Whatever your horse is afraid of it, your general approach will be similar, so it's good to know what to do!
Firstly you need to recognize that your horse is feeling scared - this sounds obvious, but the early signs are quite subtle. Then you need a clear approach to resolving her fear.
The early signs of tension, before the horse's head comes up and she starts trying to escape or defend herself, are quite subtle, but easy to recognise once you know what you're looking for.
The lip line going straight - usually there is a little downward curve at the corner of the lips, this disappears as the horse becomes tense.
The chin looking tight or 'tucked' - in a very relaxed horse, the lower lip loose, sometimes floppy and protruding, and the profile of the muzzle looks sort of square. As they become tense, the chin tightens and goes back a little, so the lower lip looks more tense, and the profile is not square any more, the chin slopes backwards from the upper lip.
The look of the nostrils may also change - the 'piping' round the edge of the nostrils is round and soft looking when the horse is relaxed, but the edges will become long and thin when the horse is tense.
In the picture on the left, the nostril edges are thin (you can see them clearly), the lip doesn't curl down at the corner, and the chin is tucked back just a little - the early signs of tension.
The more obvious signs, such as the head coming up, have not yet appeared. In comparison, in the picture on the right, the nostrils edges are soft, chin relaxed (see the bottom lip protrudes a little), and there is a clear downwards curl at the corner of the lips.
Trainers, but it is helpful to know what the underlying processes are and the effects they have on your horse.
This involves exposing the subject (be it human, horse or any other animal) to the source of their fear, with no option to get away from it, for the entire training session. This can be a very fast method, and is often used (although trainers sometimes don't explain what they are doing!).
Welfare - exposing the horse to fear and causing unnecessary stress.
Safety - for the person and the horse - risking provoking extreme reactions in the horse is clearly unsafe!
Practical reasons - the horse may succeed in escaping what it is afraid of, learning exactly the wrong thing.
There are other reasons, such as the horse's reduced ability to learn when scared and that they do not necessarily become comfortable with their source of fear, but only learn that resistance is useless, but if the first three don't convince you this is a poor option, probably nothing will! There are a few situations where there is no option but to use this technique, such as a medical emergency where treatment has to be applied instantly and sedation is not possible.
This is at the other end of the scale and, sometimes with minor modifications, would be my preferred option. The exact stimulus causing the horse's fear is identified and the horse is then exposed to a much reduced version of this stimulus, at a level where it only causes mild worry. For example, if the horse is scared of the sound of clippers, record the noise of the clippers, and play it at low volume for just one second so she is just a little worried. Repeat until the horse in no longer worried at all, then repeat some more! Then increase the volume, or lengthen the time the noise is played, and so on. This can take quite a long time, and you may need some help to identify exactly what your horse is scared of, and to put together an effective training plan to increase the stimulus at the right speed, but it is very safe, effective, kind to the horse, and provides a long lasting cure.
Other proposed methods of dealing with fear, such as advance and retreat, basically use a combination of thesetechniques and have varying degrees of success. I would always go for the option which is safest, kindest to the horse, and most effective in the long term - systematic desensitisation.


Autumn is a time of transition. The days have begun to shorten, the air has started to cool, leaves have begun to shed their leaves in preparation for the colder climates, and a crispness in the air is present when we leave the warmth of our homes. It has become clear; summer has made its exit through the small back door. As this shift in season brings many changes, it can be tough to think that the long, bright days have gone by once again.
We tend to hold on to these joyful days of sunshine, the BBQs in the park, and the warm social gatherings making this change feel as though we are peering into a dark time ahead. The schedule begins to fill once again as colleagues and friends return from summer holiday, and we begin to wonder how we will fit it all into our days. Autumn is a time of transition.
It is during these times, these times of change, when it is most important to check in and ask how to make things a bit softer. How can we make this change a bit gentler? How can we embrace this seasonal shift in a positive light?
Let's shift our mindsets and begin looking forward. The warm, bright season of summer may be behind us, but the season of knitted sweaters, woolen socks, and warm tea has arrived. Let's get out the soup recipes, light some candles, and turn on the kettle. Although it may be cold outside, we can still create warmth within our bodies, within our hearts, within our minds.
Instead of regretting the items left unchecked on our summer bucket list, why don't we make a new list of how to finish the year strong? Change our perspective from times of loss, to times of abundance, making the most of these next few months.
In order to make this transitional time more appealing, Hot Yoga Copenhagen is inviting you to a 95 minute Candlelit Yin Workshop followed by soup during the evening of Sunday October 8th. From 18.30-20.00 our very own instructor, Maiken, will be guiding us through a series of poses focusing on stimulating the lung and large intestine meridians. These meridians govern the respiratory and eliminatory functions of the body, which both Chinese medicine and Yin Yoga focus on during this time of year. We will listen to gentle music as Maiken guides our bodies and minds through ways to deal with this seasonal shift in the most positive way. We will turn our focus inwards as the dancing lights of candles and soft tunes of gentle music calm us. Our intention is to leave you soothed, refreshed and ready for this new exciting season ahead.
After the workshop, we will warm our bellies with a tasty, warm veggie soup. The workshop is free of charge for members and you can book it in our Hot Yoga Copenhagen App. Make sure to select Central as the location.


Peavey make guitar, bass, acoustic, keyboard and pedal-steel amps being best known for their range of 'metal' guitar amps.
Their range of guitar amps includes the iconic 6505, Classic, Invective, Vypyr, TransTube and Mini Heads.
Established in 1965, Peavey are an innovation company with a total dedication to quality and reliability. Today Peavey Electronics Corporation is one of the largest makers and suppliers of musical instruments, amplifiers and professional audio systems in the world—distributing more than 2,000 products to more than 130 countries.
Well known for their amps they also make electric, bass and acoustic guitars, speakers and pro-audio equipment.


Hotels are amongst the most visible and important aspects of a country's infrastructure. Hotel industry is a closely linked one to the tourism industry. A number of factors like promotion of tourism and rapid industrial progress have given a boost to hoteliering. The recent liberalization of trade and opening up of economy will further lead to revolutionary growth in this sector.
With increasing globalization, career opportunities in this field are not only limited within the country but there are chains of hotels which operate internationally providing scope of a career abroad. It is a glamorous profession which has a bright future. With the growth of hotel industry propelled by foreign and domestic tourism and business travel, the demand for well trained quality personnel too has grown impressively.
As the wish to travel on holiday has made the hotel industry a very competitive one.
Hospitality and travel organizations are on the lookout for talent to manage their hotels, restaurants, resorts, spas, travel agencies and airline companies worldwide. The hospitality industry is one of the largest and most dynamic industries in the world. With the fast growth and tremendous opportunities throughout Asia, especially in India, currently the hospitality industry in this region has great demand for professionals with management skills, industry expertise, and international experience and perspective.
With the right attitude, an international qualification from a reputed hospitality school, and global work experience, the sky is the limit.


Like spring, fall is one of the best times of the year to build a deck in your backyard because the outside temperatures aren't extreme.
But despite how great the weather is for a new home project and how much you like the look of a real wood deck, you've probably heard a few things about wood decks that are keeping you from realizing your backyard dream.
So are the claims you've heard about wood decks really true? Or are they a work of fiction? Here are five myths about wood decks officially busted.
Real wood decks aren't eco-friendly. If hardwood trees stopped growing today and harvesting continued at the same rate it is now, hardwood timber supply would last more than 75 years. Lumber companies in North America are committed to sustainability and make sure to regenerate the trees that are harvested for wood.
Wood isn't an energy-efficient product. Wood is one of the most energy-efficient materials you can use to make products such as wood slabs for your deck. In fact, companies that make products from aluminum, cement, brick, glass, and plastic use up to 126 times more energy than those who make products from wood.
Wood decks are difficult to maintain. A wooden deck isn't any more challenging to maintain than any other deck. Every deck you install will need maintenance and you'll need to clean and seal the surface of your deck at regular intervals. This keeps dirt from getting a chance to build up.
Wood decks have splinters. It's true that repeated expansion and shrinkage due to the humidity in the air can cause splintering, but a periodically applied sealer can prevent cracks and splinters from developing.
Wood decks don't last for very long. Real wood decks can last up to 20 years with regular sealing and cleaning. But it ought to be noted that certain factors such as lack of maintenance and poor ventilation can cause your deck to age prematurely.
Where can I find wood slabs for sale?
The average single-family home contains over 13,000 board feet of lumber. A wooden deck naturally fits in with the architecture.
If you're looking for wood slabs for sale to build your deck or another new home project, Horizonwood has the slab lumber and more specialty wood products you need. To learn more about our hardwood lumber and timber prices, contact Horizonwood today.
Which Types Of Wood Are Recommended For A Desk?


