-to-date information about astrological happenings.
Great for relationship predictions
Additional features come at a cost.
Nebula is well-known for providing accurate zodiac forecasts. It contains everything you need to be your one-stop horoscope and astrology shop and one of the best astrology apps.
Based on zodiac signs like Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, and Aquarius, the app can help you gain a better knowledge of yourself and what kinds of things are likely to come your way.
The 28-day relationship advice is one of the most noticeable aspects of this highly-rated horoscope app for iOS and Android. It may go a long way toward strengthening your relationship.
Pretty accurate zodiac forecasts
You have the option of checking your zodiac compatibility with your mate.
It demonstrates compatibility in marriage.
Reading your natal chart on a one-on-one basis.
The user interface is a little clumsy.
The premium version is more costly, costing $9.99 each week.
Best Astrology Apps for Birth Charts
Are you looking for a more tailored astrological reading? Co-Star is the best astrology app that caters to your preferences by providing horoscopes specifically tailored to your needs.
Unlike other astrology applications, its predictions are based on a sky map and your birth's precise time and location. In addition, the program uses NASA data to follow the movement of the planets.
Co-Star was the first A.I.-powered astrology software accessible, and it is perhaps the most well-known astrological app on our list. The program produces daily insights based on your astrological chart on your time, date, and birth location. You may also compare your horoscope to your friends to learn about your compatibility with their various signs.
Personalizes astrology by displaying planet-by-planet compatibility.
Forecasts for personality analysis are based on a sky map.
Daily reminders appear to be a touch excessive.
The freemium version is straightforward.
AstroTalk
AstroTalk is one of the best astrology apps, especially when making accurate forecasts. Because it employs sophisticated astrological techniques such as Nadi astrology, Vedic astrology, Hindu astrology, KP astrology, Prashant chart, and others, it is thought to be more accurate.
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Advanced astrological techniques are used.
Answers are well-researched.
Suggestions for child names
Correction of the time of birth
Predictions might be off-target at times.
Kundali analysis takes longer for astrologers.
AstroFuture
If you're looking for the best astrology app to help you figure out love compatibility, I recommend "Astro Future." It's a straightforward yet effective horoscope app for iOS. You may add as many people as you wish to achieve a percentage of compatibility.
A fantastic synastry chart shows how two people's horoscopes may impact each other. You may also look at the birth charts, which include complete horoscopes.
You will be able to comprehend the significance of each feature because it is described in detail.
Compatibility in love
Detailed horoscopes and natal charts
Horoscope reports in PDF format
Chart of synastry in its entirety
The information on a synastry chart isn't always correct.
The freemium edition has a lot of limitations.
Best Astrology Apps for iPhone
Zodiac Touch – Psychic Reading
With our Psychic Reading app for iPhone, you can get down-to-earth advice and honest guidance from some top spiritual gurus and life coaches globally.
You may ask your questions about life, love, relationships, work, and more with the Zodiac app for iPhone. Psychic, Tarot, Astrology, Medium, Clairvoyance, Numerology, and more specialties and methodologies are all available via Zodiac Psychics.
Why it stands out
Safe, safe, and private
In more than ten nations, it is the most popular lifestyle app.
In 126 countries, it was ranked in the top 100 overall applications.
There are around 2,300 talented advisers.
Text, chat, or phone from anywhere, 24 hours a day, seven days a week.
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TimePassages
"TimePassages" is a comprehensive horoscope app, whether you are new to astrology or have been a long-time devotee. The complete astrological dictionary is the most prominent feature of this program, as it assures you never to get confused by an unfamiliar language. It also plays an integral part in assisting you in comprehending all that is written on the cards for you.
Another area where TimePassages appears to have an advantage over many of its competitors is the clean and user-friendly layout, which makes navigating a breeze.
You may use this best astrology app to obtain daily horoscopes and learn about the meanings of the sun, rising, and moon signs, among other things.
Clean and intuitive interface
Advanced astrology glossary
Colour charts with detailed descriptions
Birth charts
Co-Star was the first A.I.-powered astrology software accessible, and it is perhaps the most well-known best astrological app on our list.
The program produces daily insights based on your astrological chart on your time, date, and birth location. You may also compare your horoscope to your friends to learn about your compatibility with their various signs.
The Daily horoscope
Whether your zodiac sign is Libra, Gemini, or Capricorn, "Daily Horoscope" can help you quickly learn about your zodiac characteristics.
The best astrology app has a friendly interface and lets you look at different druid horoscopes. Depending on your needs, you may see your daily, weekly, and monthly projections.
And, just in case you miss a forecast, you'll be able to go back and review previous forecasts. Because I enjoy customizing things, I appreciate its collection of unique colours and fonts, which allows for easy modification.
There's also the ability to save something as a favourite and return to it later, which makes this my favourite best astrology app on this list.
Custom colors and fonts
Option know your zodiac characteristics
Ability to explore alternative Druid horoscope
Plenty of customization
Having your future predicted by a fortune-teller using their mystical crystal ball and abilities may be frightening and fascinating. Not only will you be able to see into your future, but you will also be curious about how these fortune-tellers were able to predict your fortunes accurately.
Thanks to contemporary technology, knowing your destiny will never be a problem since there are apps that can tell you your fate. You can also use these apps to learn about specific things about your body, like sanpaku eyes and what it means.
Whether you're looking for daily horoscopes, compatibility chakras, or Tarot card readings, it will always be available and easily with these best astrology apps.
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What is the best daily horoscope app?
While several applications claim to be the most excellent horoscope app, we believe Astrology zone to be somewhat better than many of its competitors.
Is it safe to use astrological apps?
Not all, but several reputable and best astrology apps, such as TimePassages and AstroMatrix, provide a secure environment for your data.
What are the most excellent and most accurate free astrology websites?
Susan Miller's "AstrologyZone and Astrolis Horoscopes and Tarot are, without a doubt, the top astrology sites, owing to their reasonably accurate forecasts.
What Are Astrology Apps and How Do They Work?
Daily horoscopes, an in-depth examination of your birth chart, and compatibility readings are available through astrology apps. This data is based on your date of birth, time of birth, and geographic location.
How much do Astrology Apps cost?
The most basic functions, such as horoscopes and birth charts, are accessible on astrology applications, but other add-ons require money. The cost of a membership varies, but expect to spend roughly $12 per month or $3 per minute for a live birth chart reading.
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\section{INTRODUCTION}

Heat conduction in bulk materials at room temperature is usually described by the Fourier's law of thermal conduction, which implies that the heat carriers (phonons) undergo a diffusive process~\cite{kaviany_2008,ChenG05Oxford,liu_anomalous_2014}.
However, as the system size or dimension decreases, the phonons no longer transport diffusively~\cite{RevModPhysJoseph89,nanoscaleenergy2020,RevModPhysLibaowen,RevModPhys.90.041002,chang_breakdown_2008,PhysRevB.100.085203,PhysRevLett.107.095901,ZHANG2020,PhysRevB.101.075303,de_tomas_kinetic_2014,WangMr15application} due to the existence of the ballistic~\cite{volz1999,MajumdarA93Film} or hydrodynamic~\cite{beck1974,PhysRevLett.28.1461,PhysRevLett_secondNaF,PhysRevLett_ssNaf,machida2018,PhysRev_GK,lee_hydrodynamic_2015,cepellotti_phonon_2015,huberman_observation_2019} phonon transport.

