A View of My Philosophy :: Teaching Education Teachers Essays

A View of My Philosophy Its hard to decide on a major as a high school student preparing to enter college. There are so many possibilities and it seems impossible to choose nevertheless one. It was a major decision for me, and I was terrified I would work the wrong choice. What if I didnt like my major? What if I found aught that I liked? I attended West Virginia Wesleyan College my freshman year with no decided major. Education had always been in the back of my mind, scarcely I wanted to make sure that there was nothing else that would catch my attention. It turned out that nothing else did. During my freshman year I took a manakin of classes in many different subject areas, but nothing seemed to spark a major interest. I was beginning to worry that I would never make up my mind until I started volunteering at an after-school weapons platform at a local elementary school near WVWC in Buckhannon, WV. It was a program designed as a place for stud ents to do their homework while they waited for their parents to pick them up. From the first day volunteering, I loved it. I loved the beat hold backs on the faces of the young children as they tried to work through an arithmetic problem. I loved their smiling faces and their eagerness to learn. Most of all, I loved the look of accomplishment when a struggling student finally understood. That was how I knew I was meant to be a teacher. Just to know that I made a deflexion in the lives of the children I helped was better than almost anything Ive ever accomplished. During the program I truly realized I wanted to teach, and not just on a temporarily volunteering basis. The idea of having my own classroom and my own group of students to influence became more and more appealing. My philosophy of teaching is a combination of nearly of the most well known philosophies. One philosophy I support is progressivism.

Islam :: essays research papers

The five pillars      The Five Pillars of Islam are the structure of Muslim religion. They are the testimony of faith, prayer, giving, refrain during the month of Ramadan, and if you are lucky the excursion to Makkah once in a lifetime. The first pillar, entitled the testament of faith, exploits the Belief in one God, Allah, whom constitutes the very cosmos of Islam. Prayer is the next pillar of Islam. Prayers do not take more than five minutes and are required five times daily. In prayer, a person feels inner happiness, peace, and a direct connection between the worshipper and Allah. Giving Zakat, other known as supporting the needy, is the third pillar of Islam. Initially, Zakat was the yield that altered Muhammads views of the world and turned him into a political figure. Fasting for the month of Ramadan is the fourth pillar of Islam. Even the Quran states, "O you who believe desist is prescribed to you as it was prescribed to those before you so that you can learn Taqwa" (Quran 2183). Ramadan consists of fasting from dawn until sundown while concurrently abstaining from food, drink, and sexual contact.      Finally, the pilgrimage to Mecca is the last pillar.Although these conditions seem intense, there are actually many benefits to these submitters. The temptations of the world seem to erase our understanding for religion and g-d. Fasting will remind you of what has been give down to us. Ramadan can draw one closer to g-d by seclusion from the surrounding world. Ramadan will help achieve a willingness to give to the needy. Now you ease up been in their shoes and understand how it feels. Understanding discipline is important, since g-d has given us a free will, and the torture of Ramadan is a great reminder of how to view life. In the Quran one can find references to ritual prayer or salat. Holy Quran 2149 "From whencesoever thou startest forth turn thy face in the vigilance of the Sacred Mosque t hat is indeed the truth from thy Lord.

Mythological References in Hamlet Essay -- Essays on Shakespeare Hamlet

Mythological References in Hamlet Whats in a secernate? Hamlets good friend and confidant Horatio is doomed by the etymology of his nomenclature to give good speech. Shakespeare has gifted Horatio with an elegant lucidty that, when inspected closely, enables the reader to better comprehend the nature of the escape one of his first addresses is key in setting the tone of what James Joyce called the grave and constant in human suffering (Campbell 8). This is also a principal chemical group of classical mythology, and to fully understand Hamlet as a tragic hero, a comprehension of the mythological references at the beginning of the play must be foremost in the readers mind. These metaphoric intimations of tragedy leaked in Hamlets and Horatios earlier soliloquies deliver the fundamental clues to unlocking Hamlets enigmatic madness and foreshadow its violent emotional, physical and supernatural battles. The early Greeks believed that the universe created the gods, not .he other wa y around(Hamilton 24). They created their myths to explain the order of things how the sun sets, why the moon rises, the tides coming in and out, etc. When these patterns were interrupted, people assumed it was the exasperation or folly of the gods and went on making up more stories. Shakespeare has given his characters a heritage influenced by the Teutonic and Nordic races. Both cultures developed a collateral paganish belief shared by the early Greeks, and this parallel helps offer an explanation towards the choice of metaphor in the text. This is most important in the by-line excerpt from Horatios second soliloquy. After seeing the ghost of Hamlets father, he remarks to Bernardo Disasters in the sun and the moist star, Upon w... ...is heroic obligation. Claudius questions Hamlets mood after a month of trouble for his father CLAUDIUS How is it that the clouds still hang on you? HAMLET Not so, my lord, I am too much i the sun.(I.ii.65-66) The reader is reminded of Horatio s portentous thoughts of misfortune and simultaneously called to bed Hamlet as the center of future woes, around whom all the disasters at Elsinore revolve like satellites of the Fates is he too much like his father or not? If Hamlet truly embodies the Promethean essence, then he does know what is to happen Prometheus means foresight. What is in a name? Works Cited Campbell, Joseph. The Power of Myth. New York Doubleday, 1988. Dukore, Bernard F. Shaw on Hamlet. Educational Theatre Journal 23 (1971) 152-59 Hamilton, Edith. Mythology. New York Mentor, 1969.

Fifth Business by Robertson Davies Essay -- Essays Papers Davies Fifth

one-fifth Business by Robertson DaviesIn the essay Fifth Business, each of the main character traits is developed more and more clearly throughout their lives. Childhood characteristics are evident in the characters of Dustan Ramsay, Percy Boyd Stauton and Paul Dempster. All paranoia, and memories of the townsfolk of Deptford are resur face up in each of them after they all had left to start lives on their own. It was childhood that scared or marked them as people and the point that parents often have influence on children so, there is no question that these characters were definitely influenced by the parents in Deptford. Also, it is evident that each character seems to run forth due to the insecurities that they faced as children. Dunstan ran from his wrong-doing, Percy for his status and ego and Paul from his challenges as a misfit. Each seemed to wander like nomads throughout their lives, while these insecurities where never faced or challenged so they could change their poi nt of views. It was what seemed to be instilled within them like, the theory that the first five years of childhood shape the personalities of individuals. This is certainly evident in the novel at the beginning to the end. Dunstan ?s childhood was affected deeply through each situation he faced as a childly man, to a highly respected professor.He felt he was the center of all bad things, which occurred around him. Mrs. Demster?s accident was the main influence of his guilt because he felt that he was additionally...

Hesses Siddhartha as it Parallels Maslows Hierarchy of Needs Essay

Hesses Siddh impostureha as it Parallels Maslows Hierarchy of Needs Several parallels can be drawn between the psychologist Abraham Maslows theoretical hierarchy of needs and the spiritual expedition of Siddhartha, the eponymous main character in Herman Hesses novel. Maslows hierarchy of needs is somewhat of a pyramid that is divided into eight stages of need through which one improvementes throughout ones entire life. During the course of his lifetime, Siddharthas personality develops in a manner congruent with the stages of Maslows hierarchy. Siddharthas progress from each of the major sections of the hierarchy is marked by a sharp change in his life or behavior. Siddhartha is the story of a young mans journey in search of truth. Early in life, Siddhartha and his friend Govinda hear the teachings of the Buddha. Govinda is convinced of the validity of the Buddhas teachings and mystifys one of his followers. Siddhartha, on the other hand, was non satisfied with the Buddh as teachings because he believed that it was not possible to obtain true enlightenment through the words of others but that it must(prenominal) be experienced empirically. Siddhartha therefore rejects the life of a Brahmin to become a Samana (a wandering person who gives up solid possessions for his faith). After he tires of this life, he moves on to learn the art of love from a woman named Kamala and the art of business from a man named Kamaswami. He lives his new life for many years but then begins to feel that his mind has become stagnant and that he needs something new in his life. He abandons the surroundings he now finds decadent and becomes friends and lives with a ferry-man who he met years before. He spends the rest of his life with ... ...e of material wants and was truly happy just to exist. He realized his potential and found wonder in the world around him. At some time during life, everyone must face Siddharthas challenge. Everyone spends their life trying to attain self fulfillment and true contentment. And during that time, one must remember the importance of the journey itself, not only the substantial achievement. Although not everyone reaches that goal, it is that goal which motivates people to strive to be their very best, knowing that lasting happiness sits shining atop the pyramid. Works Cited Hesse, Herman. Siddhartha. New York New Directions produce Company, 1951. Glenn, Jerry. Monarch Notes. The Major Works of Herman Hesse. New York Monarch Press, 1973. Schultz, Duane . Theories of Personality. Monterey, California Brooks/Cole Publishing Company, 1976.

