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Irishman Shane Lowry continues his pre-U.S. Open tour in wine country
Ian Rutherford-USA TODAY Sports
By Adam Schupak September 8, 2020 5:57 pm Follow	@AdamSchupak
By Adam Schupak | September 8, 2020 5:57 pm	Follow @AdamSchupak
NAPA, Calif. – Shane Lowry's tour of the United States continues this week in wine country, though the reigning British Open champion isn't imbibing.
"I do like a glass of wine, but I don't drink the week of tournaments, so I'm taking it easy; I'm relaxing," he said. "We are going for a nice meal tonight, we're going to French Laundry, which is supposed to be very special, so I'm looking forward to that. I'll leave the drinking to (caddie) Bo. Bo will look after the wine for me this week."
Lowry squeezed into the FedEx Cup playoffs at No. 123, but missed the cut at the Northern Trust and spent the last two weeks taking a buddy trip to Newport, Rhode Island, and practicing at his U.S. base in Florida for the upcoming U.S. Open.
He arrived here on Sunday and the Irishman was greeted by temperatures reaching 114 degrees. In addition to caddie Brian 'Bo' Martin, Lowry is traveling with a childhood friend, who doubles as his physical therapist and keeps Lowry company after his wife and young daughter returned to Ireland six and a half weeks ago.
Safeway Open: Odds and best bets
"It's not been easy at times. I miss them an awful lot and to be honest, probably if it wasn't for the U.S. Open next week, I probably wouldn't have lasted the full stint over here because it's just been incredibly hard," he said. "Especially when your daughter's growing up and she's only three and a half, but it is what it is. It's not like I'm going to war or anything, I'm just going to play golf, so I just have to get on with it."
Lowry, who didn't play the week before winning the British Open last year in Northern Ireland, joked he'll probably never play ahead of that major again, but he didn't want to go into next week's U.S. Open at Winged Foot in New York having gone three weeks without competing. While some players may have been turned off by having to go from the West Coast to the East Coast for the re-scheduled national championship, Lowry took a different view.
Shane Lowry studies his putt on the first hole during the second round of the Wyndham Championship golf tournament at Sedgefield Country Club. (Rob Kinnan-USA TODAY Sports)
"I'm close, I'm in America. My family are in Ireland, so I couldn't go back to Ireland because of the quarantine — there was no point — so, yeah, I just decided to play here," he said. "I would be down in Florida this week just playing golf every day and practicing and probably with my mind too much on the U.S. Open, so it's nice to get to an event."
It also gives him a chance to test drive a slew of new equipment under the gun. Lowry arrived at yesterday's practice round with six different drivers and whittled it down to one and also said he'll be putting a fresh set of irons in play in hopes that the new grooves will spin more, "which might be helpful at Winged Foot trying to stop the ball on some of those greens," he said. "For me, this is the only week that matters right now. Then when I get to Winged Foot next Monday I'll start to think about that, but obviously this week, first and foremost, I want to do well."
quarantine, Safeway Open 2020, Shane Lowry, Golf, PGA Tour, Professional
Follow @AdamSchupak
Adam Schupak has been lugging his laptop and golf bag and following the sun to write about golf since 1997. He has worked for the PGA Tour, World Golf Hall of Fame and as a senior writer at Golfweek.
Thailand opens six resorts to international golfers on quarantine


Alcohol and Drug Use | Fuicelli & Lee, P.C.
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Colorado and Federal laws require that any time a commercial truck is involved in an accident that causes injury, a post-accident drug and alcohol test must be performed. The results of this test can be particularly incriminating to the trucking company and driver and can also lead to a claim for punitive damages.
If you or a loved one has been injured in an accident involving a commercial truck, please call or email us today. The highly skilled attorneys at Fuicelli & Lee, PC are experts in handling semi-truck cases and drug/alcohol related automobile accidents. Fuicelli & Lee, PC is here to help.


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Wednesday, July 15, 2020 | History
4 edition of Woman"s Guide to Balancing Career and Family found in the catalog.
Woman"s Guide to Balancing Career and Family
Marian Thomas
Solutions for Your Job, Your Family, and Yourself (Reading & Writing Series)
by Marian Thomas
Published January 2001 by National Press Publications .
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Being a working parent is a huge challenge, even for the most organized mum or dad. When it comes to trying to balance a career and family – juggling school drop-offs and pick ups with getting to and from the office, being able to find childcare for school breaks, and trying to plan time off to attend presentations and parent-teacher conferences – it can soon become totally overwhelming. Family and mother focused Author uses herself and family as examples a lot A Quarter-life Woman's Guide to Balance and Direction Price: $ or pondering your future career, this book.
Career Development Issues Facing Women. Despite the major advancements women have made in becoming a significant part of the workforce, they are still facing important career development issues. Although many women achieve lower and middle management positions, they seem to hit the "glass ceiling" in many. "Work-life balance" is a piece of cake compared to career-and-family balance. "Work-life balance" implies one person deciding how hard to work, how much to play, and how little to sleep. "Career-and-family balance" for a man means taking into account how his work and earning habits impact his wife and any children.
28 women at the top of their games share their secrets to kicking butt! Coco Chanel once said, "A girl should be two things: classy and fabulous." This piece of advice from one of the world's most famous fashion designers (among other tidbits) is as inspirational today as it was when she launched. Women fail to balance career and home life. most women struggle to look after their family and hold down a demanding job to make ends meet. Despite their hard work, many are up to .
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Measurement of the change orientation of vocational teachers
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Woman"s Guide to Balancing Career and Family by Marian Thomas Download PDF EPUB FB2
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Price New from Used from Hardcover "Please retry" $ — $ Hardcover $Format: Hardcover. Women today are pulled in a million different directions. Between taking care of the kids, trying to stay connected with husbands or partners and, in many cases, caring for an elderly parent, most of us don't have time to make the grocery list, let alone think about balance.
out of 5 stars This book moves the ball forward for women struggling to balance the needs of work/career, family, and self Reviewed in the United States on /5(17). 10 Tips For Balancing Your Career and Family Set Your Family and Business Goals—I believe the first step to achieving anything important is to begin by identifying your The Support From Your Spouse—The only way you are going to achieve these goals is with the support of your Blocks of Time For Work and Family—One of the first things my wife and I did when we got married was to put together a work and family Schedule with Family—After you and your spouse have negotiated the blocks of time you will allocate for your career, spend together and be with your children, then review the schedule with your children and get their Sure Your Work Time is Respected—During the blocks of time you have allocated to work, you are not to be interrupted unless there is important reason.
(more items). It has never been easy for women to juggle a thriving career and a happy family life. A Working Woman's Guide to Maintain Work-life Balance. Feel free to take leaf or two out of my book.
Mary Blair-Loy examines the career paths of women financial executives who have tried various approaches to balancing career and family. The professional level these women.
Juggling work and family has never been easy. In today's net-speed economy, it seems harder and harder to maintain a balanced lifestyle while simultaneously achieving career nirvana.
Single parents ar. Balancing A Career and A Family Essay Words6 Pages Balancing A Career and A Family Abstract Regardless of where they are in their careers, working mothers have a tough time achieving balance between work and family.
Generally, women take on or are given more responsibility for their homes than are men, leading to frustration and exhaustion. A Community For Women Balancing Caregiving and Career.
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Still can't decide on what you want. Our round-up of the best negotiation books can help you find what you're looking for.
Choice and compromise: a woman's guide to balancing family and career. [Donna N Douglass] Home. WorldCat Home About WorldCat Help.
Search. Search for Library Items Search for Lists Search for Contacts Search for a Library. Create Book\/a>, schema:CreativeWork\/a>. A rocky career start, a handful of jobs, and deciding to leave the corporate world after a few decades in it—this wasn't exactly the career path I had planned.
When I was just starting out, I had what seemed like a pretty simple career plan: get the best education I could, land the best job I could get, and then work like crazy. Period. The idea that you can balance a family, social life and a career has always seemed like a far-fetched goal, but in Vanderkam's self-help book, she teaches you the art of juggling it all.
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For example, when one family member is sick, you may need to skip a work event; or when an important deadline must be met, you might need to miss dinner at home and stay working in the office late. The dilemma of balancing family and work has no easy solution.
PepsiCo CEO Indra Nooyi's comments last week on whether working women can have it all—a fulfilling career and family life—has reignited the debate on how best to balance career and family. Balance has always been something I've struggled with.
In the last several years, I've launched a skin care company, smartphone app, and ice-cream shop. Now I'm a consultant who helps other people figure out how to turn their ideas into a viable business, from developing a marketing plan to a growth strategy to branding.
Recharge your batteries with some peace and quiet. Don't let interruptions rule your life. Solicit help from the rest of the family. When Mom is relaxed and in control, the whole family benefits. Working outside the home is a reality for most women. Balancing a career and family makes for .Women face unique challenges in balancing work and domestic duties that many men do not.
With issues such as maternity leave, having copious work at home, and the pressure to start a family, women often have to scale back on work or leave their careers altogether because of the struggle to maintain a balance.
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redpajama_set_name: RedPajamaCommonCrawl


It is very convenient that Aloft Minneapolis offers free transportation, and Aloft Minneapolis is equipped with great elevators, so you needn't walk the stairway. There are tremendous kid's activities at Aloft Minneapolis that are fun, and of course there are cribs here for your little baby. There's nothing that is more enjoyable than a spree to a zoo, Como Zoo is near Aloft Minneapolis.
These are some other hotels near Aloft Minneapolis, Minneapolis.
These outdoors activities are available near Aloft Minneapolis.