The ballistic phonon transport happens as the system characteristic length is comparable to or smaller than the phonon mean free path~\cite{MajumdarA93Film,ChenG05Oxford,kaviany_2008}.
In the ballistic regime, the phonon transport is blocked by boundary scattering so that the thermal conductivity decreases significantly as the system size decreases~\cite{volz1999,MajumdarA93Film,li_thermal_2003,ju1999phonon,xu_length-dependent_2014}.

Different from the ballistic~\cite{volz1999,MajumdarA93Film} and diffusive phonon transport, the hydrodynamic phonon transport~\cite{Gurzhi_1968,PhysRevB.10.3546,ENZ1968114,beck1974,PhysRevLett.28.1461,PhysRevLett_secondNaF,PhysRevLett_ssNaf,machida2018,PhysRev_GK,lee_hydrodynamic_2015,cepellotti_phonon_2015,huberman_observation_2019} happens as the momentum-conservation phonon-phonon scattering dominates the heat conduction.
Usually, the phonon-phonon scattering is separated into two parts: the momentum destroying umklapp (U) scattering and the momentum-conservation normal (N) scattering~\cite{PhysRevB.102.094311,PhysRevB_SECOND_SOUND,lee_hydrodynamic_2015,cepellotti_phonon_2015,PhysRevB.99.085202}.
In bulk materials at room temperature, the U scattering happens frequently so that the phonon transport recovers the Fourier's law of thermal conduction.
However, in extremely low temperature or low-dimensional materials~\cite{Gurzhi_1968,ENZ1968114,RevModPhys.90.041002,PhysRevLett_Strontium_Titanate,
huberman_observation_2019,PhysRevLett.28.1461,PhysRevLett_secondNaF,PhysRevLett_ssNaf,machida2018}, the N scattering becomes much stronger than the U scattering~\cite{beck1974,PhysRevB.10.3546,PhysRev_GK,RevModPhys.90.041002,lee_hydrodynamic_2015,cepellotti_phonon_2015,PhysRevB.99.085202} and the phonon may transport like the fluid dynamics~\cite{prandtl_fluid,liu_new_2016vortex,GuoZl13LB}.

In the past decades, three phonon hydrodynamic phenomena have been predicted both theoretically~\cite{PhysRevB_SECOND_SOUND,lee_hydrodynamic_2015,cepellotti_phonon_2015,PhysRevB.99.085202,PhysRevB.99.144303,nanoletterchengang_2018} and experimentally~\cite{Dreyer1993,machida2020,PhysRevLett_Strontium_Titanate,
huberman_observation_2019,PhysRevLett.16.789,PhysRevLett.28.1461,PhysRevLett_secondNaF,PhysRevLett_ssNaf,machida2018}, such as the second sound~\cite{RevModPhysJoseph89,Dreyer1993,huberman_observation_2019,PhysRevLett_secondNaF,PhysRevB_SECOND_SOUND,PhysRevLett.28.1461,nie2020thermal}, phonon poiseuille flow~\cite{machida2018} and Knudsen minimum~\cite{PhysRevB.99.085202,nanoletterchengang_2018}.
The second sound is that the transient heat propagates like sound wave because the N scattering recovers the temperature wave equation~\cite{nie2020thermal,lee_hydrodynamic_2015,PhysRev_GK,PhysRev.148.766,huberman_observation_2019}.
Different from the second sound, the phonon poiseuille flow is a steady problem.
Namely, under a small temperature gradient, the heat flux flows in an infinite long and finite width ribbon like the fluid flows in a long tube with parabolic distributions~\cite{machida2018,PhysRevB.99.085202}.
In addition, as the ribbon width decreases, there is a minimum value of the non-dimensional thermal conductivity, which is the phonon Knudsen minimum~\cite{PhysRevB.99.085202,nanoletterchengang_2018}.

So except above three phenomena, can we find another novel phonon hydrodynamic phenomena?
It is well known that the transport of the heat carriers including phonon, electron and gas, can be described by the Boltzmann transport equation with similar mathematical form, although their physical meanings are different~\cite{ChenG05Oxford,kaviany_2008}.
In addition, as normal scattering dominates the heat conduction, the corresponding macroscopic Guyer-Krumhansl (G-K)  equation~\cite{PhysRev_GK,PhysRev.148.766,PhysRevB.10.3546,shang_heat_2020,PhysRevX.10.011019,WangMr15application} is similar to the Navier-Stokes equation in fluid dynamics~\cite{prandtl_fluid,GuoZl13LB}.
Hence, will phonon transport like the vortexes in fluid dynamics~\cite{prandtl_fluid,GuoZl13LB,liu_new_2016vortex} or viscous electron flow in hall bar geometries~\cite{levitov_electron_2016,bandurin2016b,chandra2019a,Danz_2020}?

Recently, the heat vortexes phenomena (\cref{vortexcurl}(a)) in the phonon hydrodynamic regime have been predicted in graphene ribbon~\cite{shang_heat_2020}.
However, a systematic comparison of the heat vortexes in the phonon hydrodynamic, ballistic and diffusive regimes is lacking.
Besides, the frequency dependent N/R scattering is not accounted in previous study~\cite{shang_heat_2020}.