Hesses Siddhartha as it Parallels Maslows Hierarchy of Needs Essay

Hesses Siddhartha as it Parallels Maslows Hierarchy of Needs Several parallels can be drawn between the psychologist Abraham Maslows theoretical hierarchy of needs and the spiritual journey of Siddhartha, the eponymic main character in Herman Hesses novel. Maslows hierarchy of needs is somewhat of a pyramid that is divided into eight stages of need through which mavin progresses throughout ones ideal life. During the course of his lifetime, Siddharthas personality develops in a manner congruent with the stages of Maslows hierarchy. Siddharthas progress from each of the major sections of the hierarchy is marked by a sharp assortment in his life or behavior. Siddhartha is the story of a young mans journey in search of truth. Early in life, Siddhartha and his friend Govinda hear the teachings of the Buddha. Govinda is convinced of the boldness of the Buddhas teachings and becomes one of his followers. Siddhartha, on the other hand, was not satisfied with the Buddhas teachin gs because he believed that it was not possible to obtain true enlightenment through the words of others but that it essential be experienced empirically. Siddhartha therefore rejects the life of a Brahmin to become a Samana (a wandering person who gives up material possessions for his faith). After he tires of this life, he moves on to learn the art of love from a woman named Kamala and the art of business from a man named Kamaswami. He lives his new life for many geezerhood but then begins to feel that his mind has become stagnant and that he needs something new in his life. He abandons the surroundings he now finds effete and becomes friends and lives with a ferry-man who he met years before. He spends the rest of his life with ... ...e of material wants and was truly happy just to exist. He realized his potential and found delight in in the world around him. At some time during life, everyone must face Siddharthas challenge. Everyone spends their life trying to attain self fulfillment and true contentment. And during that time, one must remember the importance of the journey itself, not only the actual achievement. Although not everyone reaches that goal, it is that goal which motivates people to strive to be their very best, knowing that long-lived happiness sits shining atop the pyramid. Works Cited Hesse, Herman. Siddhartha. New York New Directions Publishing Company, 1951. Glenn, Jerry. Monarch Notes. The Major Works of Herman Hesse. New York Monarch Press, 1973. Schultz, Duane . Theories of Personality. Monterey, California Brooks/ kail Publishing Company, 1976.

Building Community: The Neighborhood Context of Local Social Organization Essay

Using the data in the table provided on p yearss 20 and 21, what can you hypothesize about the kinds batch exact with their neighbours and immediate conjunction?Provided is a table, which was taken from a study by the research consultancy ICM on different aspects of neighbouring. This table looks at the responses of great deal to functions on neighbouring, using different groupings. On the top, the general heading shows the gender, eon, tender class and regions. The horizontal axis identifies the answers that were give, lead by the weighted base. The header gender divides into three subheadings, the total of all the people that were asked separated into male and female.The near general heading- the age_ divides into 6 different age groups, beginning at the age of 18 and ending in 65+. From the age of 25 t here is an increase of 9. Social class, another general heading is separated into 4 subheadings, while the general heading regions divides into 5 subheadings. So, the hor izontal axis describes some characteristics of the people which were asked. What are the main patterns in this table?The row gender, here divided into male and female, doesnt reveal any huge differences in the percentage of how those wonders were answered. The percentages are quite close together. The row social class, with 4 different subheadings, reveals, that there is only one huge difference in question one I have a very heartfelt blood with my neighbours. Class AB with the highest percentage of 46% and Class C2 the lowest with 32%. A difference of 14%. Wales & South West, as one subheading of the general heading region, seems to have the best relationship with their neighbours with a percentage of 42%, examining answer one, followed by the Midlands.But here also, it is only a difference of 10% in total in comparison to Scotland, where it is 32% The biggest difference of a comfortably relationship to neighbours is to find at the general heading age. The subheading 25-34 has a better relationship to their neighbours (25%) than the age group 18-24(18%). But this increases remarkable up to the age group 65+ with a percentage of 59%. What are the interesting features?The pre-given answers are divided into positive answers(6), such asI have a very good relationship with my neighbours and negative answers(6) such asI dont have a very good relationship to my neighbours. except one answer is kind of neutral.Throughout all subheadings, the answerI spend a lot of time with my neighbours are answered quite similar, the biggest difference here is 6%, at the age subheading again. Also, the neutral answer has similarities in terms of percentage. The table shows, that even that you have a very good relationship with your neighbours, does not necessarily mean, you spend a lot of time with your neighbours.References SourceICM(2011) Good Neighbours SurveyPrepared on Behalf of Band and Brown by ICM query.London,ICM Research Ltd.End of TMA03 part oneWord count 497TMA03 Part 2Examine the argument thatGood indicates harbor good neighbours1.Social identity2.Relationship with neighbours3.Good fences do make good neighbours1. What is a social identity?According to Taylor(2009), the term identity is widely used but rather difficult to pin down. People have many different identities. A different identity (a group or collective identity is also given by difference from other groups),for example, world a woman,not a man and so on. A group or collective identity is both individual, saying something about a particular person, and social, because it refers to others who are similar or different. An identity given by connections to other people and social situations is social identity. Those different social identities can all overlap, which means, the definitions are not in return exclusive. Some kind of a relationship for example can be people in the same street that see each other and say hello to each other-they share a relational identity as neighbou rs, a collective identity as local residents.In most situations, people understand identities in terms of what people do rather than what they are. The sociologist Harold Garfinkel suggests, that social life is in constant motion. People also have the skills and knowledge to ready and maintain social order. Social order, which gets everyday social interaction between neighbours, often consists of contradictory obligations and norms, which have to be negotiated in the course of everyday social life in the street.(Byford,2009,p.267) As described by Joanna Bourke people developed for example an distance mechanism to maintain a good relationship to their neighbours.2. Everybody unavoidably good neighbours?Neighbours are anticipate to have a general disposition towards friendliness while , at the same time, respecting others need for privacy and reserve (Willmott, cited in Byford,2009,p.253) Depending on a variety of factors such as age, cultural background, socio-economic status and also personal characteristics as well as personal preferences makes up the individuals relationship to a neighbour. There is no code of conduct or a manual, how to behave as a good neighbour, however people acquire over times knowledge through socialisation, through the practice of being a neighbour. Life in a neighbourhood is ordered and structured.There are rules, habits and conventions, which regulate how people live together and interact in the street.(Byford,2009,p.262) When neighbouring goes wrong usually communications breaks down. In todays society a mediator is brought in, to re-establish communication, and neighbours should resolve their problems themselves.3.Why does a fence makes a good neighbour?Part of any streets infrastructure are timber fences, hedges, walls, gates curtains and other structural artefacts that are designed to keep residents asunder rather than bring them together. The expression Good fences make good neighbours, captures the essence of a paradox th at permeates life in everyneighbourhood. Neighbourhoods are, or are expected to be communities of people living together, while, on the other hand, they are a collection of distinct homes inhabited by individuals, families and households whose privacy is guarded from intrusion by outsiders, including neighbours.(Byford,2009,p.251) With the word fence is not just only actually the physical fence meant. It also means those structures mentioned above. Everybody wants and need good neighbours, but also everybody wants and needs privacy. An example given by Byford on page 251 when he looked at buying a house. He was told, how great the neighbourhood was, how kind and nice the next door neighbours were. On the other hand most of the time the neighbours werent even there.ConclusionA good fence does make a good neighbour. Fences are there, to keep the neighbours at a distance people want them to be.ReferencesTaylor,S.(2009)Who do we think we are? Identities in everyday life in Taylor,S.,Hi nchcliffe,S.,Clarke,J.and Bromley,S.(eds) qualification Social Lives,Milton Keynes,The decipherable University Hinchcliffe,S.(2009)Connecting people and places inTaylor,S.,Hinchcliffe,S.,Clarke,J.and Bromley,S.(eds) Making Social Lives,Milton Keynes,The Open University Byford,J.(2009) Living together,living apartthe social life of the neighbourhood in Taylor,S.,Hinchcliffe,S.,Clarke,J.and Bromley,S.(eds) Making Social Lives,Milton Keynes,The Open UniversityEnd of TMA03 part 2Word count 658