I'm Sean Egan & I started my business over 35 years ago when we concentrated on small to medium building work.
In the last 10 years the company has expanded, mostly due to recommendations and we now work with a wealth of experience and skills to provide all aspects of the building industry.
Building and construction has run through the Egan family for many years and I was pleased to welcome the next generation to the firm, my son Craig, 10 years ago, who now works along with myself to provide customer satisfaction. We run from the family home in Stoneleigh, Surrey. We operate throughout Southwest London, Surrey & further if required. We work very close with our clients to ensure that all requirements are met as customer satisfaction is our main goal.
We are fully insured with Public Liability and all our Employees are fully insured.
We endeavour to embrace any construction project, whatever the building requirement from a loft conversion to a new kitchen or bathroom to a back or side extention to a complete refurbishments. Rest assures Egan Builders will complete your project to the highest of standards, making sure everything is left clean and tidy everyday and spotless on completion.
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Please contact me at anytime, I will be happy to produce you with a client list so you can make contact with our customers yourself.
Copyright © 2018 Egan Builders - All Rights Reserved.


Inflammation is a natural part of healing. When something harms our body, we mount an immune response. Think about what happens when you step on a nail. You experience pain, redness, and swelling. Inflammation can be an uncomfortable part of healing, but without inflammation our immune system would not spring into action allowing us to heal.
Inflammation comes from a word that means "setting on fire". Inflammation becomes a problem when it becomes ongoing, which can initiate a disease process. Instead of healing, chronic inflammation, even low level chronic inflammation, can become a root cause of diseases and conditions effecting the gut, brain, skin, hormones, cardiovascular and immune system.
What Common Conditions are Associated with Inflammation?
How Can Nutrition Help Quell Chronic Inflammation?
The body's natural immune function can be reset by identifying and removing the triggers and causes of inflammation, implementing a supportive diet and lifestyle and adding in targeted supplements as needed. No one diet is optimal for every body. The key is to figure out what foods restore and then support optimum conditions for the immune system to thrive in each of us.


Stem cells have the remarkable potential to develop into many different cell types with a more specialized function, such as a muscle cell, a red blood cell, or a brain cell. Rather popularly known in regenerative medicine to treat blood and immune disorder, stem cell treatments in cosmetic surgery procedures has risen over these years due to its ability to repair and improve regeneration of skin cells for more youthful appearance.
Individuals who are considering stem cell treatments may also favour Platelet Rich Plasma (PRP) treatment.


I love making pulao. Normally i use regular basmati rice. But very recently i discovered brown basmati rice and i am kind of excited because of it. Because i couldn't find any difference in taste plus i know it is healthy and rich in fiber..I will be sharing many other brown basmati rice recipe soon..
Now heat oil in a large pot. Add in whole spices and let them sizzle.
Add in onions and salt. Saute for 2 to 3 mins.
Now add in all spice powders. Mix well.
Add in veggies and toss well. cook for few mins..
Add in water, and drained rice. Add in coriander and mint leaves. Mix well. Bring it to a boil, simmer and cover the pan with a lid. Cook for 15 mins on very low heat.
Open the pan, and fluff it with a fork. Cover again and set aside for 5 mins.


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William Danforth Mulliken (August 27, 1939 – July 17, 2014) was an American competition swimmer and Olympic champion.

Mulliken represented the United States at the 1959 Pan American Games, where he won the 200-meter breaststroke, his signature event.  At the 1960 Summer Olympics in Rome, he received a gold medal for winning the men's 200-meter breaststroke with a time of 2:37.4.

He graduated from Miami University in 1961 with a bachelor's degree, and Harvard University in 1964 with a law degree.

He founded Big Shoulders, a swim event at Ohio Street Beach in 1991 and held annually since.  An avid reader, he was an active member, supporter and leader of the Caxton Club of Chicago.  He also chaired the capital campaign for Miami University's Library.

His first marriage was to Julia N. Neavolls.  They had two daughters and a son before the marriage ended in divorce.  He later married  Lorna Filippini.

See also
 List of members of the International Swimming Hall of Fame
 List of Miami University people
 List of Olympic medalists in swimming (men)

References

External links

  Bill Mulliken (USA) – Honor Swimmer profile at International Swimming Hall of Fame

1939 births
2014 deaths
American male breaststroke swimmers
Harvard Law School alumni
Miami RedHawks men's swimmers
Olympic gold medalists for the United States in swimming
People from Urbana, Illinois
Swimmers at the 1959 Pan American Games
Swimmers at the 1960 Summer Olympics
Medalists at the 1960 Summer Olympics
Pan American Games gold medalists for the United States
Pan American Games medalists in swimming
Medalists at the 1959 Pan American Games