In this work, motivated by the viscous fluid dynamics~\cite{prandtl_fluid,GuoZl13LB,liu_new_2016vortex} or viscous electron flow~\cite{levitov_electron_2016,bandurin2016b,chandra2019a,Danz_2020}, the heat vortexes in two different structures (porous~(\cref{porous}) and ribbon~(\cref{ribbon})) are studied in different regimes based on gray model~\cite{ChenG05Oxford}.
Note that the gray model is not limited by a specific material so that all variables are set to be dimensionless.
It is found that the heat vortexes can appear in both the ballistic and hydrodynamic regimes while disappear in the diffusive regime.
Namely, as long as the R scattering is not sufficient, the heat vortex can appear.
To make results more convincing, graphene~\cite{RevModPhys.90.041002,xu_length-dependent_2014,shang_heat_2020,lee_hydrodynamic_2015,cepellotti_phonon_2015} is taken as a specific example and the heat vortexes in graphene are investigated with different system sizes or temperatures accounting for the phonon dispersion and polarization.

The paper is organized as follows.
In Sec.~\ref{sec:BTE}, the phonon BTE is introduced briefly.
Then a theoretical analysis of the heat vortexes in different regimes is made in Sec.~\ref{sec:possibility}.
In Sec.~\ref{sec:results}, the heat vortexes in two-dimensional porous (\cref{porous}) or ribbon (\cref{ribbon}) structures are investigated.
Finally, a conclusion is made in Sec.~\ref{sec:conclusion}.

\section{Phonon BTE}
\label{sec:BTE}

In order to predict the thermal transport in two-dimensional materials correctly~\cite{RevModPhys.90.041002,xu_length-dependent_2014,lee_hydrodynamic_2015,cepellotti_phonon_2015}, the phonon Boltzmann transport equation (BTE) under the Callaway's dual relaxation model~\cite{PhysRev_callaway,wangmr17callaway,PhysRevB.99.085202,luo2019}  is used, i.e.,
\begin{equation}
\frac{\partial f}{\partial t }+ \bm{v} \cdot \nabla f= \frac{f^{eq}_{R}-f}{\tau_{R}} + \frac{f^{eq}_{N}-f}{\tau_{N}},
\label{eq:BTE}
\end{equation}
where $f=f(\bm{x}, \omega,p, \bm{K},t)$ is the phonon distribution function. $\bm{x}$ is the spatial position.
$\omega$ is the phonon angular frequency, $p$ is the phonon polarization, $\bm{K}$ is the 2D wave vector and assumed to be isotropic, i.e., $\bm{K}= |\bm{K}|\bm{s}$, $\bm{s}$ is the unit directional vector in 2D coordinate system, $t$ is the time.
$\bm{v} =\nabla _{\bm{K}} \omega$ is the group velocity calculated by the phonon dispersion~\cite{PhysRevLett.95.096105,PhysRevB.79.155413,ChenG05Oxford},
$\tau_{R}$ is the effective relaxation time of the resistive (R) scattering, which is a combination of all momentum destroying phonon scattering except the boundary scattering based on the Mathiessen's rule~\cite{ChenG05Oxford,kaviany_2008}, such as the umklapp (U) scattering, impurity scattering and isotope scattering~\cite{PhysRev_callaway,wangmr17callaway}.
$f^{eq}_{R}$ is the associated equilibrium state of the R scattering and satisfies Bose-Einstein distribution~\cite{ChenG05Oxford,kaviany_2008}, i.e.,
\begin{align}
f^{eq}_{R}(T)  = \frac{1}{ \exp \left( \frac{\hbar \omega }{k_B T} \right) -1 },  \label{eq:feqresistive}
\end{align}
where $k_B$ is the Boltzmann constant, $\hbar$ is the Planck constant reduced by $2\pi$ and $T$ is the temperature.
Different from the R scattering, the normal (N) scattering conserves momentum.
Its displaced equilibrium distribution function and the effective relaxation time are $f^{eq}_{N}$ and $\tau_{N}$, respectively, where
\begin{align}
f^{eq}_{N}(T,\bm{u})= \frac{1}{ \exp \left( \frac{\hbar \omega - \hbar \bm{K} \cdot \bm{u} }{k_B T} \right) -1 }, \label{eq:feqnormal}
\end{align}
where $\bm{u}$ is the drift velocity.
In addition, the phonon wave nature~\cite{PhysRevB.67.195311,ChenG05Oxford,kaviany_2008} is ignored in present study.

Usually, the phonon BTE (Eq.~\eqref{eq:BTE}) is rewritten into a deviational energy form as follows
\begin{align}
\frac{\partial e}{\partial t }+ \bm{v} \cdot \nabla e= \frac{e^{eq}_{R}-e}{\tau_{R}} + \frac{e^{eq}_{N}-e}{\tau_{N}},
\label{eq:energyBTE}
\end{align}
where the associated deviational distribution functions of energy density are~\cite{luo2019,wangmr17callaway}
\begin{align}
e &=\frac{ \hbar \omega  D (f-f_R^{eq} (T_0) ) }{2 \pi}, \\
e_R^{eq} &=\frac{ \hbar \omega  D (f_R^{eq}-f_R^{eq} (T_0) ) }{2 \pi}, \label{eq:eeqresistive} \\
e_N^{eq} &=\frac{ \hbar \omega  D (f_N^{eq}-f_R^{eq} (T_0) ) }{2 \pi}, \label{eq:eeqnormal}
\end{align}
where $D=|\bm{K}|/\left( 2 \pi |\bm{v}| \right)$ is the phonon density of states~\cite{luo2019,wangmr17callaway}.

Assuming a small temperature difference $\Delta T$ and a small drift velocity, i.e., $\Delta T/T_0 \ll 1$, $\bm{K} \cdot \bm{u} \ll \omega$, where $T_0$ is the reference temperature, then the equilibrium distribution functions (Eqs.~\eqref{eq:eeqresistive} and~\eqref{eq:eeqnormal}) can be linearized, i.e.,
\begin{align}
e^{eq}_{R}(T)  &\approx  C \left( T-T_0 \right) / 2\pi, \label{eq:feqR} \\
e^{eq}_{N}(T,\bm{u}) &\approx  C \left( T-T_0 \right) / 2\pi +CT \frac{\bm{K} \cdot \bm{u} }{ 2 \pi \omega},\label{eq:feqN}
\end{align}
where $C=C(\omega,p,T_0)$ is the mode specific heat at $T_0$, i.e.,
\begin{equation}
C(\omega,p,T_0)=2 \pi  \left.\ \frac{  \partial{ e^{eq}_{R}}}{ \partial{T}} \right|_{T=T_0} .
\label{eq:specificheat}
\end{equation}