We are now looking for motivated, energetic individuals to join our team. If you enjoy a fast paced environment with an amazing culture, look no further!
If you feel like you have what it takes to be an employee at Jimmy John's, please apply today!


I started 2017 with a long list of goals and I decided to read the book #GirlBoss to inspire myself to work towards them. I started reading the book in February however, my phone was stolen and everything was just a mess at that time. This lead to me feeling uninspired and I stopped reading the book. A lot of things have changed and recently, I decided to start reading the book again.
In the book, Sophia Amoruso, speaks about her journey as an entrepreneur and how her company, Nasty Gal, got to where it is today. The book also features a few other female entrepreneurs which I really found interesting because it shows what being a #GirlBoss means to different people.
The book contains valuable information about starting and running a business, dealing with finances, the importance of staying true to yourself and a whole lot more. Sophia shows that there is no correct way to start a business. It's all about working hard to get what you want.
I'm ready to join the #GirlBosses and take over the world!


text:
Zara se poate referi la:
 orașul Zadar din Croația
 districtul Zara din Turcia
 firma Zara (magazine de îmbrăcăminte)

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text:
Stefanos Tsitsipas beats Roger Federer to reach first Grand Slam quarter-final at Australian Open
Twenty-year-old will next play Roberto Bautista Agut after the Spaniard defeated Marin Cilic
By Mathieu Wood
Defending champion Roger Federer is out of the Australian Open after losing to Stefanos Tsitsipas in the fourth round
Stefanos Tsitsipas came of age to defeat two-time defending champion Roger Federer and reach his first Grand Slam quarter-final in thrilling fashion at the Australian Open.
The 20-year-old recovered from losing the first set on a tiebreak, held off four set points in the second set and saved all 12 break points he faced through unrelenting determination in front of a captivated Melbourne crowd.
Tsitsipas, who was facing Federer for the first time in a tour-level match, eventually knocked out the six-time winner 6-7 (11-13) 7-6 (7-3) 7-5 7-6 (7-5) after three hours and 45 minutes to become the first Greek man or woman to reach the last eight at a Grand Slam.
Tsitsipas won the Next Gen ATP Finals last season
The world No 15 will now meet another first-time Grand Slam quarter-finalist Roberto Bautista Agut after the Spaniard beat last year's runner-up Marin Cilic 6-7 (6-8) 6-3 6-2 4-6 6-4.
"There's nothing really I can say to describe this," Tsitsipas said. "I'm the happiest man on earth right now."
Tstsipas was elated after his historic victory
Federer was aiming for his third consecutive title at Melbourne Park but failed to take his chance to open a two-sets-to-love lead against one of the rising stars of the game in a gripping encounter of two similar game styles.
"Roger is a legend of the sport, [I have] so much respect for him. He showed such good tennis over the years. I've been idolising him since the age of six," Tsitsipas, who was not born when Federer made his professional debut, added.
"It was a dream come true just being on Rod Laver facing him. Winning at the end, I cannot describe it."
Federer saw four set points for a two sets to love lead saved by the Greek
The drama began in the opening game of the match as Tsitsipas was given two time violations, resulting in the loss of a first serve, but he saved two break points to stave off the threat of an immediate break.
Tsitsipas showed no signs of nerves on the big stage and took the first set to a tiebreak but, at a crucial stage, appeared to lose his focus due to someone shouting from the crowd mid-rally and Federer ended up with the initiative.
A similar pattern of serve dominating proceedings followed in the second set before Tsitsipas saved four break points across two extended service games and then a further four set points at 4-5 as Federer appeared on course to increase his command.
Federer had not lost a set in his three previous matches
Federer was made to rue those missed opportunities when Tsitsipas proved the more consistent player in the second tiebreak of the match, before he saved a further two opportunities midway through the third set.
The Swiss had misfired off the forehand wing - targeted by Tsitsipas - and the 14th seed moved ahead for the first time in the match when, having earlier missed out on his first two break-point chances, sealed the third set against the serve.
Halep sets up Serena contest
Australian Open: Women's Draw
Serena consoles opponent after win
Jamie Murray: Doubles undervalued by LTA
The Rod Laver Arena knew a huge story was unfolding and Tsitsipas, who called the trainer for a massage, refused to drop his levels, even under intense pressure from Federer, and secured a historic win following a third tense tiebreak.
Federer acknowledges the Rod Laver Arena crowd after his defeat
We have the Australian Open covered from all angles via our website skysports.com/tennis. On the move? Head to our app for mobile devices and iPad, or follow our Twitter account @SkySportsTennis to join in the conversation.
Australian Open talking points
A classic event for wheelchair tennis
VOTE: Federer, Nadal or Djokovic?
Djokovic: I want to overhaul Federer
Nadal: I don't have it yet

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Space >
The Universe >
Theories of the Universe: Chunks of Uncertainty
Chunks of Uncertainty
Theories of the Universe
Just a Little Math Won't Hurt
Born of Probability
They're Complementary After All
Universal Constants
Angular momentum can be thought of as momentum moving in a circle. It's what keeps a ball moving on the end of a rope as you spin it over your head. In classical physics it is defined by the mass and the speed at which it is spinning. In quantum mechanics, the angular momentum, like everything else, is quantized.
All quantum physicists will agree that they can't explain quantum theory and that it doesn't make sense. So why would they use something to define the fundamental structure of the universe if they don't understand what it is? Well, physicists who work in the field use the straightforward formulas handed down to them by the brilliant minds that developed them, but often don't understand why they work or even what they mean. But what is known about it has produced remarkably accurate results. And that's why it's used. Some even feel that quantum mechanics offers some of the most accurate numerical predictions that science has ever developed. And until something else comes along, the mathematical system used to define the quantum world is all that is available. For now, let's pick up where we left off in the last section and follow the unfolding of quantum cosmology.
Waves of Matter
Cosmonotes
One of the reasons de Broglie wanted to develop a mechanical model was to show that the theories developed so far could be verified by experimental means. Up until then, the Rutherford-Bohr atomic model was just theory; no one really knew whether the atoms looked like that. If he could show that his predictions could be confirmed by experiment, it would solidify the new ground on which quantum theory was developing.
Neils Bohr came up with a description of how light was radiated from inside an atom. He also has shown why an atom was stable. His explanation of what the structure of the atom was like was close to but not quite like the one envisioned by Rutherford that we touched upon earlier. Remember that he theorized that the atom's structure was very much like a planetary system with the electrons orbiting the nucleus the way planets orbit the Sun. Bohr adopted this basic configuration, but couldn't imagine the electrons orbiting the nucleus as some cosmic cloud that was indefinable. So he had arranged the orbiting electrons into layers or shells. Have you seen Russian nesting dolls or Chinese stacking boxes? In those you have one complete doll or box contained within another. Every time you open up one there's a smaller one inside. The shells Bohr described were just like that, except that each shell had a specific number of electrons.
Bohr was able to calculate mathematically the diameter of each electron orbit along with the maximum number of electrons in each shell. The angular momentum of the electron in orbit was counteracted by the attraction of the nucleus. In other words, since unlike electrical charges attract each other, the positive charge inside the nucleus attracted the negative charge of the electron. This theory explained the structure of the atom and why it remained stable.
A French Prince Discovers Matter Waves
When you create a standing wave with a jump rope, each point on the rope where there is no movement, a resting place, is called a node. It's the point at the end of each wave and also the point between the waves that are moving up and down. For example, there are two nodes on the lowest frequency standing wave (a half wave), the two endpoints of the rope. The next higher frequency, (a whole wave) has three nodes, the ones on each end and a third in the middle, which is the point that separates the crest from the trough. The next higher frequency (1 1/2 waves) has four nodes, the two end points and two in the middle and so on. Get the idea? As the number of nodes on the rope increases, the frequency of the standing wave increases. If this were a vibrating violin string, the pitch would also increase.
In 1923, a graduate student at the Sorbonne in Paris, Prince Louis de Broglie (1892-1987), introduced the remarkable idea that particles may exhibit wave properties. He had been strongly influenced by Einstein's arguments that light had a dual nature. He was also deeply impressed by Einstein's particles of light that could cause the photoelectric effect (knock electrons out of metal) while also producing the interference patterns caused by waves as in the double slit experiment. He proposed one of the great unifying principles in quantum physics. He was convinced that the wave/particle duality discovered by Einstein in his theory of light quanta was a general principle that extended to all forms of matter. In other words, the propagation of a wave is associated with the motion of a particle of any kind—photon, electron, proton, or any other.
De Broglie wished to develop a mechanical explanation for the wave/particle duality of light and to extend this to all forms of matter. He needed to find a mechanical reason for the photons in the wave to have an energy that was determined by the frequency of that wave.
De Broglie noticed a connection between the angular momentum of the electron in a Bohr orbit and the number of nodes in a standing wave pattern (remember you created those with the jump rope in the last section). The orbiting electrons could only have one unit of h (Planck's constant) or two units, etc. Could these discontinuous changes in the electron's angular momentum, these changes in the amount of h allowed, be due somehow to a similar change in standing wave patterns?
De Broglie realized that the Bohr orbit could be seen as a circular violin string, like a snake swallowing its own tail. Would the orbit size predicted by his standing matter waves correspond to Bohr's calculated electron shells? What would his wave do if confined to a circle? Well, what he discovered was that his matter waves fit Bohr's orbits exactly. And when he calculated the wavelength of the lowest orbit, he discovered another astonishing mathematical connection between the wave and the particle. The momentum of the orbiting electron equaled Planck's constant divided by the wavelength.
Excerpted from The Complete Idiot's Guide to Theories of the Universe © 2001 by Gary F. Moring. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
Theories of the Universe: Just a Little Math Won't Hurt