redpajama_set_name: RedPajamaWikipedia


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← Ann Rutherford (1917-2012)
Lara at a Master Class for 1 Week →
Queer Film Blogathon–Queen Christina (1933)
I have been friends with Caroline, the author of Garbo Laughs, since we were classmates at Mills College, a whopping 6 years ago. We started our blogs within 5 months of each other (Garbo Laughs is 5 months older!), and since then, because of our personal friendship and blogging style, Garbo Laughs and Backlots have been intertwined in many ways. Caroline is a gifted writer, and I'm certainly honored to be associated with a blog of this caliber.
For the second year in a row, Garbo Laughs is hosting its Queer Film Blogathon, a celebration of, in Caroline's words, "lesbian, gay, bisexual, trans, or otherwise non-heterosexual, non-gender-binary depictions or personages in film." Last year I submitted a profile of Judy Garland, because really, what would a queer blogathon be without any mention of Judy Garland? This year, I will try my luck with Queen Christina, a riveting historical account of a monarch, a fascinating study of lesbian depictions onscreen, and of what movies could get away with before the implementation of the Hays Code.
Based loosely on the life of Queen Christina of Sweden, the film takes particular care to hint subtly at the queen's real-life sexual identity, alleged to be transgender or intersex (the queen often wore men's clothes, had a strong, masculine voice and masculine features, never expressed an interest in marriage, and displayed a general manner that was described as being manly. The historical evidence mounted so high that in 1965 an archaeological undertaking was commissioned to explore the possibility of Queen Christina being intersex. The results were inconclusive.) It is particularly interesting that this part of the queen's life was kept intact, while other, less controversial details of her life were fabricated or exaggerated for the movie.
Portrait of the real-life Queen Christina of Sweden.
The most famous scene in Queen Christina is one in which the queen kisses a duchess firmly on the lips in a good morning greeting. Though the textual layer of the film suggests that they're just friends, when one reads into the subtext, in this case the facial expressions and acting technique of Greta Garbo, one is enticed to question exactly what this relationship is.
Another scene that toys with the audience's perception and interpretation of the film is this exchange, between the Chancellor and the queen:
CHANCELLOR: But your majesty! You cannot die an old maid!
QUEEN CHRISTINA: I have no intention to, Chancellor. I shall die a bachelor!
In the days before the implementation of the revised production code in 1934, filmmakers had a much freer reign and tighter artistic control over their work, and thus many plots, scenes, and situations that we today would consider racy or controversial were frequent occurrences in the era before the Code. After the Hays Code went into effect, we began to see far less outward expression of homosexuality, as the Hays Code stipulated that "sex perversion [homosexuality] or any inference to it is forbidden." The first known lesbian kiss onscreen occurred in 1930 in the Marlene Dietrich film Morocco:
In Queen Christina, Greta Garbo's portrayal of the queen is striking. Her physicality in conveying the character is masterfully executed–walking with a cat-like stride, with slumped shoulders and decidedly masculine intonation in her voice, this is not the Garbo that audiences were used to. Interestingly enough, however, it seems that this was an outward expression of Garbo's own bisexuality, which was scarcely a secret in Hollywood. She had affairs with many women over the years, notably playwright and poet Mercedes de Acosta, whose correspondence with Garbo was released in 2000. Though the Rosenbach Museum possesses love letters between them, they have not been made available to the public.
Queen Christina is a unique film with a unique place in cinema history. It is important not only as an early expression of sexuality on film, but also as a veiled hint at the sexuality of an explosive Hollywood star, and, apart from the LGBT aspect of this film, it is considered one of the best of the 1930's and an essential classic in the Greta Garbo pantheon of films.
9 responses to "Queer Film Blogathon–Queen Christina (1933)"
Margaret Perry Movies | June 16, 2012 at 10:58 am |
I find it terribly interesting how films about strong female monarchs provided an arena for the great actresses of the 1930s to express both feminist and LGBT tendencies on the screen. I am thinking also of Bette Davis as Elizabeth and of course Katharine Hepburn in the less successful MARY OF SCOTLAND. Thank you for writing about Garbo and Dietrich, two of the great progressive female screen stars!
http://thegreatkh.blogspot.com/
Paul Lambertson (@LassoTheMovies) | June 18, 2012 at 8:12 am |
Thanks for your thoughts on this movie. I have not seen it yet, but it has been sitting on my shelf waiting for my for a while and I think I am going to bump it to the front of the line now. Thanks for convincing me.
Kendra Bean (@vivandlarry) | June 18, 2012 at 10:53 am |
Great post, Lara! I just watched Morocco for the first time not too long ago and was struck by Deitrich's androgynous allure. I have yet to see Queen Christina but Ill never forget the story of Garbo sacking Laurence Olivier in favour of John Gilbert, haha!
R. D. Finch | June 18, 2012 at 9:57 pm |
Lara, a thoughtful examination of Garbo's sexual ambiguity in what for me is one of her two greatest performances. The other is "Camille," in which she behaves in an unambiguously feminine manner and even uses her feminine allure professionally. What a contrast she is in "Queen Christina," where she seems to be able to modulate her level of femininity as the situation requires. When she is in her masculine mode, she conveys this, as you note, with all her being–her manner, voice timbre, and especially her body language–that swaggering masculine gait you refer to and the way she uses her upper body, arms, and hand gestures. The way she doesn't just give that lady-in-waiting a chaste peck but grasps her face in her hands and gives a full-bodied kiss definitely suggests something more than just platonic interest!
Caroline | June 19, 2012 at 10:53 pm |
Goodness, the kiss is like the least gay part of that whole interaction! Thank you for this excellent contribution — I was so shocked when nobody covered Queen Christina last year, I figured I'd end up just having to do it myself, but you nailed it! And thank you so much for the very sweet words about the blog. We need to come up to the Bay soon — I miss all my Mills ladies!
Backlots | June 20, 2012 at 10:08 pm |
Nobody did Queen Christina last year?? That really surprises me! Thanks for hosting, and you really do need to come back to the Bay for a visit. You and Molly are welcome to stay at my house, I have an extra bed!
Joseph | June 21, 2012 at 10:15 am |
Actually, Morocco was released a little before 1931–in November 1930. About Queen Christina, the conversation between Christina and Ebba after that kiss always reminds me of the now-canned movie & T.V. dialogues between a busy doctor (or C.E.O.) and his neglected wife ("You're surrounded by musty old papers and musty old men, and I can't get near you."). Here, I find the exchange titillating precisely because–on the surface–these two are just friends.
Lê | June 22, 2012 at 12:48 pm |
This is a great movie and a great performance by Garbo. Did you know that the time in the seduction scene had to be meticulously followed?
For sure Queen Christina is an essential movie when it comes to queer films.
Backlots | June 26, 2012 at 9:45 pm |
Oops, typo! Of course Morocco was released in 1930, what was I thinking? LOL. Thanks for alerting me to that! About Queen Christina, I feel exactly the same way. The dialogue seems to be teasing us!


text:
\section{Introduction}
\label{s1}
Fluids in extreme confinement between two parallel plates, have many peculiar properties as compared with the behavior of  bulk systems far away from the boundaries.
This includes both equilibrium \cite{TGyR92,DyH95,RSLyT97,FLyS12} and non-equilibrium situations  \cite{KyD89,LBOHFyS10} of molecular gases, and also  granular gases, i.e. systems composed of macroscopic particles whose interactions are inelastic \cite{Metal05,MyS16}.  For instance, in this latter case, it has been experimentally observed  that when energy is continuously injected into the system to compensate the energy dissipation in collisions, the system reaches a spatially homogeneous steady state for a wide range of values of the density. Nevertheless, for certain densities, which depend on the values of the magnitudes characterizing the energy injection, the final state reached by the system exhibits two coexisting phases, one of them having a smaller number of particles density and a larger kinetic energy density than the other \cite{PEyU02,RCByH11,CMyS12,CMyS15}. Although several effective models have been proposed to describe this phenomenology with more or less success \cite{BRyS13,CLyH00,BBGyM16,RSyG18,MGyB19a},  a satisfactory explanation, based on a well established   description of the dynamics of the system is still lacking. One of the main reasons for this is that the form of the macroscopic hydrodynamic equations describing the evolution of the system is unknown. For strongly confined systems, the effect of the physical boundaries occurs not only through the boundary conditions to be imposed to the macroscopic evolution equations, but it is expected to modify also the structure of the transport equations themselves. The changes will be deeper  if the dynamics observed by projecting on a plane parallel to the plates is considered, and   the interest focusses in formulating macroscopic transport equations for that quasi-two-dimensional dynamics.

Kinetic theory provides an intermediate level of description of a system of particles in which the details and basis for macroscopic balance equations have their origins. To isolate the more important distinguished features of the extreme confinement, the simplest case of inelastic smooth hard spheres at low density is considered here. The bulk macroscopic hydrodynamic equations for a system composed of these particles have been extensively investigated, their structure has been established and explicit expressions for the coefficients appearing in them has been obtained. The  theoretical predictions following  from them   agree quite well with molecular dynamics simulations results \cite{BDKyS98,ByC01,Go03}. The starting point for this research was the Boltzmann kinetic equation modified to account for inelastic two-particle collisions. From it, exact balance equations were derived for the mass, momentum, and energy. Then, to close those equations, a modified Chapman-Enskog method was employed to obtain a ``normal solution''  of the kinetic equation as an expansion in spatial gradients of the macroscopic fields. The analysis is something more complex that in the case of elastic particles, due to the inelasticity of collisions that implies that there is no equilibrium state.  Instead, a particular solution describing a homogeneous state in which the temperature decreases monotonically in time (the so-called homogeneous cooling state), was used as the  reference state to carry out the expansion.

In order to implement a similar program for strongly confined systems, the first issue is to formulate a kinetic equation describing the dynamic of the system. Recently, a Boltzmann-like kinetic equation has been derived for a system of hard spheres confined between two infinite parallel plates at rest, separated a distance smaller that two particles diameters.  The equation can be applied, to both elastic \cite{BMyG16,BGyM17} and inelastic particles \cite{MGyB19,BGyM19a}. The only difference is in the collision rule used to determine the change of the velocities  when two particles collide. For elastic systems, the kinetic equation has been proven to obey an H theorem, implying the tendency of any solution to a final equilibrium state. This state coincides with the one predicted by equilibrium statistical mechanics: the velocity distribution is Maxwellian with a uniform temperature and the density profile is inhomogeneous along the direction perpendicular to the plates. In the case of inelastic hard spheres, if no external energy is continuously injected into the system, trivially there is no equilibrium state. Instead there is again a particular state showing a uniform temperature that decreases monotonically in time, similarly to what happens in the homogeneous cooling state. On the other hand, as in the equilibrium state, the density is not uniform along the direction perpendicular to the plates. For this reasons, this state has been referred to as the inhomogeneous cooling state \cite{BGyM19a}

In this work, balance equations for the density, momentum, and energy are derived from the Boltzmann equation for the confined system. This is done in the more general case of inelastic hard spheres, but the results for the elastic case are easily obtained by taking the appropriate limit. As a consequence of the extreme confinement, the pressure tensor and the heat flux have, in addition to the kinetic contributions associated to the motion of the particles, collisional transfer contributions that, in the present case, are due to the restriction on the allowed collisions imposed by the boundaries. An analysis of the structure of the collisional transfer contributions leads to a result that generalizes the ``contact theorem'' of equilibrium statistical mechanics \cite{HyMc06} to arbitrary non-equilibrium states and also to inelastic particles for extreme confinement. Actually, this  is a particular case of a completely general property following from the dynamics itself of  confined hard spheres \cite{MGyB18}.