The local temperature $T$ and heat flux $\bm{q}$ are calculated by taking the moments of the distribution function, i.e.,
\begin{align}
T  &=T_0+ \frac{ \sum_{p} \int \int e d\Omega d\omega } { \sum_{p} \int C d\omega  },   \label{eq:T}  \\
\bm{q} &=  \sum_{p} \int \int \bm{v} e d\Omega d\omega ,
\label{eq:heatflux}
\end{align}
where $p$ is the phonon polarization, $d\Omega$ and $d\omega$ are the integral over the whole 2D solid angle space and frequency space.
In addition, based on the conservation properties of the R (only energy conservation) and N (energy and momentum conservation) scattering, we have
\begin{align}
0  &=  \sum_{p} \int \int  \frac{e^{eq}_{R}(T_R) -e}{\tau_{R}}   d\Omega d\omega , \label{eq:Rconsertvation} \\
0  &=  \sum_{p} \int \int    \frac{e^{eq}_{N} (T_N)-e}{\tau_{N}}   d\Omega d\omega , \label{eq:Nconsertvation}  \\
0  &=  \sum_{p} \int \int  \frac{\bm{K} }{\omega } \frac{e^{eq}_{N} (T_N,\bm{u})-e}{\tau_{N}}   d\Omega d\omega , \label{eq:NMconsertvation}
\end{align}
where two local pseudo-temperatures $T_R$ and $T_N$ are introduced to ensure the conservation principle of the R and N scattering~\cite{WangMr15application,wangmr17callaway}.

By choosing specific heat $C$, reference temperature $T_0$, system characteristic length $L$ and group velocity $\bm{v}$ as the reference variables, the stationary phonon BTE (Eq.~\eqref{eq:energyBTE}) can be normalized as
\begin{align}
\frac{\bm{v} }{ |\bm{v}|} \cdot \nabla_{\bm{x^*}} e^* = \beta_R (e^{eq,*}_{R}-e^*) + \beta_N (e^{eq,*}_{N}-e^*), \label{eq:dimensionlessBTE}
\end{align}
where
\begin{align}
e^* &=\frac{e}{ C T_0 }, &\quad x^* &=\frac{x}{L},&\quad e^{eq,*}_R &=\frac{e^{eq}_R }{ C T_0 } , \notag \\
e^{eq,*}_N &=\frac{e^{eq}_N }{ C T_0 },&\quad \beta_N &=\frac{L}{|\bm{v}| \tau_N},& \quad \beta_R &=\frac{L}{|\bm{v}| \tau_R}.
\end{align}
It can be found that the heat conduction is totally determined by $\beta_N$ and $\beta_R$, which represent the ratio between the system size $L$ to the phonon mean free path of N and R scattering, i.e., $|\bm{v}|\tau_N$ and $|\bm{v}|\tau_R$, respectively.

\section{Theoretical analysis of heat vortexes in the ballistic, diffusive and hydrodynamic regimes}
\label{sec:possibility}

\begin{figure}
 \centering
 \includegraphics[scale=0.30,viewport=0 100 900 500,clip=true]{vortexcurl.pdf}
 \caption{(a) Schematic of the heat vortex~\cite{shang_heat_2020}, where $\bm{q}$ is the heat flux. (b) Given an arbitrary closed curve $l$, the curl of heat flux is $
\int_l \bm{q} \cdot d \bm{r} $~\cite{liu_new_2016vortex,prandtl_fluid}, where $d \bm{r}$ is the unit tangent vector of the closed curve $l$ in a clockwise direction.}
 \label{vortexcurl}
\end{figure}
The heat vortexes (\cref{vortexcurl}(a)) in the diffusive ($\beta_R \rightarrow \infty $, $\beta_N=0$), ballistic ($\beta_R=0$, $\beta_N=0$) and hydrodynamic ($\beta_R=0$, $\beta_N \rightarrow \infty$) limits are analyzed, respectively.
So that the separate thermal effects of R scattering, boundary scattering and N scattering can be found clearly.

For simplicity, the phonon gray model and Debye approximation~\cite{ChenG05Oxford,kaviany_2008} are used, i.e., $\omega= |\bm{v}| |\bm{K}|$.
Taking zero- and first- order moments of phonon BTE (Eq.~\eqref{eq:energyBTE}) leads to~\cite{WangMr15application,PhysRev_GK,PhysRev.148.766,shang_heat_2020}
\begin{align}
\frac{\partial E }{\partial t }+ \nabla \cdot   \bm{q}  &=0, \label{eq:btezero}  \\
\frac{\partial \bm{q}  }{\partial t } +  \nabla   \cdot   \bm{Q}  &= -\frac{\bm{q} }{\tau_R} + 0, \label{eq:rbtefirst}
\end{align}
where
\begin{align}
E & = \sum_{p} \int \int  e  d\Omega d\omega,  \\
\bm{Q} &=  \sum_{p} \int \int  \bm{v} \bm{v} e  d\Omega d\omega,
\end{align}
are the local deviational energy and the macroscopic flux of heat flux, respectively.

In the diffusive limit ($\beta_R \rightarrow \infty$, $\beta_N=0$), the R scattering dominates the phonon transport and causes heat dissipations based on Eq.~\eqref{eq:rbtefirst}.
At steady state, the distribution function can be approximated as
\begin{align}
e \approx e_{R}^{eq}- \tau_R \bm{v} \cdot \nabla e_{R}^{eq} .
\end{align}
So that the Fourier's law of heat conduction can be derived~\cite{GuoZl16DUGKS,ChenG05Oxford,MajumdarA93Film}, i.e.,
\begin{align}
\bm{q} = - \kappa_d \nabla T,
\label{eq:fourier2D}
\end{align}
where $\kappa_d=\frac{1}{2} C | \bm{v} |^2  \tau_R $ is the thermal conductivity in the diffusive limit.