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Glasgow Caledonian University (GCU) is one of Scotland's largest universities.
Tracing its origins back to 1875, this modern, innovative and forward-thinking university has a large, self-contained campus in the city centre, with many purpose built facilities making it one of the best learning environments in the UK.
Courses and students
The university has three academic schools:
Glasgow School for Business and Society
School of Computing, Engineering and Built Environment
School of Health and Life Sciences.
Glasgow Caledonian University is the only institution in Scotland to offer courses in Orthoptics, Optometry and Ophthalmic Dispensing Management. The university has its own eye clinic providing clinical training facilities. The clinic is open to the public, and patients are examined by students under the direct supervision of experienced and qualified clinical staff.
The university has 1,600 staff teaching over 20,000 students from over 100 countries.
You could spend part of your undergraduate course abroad through the European Erasmus+ programme or on exchange with a university in North or South America, Australia, Hong Kong, Malaysia or Vietnam.
he Sir Alex Ferguson Library is home to the university library and student support services hub, based in the Saltire Centre it offers students: 1,800 internet-connected study spaces; 600-seat learning cafe; 370,000 RFID-tagged books; 500 computers; laptop borrowing facility; and free WI-FI.
The Arc is the modern health and fitness centre which has fitness classes, a well equipped gym, sports clubs and a hair salon and spa.
The university has a simulated hospital ward, a high-tech broadcasting suite and a fashion factory.
The university is ranked 3rd in the UK for Optometry, Opthalmology and Orthoptics in the Complete University Guide 2022.
In the same guide, the university is in the top 10 for seven other subject areas: Biomedical Sciences; Counselling, Psychotherapy and Occupational Therapy; Information Technology and Systems; Medical Technology and Bioengineering; Paramedic Science; Physiotherapy; and Tourism, Transport, Travel and Heritage Studies.
GCU is ranked 2nd for Social Work in the Guardian University Guide 2021.
The university climbed 21 places in the Times and Sunday Times Good University Guide (2021).
Courtesy of Glasgow Caledonian University
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Be at least 15 years of age by the last day of the course attended.
Successfully complete a pre-course screening.
Pass a written test with the score of at least 80%.
100 yards front crawl or breaststroke or combination of both.
Brick Retrieval Test - Swim 20 yards (front crawl or breast stroke), surface dive to a depth of 7-10 feet, retrieve a 10-pound object, and return with it to the surface, and swim back 20 yards.
Tread water for 2 minutes using your legs only.


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Q: Stipulating how Excel treats data when I export it into R? I am using huxtable to process data so that it is formatted in the same way to a pre-existing publication. When I export the data to Excel, however, my data appears with decimal points (it appears correctly in r). How do I get Excel to read the data exactly as it appears in r? Is there a way of coding in stipulations for how Excel reads your data into r?
Thanks and sorry if this has been asked before, I had a look and couldn't find anything.

A: Have you tried using number_format? If that doesn't work you might want to file an issue at https://github.com/hughjonesd/huxtable/issues. (Disclaimer: I wrote the package.)
Example:
library(huxtable)
foo <- as_huxtable(matrix(rnorm(4), 2, 2))
number_format(foo) <- 0   # zero decimal places
as_Workbook(foo)          # should have the right formatting

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\section{Introduction}
Saturated liquid $^4$He transits to the superfluid phase (known as He II) below about 2.17 K \cite{Tilley-book}. In He II, two miscible fluid components co-exist: an inviscid and zero-entropy superfluid component (i.e., the condensate) and a viscous normal-fluid component (i.e., the collection of thermal excitations). This two-fluid system possesses many fascinating thermal and mechanical properties \cite{landau-1987}. For instance, He II supports two distinct sound modes: an ordinary pressure-density wave (i.e., the first sound) where the two fluids move in phase, and a temperature-entropy wave (i.e., the second sound) where the two fluids move oppositely. Furthermore, heat transfer in He II is via an extremely effective counterflow mode instead of convection \cite{vansciver-2012}: the normal fluid carries the heat away from a source at a velocity $v_n$=$q/\rho sT$, where $q$ is the heat flux, and $\rho$ and $s$ are the He II density and specific entropy, respectively; while the superfluid moves in the opposite direction at a velocity $v_s$=$-v_n\rho_n/\rho_s$ to balance the mass flow (here $\rho_n$ and $\rho_s$ are the densities of the respective fluid components). When the relative velocity of the two fluids exceeds a small critical value, a chaotic tangle of quantized vortex lines can be spontaneously created in the superfluid, each carrying a quantized circulation $\kappa\simeq10^{-3}$ cm$^2$/s around its angstrom-sized core~\cite{donnelly-1991}. A mutual friction force between the two fluids appears due to the scattering of the thermal excitations off the quantized vortices \cite{vinen-1957_Proc.R.Soc.Lond.A_II}. This mutual friction can profoundly affect the heat transfer and turbulence characteristics in both fluids \cite{marakov-2015_Phys.Rev.B, Gao-2016-JETP, Gao-2017-PRB, Gao-2017-JLTP, Gao-2018-PRB, Bao-2018-PRB,Mastracci-2018-PRF}.

Due to its low temperature and extraordinary heat-transfer capability, He II has been widely utilized in scientific and engineering applications such as for cooling superconducting particle accelerator cavities, superconducting magnets, and satellites \cite{vansciver-2012}. Many of these applications involve transient heat transfer in He II, a process that is complicated due to the interplay of counterflow, second-sound emission, and vortex nucleation. There have been extensive experimental and numerical studies of one-dimensional (1D) transient heat transfer of He II in a uniform channel driven by a planar heater, due to the simplicity of this geometry~\cite{fiszdon-1989_J.LowTemp.Phys.,fiszdon-1990_J.FluidMech.,shimazaki-1995_Cryogenics,hilton-2005_J.LowTemp.Phys.,zhang-2006_Int.J.HeatMassTransf.}. These studies have revealed that the transient heating from the heater generates a second-sound pulse that propagates in He II. A counterflow establishes in the pulse, which produces tangled quantized vortices. These vortices then attenuate the second-sound pulse, converting the energy carried by the pulse to the internal energy of He II. This heated region in front of the heater is termed as the thermal layer. When the heat flux is relatively high, the continuous attenuation eventually curtails the second-sound pulse to a limiting profile \cite{shimazaki-1995_Cryogenics}, and the heat produced by the heater largely gets deposited in the thermal layer which gradually diffuses along the channel following the second-sound pulse.