The complexity of the collision term makes quite difficult to apply the kinetic equation to realistic problems, beyond situations for which a linearization around the reference state  (equilibrium or the ICS) can be done. For this reason, it is useful to formulate a kinetic model that permits analytical studies in fully nonlinear problems. A model kinetic equation is obtained by replacing the collision term with a much simpler form that preserves the most important properties. These include normalization and the local conservation laws for the number of particles, momentum, and energy. By preserving the equations, it is meant that, not only the structure of the equations is the same, but also that the expressions giving the pressure tensor and the heat flux, and in the inelastic  case the energy sink term, are the same functionals of the distribution function as in the original description. The model proposed here is inspired in the well-known Bhatnagar-Gross-Kook  (BGK) model for the usual Boltzmann equation \cite{Ce75}. The BGK model has been extensively used and studied. Its limitations are well established, but also the means to correct them are known. As an illustration, the solution of the model kinetic equations describing the ICS  is investigated and the results compared with the description provided by the original Boltzmann equation. The agreement can be considered as quite satisfactory.

The remaining of the paper is organized as follows. In the next section, le Boltzmann equation for the confined system of hard spheres is reviewed.  The local balance laws  are derived from it by taking velocity moments. The collisional transfer contributions to the pressure tensor and the heat flux, due to the extreme confinement, are identified.  The model kinetic equation is formulated in Sec.\, \ref{s3}, where its relation with the original Boltzmann equation is stressed. As an initial relevant test of the model, it is shown to have a solution describing the same macroscopic ICS as the original equation, namely uniform decreasing temperature and density gradient along the direction perpendicular to the plates. The comparisson of the cooling rates and the density profiles shows a good qualitative and quantitative agreement. The final section of the paper contains a short summary and some analysis of the possible applications of the model. Details of the calculations are provided in the Appendices.

\section{description of the system and balance equations}
\label{s2}
The Boltzmann equation, describing the time evolution of the one-particle distribution function, $f ({\bm r},{\bm v},t)$ for a dilute gas of inelastic hard spheres of mass $m$ and diameter $\sigma$, confined between two large hard parallel plates separated a distance $h$ smaller than  twice the diameter of a particle, $\sigma<h < 2 \sigma$, has the form \cite{BMyG16,BGyM17,MGyB19}
\begin{equation}
\label{2.1}
\frac{\partial f}{\partial t}+{\bm v} \cdot \frac{\partial f}{\partial {\bm r}}= J[{\bm r},{\bm v}|f],
\end{equation}
with thw binary collision term given by
\begin{eqnarray}
\label{2.2}
J[{\bm r},{\bm v}|f] & = & \sigma  \int d{\bm v}_{1} \int_{0}^{2\pi} d \varphi \int_{\sigma/2}^{h-\sigma/2} dz_{1}\,   | {\bm g} \cdot \widehat {\bm \sigma} | \nonumber \\
&& \times | \left[ \Theta ({\bm g} \cdot \widehat {\bm \sigma})  \alpha^{-2} b_{\bm \sigma}^{-1} - \Theta (-{\bm g} \cdot \widehat{\bm \sigma})\right]f(x,y,z_{1},{\bm v}_{1} ,t) f({\bm r},{\bm v},t).
\end{eqnarray}
Here ${\bm r}$ is the position vector of components  $\{x,y,z \}$, $ {\bm r}_{1z} \equiv \{x,y,z_{1} \}$, ${\bm g} \equiv {\bm v}_{1}-{\bm v}$, $\Theta$ is the Heaviside step function, and $b_{\bm \sigma}^{-1}$ is an operator changing all the velocities ${\bm v}$ and ${\bm v}_{1} $ to its right into their pre-collisional values for a collision defined by the unit vector $\widehat{\bm \sigma}$,
\begin{equation}
\label{2.3}
{\bm v}^{*} \equiv b_{\bm \sigma}^{-1} {\bm v} = {\bm v} + \frac{1 + \alpha}{2 \alpha} \left( {\bm g} \cdot \widehat{\bm \sigma} \right) \widehat{\bm \sigma},
\end{equation}
\begin{equation}
\label{2.4}
{\bm v}^{*}_{1} \equiv b_{\bm \sigma}^{-1} {\bm v}_{1} = {\bm v} -  \frac{1 + \alpha}{2 \alpha} \left( {\bm g} \cdot \widehat{\bm \sigma} \right) \widehat{\bm \sigma}.
\end{equation}
In the above expressions, and also in Eq.\ (\ref{2.2}), $\alpha$ is the coefficient of normal restitution, defining the inelasticity of collisions and defined in the interval $0<\alpha \leq 1$.
The value $\alpha=1$ corresponds to the limit of elastic collisions. The unit vector $\widehat{\bm \sigma}$ is given by
\begin{equation}
\label{2.5}
\widehat{\bm \sigma} \equiv  \left\{ \sin \theta \sin \varphi, \sin \theta \cos  \varphi, \cos \theta \right\}
\end{equation}
with $\varphi$ being an azimuth angle and
\begin{equation}
\label{2.6}
\cos \theta = \frac{z_{1}-z}{\sigma}, \quad \sin \theta\geq 0 .
\end{equation}
A scheme of the coordinates used to describe the collision of two particles is given in Fig. \ref{fig1}.

\begin{figure}
\includegraphics[scale=0.8,angle=0]{BMyG19af1.eps}
\caption{Coordinates used for the description of the collision of two hard spheres of diameters $\sigma$,  confined in a quasi-two-dimensional system by means of two hard parallel plates separated a distance $h$,  $\sigma< h< 2 \sigma$.}
\label{fig1}
\end{figure}

The kinetic equation (\ref{2.1}) holds for $\sigma/2 < z< h-\sigma/2$ and it has to be solved with the appropriate boundary conditions \cite{BGyM17}. These conditions guarantee that if the initial one-particle distribution function vanishes outside the system, this property is kept in time by the solution of the kinetic equation. For an arbitrary function $\phi ({\bm v})$, the collision term has the useful property
\begin{eqnarray}
\label{2.7}
\int d{\bm v}\,  \phi ({\bm v}) J[{\bm r},{\bm v}| f]   &= & \sigma \int d{\bm v} \int d{\bm v}_{1} \int_{\sigma/2}^{h-\sigma/2} dz_{1}   \int_{0}^{2\pi} d \phi\, |  {\bm g} \cdot \widehat{\bm \sigma}| \Theta (-{\bm g} \cdot \widehat{\bm \sigma})  \nonumber \\
&& \times  f({\bm r}_{1z},{\bm v},t) f({\bm r},{\bm v},t) \left( b_{\bm \sigma} -1\right) \phi({\bm v}),
\end{eqnarray}
where $b_{\bm \sigma}$ is the inverse operator of $b_{\bm \sigma}^{-1}$, i.e. it changes all the velocities ${\bm v}$ and ${\bm v}_{1}$ to its right into the post-collisional values given by
\begin{equation}
\label{2.8}
 {\bm v}^{\prime} \equiv b_{\bm \sigma} {\bm v}= {\bm v} + \frac{1+\alpha}{2} \left( {\bm g} \cdot \widehat{\bm \sigma} \right) \widehat{\bm \sigma},
\end{equation}
\begin{equation}
\label{2.9}
 {\bm v}^{\prime}_{1} \equiv b_{\bm \sigma} {\bm v}_{1} = {\bm v}_{1}- \frac{1+\alpha}{2} \left( {\bm g} \cdot \widehat{\bm \sigma} \right) \widehat{\bm \sigma}.
\end{equation}
Macroscopic density, $n({\bm r},t)$, velocity, ${\bm u}({\bm r},t)$,  and granular temperature, $T({\bm r},t)$  fields are defined in the usual way,
\begin{equation}
\label{2.10}
n({\bm r},t) \equiv \int d{\bm v}, f({\bm r}, {\bm v},t),
\end{equation}
\begin{equation}
\label{2.11}
n({\bm r},t) {\bm u}({\bm r},t) \equiv \int d{\bm v}\, {\bm v} f({\bm r},{\bm v},t),
\end{equation}
\begin{equation}
\label{2.12}
\frac{3}{2} n({\bm r},t) T({\bm r},t) \equiv  \frac{m}{2} \int d{\bm v}\, V^{2}({\bm r},t) f({\bm r},{\bm v},t).
\end{equation}