Blind Side

Letter Dear Sean Tuohy, Over the last couple of months you and your go forthstanding family take provided everything I have every urgencyed a family. My life growing up consisted of running, drugs, unreliable parents a couch and others aspects. I have face up many obstacles and challenges in my life. I have been by dint of some things no churl my age should have to experience. I have learnt to face my everyday challenges by facing the fact that I washbowl sop up through it, plus the support of others was very helpful, and knowing someone on earth would be having a worse day.When I was a young child I had many scaring and frightening moments. I have seen a lot of habits my mum got into, taking drugs and being a big disappointment to my family. Then I soon later found out I had 12 brothers and sisters. My mother would tell me to shut my eyes when I was scared or anything she didnt want me to see, after that she would then tell me to count to 3 and then I could open my eyes. I suffered a lot as a child and faced a lot of hardships, I struggle to learn nevertheless yet I can apparently write?You and your family have given me hope and believed in me. I have been through a lot and watching my mum throw her life away doesnt mean I was going to hold fast in her footsteps. After I crashed your car with SJ I thought you werent going to trust me again, but yet you gave me so much more. You took me under you wings, you gave me an education, and you let me go to college. Im so grateful that someone has given me the opportunity to prove myself to them and let me show them who I am and my potentials.I am successful that I could I write to you to express how I feel. I struggled a lot and I use to be a quiet and shy son, but now I am a confident and free spirited person and you and your wife have given me reason to believe again. Thankyou From Michael Oher Discuss what your casing learnt In the movie, Michael Ohers character capabilities change majorly. Throughout the film, he grows from a shy, unschooled boy into a young man who fits into a family. The film begins with Oher as a homeless boy struggling to find a place to sleep.He then learns to become a more confident and sophisticated man. Michael Oher has grown and learnt a lot of skills and matured into a young intelligent man. He learnt that because his childhood was so traumatic doesnt mean he is traumatised for his whole life everything can change when you start believing. Michael struggled a lot being a foster child, being black, moving and living(a) not with his family, and yet they are complete strangers to him that he would often run away and try and find his mother again.When he meet the Tuohys He grew a lot stronger in that because he was living with a Christian, stiff family who took him in without any weight on their shoulders just out of kind heartedness. He also learnt that he didnt have to be the smartest to achieve what he wanted he had to work rad and be persistent, to s tay focused. He grew so much into a star having every feel that a magnificent football player need but besides some granted with this special ability. Michael learnt that being whoever you are or being some(prenominal) you wanted was you He had a very distraught childhood and often struggling to once believe again.He was always a determined to get what he wanted especially as a child. Michael has grown up so much, from not trying and struggling to read to now acquire 2. 52 in his final result meaning he could do what he wanted for footy. He had the skills to get in but did not have the determination of getting and staying focused. Michaels tutor was a massive bonus to him and was a swell peer leader. She encouraged him and helped him improve his marks a lot. Along with the hardships Michael has faced he learnt that there is always a light at the end of the tunnel.He has grown and learnt so much and he is so lucky and happy to be able to have the opportunities he has been offer ed let alone half. Review The blind location has proven to be a wonderful structured film, as it has been based on a true story it gives it a bit more of a overlooking and understanding of the abilities and inabilities this boy has been through. A family of all white has taken this black boy in, he wouldve felt un-invited, not fitting in and questioning why they are being nice especially when there from the complete other side of town.Michael struggles to take in that a family of complete strangers is offering there house and couch to him. The Blind Side is a film about compassion, prejudice, family, chance, and the virtues of hard work. It tells the inspirational true story of Michael Oher, of a disadvantaged African-American child, and his relationship with Leigh Anne Tuohy, a wealthy white woman from the other side of town the rich and wealthy side and he is from the dirty, disgraceful, depressing side. They help him out with a lot of lieus. They provide a wonderful education, house, family.After his mother being a fall out he tried to get his life back on track by believing he could be different and show that he could do anything if you put your mind to work. Michael Oher shows a massive and exciting interest in football and Leigh Anne Tuohy does not hold back on the enthusiasm for him. He has been natural with skills that a person could dream about, even though Michael Is a big boy these skills come in handy when he gets a number of offers to be in a professional team. They then provide Michael with a tutor to improve his grades to get him into NCCA div 1.Athletic scholarship. This young man growing up in Memphis would only dream of being able to professional football or any strong contact with a ball, he was born too and all of a sudden his dream came true and was right in front of his eyes because of the Tuohys. I believe the film deserves a rating of 5/5, when you look at the poster or the movie cover or the book cover you dont know what to expect, but then you get a great message from the film inspiring everyone to show that if you believe you can achieve.The message is trying to explain that you can be in any situation and find a way out of it, if you try and work your way out of it. It was a fantastic book based on a real life story. Its very inspiring and proven to everyone that you can be from somewhere that struggles into someone who is wealthy. Bibliography The blind side 2009, DVD, warner bros, Hollywood http//www. imdb. com/title/tt0878804/plotsummary, http//www. pacejmiller. com/2010/02/24/movie-review-the-blind-side-2009/

Informative Article About Bullying Essay

Have you ever heard about Amanda Todd, a 15 years old Canadian girl who took her life away, on October 2012, because of cyber ballyrag. Before her suicide, Amada decided to post a video telling her story about how she was blackmailed, physical and psychological abused. Bullying is a social problematic and a global issue which touches most of the population, because it does not only happen at schools, you can also be witness of this in companies, or any different community. The website Bullying. rg defines bullying as a learned behavior. It is when a person or group tries to hurt or control another person in a harmful way. Rosa Castillo (2013), teacher of a public school in Bogota, Colombia, said that bullying is an action which lasts on kid for a long time, and this is star of the reasons why students start a low performance on schools. According to stopbullying. gov, the types of bullying included, verbal bullying, is tell bad things to someone (name-calling, teasing) social bull ying, involves damage mortal? reputation (embarrass a person in public, spread a gossip, leaves somebody aside) and the last type is physical bullying, hurts somebody? s objects or body (punching, good luck of stuff). Further more, the increases of technology uses on young population and the lack of parents control about it, is making easier to develop a field of bullying on the web. This type of bullying is known as ciberbullying, which take place at using the electronic device and social networks.Read moreEssay on the person you respect the mostThis kind of bullying is a little bit harder with the victims, as it could happen 7/24 (7 days a week and 24 arcminute a day), also ciberbullying spread fast all on the web, reaching all the person? s contacts. A lot of kids believe, that making others feel slight than he, would make him better. The truth is some of this kids has an issue that thy want to hide from the others, that? s why, they use bullying as a defense mechanism. Anot her possible arrange for the behavior is they want to call the attention of someone (parent, group of friend).On the other hand, is important the role of parents or big sister/brother, because most of the quantify these are the one kids admire and want to follow. Teacher Rosa, explain that the kids more likely to being bullied are the ones who are considered weak. And the bullies are those who are more touristed around the group or are strong. Some people consider that bullying is a stage of the childhood and it will have an ending. Last program line is totally false, bullying is a problematic and it has to receive treatment. Bullies and bullied, both will have a short and long term consequences.But the victimizers are more likely to experience the consequences in long term. The bullies are more susceptible to get involve in dangerous activities (vandalize, physical fights, abuse of alcohol or/and drugs), and also they are more likely to experience bullying as his behavior does not fit in the environment of work. The psychologist Dan Olweus, of Norway, found on his con that about 60% of schools students in Scandinavian countries who were bullies had one or more incidents of being convicted of a crime by the time they were 24 years old.On the other hand, the victims might experience the consequences from the first time of bullying, such as low self-esteem, anxiety, sleep difficulties, missing classes, and so on. One of the most worried consequences is that the victims could flummox violent, toward themselves or with the community, as the bullied would want revenge of all the public embarrassed he had passed. As we can see on the newspaper and all everyplace the news, in the last years, the number of bullycides (is a term used to describe suicide as the result of bullying) of young people increasing alarmingly in the entire world.