It has been recognized that the heat transfer of He II in nonhomogeneous geometries can exhibit new features. For instance, Fiszdon \emph{et al.} conducted transient heat transfer experiments in He II using cylindrical heaters~\cite{fiszdon-1990_J.FluidMech.}. They found that a rarefaction tail of the second-sound pulse can develop, which exhibits a drop in temperature. The thermal layer in this geometry can be significantly suppressed as compared to that in the planar geometry. These observations were examined and reproduced in numerical simulations by Kondaurova, \emph{et al.}~\cite{kondaurova-2008_J.LowTemp.Phys.,kondaurova-2017_J.LowTemp.Phys.,kondaurova-2020_LowTemp.Phys.}. Nevertheless, there lacks a systematic characterization of the thermal-layer dynamics and how the heat energy is divided between the thermal layer and the propagating second sound. Producing this knowledge could benefit applications pertinent to cylinder shaped systems cooled by He II, such as superconducting transmission lines and magnet coils~\cite{maksoud-2010_IEEETrans.Appl.Supercond,xavier-2019_IEEETrans.Appl.Supercond}. An emergent effort in developing hot-wire anemometry for studying quantum turbulence in He II~\cite{duri-2015_Rev.Sci.Instrum} has further strengthened this need.

Besides the cylindrical geometry, transient heat transfer of He II in spherical geometry is also relevant to practical applications. In particular, it has been known that superconducting accelerator cavities cooled by He II can quench due to transient heating from tiny surface defects~\cite{conway-2017_Supercond.Sci.Technol.}. Locating these surface hot spots for subsequent defect removal is the key for improving the cavity performance. Our team has recently developed an innovative molecular tagging technique for locating surface hot spots via tracking thin lines of He$^*_2$ molecular tracers~\cite{bao-2019_Phys.Rev.Applied, bao-2020_Int.J.HeatMassTransf.}. These tracers move with the normal fluid~\cite{Guo-2009-PRL,Guo-2010-PRL,Gao-2015-RSI,Guo-2014-PNAS}, and therefore the transient radial heat transfer from a hot spot can lead to line deformations that contain accurate information about the spot location. In order to extract this information, it is critical to understand how the heat energy is partitioned between the thermal layer and the second-sound pulse~\cite{bao-2019_Phys.Rev.Applied}. However, despite some limited studies on steady-state vortex distribution near a spherical heater \cite{inui-2020_Phys.Rev.B,Varga-2019-JLTP}, the transient behaviors of the thermal layer and its interaction with the second sound in this geometry have remained largely unexplored~\cite{kondaurova-2008_J.LowTemp.Phys.}.

In this paper, we present a numerical study of transient heat transfer in all three (i.e., planar, cylindrical, and spherical) geometries in He II. Our goal is to examine and compare the heat-transfer characteristics in these geometries so that a better understanding of the energy partition and thermal-layer dynamics can be achieved. The paper is organized as follows. In Sec.~\ref{sec:model}, we introduce our model, which is based on the governing equations of the two-fluid system and the Vinen's equation for vortex-density evolution. In Sec.~\ref{sec:validation}, we validate our model by comparing the simulation result of transient heat transfer in planar and cylindrical geometries with the experimental measurements by Fiszdon \emph{et al.}~\cite{fiszdon-1990_J.FluidMech.}. The systematic study of the heat transfer in all geometries is discussed in Sec.~\ref{sec:result}. We first present in Sec.~\ref{sec:wave} the calculated spatial profiles of the second-sound wave, the vortex-line density, and the thermal energy in the three geometries under the same heating conditions. This comparison clearly shows the unique features of He II heat transfer in nonhomogeneous geometries. We then discuss the thermal-layer dynamics in the cylindrical and the spherical geometries and the effects of various heat-pulse parameters in Sec.~\ref{sec:tl}. In Sec.~\ref{sec:peak}, we illustrate how our model can also be used to determine the peak heat flux for the onset of boiling in He II in different geometries. A summary is included in Sec.~\ref{sec:conclusion}.

\begin{figure*}[htb]
\center
\includegraphics[width=0.9\linewidth]{Fig1.eps}
\caption{Schematic diagrams showing the transient heat transfer in He II from (a) a planar heater, (b) a cylindrical heater, and (c) a spherical heater.} \label{Fig1}
\end{figure*}

\section{\label{sec:model} Numerical model}
To study the flow field and the heat transfer in He II, we adopt the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model~\cite{Nemirovskii-1995-RMP}, which is based on the conservation equations for the fluid mass, momentum, and entropy, as listed below:
\begin{equation}
	\frac{\partial \rho}{\partial t}+\mathbf{\nabla} \cdot(\rho \mathbf{v})=0
	\label{eq:mass}
\end{equation}
\begin{equation}
	\frac{\partial(\rho s)}{\partial t}+\mathbf{\nabla} \cdot(\rho s\mathbf{v_n})=\frac{\mathbf{F_{ns}}\cdot\mathbf{v_{ns}}}{T}
	\label{eq:energy}
\end{equation}
\begin{equation}
	\frac{\partial \mathbf{v_s}}{\partial t}+\mathbf{v_s} \cdot \mathbf{\nabla} \mathbf{v_s}+\mathbf{\nabla} \mu=\frac{\mathbf{F_{ns}}}{\rho _s}
	\label{eq:vs}
\end{equation}
\begin{equation}
	\frac{\partial (\rho \mathbf{v})}{\partial t}+\mathbf{\nabla} (\rho_sv_s^2+\rho_nv_n^2)+\mathbf{\nabla} p=\eta_n\mathbf{\nabla} v_n^2
	\label{eq:v}
\end{equation}
The definitions of the relevant parameters are provided in the Nomenclature table. In the above equations, $\rho \mathbf{v}=\rho_s \mathbf{v_s}+\rho_n \mathbf{v_n}$ represents the total momentum density. The Gorter-Mellink mutual friction $\mathbf{F_{ns}}$ per unit fluid volume depends on the vortex-line density $L$ and the relative velocity $\mathbf{v_{ns}}=\mathbf{v_{n}}-\mathbf{v_{s}}$ between the two fluids as~\cite{vinen-1957_Proc.R.Soc.Lond.A_II,vinen-1957_Proc.R.Soc.Lond.A_III}:
\begin{equation}
	\mathbf{F_{ns}}=\frac{\kappa}{3}\frac{\rho_s\rho_n}{\rho}B_LL\mathbf{v_{ns}}
	\label{eq:Fns}
\end{equation}
where $B_L$ is a known temperature-dependent mutual friction coefficient~\cite{Donnelly-1998-JPCRD}. The chemical potential $\mu(P,T,v_{ns})$ of He II includes a correction due to the counterflow velocity $v_{ns}$, as proposed by Landau~\cite{landau-1987}:
\begin{equation}
\mu(P,T,v_{ns})=\mu(P,T)-\frac{1}{2}\frac{\rho_n}{\rho}v_{ns}^2
\label{eq:landau_mu}
\end{equation}
This HVBK model represents a coarse-grained description of the two-fluid hydrodynamics, since the action of individual vortices on the normal fluid \cite{Yui-2020-PRL, Mastracci-2019-PRF} is smoothed out. When the vortex-line density is relatively high, this model has been shown to describe non-isothermal flows in He II very well even in nonhomogeneous geometries \cite{Sergeev-2019-EPL, zhang-2006_Phys.Rev.B}.

To provide a closure to the above HVBK model, we adopt a modified version of the Vinen's phenomenological equation to determine the temporal and spatial variations of the vortex-line density $L(\mathbf{r},t)$~\cite{vinen-1957_Proc.R.Soc.Lond.A_II,vinen-1957_Proc.R.Soc.Lond.A_III}:
\begin{equation}
	\frac{\partial L}{\partial t}+\mathbf{\nabla} \cdot (\mathbf{v_L}L)=\alpha_V|v_{ns}|L^{3/2}-\beta_VL^2+\gamma_V|v_{ns}|^{5/2}
	\label{eq:l}
\end{equation}
where $\alpha_V$, $\beta_V$ and $\gamma_V$ are temperature-dependent phenomenological coefficients introduced by Vinen~\cite{vinen-1957_Proc.R.Soc.Lond.A_II}. The term $\mathbf{\nabla}\cdot(\mathbf{v_L}L)$ accounts for the drifting of the vortices~\cite{schwarz-1988_Phys.Rev.B, nemirovskii-2019_LowTemp.Phys.}, where the vortex mean velocity $\mathbf{v_L}$ is taken to be the local superfluid velocity $\mathbf{v_s}$, as originally proposed by Vinen~\cite{vinen-1957_Proc.R.Soc.Lond.A_II,vinen-1957_Proc.R.Soc.Lond.A_III} and later utilized by many others~\cite{wang-1987_Phys.Rev.B,kondaurova-2014_Phys.Rev.B}. The first two terms on the right-hand side of Eq.~\ref{eq:l} respectively account for the generation and the decay of the vortices, and the third source term serves to trigger the initial growth of the line density~\cite{vinen-1957_Proc.R.Soc.Lond.A_II}.