Here, ${\bm V} ({\bm r},t) \equiv {\bm v}-{\bm u},({\bm r},t)$ is the peculiar velocity of the particle  relative to the local flow velocity. Notice that, as it is usual in the literature of granular gases, the Boltzmann constant does not appear in the definition of the granular temperature. This is done to emphasize the absence of any thermodynamic meaning in such definition. From Eq. (\ref{2.1})), balance equations can be derived for the macroscopic fields by taking velocity moments. They have the form
\begin{equation}
\label{2.13}
\frac{\partial n({\bm r},t)}{\partial t} + \frac{\partial}{\partial {\bm r}} \cdot \left[ n({\bm r},t) {\bm u} ({\bm r},t )\right] =0,
\end{equation}
\begin{equation}
\label{2.14}
m n({\bm r},t) \frac{\partial {\bm u} ({\bm r},t)}{\partial t} + m n ({\bm r},t) {\bm u}({\bm r},t) \cdot  \frac{\partial {\bm u}({\bm r},t)}{\partial {\bm r}} + \frac{\partial}{\partial {\bm r}} \cdot {\sf P}({\bm r},t)=0,
\end{equation}
\begin{eqnarray}
\label{2.15}
\frac{3}{2} n({\bm r},t) \frac{\partial T({\bm r},t)}{\partial t} & + & \frac{3}{2} n({\bm r},t)) {\bm u}({\bm r},t) \cdot \frac{\partial T({\bm r},t)}{\partial {\bm r}} +{\sf P}({\bm r},t) : \frac{\partial {\bm u}({\bm r},t)}{\partial {\bm r}} \nonumber \\
&+& \frac{\partial}{\partial {\bm r}}\, \cdot {\bm J_{q}}({\bm r},t) = \left(1-\alpha^{2} \right) \omega ({\bm r},t).
\end{eqnarray}
The pressure tensor ${\sf P}({\bm r},t)$, and the heat flux ${\bm J}_{q}({\bm r},t)$ have both ``kinetic'' and ``collisional transfer'' contributions,
\begin{equation}
\label{2.16}
{\sf P}({\bm r},t) = {\sf P}^{(k)}({\bm r},t) + {\sf P}^{(c)}({\bm r},t),
\end{equation}
\begin{equation}
\label{2.17}
{\bm J}_{q}({\bm r},t)= {\bm J}_{q}^{(k)}({\bm r},t) + {\bm J}_{q}^{(c)}({\bm r},t).
\end{equation}
The kinetic contributions are given by
\begin{equation}
\label{2.18}
{\sf P}^{(k)}({\bm r},t)= m\int d{\bm v}\, {\bm V}({\bm r},t) {\bm V} ({\bm r},t) f({\bm r},{\bm v},t),
\end{equation}
\begin{equation}
\label{2.19}
{\bm J}_{q}^{(k)} ({\bm r},t)) = \frac{m}{2} \int d{\bm v}\, V^{2}({\bm r},t) {\bm V}({\bm r},t) f({\bm r},{\bm v},t).
\end{equation}
The collisional transfer parts of the fluxes verify
\begin{equation}
\label{2.20}
\frac{\partial}{\partial z} {\sf P}_{zi}^{(c)}({\bm r},t) = - m  \int d{\bm v}\,  v_{i} J[{\bm r},{\bm v}|f],
\end{equation}
\begin{equation}
\label{2.21}
\frac{\partial}{\partial z} J_{q,z}^{(c)}({\bm r},t)= - \frac{m}{2} \int d{\bm v} v^{2} J[{\bm r},{\bm v} |f] - \frac{\partial}{\partial z}  \left[ {\bm u}({\bm r},t) \cdot {\sf P}^{(c)}({\bm r},t) \right] -(1-\alpha^{2}) \omega ({\bm r},t)
\end{equation}
Using  these relations, the following explicit expressions can be identified
\begin{eqnarray}
\label{2.22}
{\sf P}_{ij}^{(c)} ({\bm r},t) &=& \delta_{iz} \frac{1+\alpha}{4}\, \sigma^{2} m \int d{\bm v} \int d {\bm v}_{1} \int_{\sigma/2}^{h-\sigma/2} d z_{1} \int_{0}^{2\pi} d \varphi \int_{0}^{1} d\lambda\, \lambda^{-1} |{\bm g} \cdot \widehat{\bm \sigma}_{\lambda}|^{2}\widehat{\sigma}_{\lambda z} \widehat{\sigma}_{\lambda j}  \nonumber \\
& & \times \Theta \left( -{\bm g} \cdot \widehat{\bm \sigma}_{\lambda} \right) f({\bm r}_{1z},{\bm v}_{1},t)) f({\bm r}_{\lambda z},{\bm v},t),
\end{eqnarray}
\begin{eqnarray}
\label{2.23}
J_{qi}^{(c)}({\bm r},t)& = & \delta_{iz} \frac{1+\alpha}{4}\, \sigma^{2} m \int d{\bm v} \int d {\bm v}_{1} \int_{\sigma/2}^{h-\sigma/2} d z_{1} \int_{0}^{2\pi} d \varphi \int_{0}^{1} d\lambda\, \lambda^{-1} |{\bm g} \cdot \widehat{\bm \sigma}_{\lambda}|^{2}\widehat{\sigma}_{\lambda z} \cal{\bm G}\cdot \widehat{\bm \sigma}_{\lambda} \nonumber \\
& & \times \Theta \left( -{\bm g} \cdot \widehat{\bm \sigma}_{\lambda} \right) f({\bm r}_{1z},{\bm v}_{1},t)) f({\bm r}_{\lambda z},{\bm v},t).
\end{eqnarray}
Here,
\begin{equation}
\label{2.24}
\mathcal{\bm G}({\bm r},t) \equiv  \frac{{\bm V}({\bm r},t)+ {\bm V}_{1} ({\bm r},t)}{2} = \frac{{\bm v}+{\bm v}_{1}}{2}\, - {\bm u}({\bm r},t),
\end{equation}
i.e. it is the center of mass velocity of the colliding particles relative to the macroscopic flow. Moreover
\begin{equation}
\label{2.25}
{\bm r}_{\lambda z} \equiv \left\{ x,y,z_{\lambda} \right\},
\end{equation}
\begin{equation}
\label{2.26}
z_{\lambda} \equiv  \frac{z-(1-\lambda)z_{1}}{\lambda},
\end{equation}
and
\begin{equation}
\label{2.27}
\widehat{\bm \sigma}_{\lambda} \equiv \left\{ \sin \theta_{\lambda} \sin \varphi, \sin \theta_{\lambda} \cos \varphi, \cos \theta_{\lambda} \right\},
\end{equation}
where
\begin{equation}
\label{2.28}
\cos \theta_{\lambda} = \frac{z_{1}-z}{\lambda \sigma}\, ,  \quad \sin \theta_{\lambda} = + \sqrt{1- \cos^{2} \theta_{\lambda}}\, .
\end{equation}

Finally, the source term on the right hand side of the energy balance equation, Eq.\, (\ref{2.15}), describes the kinetic energy dissipation in collisions due to inelasticity, and it can not be written as the divergence of a flux. Its expression is
\begin{equation}
\label{2.29}
\omega ({\bm r},t )= \frac{m \sigma}{8} \int d{\bm v} \int d {\bm v}_{1} \int_{\sigma/2}^{h-\sigma/2} d z_{1} \int_{0}^{2\pi} d \varphi\,  |{\bm g}  \cdot \widehat{\bm \sigma} |^{3} \Theta  \left( - {\bm g}  \cdot \widehat{\bm \sigma} \right) f({\bm r}_{1z},{\bm v}_{1},t) f({\bm r},{\bm v},t).
\end{equation}