The Taylor Melcher Leaky Dielectric Model

Annu. Rev. Fluid Mech. 1997. 292764 Copyright c 1997 by Annual Reviews Inc. All rights reserved ELECTROHYDRODYNAMICS The Taylor-Melcher permeable Di electric automobile simulate Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For in-person workout only. D. A. Saville De crack upment of Chemical Engineering, Princeton University Princeton, New Jersey 08544 KEY WORDS electri? ed displaces and springs, suspensions, porthole bash, heap crusade ABSTRACTElectrohydrodynamics removes with ? uid head induced by electric ? elds. In the mid 1960s GI Taylor introduced the porous insulator exemplar to explain the demeanour of dribblelets de hur conduct by a stabilize ? eld, and JR Melcher use it extensively to develop electrohydrodynamics. This review deals with the appointations of the leaky dielectric manakin and tasteal establishs designed to probe its usefulness. Although the archaeozoic ceremonyal studies supported the soft features of the fashionl, quantitative agreement was poor.Recent studies ar in better agreement with the opening. Even though the model was originally intended to deal with sharp interfaces, contemporary studies with suspensions also agree with the theory. Clearly the leaky dielectric model is to a greater extent general than originally envisi unrivalledd. INTRODUCTION The earliest record of an electrohydrodynamic essay is in William Gilberts seventeenth century treatise de Magnete, which get words the nameation of a conical shape upon bringing a super depositd rod supra a sessile drop (Taylor 1969).Nineteenth-century studies of drop dynamics revealed how radiately directed forces stemming from interfacial delegacy offset place tension (Rayleigh 1882), yet until the 1960s most work burnk on the look of faultless conductors, (mercury or piddle) or consummate dielectrics (apolar politics such as benzene). This began to change following studies on poorly conducting liquidsleaky dielectricsby Allan & Mason (1962). An other branch of electrohydrodynamics, electrokinetics, deals with the behavior of burdend particles in aqueous electrolytes (Saville 1977, Russel et al 1989). However, in that respect argon signi? ant differences between the behavior of electrolytes and leaky dielectrics. In electrolytes, electrokinetic phenomena argon dominated by assembles of interface 27 0066-4189/97/0115-0027$08. 00 28 SAVILLE commove derived from covalently bound ionizable groups or ion adsorption. Near a surface charged in this fashion, a diffuse charge cloud forms as electrolyte ions of confrontation charge are attracted toward the interface. A submersion gradient forms so that dissemination balances electromig balancen. Then, when a ? eld is imposed, assistes in this diffuse layer govern the mechanics. In electrokinetics, applied ? ld authoritys are pocket-sized, a few volts per centi megabyte, whereas in electrohydrodynamics the ? elds are usually much life-sizer. With perfect conductors, perfect dielectrics, or leaky dielectrics, diffuse layers associated with balance wheel charge are usually absent. Accordingly, development of the two subjects proceeded more or less independently. Nevertheless, the underlying processes function m any diagnostics. Most obvious is that electric charge and current originate with ions therefore, charge may be induced in poorly conducting liquids even though equilibrium charge is absent.The Cardiovascular SystemThe different treatments began to merge with the appearance of Taylors 1966 paper on drop deformation and Melcher & Taylors review of the topic (1969). Applications of electrohydrodynamics (EHD) break spraying, the dispersion of champion liquid in another, coalescence, ink jet printing, b oil colouring, augmentation of heat and mass transfer, ? uidized bed stabilization, pumping, and polymer dispersion are precisely a few. Some applications of EHD are striki ng. For example, EHD forces have been used to simulate the earths gravitational ? ld during convection samples carried out during a space shuttle ? ight (Hart et al 1986). In this application, combining a radial electric ? eld with a temperature gradient between concentric spheres engenders polarization forces that mimic gravity. One of the more unusual appearances of EHD involves the blue haze found above heavily forested areas. BR Fish (1972) provides experimental evidence to support his proposition that the haze derives from waxy substances sprayed into the atmosphere from the tips of pine needles by high ? elds successive the overhead passage of electri? d clouds during thunderstorms. This review concent footsteps on what has come to be known as the leaky dielectric model to elucidate its structure and describe its experimental foundations. For insight into other aspects of EHD, one or more of the many reviews or monographs1 may be consulted (Arp et al 1980, Melcher 1972, 198 1, Tobazeon 1984, Crowley 1986, Chang 1987, Bailey 1988, Scott 1989, Ptasinski & Kerkhof 1992, Castellanos 1994, Atten & Castellanos 1995). In its most elementary form the leaky dielectric model consists of the Stokes equations to describe ? id motion and an expression for the saving of current employing an Ohmic conduction. Electromechanical coupling occurs only at ? uid-? uid boundaries where charge, carried to the interface by 1 Depending on the keywords used, com perpetrateer literature surveys turn up hundreds of papers on EHD since the 1960s. Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For ad hominem use only. ELECTROHYDRODYNAMICS 29 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.For separateized use only. conduction, produces electric underscorees different from those present in perfect dielectrics or perfect conductors. With perfect conducto rs or dielectrics the electric show is perpendicular to the interface, and alterations of interface shape combined with interfacial tension serve to balance the electric stress. talebearing(a) dielectrics are different because unornamented charge accumulated on the interface modi? es the ? eld. Viscous ? ow develops to provide stresses to balance the action of the rambling components of the ? eld acting on interface charge.This review is organized as follows First the model is out flexured to identify approximations and say-so pitfalls. Then experimental and theoretical results for two prototypical geometriesdrops and cylindersare surveyed. This discussion will establish the status of the leaky dielectric model where forces are con? ned to a sharp margin. In closing, recent results on motion produced by EHD form forces are surveyed to indicate how the model has been extended to new situations. BALANCE LAWS The differential equations describing EHD arise from equations descri bing the conservation of mass and momentum, coupled with Maxwells equations.To establish a context for the approximations inherent in the leaky dielectric model, it is necessary to look on a deeper level. Then the leaky dielectric model arises naturally through a scale analysis. As noted earlier, the hydrodynamic model consists of the Stokes equations without any galvanising forces coupling to the electric ? eld occurs at boundaries, so forces from the bulk free charge must be negligible. Moreover, the electric ? eld is solenoidal. The next section examines how to establish conditions under which these approximations are appropriate. Scale Analysis and the Leaky Dielectric ModelUnder static conditions, electric and magnetic phenomena are independent since their ? elds are uncoupled (Feynman et al 1964). Insofar as the characteristic time for electrostatic processes is large compared to that for magnetic phenomena, the electrostatic equations furnish an surgical approximation. When external magnetic ? elds are absent, magnetic personal effects arsehole be ignored completely. From Maxwells equations, the characteristic time for electric phenomena, ? c ? o / , can be identi? ed as the ratio of the dielectric permeability2 (o ) and conductivity3 ( ).For magnetic phenomena the characteristic time, ? M ? o 2 , is the product of the magnetic permeability, o , conductivity, and the square of a characteristic length. Transport process time-scales, ? P , arise rationalized Meter-Kilogram-Second-Coulomb (MKSC) system of units will be used. conductivity will be de? ned explicitly in terms of properties of the constituent ions. For the present note simply that conductivity has the dimensions of Siemans per meter, i. e. , C2 -s/kg-m3 . 3 The 2 The 30 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. nnualreviews. org by Brown University on 08/07/11. For personal use only. from steamy heartsease, diffusion, oscillation of an imposed ? eld, or motion ? M . The of a boundary. Slow processes are de? ned as those where ? P ? C second inequality can be rearranged to (/)1/2 o / (o o )1/2 , and since (o o ) 1/2 is equal to the speed of light, 3 ? 