{\color{black}{Note that the vortex definition and identification have
been a longstanding issue with many disputes~\cite{liu_new_2016vortex}, so that here the vorticity is introduced and discussed simply, which is defined as the curl of heat flux in present work~\cite{prandtl_fluid}.
Considering an arbitrary closed curve $l$ inside the thermal system (boundaries are not included), according to Eq.~\eqref{eq:fourier2D}, the curl of heat flux is (\cref{vortexcurl}(b))
\begin{align}
\int_l \bm{q} \cdot  d \bm{r} = -\kappa_d \int_l \frac{dT}{d \bm{r}} \cdot  d \bm{r} =0,   \label{eq:circulationzero}
\end{align}
where $d \bm{r}$ is the unit tangent vector of the closed curve $l$ in a clockwise direction.
This indicates that the curl of heat flux for an arbitrary closed curve inside the thermal system is always zero.
So that it can be concluded that there are no heat vortexes inside the system in the diffusive regime.}}

However, different from diffusive phonon transport, there is rare phonon-phonon scattering in the ballistic limit ($\beta_R=0$, $\beta_N=0$), so that Eqs.~\eqref{eq:energyBTE} and~\eqref{eq:rbtefirst} become,
\begin{align}
\frac{\partial  e   }{\partial t } + \bm{v} \cdot \nabla e  &=0 , \label{eq:bteballistic} \\
\frac{\partial \bm{q} }{\partial t } +  \nabla   \cdot   \bm{Q} &=  0. \label{eq:conserved}
\end{align}
Equation~\eqref{eq:conserved} indicates that there are no heat (or momentum) dissipations.
Besides, Eq.~\eqref{eq:conserved} is also satisfied in the phonon hydrodynamic limit ($\beta_N \rightarrow \infty ,~\beta_R=0$)~\cite{huberman_observation_2019,li2018a,cepellotti_phonon_2015,lee_hydrodynamic_2015}.
Because the N scattering conserves momentum and causes no thermal resistance~\cite{shang_heat_2020,PhysRev.148.766,PhysRev_GK}.
In other words, in both hydrodynamic and ballistic limits, the heat flux (or momentum) is conserved in the interior domain~\cite{Nanalytical,cepellotti_phonon_2015,lee_hydrodynamic_2015}, which is different from that in the diffusive limit.

Although Eq.~\eqref{eq:conserved} is satisfied in the ballistic and phonon hydrodynamic limits, there is something different.
In the ballistic regime~\cite{PhysRevB.99.085202}, the momentum conservation is satisfied due to rare phonon-phonon scattering.
The phonon mean free path is much larger than the local characteristic length.
At steady state, Eq.~\eqref{eq:bteballistic} becomes
\begin{align}
\bm{v} \cdot \nabla  e =0,
\end{align}
which indicates that for a given group velocity, the phonon distribution function does not vary with the spatial position until scattering with the boundaries.
Namely, the heat flux distributions inside the domain, which are obtained by taking the first-order moment of distribution function (Eq.~\eqref{eq:heatflux}), depend on the boundary conditions.
Therefore, whether the heat vortexes appear in the ballistic regime depends on the geometry settings or boundary conditions.

However, in the phonon hydrodynamic regime, the phonon mean free path is much smaller than the local characteristic length due to the frequent N scattering~\cite{Gurzhi_1968,beck1974,PhysRevLett.28.1461,PhysRevLett_secondNaF,PhysRevLett_ssNaf,PhysRev_GK,lee_hydrodynamic_2015,cepellotti_phonon_2015,huberman_observation_2019}.
At steady state, the phonon distribution function can be approximated as~\cite{PhysRev_GK,PhysRev.148.766,WangMr15application,shang_heat_2020}
\begin{align}
e \approx e_{N}^{eq}- \tau_N \bm{v} \cdot \nabla e_{N}^{eq}.
\label{eq:firstCEhydrodynamic}
\end{align}
It can be found that the phonon distribution function varies with spatial position unless $\tau_N =0$ according to Eq.~\eqref{eq:firstCEhydrodynamic}, which is different from that in the ballistic regime.

In addition, the macroscopic Guyer-Krumhansl (G-K) equation of hydrodynamic phonon transport in three-dimensional bulk materials has been derived based on the phonon BTE as $\beta_N \gg 1.0 \gg  \beta_R$ (Eq.~\eqref{eq:dimensionlessBTE})~\cite{PhysRev_GK,PhysRev.148.766,PhysRevB.10.3546,PhysRevX.10.011019,WangMr15application}.
A recent study has also extended it from three-dimensional into two-dimensional materials, i.e.,
(Eq.(33) in Ref~\cite{shang_heat_2020})
\begin{align}
\frac{\partial \bm{q} }{\partial t} + \frac{1}{2} C | \bm{v} |^2  \nabla  T + \frac{\bm{q}}{\tau_R} = \frac{1}{4}  | \bm{v} |^2  \tau_N \left( \nabla ^2 \bm{q}  \right) . \label{eq:GKshang}
\end{align}
If the R scattering is totally ignored, at steady state, Eq.~\eqref{eq:GKshang} becomes
\begin{align}
C   \nabla  T  = \frac{1}{2}  \tau_N \left( \nabla ^2 \bm{q}   \right). \label{eq:steadyGKshang}
\end{align}
It can be found that the right hand side of Eq.~\eqref{eq:steadyGKshang} is similar to the viscous term in Navier-Stokes equation~\cite{prandtl_fluid,GuoZl13LB}, so that it is possible to predict the heat vortexes in the phonon hydrodynamic regimes.
Besides, based on Eq.~\eqref{eq:steadyGKshang}, whether the heat vortexes appear in the phonon hydrodynamic regimes depends on the boundary conditions, too.

In a word, the heat vortexes disappear in the diffusive regime.
But they may appear in both ballistic and hydrodynamic regimes, which depends on the boundary conditions or geometry settings.

\section{Numerical results and discussions}
\label{sec:results}

In order to validate above theoretical analysis, the heat vortexes in two-dimensional porous and ribbon structures are introduced and investigated numerically in different regimes, respectively.
The implicit discrete ordinate method~\cite{ZHANG20191366,wangmr17callaway} is used to solve the phonon BTE, which can refer to Ref~\cite{wangmr17callaway}.
Detailed numerical solutions of the phonon BTE and associated boundary conditions~\cite{luo2019,ZHANG20191366} can be seen in Appendix~\ref{sec:callaway}.

\subsection{Heat vortex in two-dimensional porous materials}

\subsubsection{Problem description}

\begin{figure}
 \centering
 \includegraphics[scale=0.30,viewport=100 0 800 520,clip=true]{2D_porous.pdf}
 \caption{Schematic of a unit 2D square cell with a square pore in the center. The side lengths of the unit cell and square pore are $L$ and $d=L/2$, respectively. The boundaries of the unit cell and the square pore are periodic and adiabatic, respectively. A small temperature gradient is added across the $x$ direction so that the heat flows from the left to the right around the square pore, where $T_h=T_0 +\Delta T/2$, $T_c= T_0 -\Delta T/2$, $\Delta T/T_0=0.01$.}
 \label{porous}
\end{figure}
The heat conduction in periodic two-dimensional porous materials with aligned square pores is studied first.
Figure~\ref{porous} shows one of the unit 2D square cells with side length $L$.
In its center, there is a square pore with side length $d=L/2$.
The temperatures of the left and right sides of the cell are $T_h =T_0 +\Delta T/2$ and $T_c=T_0 - \Delta T/2$, respectively, where $T_0 = \left(T_h +T_c  \right)/2$ and $\Delta T= T_h - T_c$ are the average temperature and temperature difference in the domain.
In what follows, we set $\Delta T/T_0 =0.01$, which leads to a small temperature gradient along the $x$ direction.
The initial temperature in the interior domain is $T_0$.
The boundaries of the cell and the square pore are periodic (Eq.~\eqref{eq:BC2}) and adiabatic (Eq.~\eqref{eq:BC1}), respectively.
When phonons encounter the pore boundaries, they suffer from diffusely reflecting scattering due to the roughness of the pore surface (Eq.~\eqref{eq:BC1}).
So the heat flows from the left to the right around the square pore, which is like the gas flow around a cylinder in fluid dynamics~\cite{GuoZl13LB}.