Some details of the derivation of the balance laws and the expressions of the fluxes and the source term are given in Appendix \ref{ap1}.  The collisional transfer contributions to the pressure tensor and the heat flux are due to  the delocalization of the centres of the colliding pair of particles and, more specifically, to the different values of the two $z$ coordinates. This is  a consequence of the confinement of the system that restricts the possible values of the collision vector. The explicit form of this restriction for a particle, depends on the vertical coordinate of the particle  and, for this reason, the vertical coordinates of the pair of colliding particle are relevant in describing the dynamics of the system even at low densities.  When considering the   expressions of the collisional transfer contributions to the pressure tensor and to the heat flow, Eqs. (\ref{2.22}) and (\ref{2.23}), it must be kept in mind that that, as already indicated, for physical initial conditions, the one-particle velocity distribution, $f({\bm r},{\bm v},t)$ vanishes for positions outside the system, i.e. for both $z<\sigma/2$ and $z>h-\sigma/2$. As a consequence,  the effective range of the $\lambda$-integration is restricted to values such that
\begin{equation}
\label{2.30}
\frac{\sigma}{2} < \frac{z-(1-\lambda)z_{1}}{\lambda} <h-\frac{\sigma}{2}\, .
\end{equation}
This eliminates the apparent divergence of the integrands for $\lambda=0$, and also guarantees that $-1 \leq \cos \theta_{\lambda} \leq 1$.  The previous comment is closely related with an important general property of a system of hard spheres or disks confined by means of hard walls. Consider $z=\sigma/2$, so that
\begin{equation}
\label{2.31}
z_{\lambda}= \frac{\sigma/2 -(1-\lambda)z_{1}}{\lambda}\, .
\end{equation}
For $z_{1}=\sigma/2$, it is $z_{\lambda} (z=\sigma/2,z_{1}=\sigma/2)= \sigma/2$. Besides
\begin{equation}
\label{2.32}
\left( \frac{\partial z_{\lambda}}{\partial z_{1}} \right)_{z} = - \frac{1-\lambda}{\lambda}\,  ,
\end{equation}
that is negative for $0<\lambda<1$. It follows that, as $z_{1}$ increases from $\sigma/2$, $z_{\lambda}$ decreases from the same vale. Let is stress that this holds for the particular value $z=\sigma/2$. Consequently, $z_{\lambda}$ always takes values outside the system for $z=\sigma/2$, and $f({\bm r}_{\lambda z},{\bm v},t)=0$, implying that
\begin{equation}
\label{2.33}
{\sf P}_{ij}^{(c)}({\bm r},t)=0
\end{equation}
and
{\begin{equation}
\label{2.34}
{\bm J}_{q}^{(c)} ({\bm r},t)=0
\end{equation}
for $z=\sigma/2$. A similar argument shows that the collisional transfer contributions to the pressure tensor and the heat flux also vanish at the other plate, i.e. for $z=h-\sigma/2$. The property that the collisional part of the pressure vanishes at a hard wall for a system of hard spheres or disks at equilibrium is known asas the ``contact theorem'' \cite{HyMc06}. Recently, it has been extended to arbitrary non-equilibrium states and shown that, actually,  all the components of the collisional transfer contribution to the pressure tensor vanish \cite{MGyB18}. Here, it has been shown that the same property applies in the case of a strongly confined gas of inelastic hard spheres, and that the collisional transfer contribution to the heat flux also vanishes at the hard boundary.

\section{Kinetic model}
\label{s3}
The Boltzmann equation (\ref{2.1}) is technically more involved than the usual Boltzmann equation for bulk systems, due to the space dependence of the collision term. For this reason, it is appealing to consider kinetic equations obtained as approximated representations of the original Boltzmann equation, that allow a more detailed analytical study. The basic idea of a kinetic model is to look for the maximum simplicity while retaining the  most relevant physical and mathematical properties of the original equation. In the present case, these include the existence of the equilibrium state in the elastic limit and the exact form of the conservation laws. This will accomplished by considering the collision term $J$ as expanded in a complete set of velocity polynomials with a scalar product weighted with the local equilibrium distribution function. The contribution from the subspace leading to the balance equations for the macroscopic fields  is kept exactly, while the   rest is approximated by a single relaxation term. Similar models have been successfully formulated for the revised Enskog theory of a non-confined system of hard spheres, both for elastic \cite{DSyB96,SMDyB98} and dissipative  \cite{BDyS97} collisions.
To derive the kinetic model, it is useful to introduce a Hilbert space of real functions of the velocity ${\bm v}$ by means of the scalar product
\begin{equation}
\label{3.1}
\langle g|h\rangle \equiv \int d{\bm v}\, \psi_{l}({\bm v} ) g({\bm v}) h({\bm v}),
\end{equation}
where $\psi _{l}({\bm v})$ is the Maxwellian defined with the actual values of the local velocity and granular temperature at position ${\bm r}$ and time $t$,
\begin{equation}
\label{3.2}
\psi _{l}({\bm v}) \equiv \left[ \frac{m}{2 \pi T({\bm r},t)} \right]^{3/2} \exp - \frac{mV^{2}({\bm r},t)}{2 T ({\bm r},t)}\, .
\end{equation}
This function is related with the usual local equilibrium distribution function $f_{l}({\bm r},{\bm v},t) $ by
\begin{equation}
\label{3.3}
f_{l}({\bm r}, {\bm v},t ) = n({\bm r},t) \psi_{l}({\bm v}).
\end{equation}
Notice that both $\psi_{l}$ and $f_{l}$ depend on position and time through their functional dependence on the macroscopic fields, although it is not explicitly indicated for the former. In this sense, the Hilbert space is defined for given values of ${\bm r}$ and  $t$, namely those corresponding to the position and time at which the distribution function will be evaluated.
Consider next the set of functions
\begin{equation}
\label{3.4}
\left\{ \phi_{\beta} \right\} \equiv \left\{ 1, \left( \frac{m}{T} \right)^{1/2} {\bm V}, \left( \frac{2}{3} \right)^{1/2} \left( \frac{mV^{2}}{2T}-\frac{3}{2} \right) \right\}.
\end{equation}
These functions are orthogonal withe the scalar product definition in Eq.\ (\ref{3.1}),
\begin{equation}
\label{3.5}
\langle \phi_{\beta} | \phi_{\beta^ \prime} \rangle = \delta_{\beta, \beta \prime}.
\end{equation}
In the following the two notations
\begin{equation}
\label{3.6}
\left\{ \phi_{\beta} \right\} \equiv \left\{ \phi_{1},{\bm\phi}_{2}, \phi_{5} \right\}
\end{equation}
and
\begin{equation}
\label{3.7}
\left\{ \phi_{\beta} \right\} \equiv \left\{ \phi_{1},\phi_{2}, \phi_{3},\phi_{4},\phi_{5} \right\}
\end{equation}
will be employed indistinctly. A projection operator $\mathcal{P}$ over the subspace spanned by the functions $\phi_{\beta}$ is defined by
\begin{equation}
\label{3.8}
\mathcal{P} h({\bm v}) \equiv \psi_{l}({\bm v}) \sum_{\beta} \phi_{\beta}({\bm v}) \langle \phi_{\beta} | \psi_{l}^{-1} h \rangle = \psi_{l}({\bm v}) \sum_{\beta} \phi_{\beta} ({\bm v})  \int d {\bm v}^{\prime} \phi_{\beta}({\bm v}^{\prime}) h({\bm v}^{\prime}).
\end{equation}
By means of this operator, the collision term $J$ in the kinetic equation (\ref{2.1}) is decomposed into the two parts
\begin{equation}
\label{3.9}
J[{\bm r},{\bm v}|f]  =\mathcal{P} J[{\bm r},{\bm v}|f] + (1-\mathcal{P} )J[{\bm r},{\bm v}|f] .
\end{equation}
It is
\begin{equation}
\label{3.10}
\int d{\bm v}\, \phi_{\beta}({\bm v}) \mathcal{P} h({\bm v}) = \int d{\bm v} \phi_{\beta}({\bm v}) h({\bm v}),
\end{equation}
for arbitrary $h({\bm v})$ and, consequently,
\begin{equation}
\label{3.11}
\int d{\bm v}\, \phi_{\beta}({\bm v})  \left( 1- \mathcal{P} \right) J[{\bm r},{\bm v}|f]=0.
\end{equation}
Then, the second term on the right hand side of Eq.\ (\ref{3.9}) does not contributes to the form of the balance equations. Here it will be approximated by a single exponential relaxation term,
\begin{equation}
\label{3.12}
(1-\mathcal{P} ) J[{\bm r},{\bm v}|f] \rightarrow  -\nu \left[ f({\bm r},{\bm v},t) - f_{l}({\bm r}, {\bm v}, t) \right],
\end{equation}
where $\nu$ is a velocity independent characteristic frequency that can be a functional of the density and granular temperature fields. It can be chosen to optimize the agreement between the kinetic model and the original Boltzmann equation with regards to some property of interest. From the definition of the local equilibrium distribution it follows that
\begin{equation}
\label{3.13}
\int d{\bm v}\, \phi_{\beta}({\bm v}) f({\bm r},{\bm v}.t) = \int d{\bm v}\, \phi_{\beta}({\bm v}) f_{l}({\bm r},{\bm v},t),
\end{equation}
$\beta=1,\ldots,5$, that implies
\begin{equation}
\label{3.14}
\mathcal{P} \left[-\nu (f-f_{l}) \right] =0,
\end{equation}
showing the consistency of the approximation made in Eq.\ (\ref{3.12}).  Then, the model kinetic  equation reads
\begin{equation}
\label{3.15}
\frac{\partial}{\partial t}\, f({\bm r},{\bm v},t) +{\bm v} \cdot \frac{\partial}{\partial {\bm r}}\, f({\bm r},{\bm v},t) = -\nu \left[ f({\bm r},{\bm v},t)-f_{\l}({\bm r},{\bm v},t) \right] + \mathcal{P} J[{\bm r},{\bm v}|f].
\end{equation}
By construction, this model equation leads to the same balance equations as the original Boltzmann equation (\ref{2.1}), since the part of the collision term that has been approximated does not contribute to those equations. The term $\mathcal{P} J$ can be written in a more explicit way in terms of the collisional transfer contributions to the fluxes, ${\sf P}^{(c)}$ and ${\bm J}_{q}^{(c)}$, and the source term, $\omega$. By using Eqs. (\ref{2.20}) and (\ref{2.21}), the resulting kinetic equation is
\begin{equation}
\label{3.16}
\frac{\partial f}{\partial t} + {\bm v} \cdot \frac{\partial f}{\partial {\bm r}} =  -\nu \left( f-f_{l} \right)- \frac{f_{l}}{nT}  \left[ V_{i}   \frac{\partial}{\partial z} {\sf P}_{zi}^{(c)} + \left( \frac{mV^{2}}{3T}-1 \right)
\left( \frac{\partial}{\partial z}\, J_{q,z}^{(c)} - (1-\alpha^{2}) \omega + {\sf P}_{zi}^{(c)} \frac{\partial u_{i}}{\partial z} \right) \right],
\end{equation}
where repeated indices are implicitly summed over. This equation is a highly non-lineal integro-differential equation. The local equilibrium distribution, the collisional transfer contributions of the pressure tensor and the heat flux, and the source energy term appearing in the equation, are given by the same functional of $f$ as in the original Boltzmann equation, i.e. by Eqs.  (\ref{2.22}), (\ref{2.23}), and (\ref{2.29}), respectively. The main and essential mathematical advantage of Eq. (\ref{3.16}) as compared with the original  Boltzmann equation (\ref{2.1}), is that the velocity dependence of the collision term on the right hand side is much simpler now. It is given by a Gaussian times polynomials of second degree. A relevant example of application of the model is given in the next section.