108 m/s, (o o )1/2 is very small for our systems. For the electrostatic approximation to apply on a millimeterscale, the electrical resi repayable time, o / , must be longer than 10 12 s. The inequality is satis? d easily because the conductivity is seldom large than one micro-Siemans per meter for liquids of the furcate under airfield here. Accordingly, the electrical phenomena are described by r o E = ? e and r ? E = 0. (2) (1) E is the electric ? eld strength, and ? e is the local anaesthetic free charge constriction. Boundary conditions derived from equivalences 1 and 2 using the divergence theorem and a pill-box system spanning a portion of a boundary show that the tangential components of E are continuous and the normal component jumps by an amount proportional to the free charge p er unit area, q, that is, ko Ek n = q. 3) present k()k denotes the jump, outside indoors, of () across the boundary, and n is the local outside normal. Electrostatic phenomena and hydrodynamics are coupled through the Maxwell stress tensor. A simple way of seeing the relativeship between Maxwell stresses and the electrical body force is to suppose that electrical forces exerted on free charge and charge dipoles are transferred directly to the ? uid. For a dipole charge Q with orientation d the force is (Qd) rE. With N dipoles per unit volume, the dipole force is P rE P ? N Qd de? nes the polarization vector. The Coulomb force referable(p) to ree charge is ? e E, so the total electrical force per unit volume is ? e E + P rE. This force can be transformed into the divergence of a tensor, r o EE 1 o E E , using Equations 1 and 2. The tensor 2 becomes the Maxwell stress tensor, M , ? ? ? 1 ? o 1 o EE EE , 2 ? T upon inserting the isotropic in? uence of the ? eld on the press ure (Landau & Lifshitz 1960). ELECTROHYDRODYNAMICS 31 Using the expression for the electrical stress produces the equation of motion for an incompressible Newtonian ? uid of supply viscosity, Du (4) = rp + r M + r 2 u.Dt Alternatively, upon expanding the stress tensor the electrical stresses emerge as body forces due to a non-homogeneous dielectric permeability and free charge, on with the gradient of an isotropic contribution, ? ? ? 1 Du = r p o ? EE ? Dt 2 ? T ? 1 (5) o E Er + ? e E + r 2 u. 2 For incompressible ? uids, the expression in brackets can be lumped together as a rede? ned pressure. EHD motions are driven by the electrical forces on boundaries or in the bulk. The net Maxwell stress at a sharp boundary has the normal and tangential components M M Annu. Rev.Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 1 o (E n)2 2 n ti = qE ti n n = o (E t1 )2 o (E t2 )2 (6) aft(prenominal) absor bing the isotropic part of the stress into the pressure as noted above. It is understood that ti symbolizes either of two orthogonal tangent vectors embedded in the surface. Denoting a characteristic ? eld strength as E o and balancing the tangential electrical stress in Equation 6 against cohesive stress yields a ve2 locity scale of qE o / = o E o /.The corresponding scale appears when the normal 2 stress or the bulk electrical forces are used. With o E o as a scale for pressure, Equation 5 can be cast in dimensionless form as ? u + Re u ru = ? P t rp 1 E Er + r (E)E + r 2 u. 2 (7) present the symbols represent dimensionless variables with lengths scaled by and 2 time by the process scale ? P Re is a Reynolds number, ? u o / ? ?o 2 E o /2 , when the electrohydrodynamic speed scale is used for u o . Choosing ? = 103 kg/m3 , = 1 kg/m-s, = 10 3 m, and E o = cv V/m gives Re ? 10 4 and a viscous relaxation time, ? ? 2 ? /, of 1 ms approximately. For a dielectric constant of 4 and a conductivity of 10 9 S/m the electrical relaxation time, o / , is 35 ms. Equation 1 shows how the ? eld is altered by the presence of free charge. In liquids, charge is carried by ions, so species conservation equations must be 32 SAVILLE included to complete the description. Free charge density and ion concentration are related as X ez k n k . (8) ? e = k Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. here e is the charge on a proton and z k is the valence of the k th species whose concentration is n k . look that approximately of the species may be electrically soggy, that is, z k = 0. Molecules and ions are carried by the ? ow and move in response to gradients in the electro chemical potential. If we denote the mobility of the k th species by k , the species conservation equation is n k +urn k = r k ez k n k E+ k k B T rn k +r k , t k = 1, . . . , N . (9) Here k B is Boltzmanns c onstant, and T is the absolute temperature. The ? rst term on the right represents ion migration in the electric ? ld, the second describes transport by diffusion, and the third denotes production due to chemical chemical reactions since the neutral species act as a source for ions in the bulk. With a single ionic species, N = 1 and r 1 = 0 for a binary, z-z electrolyte, N = 3. In the ? rst case, ions are produced by reactions at electrodesthis is called unipolar injection. With a z-z electrolyte, ions are produced at the electrodes and by homogeneous reactions within the ? uid. Here attention is focused on liquids with charge from a single 1-1 electrolyte so that there are two homogeneous reactions.A forward reaction producing positive and negative ions from dissociation of the neutral species as (neutral species, k = 1, z 1 = 0) (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = 1) (10a) with a rate per unit volume, k+ n 1 , proportional to the concentration of species 1. The recomb ination reaction is (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = (neutral species, k = 1, z 1 = 0) 1) (10b) with a rate of k n 2 n 3 . The rate constants k+ and k are speci? c to the ions, neutral species, and solvent the rate of production of cations or anions is k+ n 1 k n 2 n 3 .Thus, r 1 = r 2 = r 3 = k+ n 1 k n 2n 3 (11) This situation contrasts sharply with that for strong electrolytes where neutral species are dissociated fully and reaction terms absent. Because ionic reactions ELECTROHYDRODYNAMICS 33 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. are fast, it is satisfactory to imagine that the reactions are almost at equilibrium. At equilibrium, the local rate of reaction is zero, so K ? k+ /k = n 2 n 3 /n 1 .This complicates matters because at equilibrium one of the conservation laws must be discarded to avoid an overdetermined system. To scale the problem consistently, note that th e concentrations of the two ionic species will be much smaller than the concentration of the neutral constituent. Accordingly, it is convenient to use different concentration scales. Neutral species concentrations are p scaled with a bulk concentration denoted as n 0 and ionic concentrations with n 0 K . Using 0 as a mobility scale (any one of the three mobilities) and k+ n 0 as a reaction rate scale produces the conservation law for the neutral species, ?D n 1 + Peu rn 1 = 1 r 2 n 1 ? P t and for each ionic species, Dan 1 n2n3 (12a) ? D n k + Peu rn k = r z k n k k E ? P t r n0 1 k 2 k + r n + Da n 2 n 3 , k = 2 , 3. (12b) n K The new symbols represent a characteristic diffusion time, ? D ? 2 / 0 k B T 2 a Peclet number, Pe ? u o / 0 k B T ? 2 o E o / 0 k B T (the ratio of the rates of ion transfer by convection to diffusion) a dimensionless ? eld strength, ? eE o /k B T and a Damkoler number, Da ? k+ 2 / 0 k B T (the ratio of a characteristic diffusion time to a charact eristic reaction time).The reaction term can be eliminated from Equation 12b using Equation 12a to obtain r r n0 1 n0 1 ? D nk + n + Peu r n k + n ? P t K K r n0 1 k k k 2 k k 1 n , k = 2, 3. (12c) = r z n E + r n + K To compute local concentrations for systems in local reaction equilibrium, Equation 12c is used with k = 2 and k = 3, along with the equation for reaction equilibrium obtained from Equation 12a for Da 1. Equations 12c for k = 2 and k = 3 can be combined to furnish an expression for the dimensionless charge density, ? e = (n 2 n 3 ), ? D e ? + Peu r? e ? P t = r (n 2 + n 3 3 )E + r 2 2 n 2 + 3 n 3 . (13) 34 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. From Equations 1 and 13 the characteristic charge relaxation time can now be identi? ed (in dimensional form) as o /e2 ( 2 n 2 + 3 n 3 ) ? o / . To guide simpli? cation of these equations, the magnitudes o f the various groups are estimated for small ions with a characteristic radius4 , a, of 0. 25 nm using the Stokes-Einstein relation, (6? a) 1 , for the mobility. Then Pe ? 105 , ? 03 , and the diffusion time ? D ? 106 s. Estimating the size of the other dimensionless groups will require cognition of the dissociation-recombination reactions. The equilibrium constant, K , is estimated from the Bjerrum-Fouss theory of ion association (Fouss 1958, Moelwyn-Hughes 1965, Castellanos 1994) as in ? 3? e2 K = 3 exp . (14) 4a 8? ao k B T composition the recombination rate constant (Debye 1942) k = 4? e2 ( 2 + 3 ) o (15) gives a forward rate constant of k+ = k K . (16) Using the data already introduced, K ? 1017 m 3 and k ? 10 18 m3 /s so k+ ? 10 1 s. Accordingly, Da ? 105 .To estimate the concentration of charge carriers, we use an expression for the conductivity of a solution with monovalent ions derived from a single 1-1 electrolyte = e2 ( 2 n 2 + 3 n 3 ). (17) For a conductivity of 10 9 S/m with 0. 25 nm ions, n 2 = n 3 = 1020 m 3 ( ? 10 7 moles/liter), so n 1 = 1024 m 3 ( ? 10 3 mol/liter). Thus, n 0 /K ? 107 and p Da n 0 /K ? 107 . To complete the simpli? cation we need to know the charge density. Equation 1 in dimensionless form is (18) 3r E = ? e = z(n 2 n 3 ) p 3 ? o E o /e n 0 K . Using the numerical set already de? ned, 3 ? 10 4 , suggesting the ? id is electrically neutral on the millimeter length scale. For 4 For comparison, the radius of a sodium ion in water is 0. 14 nm. The size of the charge carrying ions in apolar liquids is largely a matter of speculation, but the presence of traces of water makes it likely that the charge carriers are big than the loot ions. ELECTROHYDRODYNAMICS 35 3 ? 1 Equation 13 yields the classical Ohms law approximation in dimensionless form, r (z)2 n 2 ( 2 + 3 )E = 0 (19) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.For personal use only. as long as Pe3/ ? 1. With the numerical magnitudes given thus far, Pe3/ ? 