\subsubsection{Gray model}

The phonon gray model and Debye approximation~\cite{ChenG05Oxford,kaviany_2008} are used. It is not limited by a specific material so that all physical variables are dimensionless.
Based on Eq.~\eqref{eq:dimensionlessBTE} and dimensional analysis~\cite{barenblatt1987dimensional}, as long as $\beta_N$ and $\beta_R$ are fixed, the results predicted by the phonon BTE will be fixed.
For simplicity, in the following simulations, only $\tau_R^{-1}$ or $\tau_N^{-1}$ is changed and all other variables are fixed, i.e., $L=2d=1$, $C=1,~|\bm{v}|=1,~T_0=1$.

Numerical results are shown in~\cref{porousBDH}.
It can be observed that heat vortexes appear in both ballistic and hydrodynamic regimes but disappear in the diffusive regime.
The results are in consistent with our theoretical analysis as mentioned in last section.
Besides, the vortex sizes of the ballistic phonon transport are larger than those in the hydrodynamic regime.

\begin{figure}
 \centering
 \includegraphics[scale=1.1,viewport=50 150 500 300,clip=true]{porousfig1.pdf}
 \caption{Normalized temperature contour $(T-T_c)/\Delta T$ and heat flux streamline in different regimes in porous structures (\cref{porous}) based on gray model. It is not limited by a specific material so that all physical variables are set to be dimensionless, where $L=2d=1,~C=1,~|\bm{v}|=1,~T_0=1$. $X$ and $Y$ are the coordinates. (a) Diffusive ($\tau_R=0.01,~\tau_N^{-1}=0$), (b) ballistic ($\tau_R^{-1}=0,~\tau_N^{-1}=0$) and (c) hydrodynamic ($\tau_R^{-1}=0,~\tau_N=0.01$).}
 \label{porousBDH}
\end{figure}

\subsubsection{Porous graphene}

\begin{figure}
     \centering
     \includegraphics[scale=1.1,viewport=50 150 500 300,clip=true]{porousfig2.pdf}
     \caption{The normalized temperature contour $(T-T_c)/\Delta T$ and heat flux streamline in porous graphene (\cref{porous}) with different $T_0$, where $L=2d=10~\mu$m, $X$ and $Y$ are the coordinates. (a) $T_0=30~\text{K}$, (b) $T_0=100~\text{K}$, (c) $T_0=300~\text{K}$.}
     \label{porousL10um}
\end{figure}
\begin{figure}
     \centering
     \includegraphics[scale=1.1,viewport=50 150 500 300,clip=true]{porousfig3.pdf}
     \caption{The normalized temperature contour $(T-T_c)/\Delta T$ and heat flux streamline in porous graphene (\cref{porous}) with different system sizes $L$, where $T_0 =300~ \text{K} $, $X$ and $Y$ are the coordinates. (a) $L=20$ nm, (b) $L=200$ nm, (c) $L=2~\mu $m.}
     \label{porousT300KL}
\end{figure}

To make the results more convincing, the porous graphene (isotopically pure) is taken as a specific example accounting for the phonon dispersion and polarization~\cite{PhysRevLett.95.096105,PhysRevB.79.155413,PhysRevB.93.235423}.
The detailed mathematical formulas of the phonon dispersion curves and scattering can refer to Ref~\cite{luo2019}.

First, the heat vortexes phenomena with different temperatures $T_0$ are predicted first for a given system size $L=10~\mu$m, as shown in~\cref{porousL10um}.
It can be observed that at $T_0 =30~\text{K}$, there are huge heat vortexes in front of the pore.
With the increase of $T_0$, the heat vortexes fade away gradually.
Because at low temperature, there are strong N scattering and weak R scattering in graphene~\cite{lee_hydrodynamic_2015,cepellotti_phonon_2015}.
As the temperature increases, the R scattering happens very frequently and dominates heat conduction, which recovers the Fourier's law of thermal conduction without heat vortexes.

Second, the heat vortexes phenomena with different system sizes $L$ are predicted at $T_0=300~\text{K}$.
From the results shown in~\cref{porousT300KL}(a)(b), it can be observed that as system size is smaller than or comparable to the phonon mean free path, the heat vortexes appear because boundary scattering (or ballistic phonon transport) dominates heat conduction and R scattering is weak in this length scale~\cite{RevModPhys.90.041002,xu_length-dependent_2014}.
As the system size increases, the R scattering increases so that the vortex sizes become small till zero at micron scale (\cref{porousT300KL}(c)).

In other words, the heat vortexes can appear in porous graphene at low temperature or nanostructures, where the R scattering is not sufficient~\cite{kaviany_2008,lee_hydrodynamic_2015,cepellotti_phonon_2015,RevModPhys.90.041002,xu_length-dependent_2014}.
And it is not easy to clearly distinguish the ballistic or hydrodynamic phonon transport in porous graphene only by the heat vortexes because the heat flux profiles in these regimes are very similar as shown in~\cref{porousT300KL}(a) and \cref{porousL10um}(a).

\subsection{Heat vortex in a 2D ribbon}

In previous studies~\cite{shang_heat_2020}, the heat vortexes in a graphene ribbon have been studied in the diffusive and hydrodynamic regimes with frequency independent relaxation time.
The heat vortexes appear in the hydrodynamic regime but disappear in the diffusive regime.
However, no comparison is made between the heat vortexes in the ballistic and hydrodynamic regimes.
And what will happen if considering the frequency dependent relaxation time and group velocity?