If one ignores the vortices and linearizes Eqs.~(\ref{eq:mass})-(\ref{eq:v}) assuming small-amplitude wave-form variations of the entropy and the counterflow velocity, it is straightforward to derive a temperature-entropy wave mode (i.e., the second sound)~\cite{landau-1987}. A transient heating from a heater surface then generates a second-sound pulse in He II whose amplitude $\Delta T$ is determined by the heat flux. When this amplitude is relatively high, the second-sound speed $c_{2}$ can be written as $c_2=c_{20}[1+\varepsilon(T)\Delta T]$, where $c_{20}$ is the speed in the zero-amplitude limit and the nonlinear coefficient $\varepsilon(T)$ takes the form~\cite{landau-1987}:
\begin{equation}
\varepsilon(T)=\frac{\partial}{\partial T}\mathrm{ln}\left(\frac{c_{20}^{3}C_p}{T}\right)
\label{eq:c2}
\end{equation}
At $T<1.88$ K where $\varepsilon(T)$ is positive, the second-sound wave with a higher amplitude travels faster. Therefore, a front shock can appear at the leading edge of the second-sound pulse at sufficiently large $\Delta T$. At $T>1.88$ K where $\varepsilon(T)$ is negative, a rear shock can form at the tail of a second-sound pulse. This physical picture gets complicated when vortices are present, which can attenuate and distort the second-sound pulse profile.

Here we consider the transient heat transfer from planar, cylindrical, and spherical heaters in He II based on all the coupled governing equations (i.e., Eqs.~\ref{eq:mass}-\ref{eq:l}), as shown schematically in Fig.~\ref{Fig1}. For simplicity, we ignore small-scale turbulent fluctuations and assume 1D flow in all three geometries, i.e., 1D flow perpendicular to the heater in the planar case and along the radial direction in the cylindrical and the spherical cases. For a rectangular heat pulse with a surface heat flux $q_h$ and a duration $\Delta t$, we set the boundary conditions at the heater surface to be $v_n=q_h/\rho sT$ for the normal fluid and $v_s=-v_n\rho_n/\rho_s$ for the superfluid during $0<t<\Delta t$ and $v_n=v_s=0$ at $t>\Delta t$. All the thermodynamic properties of He II are calculated using the Hepak dynamic library~\cite{hepak-2005}. The values of the coefficients $\alpha_V$ and $\beta_V$
as recommended by Kondaurova \emph{et al.} are used in Eq.~\ref{eq:l}, which appear to produce simulation results in good agreement with experimental observations~\cite{kondaurova-2017_J.LowTemp.Phys.}. We then evolve the governing equations using the MacCormack's predictor-corrector scheme, which is accurate to the second order in time and space~\cite{fletcher-2003}. A flux-corrected transport approach is also adopted to suppress the numerical oscillations due to the discontinuity at the shock front~\cite{fletcher-2003}. We have tested various spatial steps $\Delta r$ and time steps $\Delta t_s$ and found that the calculated results converged well when $\Delta r<2\times10^{-5}$ m and $\Delta t_s<2\times10^{-8}$ s. In order to balance the result fidelity and the computational cost, $\Delta r=10^{-5}$~m and $\Delta t_s=10^{-8}$ s are used in all the reported simulations.

\section{\label{sec:validation}Model Validation}
For model validation purpose, we first performed numerical simulation on transient heat transfer of He II under the same conditions as in the experiments conducted by Fizdon \emph{et al.}~\cite{fiszdon-1990_J.FluidMech.}. These authors examined the transient heat transfer from both a planar heater and a cylindrical heater (radius $r_h=2.5$ mm) immersed in He II at 1.4 K. For the planar heater case, they used heat pulses with a duration $\Delta t=1$ ms at a repetition rate of 0.2 Hz, and the heat flux on the heater surface was $q_h=5$ W/cm$^2$. They measured the time variations of the He II temperature at distances $r=1$ mm, 2 mm, and 5.4 mm from the heater surface using a movable superconducting bolometer. In their experiment, the vortices generated by one heat pulse did not have enough time to decay when the next heat pulse arrived. Therefore, the initial vortex-line density $L_0$ seen by a given heat pulse was relatively high, which is often treated as a tuning parameter in past numerical works~\cite{fiszdon-1989_J.LowTemp.Phys.,fiszdon-1990_J.FluidMech.,kondaurova-2008_J.LowTemp.Phys.}. In our calculation, we set $L_0=8\times10^5$ cm$^{-2}$ to achieve the best match with the experimental observations. Fig.~\ref{fig:vali1D}(a) shows the measured temperature profiles together with our simulation results. Since $\varepsilon(T)>0$ at 1.4 K, the temperature at each location first spikes up sharply upon the arrival of the shock front of the second-sound pulse. A gradual temperature overshot is then observed, which is due to the spreading of the thermal layer as we shall discuss in details in Sec.~\ref{sec:result}. All these observations are well reproduced in our simulations.

\begin{figure}[h]
	\centering
	\includegraphics[width=1\linewidth]{Fig2.eps}
	\caption{Experimental and simulated temporal profiles of the temperature increment $\Delta T=T-T_\infty$ at (a) 1, 2, and 5.4 mm from the surface of a planar heater with $q_h$=5 W/cm$^2$, $\Delta t$=1 ms , and a repetition rate of 0.2 Hz; and (b) at 1, 2, and 3 mm from the surface of a cylindrical heater with $q_h$=6 W/cm$^2$, $\Delta t$=1 ms, $r_h$=2.5 mm, and a repetition rate of 2 Hz. The bath temperature is $T_\infty=1.4$ K.} \label{fig:vali1D}
\end{figure}

In the cylindrical heater case, the measured and the simulated temperature variations at distances $r-r_h=1$ mm, 2 mm, and 3 mm from the heater surface are shown in Fig.~\ref{fig:vali1D}(b). In this case, heat pulses with a duration $\Delta t=1$ ms and a surface heat flux $q_h=6$ W/cm$^2$ were applied at 2 Hz repetition rate. Due to the nonhomogeneous geometry, a radial dependance of the initial line density $L_0(r)=L_h(r_h/r)^2$ as recommended by Kondaurova \emph{et al.}~\cite{kondaurova-2008_J.LowTemp.Phys.} was adopted in our calculation, where the line density at the heater surface $L_h$ was set to 8$\times10^6$ cm$^{-2}$ due to the increased repetition rate. Again, all the key features of the observed temperature curves are reproduced. This excellent agreement between the experimental measurements and our simulation results has thereby validated the fidelity of our model calculation.

\section{\label{sec:result}Simulation results and discussion}
In this section, we first present the simulation results to compare the key features associated with the transient heat transfer in different heater geometries. We then examine the time evolution of the thermal layer in the cylindrical and the spherical heater cases under various heating conditions. Since our focus is the heat transfer following a single heat pulse, a small initial vortex-line density $L_0=10^2$ cm$^{-2}$ is assumed in the calculations. This $L_0$ is comparable to the typical density of remnant vortices pinned to He II container walls~\cite{Awschalom-1984-PRL}. Indeed, it has been shown that in relatively high flux counterflow, the simulated temperature profile in He II is nearly independent of $L_0$ when $L_0$ is smaller than about $10^5$ cm$^{-2}$ due to the source term in Eq.~\ref{eq:l}~\cite{kondaurova-2020_LowTemp.Phys.}. To avoid the complication of possible boiling in He II near the heater surface, we have also assumed that the heater is placed at a 1-meter depth below the He II free surface in all the cases. We will discuss in the last subsection how this hydrostatic head pressure ensures the helium to be always in the He II state during the transient heat transfer. This discussion also provides a foundation for our future study of the peak heat flux for the onset of boiling in He II.