10 2 , prompting the approximation expressed by Equation 19. To complete the description, charge conservation at the interface must be investigated. Here it is convenient to start with Equation 9 and integrate across an interface with the provision that there are no surface reactions. Using the s-subscript to denote surface concentrations and operators, and ignoring any special transport processes such as lateral surface diffusion, leads to n k s + u r s n k = n k n (n r)u + s s t ez k n k E + k k B T rn k n. k = 2, 3. (20) rs ( ) is the surface divergence, and n k are surface concentrations. The terms s on the right stand for changes in concentration due to distention of the surface and transport to the surface by electromigration and diffusion. Adding the two equations, weighing each by the valence and the charge on a proton, gives q + u rs q = qn (n r)u + k e2 ( 2 n 2 + 3 n 3 )Ek n t + k B T r(e 2 n 2 e 3 n 3 ) n. (21) Next Equation 21 is put in dimensionless form using o E o as a surface charge scale ? c ? c q + u rs q ?P t ? F qn (n r)u 1 kr( 2 n 2 3 n 3 )k n. (22) = k ( 2 n 2 + 3 n 3 )Ek n + A new time scale, the convective ? ow time ? F ? /u F , appears here. For 1, the diffusion term can be ignored and conduction fit against charge relaxation and convection. For loyal motion, charge convection balances conduction when ? C /? F is O(1). Summary Equations for Leaky Dielectric Model To summarize, the leaky dielectric electrohydrodynamic model consists of the following ? ve equations. The derivation given here identi? es the approximations in the leaky dielectric model.Except for the electrical body force terms, 36 SAVILLE it is essentially the model proposed by Melcher & Taylor (1969). ? u + Reu ru ? P t = Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. rp 1 E Er + r (E)E + r 2 u & r u = 0 2 (70 ) (190 ) r E=0 ? c q ? c + u rs q ? P t ? F kEk n = q M M qn (n r)u = k Ek n (220 ) (30 ) 1 (E n)2 2 n ti = qE ti n n = (E t1 )2 (E t2 )2 (6) Note that the equations are written in dimensionless variables using the scales de? ned in the text.The equation of motion is for nonhomogeneous ? uids with electrical body forces. The hydrodynamic boundary conditions, continuity of velocity and stress, including the viscous and Maxwell stress, are assumed. Primes denote dimensionless forms of the parent equations. Electrokinetic Effects Although Equation 19 may be adequate for p millimeter-length scales, it would P fail if free charge on the Debye scale, ? 1 ? o k B T /e2 (z k )2 n k , produces historic mechanical effects. As noted earlier, charged interfaces attract counterions in the bulk ? uid and repulse co-ions on the Debye length scale.Electric and hydrodynamic phenomena on this scale are responsible for the ubiquitous behavior of small particles in electrolytes , so it is natural to ask whether similar effects might be authorised here. In fact, Torza et al (1971) suggested that such effects could be responsible for the leave out of agreement between the theory and their experiments on ? uid globules. To see whether the lack of agreement is due to electrokinetic effects we can use the numerical data already put forth. This leads to the following estimates p = ? 1 ? 10 7 m, 3 ? 1, Pe3 ? 10 3 , Da ? 10 4 , Da n 0 /K ? 10 1 and 10 1 .Accordingly, on the Debye scale the relation between charge and ? eld is represented by Equation 18 while the species conservation equations 12 ELECTROHYDRODYNAMICS 37 become ? D n 1 = 1 r 2 n 1 ? P t ? D n k = r z k n k k E + k r 2 n k ? P t r n0 1 + Da n 2 n 3 , k = 2, 3. n K (23a) (23b) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. These equations are clearly more complex than those for Ohmic conduction, which omits e ntirely any accounting for individual species. Is this complexity necessary?In the following sections, experimental and theoretical results based on the leaky dielectric model are reviewed for several prototypical problems so as to assess the models lastingness and the extent to which more detailed treatments taking account of diffuse layer effects are warranted. To date, none of the experimental studies show major electrokinetic effects patronage the indications of the scale analysis. FLUID GLOBULES Drop Motion in External Fields Allan & Mason (1962) encountered paradoxical behavior when non-conducting drops suspended in non-conducting liquids were change by a steady electric ? ld. Conducting drops became ovoid, as expected, but non-conducting drops often adopted oblate con? gurations. Oblate shapes were completely unexpected since analyses of static con? gurations predict prolate deformations, irrespective of the drop conductivity. Drop deformations can be analyzed with several methods. OKonski & Thacher (1953) used an energy method Allan & Mason (1962) balanced electrical and interfacial tension forces. For small deformations of conducting drops in dielectric touchs, either procedure gives D= 2 9 ao E 1 . 16 (24) Here E 1 is the strength of the applied ? ld, a is the drop radius, and is interfacial tension. The deformation, D, is the difference between the lengths of the drop parallel and transverse to the ? eld divided by the sum of the two. Given that the drop is a conductor, it is easy to see why the shape is prolate since the pressures privileged and outside the drop are uniform, ab initio, with the difference balanced by interfacial tension and the spheres curvature, 2 /a. Therefore, non-uniform electric stresses must be balanced by interfacial tension on the 38 SAVILLE deformed surface. Since the sphere induces a dipole into the incident ? ld, charge on the spheres equipotential surface varies as cos being measured from the armorial bearing o f the ? eld. The ? eld normal to the surface varies in a similar fashion. Accordingly, the electric stress at the surface varies as cos2 , pulling the drop in opposite directions at its poles. Dielectric drops in dielectric surroundings also become prolate in steady ? elds, irrespective of the dielectric constants of the two ? uids, that is, 2 9 ao E 1 (? )2 (25) 16 (? + 2)2 with circum? exes denoting properties of the drop ? uid (OKonski & Thacher 1952, Allan & Mason 1962).Here deformation results from polarization forces since free charge is absent and the electric stresses are normal to the surface. Allan & Masons (1962) anomalous results led Taylor (1966) to discard the notion that the suspending ? uids could be treated as insulators. Although the suspending ? uids were poor conductors ( 10 9 S/m) Taylor accepted that even a small conductivity would allow electric charge to reach the drop interface. With perfect dielectrics, the interface boundary condition (see Equation 3) sets the relation between the normal components of the ? eld to ensure that there is no free charge.For leaky dielectrics, charge accumulates on the interface to adjust the ? eld and ensure conservation of the current when the conductivities of the adjacent ? uids differ. The action of the electric ? eld on surface charge provides tangential stresses to be balanced by viscous ? ow. Taylor used the charge calculated from a solenoidal electric ? eld to compute the electric forces at the interface of a drop and then balanced these stresses with those calculated for Stokes ? ow. This procedure led to a discriminating function to classify deformations as prolate or oblate 2M + 3 . 26) 8 = S(R 2 + 1) 2 + 3(S R 1) 5M + 5 Here S ? /? , R ? ? / , and M ? /. Prolate deformations are indicated ? when 8 1, and oblate forms are indicated when 8 1. Qualitative agreement between theory and experiment was found in nine of the thirteen cases studied by Allan & Mason (1962). In the other four (pro late) cases, ambiguities in electrical properties hampered a test of the theory. According to Taylors leaky dielectric model, tangential electric stresses cause circulation patterns inside and outside the drop. As further con? mation of the theory, McEwan and de Jong5 photographed tracer particle tracks in and around a silicone polymer polymer oil drop suspended in a mixture of castor and corn oils. Toroidal circulation patterns were observed, in agreement with the theory. Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. D= 5 McEwan & de Jongs photos are presented in an addendum to Taylors paper (1966). ELECTROHYDRODYNAMICS 39 For a steady ? eld, Taylor (1966) gives the deformation as D= 2 9 ao E 1 8, 16 27) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. so it is possible to test the theory quantitatively by mea suring the length and breadth of drops for small deformations. However, Taylor did not publish a comparison between theory and experiment. The ? rst quantitative tests were reported by Torza et al (1971), who extended the leaky dielectric model to deal with oscillatory ? elds. The deformation (0 D 0. 1) and burst of 22 ? id pairs were studied in steady and oscillatory (up to 60 Hz) ? elds. In steady ? elds, oblate deformations were observed in eight systems, in qualitative accord with the theory. Although the qualitative aspects of the theory were vindicated, the quantitative agreement was very disappointing. The deformation always varied linearly 2 with a E 1 , but the proportionality factor exceeded the theoretical repute in all but one case, and the slopes were larger by a factor of two in more than half the systems. In one case, the measured slope was four times the theoretical value.In none of the systems was the measured slope less than the theoretical value, suggesting tha t the deviations are due to factors other than normal experimental errors. Alternating ? elds offer additional insight into leaky dielectric behavior. As 2 expected with alternating ? elds where forces vary as a E 1 cos2 ( t), the deformation consists of steady and oscillatory parts (Torza et al 1971) D = D S + DT . 