\subsubsection{Problem description}

In order to answer above questions, the heat vortexes in a 2D ribbon are investigated in all regimes.
The schematic of 2D ribbon is shown in~\cref{ribbon}, where its side length and width are $L_x$ and $L_y$, respectively.
The heat source and heat sink are set at the bottom and top of the ribbon with width $w=0.4 L_y$, respectively.
$d=2.5 L_y$ is the distance between the heat source (sink) and the left boundary of geometry.
The temperatures of heat source and heat sink are $T_h$ and $T_c$, respectively, where $T_h=T_0 +\Delta T/2$, $T_c= T_0 -\Delta T/2$, $\Delta T/T_0=0.01$.
The initial temperature inside the domain is $T_0$.
The isothermal boundary conditions (Eq.~\eqref{eq:BC3}) are used for the heat source and sink.
For other boundaries, the diffusely reflecting boundary conditions (Eq.~\eqref{eq:BC1}) are implemented.

\begin{figure}
 \centering
 \includegraphics[scale=0.30,viewport=100 130 800 450,clip=true]{ribbon2D.pdf}
 \caption{Schematic of heat conduction in a 2D ribbon, where the length and width of the geometry are $L_x$ and $L_y$, respectively. The heat source and heat sink are set at the bottom and top of the ribbon with width $w=0.4 L_y$, respectively. $d=2.5 L_y$ is the distance between the heat source (sink) and the left boundary of geometry. The temperatures of heat source and heat sink are $T_h$ and $T_c$, respectively, where $T_h=T_0 +\Delta T/2$, $T_c= T_0 -\Delta T/2$, $\Delta T/T_0=0.01$. So that the heat flows from the bottom to the top~\cite{shang_heat_2020}.}
 \label{ribbon}
\end{figure}
\begin{figure}
 \centering
 \includegraphics[scale=1.2,viewport=0 450 280 730,clip=true]{viscous_length_BH.pdf}
 \caption{Heat vortexes in a ribbon (\cref{ribbon}) with different $L_x$ and fixed $L_y$ based on gray model~\cite{ChenG05Oxford,kaviany_2008}. It is not limited by a specific material so that all physical variables are set to be dimensionless, where $L_y=1.0$, $C=1,~|\bm{v}|=1,~T_0=1$, $X$ and $Y$ are the coordinates.
  Colored background is the normalized temperature field ($(T-T_c)/ \Delta T$) and black line is the heat flux streamline. (a)(c)(e)(g) Ballistic phonon transport ($\tau_N^{-1}=0$, $\tau_R^{-1}=0$). (b)(d)(f)(h) Hydrodynamic phonon transport ($\tau_N=0.01$, $\tau_R^{-1}=0$). (a)(b) $L_x=3.0$. (c)(d) $L_x=4.0$. (e)(f) $L_x=5.0$. (g)(h) $L_x=6.0$.}
 \label{viscous_length_BH}
\end{figure}
\begin{figure}
 \centering
 \includegraphics[scale=1.2,viewport=0 360 300 710,clip=true]{viscous_length_GNR.pdf}
 \caption{Heat vortexes in graphene ribbon (\cref{ribbon}) with different temperature $T_0$ and fixed system size $L_x=3 L_y =30 \mu$m. $X$ and $Y$ are the coordinates. Colored background is the normalized temperature field ($(T-T_c)/ \Delta T$) and black line is the heat flux streamline.}
 \label{viscous_length_GNR}
\end{figure}

\subsubsection{Gray model}

The gray model~\cite{ChenG05Oxford,kaviany_2008} is used here to investigate the ballistic ($\tau_R^{-1}=\tau_N^{-1}=0$) and hydrodynamic ($\tau_R^{-1}=0,~\tau_N=0.01$) phonon transport, where the width is fixed $L_y=1$ and the system length $L_x$ is changed.
Besides, $C=1$, $|\bm{v}|=1$, $T_0 =1$ and the thermal effects of R scattering is totally ignored.

Figure~\ref{viscous_length_BH} shows the heat vortexes in 2D ribbon of ballistic and hydrodynamic phonon transport with different system length $L_x$ ($L_x=3.0,~4.0,~5.0,~6.0$).
It can be observed that heat vortexes appear on the left side of heat source (sink) in the ballistic regime (\cref{viscous_length_BH}(a)(c)(e)(g)).
And as system length $L_x$ increases, the heat vortexes appears on the right side of heat source, too.
It indicates that the appearance of heat vortexes is related to the geometric configurations (or boundary conditions).
While in the hydrodynamic regime (\cref{viscous_length_BH}(b)(d)(f)(h)), there are heat vortexes, too.
Besides, the vortexes sizes are smaller than those in the ballistic regime with the same $L_x$.
As system length $L_x$ increases, the number of vortexes increases from one to two on the left side of heat source.

{\color{black}{Furthermore, it is also interesting to find that the biggest vortex size in the hydrodynamic regime is approximated as $L_y$ (\cref{viscous_length_BH}(b)(d)(f)), namely, the width of 2D ribbon.
While in the ballistic regime~\cite{MajumdarA93Film}, the biggest vortex size increases with the ribbon length $L_x$.
It may attribute to that in the ballistic regime, the phonon mean free path is much larger than the system size ($L_x$ or $L_y$).
In other words, the phonons can transport a very long distance without any change so that the information exchange in the whole physical space is very efficient.
However, in the phonon hydrodynamic regime~\cite{PhysRevB.102.094311,PhysRevB_SECOND_SOUND,lee_hydrodynamic_2015,cepellotti_phonon_2015,PhysRevB.99.085202}, frequent N scattering blocks the efficient information exchange in the physical space and the phonon distribution function approximately reaches a local equilibrium (Eq.~\eqref{eq:firstCEhydrodynamic}), not globally.
According to Eq.~\eqref{eq:steadyGKshang}, the boundary conditions or confined structures affect the vortex size, too.
So that the vortex size in the hydrodynamic regime does not increase with length $L_x$ endless but confined by the ribbon width $L_y$.
Maybe the vortex size in the ballistic or hydrodynamic limits can be theoretically derived based on the phonon BTE, which is beyond the scope of this work.}}

\subsubsection{Graphene ribbon}

The heat vortexes in graphene ribbon are studied and the frequency dependent relaxation time and group velocity are both considered~\cite{PhysRevLett.95.096105,PhysRevB.79.155413,PhysRevB.93.235423,luo2019}.
Different from last simulations without consideration of R scattering, the heat vortex in graphene ribbon is a competition result of boundary scattering, R scattering and N scattering.