 \section{The inhomogeneous cooling state}
 \label{s4}
 A special idealized state of the confined system we are considering here is that of inhomogeneous cooling state (ICS), for which all the time dependence of the macroscopic dynamics occurs through the time dependence of the granular temperature, while there is a spatial dependence through the density, that is a function of the distance to the confining walls \cite{BGyM19a}. The ICS occurs in freely evolving systems, i.e. without any kind on external energy injection. It  is an extension to confined quasi-two-dimensional granular gases of the homogeneous cooling state exhibited by bulk systems \cite{Ha83}.

 The model kinetic equation (\ref{3.16}) has a solution, $f_{ICS}$, whose macroscopic description corresponds to the ICS. It is given by the Maxwellian
 \begin{equation}
 \label{4.1}
 f_{ICS}(z,{\bm v},t) = n(z) \psi_{ICS}({\bm v},t)
 \end{equation}
 with
 \begin{equation}
 \label{4.2}
 \psi_{ICS}({\bm v},t)= \left[ \frac{m}{2\pi T(t)} \right]^{3/2} \exp - \frac{mv^{2}}{2T(t)}\, .
 \end{equation}
 The density profile must satisfy the equation
 \begin{equation}
 \label{4.3}
 \frac{\partial}{\partial z} \ln n(z) = \pi (1+\alpha) \int_{\sigma/2}^{h-\sigma/2} dz_{1}\, (z-z_{1}) n(z_{1})
 \end{equation}
 and the time evolution of the granular temperature is given by
 \begin{equation}
 \label{4.4}
 \frac{\partial}{\partial t}\, T(t) = -\frac{2}{ 3 n(z)}\ (1-\alpha^{2}) \omega.
 \end{equation}
 The proof of the above results is given in Appendix \ref{ap2}. Using Eq. (\ref{4.1}) into Eq. (\ref{2.29}) it is easily obtained
 \begin{equation}
 \label{4.5}
 \omega(z,t) =  \left( \frac{\pi}{m} \right)^{1/2} \frac{N \sigma}{A}\, n(z) T(t)^{3/2}\, ,
 \end{equation}
 so that Eq.\ (\ref{4.4}) can be expressed in the usual for
 \begin{equation}
 \label{4.6}
 \frac{\partial}{\partial t}\, T(t) =- \zeta(t) T(t),
 \end{equation}
 with the cooling rate given by
 \begin{equation}
 \label{4.7}
 \zeta(t) = \frac{2}{3}\, \left( 1 -\alpha^{2} \right) \frac{N \sigma}{A} \left[ \frac{\pi T(t)}{m} \right]^{1/2}\, .
 \end{equation}
 In these expression s$A$ is the area of each of the two plates confining the system, so that $N/A$ can be seen as an effective two-dimensional number of particles density. As required by consistency with the existence of the ICS, the cooling rate does not depend on $z$.

 Let us consider the equation for the density profile, Eq. (\ref{4.3}).  Taking into account that for symmetry considerations it must be $n(z)=n(h-z)$, the equation is seen to be equivalent to
 \begin{equation}
 \label{4.8}
 \frac{\partial}{\partial z}\, \ln n(z) = \frac{\pi N}{A} (1+\alpha) \left( z-\frac{h}{2} \right),
 \end{equation}
 whose solution is
 \begin{equation}
 \label{4.9}
 n(z) = \frac{N}{Ab} \exp \left[ a \left(z-\frac{h}{2} \right)^{2}\right]
 \end{equation}
 where
 \begin{equation}
 \label{4.10}
 a \equiv \frac{\pi (1+\alpha) N}{2A}\, ,
 \end{equation}
 \begin{equation}
 \label{4.11}
 b \equiv \left( \frac{\pi}{a}\right)^{1/2}  \erfi \sqrt{a} \left( \frac{h-\sigma}{2} \right).
 \end{equation}
 Here $\erfi(x)$ is the imaginary error function defined as
 \begin{equation}
\label{4.12}
\erfi(y) \equiv \pi^{-1/2} \int_{-y}^{y} dy^{\prime}\, e^{y^{\prime2}}.
\end{equation}
 In the elastic limit $\alpha=1$, the above density profile coincides with the exact equilibrium result reported in ref. \cite{BMyG16}. For $\alpha<1$, the comparison can be made with an approximate result derived from the Boltzmann equation (\ref{2.1}) also for the ICS, and valid in the limit $(h-\sigma)/\sigma \ll 1$. The only difference is that the latter contains a factor due to the anisotropy of the velocity distribution function, an effect that is neglected in the model kinetic equation. Let us mention that the inhomogeneity of the density field in the direction perpendicular to the confining plates, and the quantitative accuracy of the theoretical predictions based on the Boltzmann equation have been verified both for  elastic \cite{BMyG16} and inelastic  \cite{BGyM19a} hard spheres by means of molecular dynamics simulations.

 Since, by construction,  the model leads to the same balance equation as the original Boltzmann equation, Eqs.\ (\ref{2.13})-(\ref{2.15}) can be directly applied to analyze some properties of the ICS, as predicted by the model. From the momentum conservation equation, it follows that the component of the pressure tensor in the ICS must verify
 \begin{equation}
 \label{4.13}
 \frac{\partial {\sf P}_{zx}^{(c)}}{\partial z} = \frac{\partial {\sf P}_{zy}^{(c)}}{\partial z} =0,
 \end{equation}
 \begin{equation}
 \label{4.14}
 \frac{\partial {\sf P}_{zz}}{\partial z} = \frac{\partial}{\partial z}\ \left( {\sf P}_{zz}^{(k)} + {\sf P}_{zz}^{(c)} \right) =0.
 \end{equation}
 Since the model alkso conserves the expresion of the pressure tensor, the general result derived in Sec. \ref{s2} that the collisional treansfer contributions to the pressure tensor vanish at the hard walls can be employed, and write
 \begin{equation}
 \label{4.15}
 {\sf P}_{zz}(t)= {\sf P}_{zz}^{(k)} (z= \sigma/2,t) = n(z=\sigma/2) T(t),
 \end{equation}
 where Eq.\ (\ref{ap2.1}) has been used. This expression gives the normal  force per unit of area exerted by the gas on the plates in the ICS. In the elastic limit, the only change to be made is tu substitute the time dependent temperature by its equilibrium value. Then, the expression agrees with the result obtained previously for the equilibrium state of an elastic gas of hard spheres \cite{SyL97,BMyG16}. The simplicity of this result is consistent with the behaviour obtained in the limit of extreme confinement also for elastic spheres \cite{FLyS12}. It is worth to emphasize that, as derived here, Eq. (\ref{4.15}) reflects a property valid not only at the hard boundaries of the system but also in the bulk.