\subsection{\label{sec:wave} Transient heat transfer characteristics in different heater geometries}
\begin{figure*}[htb]
	\centering
	\includegraphics[width=1\linewidth]{Fig3.eps}
	\caption{Profiles of (1) temperature increment $\Delta T$, (2) vortex-line density $L$, and (3) thermal energy density $W$ compensated by the ratio of the cross section area $A(r)$ to the heater surface area $A_h$ in (a) planar, (b) cylindrical, and (c) spherical geometries at 1.78 K. In all cases, $q_h$=23 W/cm$^2$, $\Delta t$=0.5 ms and $r_h$=2 mm.} \label{fig:TLq}
\end{figure*}
To compare the heat transfer characteristics in different heater geometries, we show the simulated spatial profiles of the temperature increment $\Delta T=T-T_\infty$, the vortex-line density $L$, and the thermal energy density $W=\rho C_p\Delta T$ at various time $t$ in Fig.~\ref{fig:TLq}. In this calculation, we set the He II bath temperature to $T_\infty=1.78$ K. A heat pulse with a surface flux $q_h=23$ W/cm$^2$ and a fixed duration $\Delta t=0.5$ ms is applied to the heater at $t=0$ in all three cases. The radii of both the cylindrical and the spherical heaters are set to $r_h=2$ mm. In what follows, we discuss the main features of the transient heat transfer.

\textbf{1) Second-sound pulse:} As shown in Fig.~\ref{fig:TLq}(1a-1c), a second-sound pulse with positive $\Delta T$ emerges when the heater turns on, which carries the heat energy and propagates away from the heater surface at the known second-sound speed (i.e., $c_2=19.6$ m/s at 1.78 K \cite{Donnelly-1998-JPCRD}) in all three cases. Inside the pulse profile, a counterflow establishes where the normal-fluid velocity is determined by the thermal energy flux as $v_n=c_2W/\rho sT=(c_2C_p/s)\cdot\Delta T/T$. This counterflow leads to a rapid generation of the quantized vortices. In the planar heater case, the second-sound pulse gradually evolves from a rectangular profile near the heater to a front-shock profile due to the combined effects of the vortex attenuation and the positive $\varepsilon(T)$. In the cylindrical and the spherical heater cases, as the second-sound pulse propagates outward, the cross-section area of the pulse $A(r)$ increases as $r$ and $r^2$, respectively. In regions where the vortex density is low and hence the mutual friction is negligible, the kinetic energy of each fluid component is nearly conserved. For the normal fluid, this means that $v_n$ in the second-sound pulse must drop as $1/\sqrt{r}$ in the cylindrical geometry and as $1/r$ in the spherical geometry. Since $\Delta T$ is proportional to $v_n$, it also drops in a similar fashion as the pulse propagates, which is clearly seen in Fig.~\ref{fig:TLq}(1b-1c).

\textbf{2) Quantized vortices:} The vortices are created as a consequence of the counterflow in the second-sound pulse. In the planar heater case, the thermal energy flux $W$ in the pulse remains high as the pulse propagates. Therefore, a dense tangle of vortices are created in the entire space traversed by the second-sound pulse (see Fig.~\ref{fig:TLq}(1a)), which continuously attenuate the pulse. In the cylindrical and the spherical heater cases, since the thermal energy flux drops with $r$ due to the diverging geometries, the line density $L$ is high (i.e., greater than 10$^5$ cm$^{-2}$) only in a thin layer of He II near the heater surface (see Fig.~\ref{fig:TLq}(1b-1c)). Outside this region, the second-sound pulse experiences negligible attenuation.

\textbf{3) Rarefaction tail:} A peculiar feature of the temperature profile in the cylindrical and the spherical geometries, as compared to the planar case, is the appearance of a tail region with negative $\Delta T$ following the positive second-sound pulse (see Fig.~\ref{fig:TLq}(1b-1c)). This negative $\Delta T$ tail, which emerges after the heater is switched off, is known as the rarefaction wave~\cite{efimov-1998_LowTemp.Phys.,fiszdon-1989_Phys.FluidsA,iznankin-1983_Sov.Phys.JETP}. The underlying physics can be understood as follows. The total thermal energy carried by the second-sound pulse can be evaluated as $Q_s=\int_{\Delta R}W(r)A(r)dr$, where $\Delta R\simeq c_2\Delta t$ is the thickness of the pulse. Since $W(r)A(r)$ is expected to increase as $\sqrt{r}$ in the cylindrical geometry and as $r$ in spherical geometry (confirmed in our simulation, i.e., see Fig.~\ref{fig:TLq}(3b-3c)), $Q_s$ increases as the pulse propagates. To supply this ever-growing thermal energy carried by the second-sound pulse, there must be a flow of the internal energy from the tail region towards the pulse front, which thereby leads to the formation of the negative $\Delta T$ rarefaction tail. If we integrate $Q_s$ over both the positive pulse and the rarefaction tail, the total thermal energy carried by them always equals the input heat energy, which fulfills the energy-conservation law.

\textbf{4) Thermal layer:} Near the heater surface where the vortex-line density $L$ is high, the interaction between the vortices and the second-sound pulse effectively converts the thermal energy carried by the pulse to locally deposited heat, resulting in a heated layer of He II, i.e., the thermal layer. To see this layer clearly, we plot the $\Delta T$ profile near the heater in all three cases in Fig.~\ref{fig:TTL}. As the heat pulse ends, $\Delta T$ on the heater surface reaches the highest value. In the planar heater case, $\Delta T(r_h)=170$ mK on the heater surface, which is about 7 times the $\Delta T$ in the second-sound pulse. The heat content in this thermal layer diffusively spreads out \cite{vansciver-2012}. On the contrary, in the cylindrical and the spherical heater cases, the temperature buildup in the thermal layer is much weaker. Indeed, both the layer thickness and the maximum $\Delta T$ in the spherical geometry are insignificant. Another important feature of the thermal-layer dynamics in the two nonhomogeneous geometries is that this layer dies out rapidly before it has time to undergo diffusive spreading. This prompt suppression is due to the same mechanism for the formation of the rarefaction tail: the internal energy in these nonhomogeneous geometries is actively transferred towards the second-sound pulse front in order to supply the ever-growing thermal energy carried by the pulse. The depletion of the deposited heat in the thermal layer occurs simultaneously with the formation of the rarefaction tail.

\begin{figure*}[htb]
	\centering
	\includegraphics[width=1\linewidth]{Fig4.eps}
	\caption{Profiles of the temperature increment $\Delta T$ near the heater surface in (a) planar, (b) cylindrical, and (c) spherical geometries at 1.78 K. In all cases, $q_h=23$ W/cm$^2$, $\Delta t=0.5$ ms and $r_h=2$ mm.} \label{fig:TTL}
\end{figure*}

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.9\linewidth]{Fig5.eps}
	\caption{Evolution of (a) the fraction of the heat energy $Q_{th}/Q$ deposited in the thermal layer; and (b) the temperature increment $\Delta T (r_h)$ at the heater surface at 1.78 K. In all three cases, $q_h=23$ W/cm$^2$, $\Delta t=0.5$ ms, and $r_h=2$ mm.} \label{fig:Geo}
\end{figure}
\textbf{5) Heat energy partition:} The partition of the heat energy between the thermal layer and the second-sound pulse as well as how this partition varies with time is of practical significance. To examine this partition, we calculate the heat energy in the thermal layer $Q_{th}$ by integrating $W$ from the heater surface $r_h$ to the boundary $r_b$ of the thermal layer as $Q_{th}=\int^{r_b}_{r_h}W(r)A(r)dr$. This boundary $r_b$ in the planar heater case is set to be the minimum temperature location between the heater surface and the second-sound pulse. For the cylindrical and the spherical heater cases, $r_b$ is set to the location where $\Delta T$ drops to zero. In Fig.~\ref{fig:Geo}, we show the calculated time evolution of the ratio of $Q_{th}$ to the input heat $Q=q_hA_ht$ (where $q_h=23$ W/cm$^2$ at $0\leq t\leq \Delta t$) as well as the $\Delta T (r_h)$ on the heater surface for all three cases. The fraction $Q_{th}/Q$ in the planar heater case increases to about 35\% by the end of the heat pulse, and it slowly increases even after the heater is turned off since the second-sound pulse keeps producing vortices and experiencing attenuation. Following the heat pulse, the thermal layer spreads out and therefore $\Delta T (r_h)$ decreases. In the cylindrical and the spherical heater cases, $Q_{th}/Q$ only reaches 5\% and 0.4\%, respectively, with $\Delta T (r_h)=100$ mK and 26 mK by the end of the heat pulse. Furthermore, $Q_{th}$ quickly drops to zero due to the aforementioned mechanism that occurs in nonhomogeneous geometries. In the end, all the input heat energy is completely carried away by the second-sound pulse and the rarefaction tail.