According to previous studies~\cite{huberman_observation_2019,cepellotti_phonon_2015,luo2019}, the heat conduction in graphene ribbon with micron size goes through ballistic, hydrodynamic and diffusive regimes in turn as the temperature $T_0$ increases from extremely low to room temperature.
Besides, the ballistic and hydrodynamic regimes are very close~\cite{lee_hydrodynamic_2015,cepellotti_phonon_2015}, which both happen at low temperature.
So that in the following simulations, the system size of graphene ribbon is fixed, i.e., $L_x =3 L_y =30~\mu$m, and then the temperature $T_0$ is increased from $10$ K to $300$ K gradually.

As shown in~\cref{viscous_length_GNR}, it can be observed that as temperature increases, the vortex sizes changes slightly as $T_0 \leq 80$ K.
As $T_0 \geq 100$ K, the vortex size decreases gradually until zero in room temperature.
In addition, the number of heat vortexes is always one as $T_0 \leq 200$ K.
Namely, the number of heat vortexes does not change in graphene ribbon as phonon transport through ballistic and hydrodynamic regime, which is different from the results shown in~\cref{viscous_length_BH} without the consideration of R scattering.
That's because the R scattering impedes the formation of vortexes to some extent although it is weak at low temperature.

Therefore, it can be concluded that the heat vortexes can appear at low temperature, namely, as R scattering is not sufficient.
Besides, based on the present results in graphene ribbon (\cref{viscous_length_GNR}), it is very hard to distinguish the temperature ranges of ballistic or hydrodynamic phonon transport only by the heat vortexes.

\section{Conclusion}
\label{sec:conclusion}

In this work, the heat vortexes in two-dimensional porous or ribbon structures (\cref{porous,ribbon}) at steady state are investigated based on the phonon BTE under the Callaway model.
It is found that heat vortexes can appear in both ballistic and hydrodynamic regimes but disappear in the diffusive regime.
Furthermore, as long as there is not sufficient R scattering, the heat vortexes can appear in present simulations.
In addition, the vortex size in the hydrodynamic regime is smaller than that in the ballistic regime.
In ribbon structures, as the ribbon length increases, the vortex size increases endless in the ballistic regime but is confined by the ribbon width in the hydrodynamic regime.

In real two-dimensional materials (e.g., graphene), it is very hard to distinguish the ballistic and hydrodynamic phonon transport clearly by the heat vortexes at steady state.
Therefore, more work is needed to be done to distinguish the ballistic and hydrodynamic phonon transport clearly by the vortexes in the future, which may be inspired by recent studies of unsteady viscous electron flow~\cite{chandra2019a}.

\section*{Acknowledgments}

This study was supported by the National Key R$\&$D Program of China (No. 2018YFE0180900).
The authors acknowledge Jing-Tao L\"u, Manyu Shang and Rulei Guo for useful communications in phonon hydrodynamics.

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*psst* is a Power System Simulation Toolbox written completely in Python. It is:

* Free software: MIT license
* Documentation: https://nrel.github.io/psst.

## Features

 Highlights include:

* Security Constrained Unit Commitment
* Security Constrained Economic Dispatch
* Integration with Pyomo, Pandas, Numpy, Scipy, Matplotlib, NetworkX and D3.js

*psst* supports:

* Wholesale power market modeling
* A Security Constrained Unit Commitment with Look Ahead
* A Security Constrained Economic Dispatch
* DAM and Hourly RTM

## Installation

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```shell
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For full instructions, see: https://nrel.github.io/psst.

## Credits

This work was developed as an extension to work done supported by Advanced Research Projects Agency-Energy (ARPA-E) and Iowa State University, in collaboration with Sandia National Laboratories, UC Davis and ISO-NE.
We also acknowledge the U.S. Department of Energy's (DOE) Wind and Water Power Technology Office (WWPTO) under the office of Energy Efficiency and Renewable Energy (EERE) for funding the development of this open source power system market clearing tool.

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Spatialized Decision-Making of Ukrainian Refugees in Germany
Forschungsschwerpunkt: Politik und Planung
Projektleitung im IRS: Dr. Valeria Lazarenko
Förderorganisation: Alexander von Humboldt Stiftung
Since Russia's full-scale war against Ukraine began, more than 13 million people have left their homes. Six million of them left the country, with around 900 000 taking their residency in Germany. Of those who fled the country, over 700 000 found their refuge in Germany, which makes Ukrainians the biggest ethnic group of refugees in the country. However, according to the IOM data, the vast majority of refugees intend to come back home in soon perspective instead of settling down in their current places of residency.
Ukrainian migrants in Germany (as well as in the other EU countries) pose a challenge to society regarding solidarity, acceptance and integration. So, it is crucial to understand how Ukrainian migrants – as a community – perceive themselves and what plans for the future they have.
From one point of view, the experiences of Ukrainian migrants are universal. As refugees from other war-torn countries, they landed in Germany, aiming to find safety and support. On the other hand, the situation of Ukrainian migrants is different, as most of them are women with children who have gone through the disruption of their families. This fact might have a sufficient impact on their life perspectives, identification, and integration attempts, requiring detailed exploration.
Since the German government has simplified the procedure for Ukrainian migrants to gain residence permits (according to §24 of the EU Residence Act), the integration process should go more smoothly for them than for refugees undergoing standard registration procedures. On the other hand, the need to go through the lasting bureaucratic system to b receive social benefits might also impact the decision of migrant women to stay in Germany or go back to Ukraine. Such decision-making is also interconnected with identification and planning of the future (e.g. those who refuse to identify as refugees and aim to move back to Ukraine as soon as possible are also refusing integration into German society). Complex research on the spatialized decision-making of Ukrainian migrant women will help to understand better the integration process, keeping the existing identities and constructing new ones. The research will also enrich the existing knowledge about the decision-making on whether to migrate or repatriate and thus will contribute to predicting the tendencies of integration of Ukrainian migrants into German society.
The objective of this project is threefold. It seeks to explore how Ukrainian refugee women make their decision to come to German cities, how they see their stay in mid-term and long-term perspectives, and how they consider coming back to their home country. It will analyze German migration policies towards people displaced from Ukraine and its dynamics since March 2022, to understand how the policies and procedures can impact decision-making and what lessons can be learned from the case with Ukrainian migrants. And it will conceptualize the sense of temporality and "resettlement identity" in a context of displacement, decision-making and integration.
Photo: Ukranian refugees crossing into Poland, 2002 Source: Czarek Sokolowski/AP Photo via Wikimedia Commons
Philipp Schwartz-Initiative der Alexander von Humboldt Stiftung