 The distribution function of the ICS following from the Boltzmann equation  (\ref{2.1})  differs qualitatively from a Maxwellian, specially for $\alpha$ significantly smaller than unity,  while we have seen that with the kinetic model a Maxwellian is obtained for all values of $\alpha$. In spite of this difference, the kinetic model may still be useful to study  a variety of physical situations. To put this expectation on a firmer basis, the velocity distribution function of the ICS has been computed using molecular dynamics simulation techniques. The results show that the velocity distribution is anisotropic, i.e. the marginal distributions associated to the velocity parallel and perpendicular to the plates differ, and so do the associated partial granular temperatures. On the other hand, both distributions can be approximated very accurately by a Maxwellian for thermal velocities, say $|v|  \lesssim (T/m)^{1/2}$, even for relatively small values of the coefficient of normal restitution and for widths of the system close to $2 \sigma$.

 \section{Summary and Conclusions}
 \label{s4}

 One of the aims of this paper is to derive the balance equations for the local density, momentum, and energy of a system of (elastic or inelastic) hard spheres  confined between two infinite parallel plates, separated a distance smaller than two particle diameters. The dynamics of the system is assumed to be accurately described by a Boltzmann-like equation derived recently.  As a consequence of the confinement, there are collisional transfer contributions to the pressure tensor and the heat flux that, in principle, are characteristic of fluid beyond the dilute limit.  Quite interestingly, the expressions of the collisional transfer contributions imply that all them vanish at the hard boundaries confining the system. This result applies independently of the state of the system and also of the nature, elastic or inelastic, of collisions. Therefore, it constitutes a wide generalization of the ``wall theorem'' formulated in equilibrium statistical mechanics. Another aim of the present work is to formulate a model kinetic equation closely related to the Boltzmann-like kinetic equation mentioned above.  In particular, the model has, by construction, the same balance equations as the original equation. As a test of the model, it has been applied to the simpler macroscopic states corresponding to a molecular, elastic gas and to a granular, inelastic gas, respectively.  A good agreement with previous results obtained using  equilibrium methods and the Boltzmann equation itself has been found.

 In principle, the model formulated here could be used to derive hydrodynamic transport equations by using the Chapman- Enskog procedure.  Actually, this has been done for other BGK-like models formulated for the Enskog equation of dense gases of hard spheres \cite{DSyB96,BDyS97}. Nevertheless, this may be not a good strategy to describe the observed macroscopic behavior exhibited by this system and mentioned in the Introduction section.  The observed phenomenology corresponds to a two-dimensional image of the system, when looked from above or from below. The effective phase separation occurs   in the plane parallel to the plates and not in a volume region of the system. Moreover, and more important, the geometry of the system makes it difficult to suppose that there is hydrodynamic behaviour in the direction perpendicular to the plates.. It seems more plausible  that the dynamical behavior observed by projecting over a horizontal plane can be accurately described at a macroscopic level by hydrodynamic-like equations. In this context, a logical way to proceed is to derive projected balance equations from  the three-dimensional ones presented here, formulating afterwards a model kinetic equation using arguments similar to those used in this paper.

 \acknowledgments

This research was supported by the Ministerio de Econom\'{\i}a, Industria  y Competitividad  (Spain) through Grant No. FIS2017-87117-P (partially financed by FEDER funds).

redpajama_set_name: RedPajamaArXiv


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This all-levels class introduces basic striking techniques, with a specific focus on adapting striking arts for mixed martial arts competition and self-defense.
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At some point, most beard owners will need some kind of beard care product. A quick google search will surely turn up products like beard balm, beard oil, and beard shampoo. The companies that make these products want you to think that you need all of them. The truth is, just one or two might be sufficient for your situation.
Figuring out where to start experimenting with beard products is unnecessarily complicated. This post will help you answer the question: What is beard oil? and determine if it is for you.
If you are needing something to soften your facial hair, then a beard balm is probably what you are looking for. If that isn't doing the trick, then adding in a beard oil might help. However, if you are plagued by skin conditions like beard dandruff or eczema, then choosing a beard oil is the first step to getting those problems under control.
Using this ingredient list as a guideline should help you to determine which beard oil is right for you. To further help you find the right beard care product, check out our beard oil comparison chart.


Summer 2019 professional learning opportunities will be available for registration beginning on April 29, 2019.
Where leaders learn and learners lead.
The Central Kentucky Educational Cooperative (CKEC) works with the superintendents and staff of its member districts to provide opportunities for collaboration, strategic planning and professional development. CKEC delivers a wide range of customized services to our member districts, including opportunities to network with colleagues. We also work with vendors of all kinds to leverage the buying power of our member districts to ensure that they are paying the most competitive prices for products and services.
At CKEC, we love kids and families! Unfortunately, due to liability issues and professional decorum, children and other family members or friends are not permitted to attend our trainings.
The CKEC inclement weather policy is as follows. If Fayette County Schools are closed, CKEC training sessions are canceled. If Fayette County Schools are open, but YOUR school district is closed, please use discretion.
If Fayette County is on a delay, please check your email for information pertaining to your training.


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Gentoo launch apprenticeship opportunities in Sunderland
An apprenticeship can help you get real work experience, gain job skills and knowledge and study towards a relevant qualification.
Sunderland-based housing association Gentoo Group is recruiting apprentices to join its workforce with support from Sunderland College.
As part of its ongoing commitment to create employment and training opportunities in the city, Gentoo is offering 14 apprenticeships across a variety of business areas including Repairs & Maintenance, Human Resources, Procurement, Property Investment, Legal and Estate Services.
The apprenticeships take between two to four years to complete, with a starting salary of £12,500 rising by £3,000 on successful completion of each year. The positions come with a range of attractive benefits including 29 days holiday plus bank holidays, a cash healthcare plan, employee assistance programme and access to an online portal of retail discounts.
Nigel Wilson, Chief Executive Officer at Gentoo group, said: "I am delighted that we have been able to create further apprenticeship opportunities for people across Wearside as we continue to invest in our workforce of the future.
"As one of the largest employers in Sunderland we are committed to providing employment and training opportunities for residents and we look forward to supporting the new recruits in realising their potential."
Since its formation in 2001, Gentoo has played a key role in developing the North East's future workforce by creating 277 apprenticeship opportunities, with many being placed through a strategic partnership with Sunderland College which is one of the North East's largest apprenticeship providers.
Iain Nixon, Vice Principal for Partnerships and Commercial at Education Partnership North East, said: "Sunderland College and Gentoo have a long history of working in partnership together. We are delighted to be supporting Gentoo with their latest apprenticeship programme, this is a great opportunity for anyone looking to start a career with one the City's leading employers.
"Over recent years Gentoo's apprenticeship programme has seen many young people go on to develop successful careers, and we are really looking forward to working with the latest cohort of apprentices."
Applications are open now and those interested in the vacancies have until Friday 23 July to apply online www.gentoogroup.com/about-us/jobs/
College updates
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Plans submitted for world-class training centre
Detailed plans for a world-class training facility that will provide 'next generation skills for next generation homes' have been submitted this w...
Systems leadership at the heart of £20 million housing innovation inv...
The City of Sunderland is proving itself to be a place where key organisations are working together to create a brighter future.
A new partnership provides the right medicine for apprenticeships
A dynamic partnership between a local college and a Clinical Commissioning Group will see dozens of apprenticeships up for grabs over the next ye...
98% of our students progress to a positive destination
(*Destination data for Sunderland College students aged 16-18, academic year 2017/18).
Why choose Sunderland College?
Our achievement rates are well above the national average.
We are the College of the Year.
(*Top college nationally for BTECs, Pearson 2019)
100% of our HND and HNC students progressed to employment or further study.
(*Destination data, academic year 2016/17, HNC 19+)
We have recently invested £50 million in our facilities.
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