\subsection{\label{sec:tl} Effects of heating conditions and bath temperature on the thermal-layer dynamics}
In this subsection, we present more detailed studies on the thermal-layer dynamics in the two nonhomogeneous geometries under various heating conditions and bath temperatures.

\textbf{1) Effects of heating conditions:} First, we vary the heater radius $r_h$ in the range of 1 mm to 5 mm while keeping the same surface heat flux $q_h=24$ W/cm$^2$, pulse duration $\Delta t=0.5$ ms, and bath temperature $T_\infty=1.78$ K. The results are shown in Fig.~\ref{fig:r0}. It is clear that both the deposited heat energy $Q_{th}$ and the surface temperature increment $\Delta T (r_h)$ increase with the heater size. This is not surprising, since the heater surface appears flatter to the adjacent He II at larger $r_h$. Therefore, the thermal-layer dynamics is expected to evolve towards that in the planar geometry as $r_h$ increases. Fig.~\ref{fig:Q} shows the results with a varying surface heat flux $q_h$ in the range of 20 to 28 W/cm$^2$ at fixed pulse duration $\Delta t=0.5$ ms, heater radius $r_h=2$ mm, and bath temperature $T_\infty=1.78$ K. As the surface heat flux $q_h$ increases, the thermal layer starts to grow earlier and can reach a higher $\Delta T(r_h)$ with more deposited energy $Q_{th}$. Interestingly, we see that when $q_h$ is lower than a threshold $q^{(c)}_h$, i.e., about 20 W/cm$^2$ in the cylindrical geometry and about 22 W/cm$^2$ in the spherical geometry, the thermal layer does not grow at all. In the planar case, $q^{(c)}_h$ is significantly lower, i.e., about 15 W/cm$^2$ in our simulation, in agreement with the reported values~\cite{vansciver-2012, shimazaki-1995_Cryogenics}. Although the exact value of $q^{(c)}_h$ depends on other heating parameters and the bath temperature, it is always higher in nonhomogeneous geometries under given heating conditions. Finally, we show the effect of the pulse duration $\Delta t$ in Fig.~\ref{fig:th}, where $\Delta t$ is varied from 0.5 ms to 1.0 ms at fixed $q_h=22$ W/cm$^2$, $r_h=2$ mm, and $T_\infty=1.78$ K. The deposited heat energy $Q_{th}$ increases almost linearly with $\Delta t$ in both geometries.

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.95\linewidth]{Fig6.eps}
	\caption{Effect of the heater radius $r_h$ on the evolution of (a) $Q_{th}/Q$ and (b) $\Delta T (r_h)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_\infty=1.78$ K, $q_h=24$ W/cm$^2$, and $\Delta t=0.5$ ms.} \label{fig:r0}
\end{figure}
\begin{figure}[!]
	\centering
	\includegraphics[width=0.95\linewidth]{Fig7.eps}
	\caption{Effect of the surface heat flux $q_h$ on the evolution of (a) $Q_{th}/Q$ and (b) $\Delta T (r_h)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_\infty=1.78$ K, $\Delta t=0.5$ ms, and $r_h=2$ mm.} \label{fig:Q}
\end{figure}

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.95\linewidth]{Fig8.eps}
	\caption{Effect of the pulse duration $\Delta t$ on the evolution of (a) $Q_{th}/Q$ and (b) $\Delta T (r_h)$ in (1) the cylindrical and (2) the spherical heater geometries. In both cases, $T_\infty=1.78$ K, $q_h=24$ W/cm$^2$, and $r_h=2$ mm.} \label{fig:th}
\end{figure}
\begin{figure}[!]
	\centering
	\includegraphics[width=0.95\linewidth]{Fig9.eps}
	\caption{Effect of the bath temperature $T_\infty$ on the evolution of (a) $Q_{th}/Q$ and (b) $\Delta T (r_h)$ in (1) the cylindrical heater geometry with $q_h=20$ W/cm$^2$ and (2) the spherical heater geometry with $q_h=22$ W/cm$^2$. In both cases, $\Delta t=0.5$ ms and $r_h=2$ mm.} \label{fig:T}
\end{figure}

\textbf{2) Effect of bath temperature:} The effect of the bath temperature $T_\infty$ on the thermal-layer dynamics is more complicated, since nearly all the parameters in the governing equations are temperature dependent but their variations with $T_\infty$ can be quite different. Fig.~\ref{fig:T} shows our calculation results when $T_\infty$ is varied from 1.46 K to 1.96 K while all other heating conditions remain fixed, i.e., $r_h=2$ mm, $\Delta t=0.5$ ms, and $q_h=20$ W/cm$^2$ in the cylindrical geometry and $q_h=22$ W/cm$^2$ in the spherical geometry. As $T_\infty$ increases from 1.46 K, the maximum heat energy deposited in the thermal layer first decreases and becomes negligible when $T_\infty$ is in a range of roughly 1.6 K to 1.8 K. Then, as $T_\infty$ further increases, the deposited energy starts to rise. This non-monotonic behavior can be qualitatively understood as caused by the temperature dependance of $\alpha_V|v_{ns}|$ in the vortex generation term in Eq.~(\ref{eq:l}). In Fig.~\ref{fig:Teffect}, we plot $\alpha_V|v_{ns}|$ as a function of temperature, where $v_{ns}$=$q/\rho_s sT$ is evaluated at the surface heat flux $q_h=22$ W/cm$^2$. It is clear that $\alpha_V|v_{ns}|$ exhibits a non-monotonic temperature dependance and reaches a minimum value at around 1.75 K. Since the generation term in Eq.~(\ref{eq:l}) largely controls the rate of vortex production, for a fixed pulse duration $\Delta t$, the vortex-line density in the thermal layer is low at small $\alpha_V|v_{ns}|$. Consequently, the attenuation to the second-sound pulse is weak, which limits the heat energy deposited in the thermal layer. We also note that at the lowest temperature that we have examined, i.e., $T_\infty=1.46$ K, the highest $\Delta T (r_h)$ on the heater surface is achieved despite the fact that only up to 5\% of the heat energy is deposited. This pronounced temperature change is essentially caused by the small heat capacity of He II at low temperatures. As shown in Fig.~\ref{fig:Teffect}, the heat capacity drops rapidly with decreasing the temperature. At low $T_\infty$, even a small energy deposition in the thermal layer can therefore result in a large temperature increment.
\begin{figure}[htb]
	\centering
	\includegraphics[width=1\linewidth]{Fig10.eps}
	\caption{The temperature dependance of the coefficient $\alpha_V|v_{ns}|$ and the He II heat capacity $C_p$. The crosses mark the temperatures examined in Fig.~\ref{fig:T}.} \label{fig:Teffect}
\end{figure}

\subsection{\label{sec:peak} Boiling in He II during transient heat transfer}
An important parameter in He II heat transfer applications is the so-called peak heat flux $q_c$~\cite{vansciver-2012}. This $q_c$ denote the heat flux from the heater surface above which boiling occurs in the helium. There have been extensive studies of $q_c$ in the planar geometry, and existing correlations can reasonably predict the values of $q_c$ at various temperatures and heating conditions~\cite{vansciver-2012}. However, the studies on $q_c$ in nonhomogeneous geometries are very limited. Here, we discuss how our model may allow us to systematically evaluate $q_c$ for He II transient heat transfer in different geometries.