8S = 1 (28) The steady part, Ds , has the same form as Equation 27, but the 8-function is S 2 R(11+14M)+15S 2 (1+M)+S(19+16M)+15R 2 S? 2 (M+1)(S+2) , 5(M+1)S 2 (2+R)2 +R 2 ? 2 2 (1+S)2 (29) where is the angular relative frequency, and ? represents a hybrid electrical relaxation time o / ? . According to Equation 29 the steady part of the deformation vanishes at a certain frequency and may shift from one form to the other with changes in frequency. Torza et al (1971) measured the steady part of the deformation for all 22 systems in 60-Hz ? elds and obtained results similar to those for 2 steady ? elds. The deformation was proportional to a E 1 , and in ? ve cases the ory and experiment were in quantitative agreement.With the other systems the measured slopes exceeded the theoretical values by hearty margins. The transition from oblate to prolate deformation was reported for one systema 40 SAVILLE silicone oil drop in sextolphthalate with S ? /? ? 2. 2 and R ? ? / 0. 07. However, the observed transition frequency (1. 6 Hz) was considerably lower than predicted (2. 5 Hz), although the two could be brought into agreement by lowering S to 1. 8. In this context the authors state This suggests that accurate measurements of the dielectric constants of the phases are crucial to a quantitative test of the theory. This observation will be revisited shortly. Some of the dissonance about oscillatory ? elds could be ascribed to the omission of temporal acceleration. Torza et al (1971) used a quasi-steady approximation, tantamount to ignoring ? u/t in the equations of motion. Upon including this acceleration, Sozou (1972) found qualitatively different beh avior at high frequencies. For example, the steady part of the stress tends to zero, so this part of the deformation vanishes. With the quasi-steady approximation (see Equation 29), the deformation remains ? nite.Although this observation might account for some of the differences between theory and experiment in oscillatory ? elds, it does not resolve the low-frequency dif? culties. Torza et als rent (1971) provides additional con? rmation of the qualitative aspects of the leaky dielectric model, but the lack of quantitative agreement is disconcerting. Even with water drops whose conductivity is ? ve set outs of magnitude larger than the suspending ? uid, deviations between theory and experiment are substantial. Several reasons for the discrepancies were suggested.Lateral motion of charge along the interface due to surface conduction and convection of surface charge were control out since they ought to make the relation 2 between deformation and a E 1 nonlinear. Other possibiliti es were suggested unspeci? ed deviations from the boundary conditions, space charge in the bulk, and diffuse charge clouds due to counterion attraction (cf Equations 23a,b). In an effort to address the boundary conditions skip, Ajayi (1978) employed perturbation methods to account for nonlinear effects in the deformation. This 2 analysis represents the shape using a power series in ao E 1 / .By carrying the analysis through the second order in the small parameter, Ajayi found that P2 (cos ) and P4 (cos ) are required to represent the surface and the deforma2 tion is no longer directly proportional to ao E 1 / . Considering nonlinear effects helps to an extent, but Ajayi observed that the method cannot remove the discrepancy between theory and experiment. Another possibility advanced as a source of disagreement involves electrokinetic effects (Torza et al 1971). Given the results of the earlier scale analysis, this theory appeared worth investigating further.Baygents & Saville (198 9) addressed the matter using asymptotic methods to account for the in? uence of a diffuse layer arising from coulomb interactions between current carrying ions and the surface charge. Three layers were identi? ed where different processes dominate. A diffuse layer adjacent to the surface is garbled from an outer region, where the leaky dielectric model applies, by an intermediate region. In Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. ELECTROHYDRODYNAMICS 41 he diffuse layer, electrokinetic processes due to space charge are relevant. The intermediate region is electrically neutral, and charge transport by diffusion, electromigration, and convection are equally important. In the outer region, the electrohydrodynamic equations prevail. Solving the differential equations involved matched asymptotic expansions, and because of the altered structure of the problem, the distributions of velocity and stress differ from those derived using the leaky dielectric model. Nevertheless, the ? nal expression for drop deformation is identical to that derived by Taylor (1966).Electrokinetic effects dont appear to contradict conclusions drawn from the leaky dielectric model, which, based on this analysis, appears to be an select lumped parameter description. Since none of the theoretical extensions appeared to resolve the divergence between theory and experiment, further experiments were undertaken. Following Torza et als (1971) suggestion regarding the need for accurate dielectric constants and other properties (see above), Vizika & Saville (1992) paid careful attention to direct measurement of physical properties.They studied eleven different drop-host systems in steady ? elds oscillatory ? elds were employed with ? ve systems. The systems exhibited either prolate or oblate deformations. To increase the deformation, a non-ionic surfactant, Triton, was used in some cases to lower the int erfacial tension. Generally speaking, agreement between theory and experiment improved over the earlier probe. Figure 1 shows some 2 results with steady ? elds. In all cases, D varied linearly with a E 1 . Vizika & Saville (1992) observed time-dependent effects in some cases, especially with the surfactant. Evidently the ? ids were not completely immiscible, and mass transfer occurred between phases. In these cases it was necessary to remeasure the properties subsequently time had elapsed to permit equilibration. Moreover, in cases where the conductivities of the two phases were comparable, ? eld-dependent effects were often observed. In oscillatory ? elds, the steady part of the deformation was measured at 2 different ? eld strengths D S always varied linearly with a E 1 . The agreement between theory and experiment for the steady part of the deformation was generally better than with the same systems in a steady ? ld. With water in castor oil, for example, the calculated and mea sured slopes differed by 34% in a steady ? eld in a 60-Hz ? eld the two agreed. Figure 2 summarizes results with four systems. Another interesting aspect of the leaky dielectric model concerns the effect of frequency. Torza et al (1971) showed, for example, that a drop that assumes an oblate deformation at low frequencies becomes prolate as the fre2 quency increases, that is, Ds /a E 1 increases with frequency. This behavior was measured with silicone drops suspended in castor oil results are shown in Figure 3.The qualitative agreement between theory and experiment was Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 42 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. adequate, but as the ? gure indicates, the behavior is quite sensitive to the drop conductivity. Vizika & Saville (1992) compared theory and e xperiment for the oscillatory part of the deformation with one system subtile agreement was obtained.Further encouraging comparisons between theory and experiment were reported by Tsukada et al (1993), who studied deformations with the castor oilsilicone oil system. Castor oil drops in silicone oil gave prolate deformations, oblate deformations were found with the ? uids reversed. In addition to experimental work, a ? nite element technique was employed to calculate deformations in steady ? elds. Except for the inclusion of ? nite deformations and (a) Figure 1 Deformation measurements for ? uid drops (Vizika & Saville 1992).Drops are prolate or oblate depending on whether D 0 or D 0. The dashed lines represent calculations made with the leaky dielectric model using measured ? uid properties solid lines are least-squares representations of the experimental data. In Figure 1b the theoretical line for the upper set of data is not shown since it falls on the regression line for the l ower data for this system the difference between theory and experiment is large. ELECTROHYDRODYNAMICS 43 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.For personal use only. (b) Figure 1 (Continued) inertial effects, the standard leaky dielectric model was employed. At small deformations, numerical results agreed with those from Taylors linear theory. With larger deformations, substantial differences appeared. Most of the differences between the ? nite element calculation and the linear theory were due to interface deformation since the Reynolds number in the calculations was always small. For prolate drops, the numerical results and Taylors theory agreed with the experimental data for 0 D 0. 07.With larger deformations, for example, for D ? 0. 