Note that in all the cases we have studied, the highest temperature in He II is always achieved near the heater surface. Therefore, we just need to monitor the state of the He II adjacent to the heater during a transient heat transfer. If at any instance, the state in the $P$-$T$ phase diagram traverses the saturation line to the vapor phase or the He I phase, boiling is deem to occur. Fig.~\ref{fig:PT} shows representative $P$-$T$ curves of the He II near the heater surface at $T_\infty=1.78$ K, $q_h=23$ W/cm$^2$, $\Delta t=0.5$ ms, and $r_h=2$ mm in three heater geometries. As the heater turns on, a sudden drop in the local pressure is seen in all three cases. Then, the state curve moves horizontally towards the saturation line in the planar heater case, while in the two nonhomogeneous geometries the pressure rises a bit before the state curve moves horizontally. Under the specified heating conditions, the horizontal move in the spherical geometry is negligible. After the heat pulse ends, the pressure spikes up in all the cases and the $P$-$T$ curves then evolve back to the start point. The complex evolution pathes in the nonhomogeneous geometries are intimately related to the rarefaction physics. From this example calculation, one can see clearly that the He II state approaches the saturation line furthermost in the planar heater case. If we increases the surface heat flux $q_h$, the state curve in the planar geometry will touch the saturation line first, which allows us to determine the $q_c$ in this geometry. The $q_c$ in the other two geometries can be determined in a similar fashion as we further increase $q_h$. It is clear that under the same heating conditions, $q_c$ is the lowest in the planar geometry and is the highest in the spherical geometry. We may then vary parameters such as the hydrostatic pressure head, the pulse duration $\Delta t$, and the heater radius $r_h$ to study their effects on $q_c$. The relevant details will be presented in a future publication.
\begin{figure}[t]
	\centering
	\includegraphics[width=1\linewidth]{Fig11.eps}
	\caption{The evolution of the He II state near the heater surface. In all cases, $T$=1.78 K, $q_h$=23 W/cm$^2$, $\Delta t$=0.5 ms, and $r_h$=2 mm. The circle and the crosses mark the start and the end of the heat pulse, respectively.} \label{fig:PT}
\end{figure}

\section{\label{sec:conclusion} Conclusion}
We have conducted numerical simulations of transient heat transfer in He II by solving the HVBK equations of motion for the two-fluid system coupled with the Vinen's equation for the evolution of quantized vortices. The characteristics of transient heat transfer from planar, cylindrical, and spherical heaters are systematically examined. Compared to the planar heater case, the heat transfer in the nonhomogeneous geometries exhibits some distinct key features. These include: 1) a rapid drop of the vortex-line density away from the heater surface; 2) the formation of a thin thermal layer near the heater beyond which the second-sound pulse experiences negligible attenuation; 3) the emergence of a rarefaction tail with a negative temperature increment following the second-sound pulse; and 4) a prompt suppression of the thermal layer upon the completion of the heat pulse such that all the input heat can be completely carried away by the outgoing second-sound pulse and the rarefaction tail. We have also examined the effects of various heating parameters and the He II bath temperature on the evolution of the thermal layer. Our result shows that the thermal layer diminishes more quickly with a smaller heater size, a lower surface heat flux, or a shorter pulse duration. When the heating conditions are fixed, the buildup of the thermal layer exhibits a non-monotonic dependence on the bath temperature. To invoke the topic in the Part II of our series of reports, we have also illustrated how our model will allow us to study the peak heat flux for the onset of boiling in He II during a transient heat transfer. These studies should provide us a solid foundation towards the development of a comprehensive understanding of He II transient heat transfer in different geometries.

redpajama_set_name: RedPajamaArXiv


text:
HomeGeneralGovernment relaxes existing timeline for submission of Life Certificate
Government relaxes existing timeline for submission of Life Certificate
September 11, 2020 IndiaTies SHM General 0
Sep 11: MOS Dr Jitendra Singh said that in a major relief to elderly persons, Government has relaxed the existing timeline for submission of Life Certificate.
All Central Government pensioners can submit Life Certificate from 1st November, 2020 to 31st December 2020. Earlier this used to be only for the month of November for maintaining continuity of Pension. However, the pensioners in the age group of 80 years and above, can submit Life Certificate from 1st October, 2020 to 31st December 2020. During this extended period, the pension will be continued to be paid by the Pension Disbursing Authorities (PDAs) uninterrupted.
Dr. Jitendra Singh said that the decision was taken in view of the ongoing Covid-19 pandemic and the vulnerability of the elderly population to the Corona Virus. In addition to the above, as per RBI notification dated January 9, 2020, which permits Video-based Customer Identification Process (V-CIP) as a consent-based alternate method of establishing the customer's identity, Pension Disbursing Banks have been also asked to explore the said methodology for obtaining a Life Certificate from the pensioner, to the extent permitted by RBI guidelines, in order to avoid the rush at the branches.
Every Central Government pensioner has to submit a life certificate in the month of November for further continuation of their pension. Pensioners can submit a life certificate by visiting the bank branches, however the Department of Pension & Pensioners' Welfare has been promoting the Digital Life Certificate which can be given from the comfort of one's home also.
In 2019, as a measure to enable additional dedicated time to very senior pensioners, the Department issued orders enabling pensioners in the age group of 80 years & above to submit Life Certificate from 1st October onward instead of 1st November, every year so that they could avoid the general rush in the month of November.
Dr.Jitendra Singh
Ministry of Personnel
Public Grievances & Pensions
Vice President calls for imparting value-based education
Need to work together on problems resulting from COVID19: Labour Minister at Saudi led G20 Meet
Dr.Jitendra Singh interacts with NERAMAC officials and FPOs to build Agri sector in North East
August 26, 2020 IndiaTies SHM Agriculture & Commodities 0
Union Minister Dr.Jitendra Singh interacts with NERAMAC officials and FPOs; Agro Sector in N-E Region is crucial for post-COVID Economy of India […]
NRA's CET to be extended to State Government Recruitments too
August 22, 2020 IndiaTies SHM Education 0
States & UTs can avail the Common Eligibility Test (CET) to be conducted by NRA for job selection: Dr. Jitendra Singh
Bamboo Skill Centres to exploit, brand, package and market Bamboo industry in India and abroad
July 30, 2020 IndiaTies SHM Agriculture & Commodities, Business 0
Union Minister of State (Independent Charge) Development of North Eastern Region (DoNER), MoS PMO, Personnel, Public Grievances, Pensions, Atomic Energy and Space, Dr Jitendra Singh has urged the Cane and Bamboo Technology Centre (CBTC) to explore setting up of Vocational Training and Skill Development Centres for Bamboo sector for its full exploitation, branding, packaging and marketing in India and abroad. Chairing a review meeting of the Ministry of DoNER today, the Minister said that CBTC in coordination with National Bamboo Mission will work in this direction to boost the Bamboo economy in the North Eastern Region. Dr. Jitendra Singh said that Skill Centres will propel Bamboo industry with new Start-ups and also enhance livelihood opportunities.
PM Modi does a VC with German Chancellor Merkel
Dr. Virander Kumar Paul concurrently accredited as Ambassador to Somalia


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Meet The Ratings Experts℠!
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Charlie knows radio from the inside out and outside in. He spent 12 years as Vice President of Research/Marketing at Eastman Radio. During that time he was also a member of the Arbitron Advisory Council and Chairman of the RAB Goals Committee. Prior to Eastman, he held positions as Client Service Representative at Arbitron for the firm's rep/network/agency division, and Account Executive at WWDC-AM/FM in Washington, D.C. You'll find Charlie involved at Research Director in sales, customer relations, and new business development. He received a B.S. in Administration and Management from Carnegie Mellon University's School of Industrial Administration.
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text:
Q: Android browser file upload only allows video files by default I'm working on a web application for mobile phones and when I tested the file upload (you know the html input type=file tag) on the Samsung Galaxy S3 it only allowed me to select a video file or to record a new video. The phone wouldn't let me choose an image or switch the camera mode to a still photo.
So, how do you get android to play nice with the  tag?

A: It's simple, you need to add the attribute Accept="image/*" like this:
<input type="file" accept="image/*" />

If you're using the asp:FileUpload like me then the only way you can add this attribute is with javascript (the only way I'm aware of). This is how I did it with jQuery:
$(document).ready(function () {
    $("input[type=file]").attr("accept", "image/*");
}


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text:
Q: Bitmap font in bbold package I'm getting a bitmap font on \mathbb{1} when I use the bbold package. After much searching on bitmap fonts specifically in relation to the bbold package, these articles are the closest I've got to a solution:
\mathbb{Z} is not displayed correctly
Image quality of pdf file generated from latex
However, they don't quite answer my problem. The best advice in the first is to use amssymb, which doesn't work in my particular case because amssymb renders \mathbb{1} as something else entirely (not a blackboard "1"). The second article is a bit general but does suggest the lmodern package, which still doesn't solve the problem.
Note also that I've already installed the cm-super font (using MiKTeX), as many other articles suggest.
Here's my code:
\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{bbold}

\begin{document}

This is a test: $\mathbb{1}$.

\end{document}

Here's a fragment (the important bit) of the output:

As the image shows, only the \mathbb{1} is a problem.