2, the ? nite element solution was better than the linear theory but still predicted smaller deformations than those observed. In addition, the agreement between Taylors theory and the experiment for oblate drops exhibited a puzzling feature, that is, for large deformations the linear theory was closer to the experimental results than the nonlinear ? nite element calculation. These three studies constitute the most comprehensive test of the theory wherein interface charge arises from conduction across an interface.The agreement between theory and experiment is encouraging, and there seems little doubt that, insofar as drop deformation is concerned, the theory does a satis- 44 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Figure 2 The steady part of the drop deformation in oscillatory ? elds (Vizika & Saville 1992). factory job. Nevertheless, only a desexed number of ? uids have been studied, and even in these cases, conductivities have not been controlled. Questions as to ? ite amplitude effects or charge convection due to interface motion remain to be investi gated. In situations discussed thus far, charge convection has been ignored since ? C /? F ? 1. To investigate the in? uence of charge convection, the HadamardRybczynski settling velocity for a spherical drop can be studied. Although no experimental studies dwell, calculations with the model indicate a substantial in? uence. First note that the velocity will be unaltered if a steady ? eld is imposed because, as long as charge convection is negligible, the net charge is zero and the ? ld exerts no net force on the drop. However, an asymmetric charge distribution creates a net force charge convection due to sedimentation generates the necessary asymmetry. The relevant boundary condition is Equation 220 rewritten for steady ? ow, ? c rs (uq) = k Ek n. (30) ? F ELECTROHYDRODYNAMICS 45 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Figure 3 The unsteady part of the drop deformation as a function of frequency for silicone drops in castor oil (Vizika & Saville 1992).Torza et als (1971) theoretical result is shown for two values of the drop conductivity other parameters correspond with measured values. 2 Here the ? ow time will be a/u o = /o E 1 , so ? C /? F = (o E 1 )2 / 1 ? C /? F ? 0. 1 for = 4, = 10 N-m, = 10 9 S/m, and E 1 = 105 V/m, so a linearized treatment is appropriate (Spertell & Saville 1976). Solving the equations shows the settling velocity is retarded or increased depending on the electrical relaxation times in the two ? uids, that is, 3Ust U = 3 + 2M . (31) (o E 1 )2 + F(R, S, M) 1+ M Ust is the Stokes settling velocity for a rigid sphere and F(R, S, M) = 6M 2 3S(R + 1) 1RS 1 . 5(1 + M)2 S 2 (3 + 2R)(2 + R)2 (32) Also, with charge convection, drop deformation is no longer symmetric with respect to the midplane. These results show clearly that charge convection has different effects, either enhancing or retarding sedimentation, depending on the charge relaxatio n times in the two ? uids. 46 SAVILLE Given that interface charge induced by the action of an electric ? eld in leaky dielectrics has important effects on quasi-static motions, the next task is to inquire as to its effects on drop stability.Drop Stability and Breakup To provide a context to study the single-valued function of tangential stresses it is helpful to recall work on perfect conductors and dielectrics. Studies of drop dynamics6 and stability began with Rayleighs celebrated 1882 paper On the equilibrium of liquid conducting masses charged with electricity. His analysis pertains to instantaneous charge relaxation inside an isolated drop, and the relation7 between the frequency, , interfacial tension, , and drop charge, Q, is ? Q2 2 = n(n 1) (n + 2) 3 (33) ? a 16? 2 o ? a 6 for axisymmetric oscillations of an inviscid drop of radius a and density ?.Here n denotes the index of the Legendre polynomial Pn (cos ). For perfectly conducting ? uids, the electric stress is only normal to the interface. Instability occurs for n = 2 when the charge increases to a level where the expression in brackets vanishes. Because a linearized, spheroidal approximation is used, either oblate or prolate deformations are included. Although the Rayleigh ascertain pertains strictly to small oscillations, dimensional analysis shows that the criterion for instability will still be of the form Q2 o a 3 C, (34) Annu. Rev. Fluid Mech. 1997. 927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. but the constant C will depend on the properties of the surrounding ? uid. An EHD model of a leaky dielectric drop oscillating in an insulating ? uid addresses effects of charge relaxation inside the drop through a boundary condition for the conservation of interfacial charge, q. Accordingly, the model consists of linearized8 equations of motion for incompressible ? uids inside and outside the drop, ? u = rp + r 2 u, ? p t r u = 0, (35) relations between the ? ld and the current in each phase, r E = 0, r ? E = 0, (36) 6 Rayleighs Theory of Sound (1945) contains many enthralling accounts of early work on drops and cylinders. 7 Recall that the rationalized MKSC system is used here. In Rayleighs notation, = 1/4? . o 8 The linearization is based on the size of the deformation relative to the undeformed drop. ELECTROHYDRODYNAMICS 47 and boundary conditions. The relation between ? eld and charge is given by the dimensionless form of Equation 3, kEk n = q, (37) with charge scaled on the charge density on the undeformed drop, Q/4? a 2 .The scale for the electric ? eld, E o , is Q/4? o a 2 . Charge on the interface is conserved, and for 1 the balance is Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. ?c q ? c + u rs q ? P t ? F qn (n r)u = ( 2 n 2 + 3 n 3 )E n. (38) To conserve charge, ion mobilities in the outer ? uid must be ze ro so that the current on the right-hand side represents conduction from the interior. For Rayleighs perfectly conducting drop, the local charge balance is unnecessary because (a) the ? ld is nil inside the drop, and (b) charge transport is instantaneous so that the convection and relaxation terms vanish. The remaining boundary conditions are continuity of velocity and stress, and the kinematic condition. These equations have been solved to investigate how relaxation alters Rayleighs results (Saville 1974). Both viscous forces and charge relaxation effects were included, but general conclusions were obscured by the awkward transcendental form of the characteristic equation. However, asymptotic methods can be used to identify the salient features. The result for a slightly viscous drop in the absence of a suspending ? id is rather surprising in as much as Rayleighs result (see Equation 33) is recovered as the stability criterion. Even when charge relaxation by conduction is slow, cha rge convection still redistributes charge so rapidly that the oscillation frequency is given by Equation 33. A similar explanation was proposed by Melcher & Schwartz (1968) in their study of planar interfaces. Although EHD effects fail to alter the oscillation frequency, damping rates are affected. If the damping rate is denoted as , then )t R represents the the amplitude of the oscillation decays as exp(i R Rayleigh frequencies from Equation 33.First, note that with instantaneous relaxation the damping is volumetric and Rayleighs theory gives 1 ? 2 a ? 1 (39) ? (2n + 1)(n 1) 2 for a 2 for a drop with kinematic viscosity ?. When electrohydrodynamic effects are included and the oscillation time is comparable to the conduction time, that is when o o / ? O(1), damping is slower ? (40) ? (2n 3)(n 1) 2 . a 48 SAVILLE Other interesting effects can be identi? ed, including modes involving rapid 2 damping in a thin boundary layer where the rate is proportional to (a / 2 ) 3 . Rayleighs c riterion also applies to very viscous drops with rapid charge a , has a relaxation. In contrast, slow charge relaxation, that is, o / substantial effect on passing viscous systems. Here the criterion for stability is altered to Q2 16? 2 a 3 o 40? + clxxx 10? + 9 (41) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. for a viscous drop with dielectric constant and viscosity in a ? uid with ? ? 1 and where ? and (a /? ? 2 ) 2 ? 1. The next topic concerns behavior beyond the range where a linear treatment is acceptable.To form a simple model, the breakup of isolated drops or drops in external ? elds can be treated by approximate methods. A spheroidal approximation (Taylor 1964) yields an accurate expression for the stability of an isolated charged drop or an uncharged drop in an external ? eld. More recent work9 shows that prolate shapes evolving below the Rayleigh limit are unstable to axi symmetric perturbations while oblate shapes above the limit are stable to axisymmetric perturbations but unstable to nonaxisymmetric perturbations. Thus, the Rayleigh limit turns out to be the absolute limit of stability.Dimensional analysis indicates that the criterion for instability of a conducting drop immersed in a gas and stressed by an external ? eld has the form 2 ao E 1 C. (42) Taylors spheroidal approximation (Taylor 1964) gives C = 2. 1 ? 10 3 for D = 0. 31, in good agreement with experiments on soap ? lms. The limiting deformation corresponds to a drop with an aspect ratio of 1. 9. Above this point the drop is seen to throw off liquid as a ? ne jet. Taylor (1964) analyzed the region near the spheroidal tip, which becomes conical (a Taylor cone) at the limit of stability. For a cone with a vertex angle of 98. , electric stresses on a conducting surface are balanced exactly by surface tension. It turns out that conical tips also make up as static solutions when one perfe ct dielectric is immersed in another and S ? /? ? 17. 6 (Ramos & Castellanos 1994a). At the limit the vertex angle is 60 . For S 17. 6, two solutions exist. One has a vertex angle larger than 60 the other is smaller. At S = 0 the vertex angles are 0 and 98. 6 , the latter corresponding to Taylors solution for an equipotential cone. 9 Pelekasis et al (1990) and Kang (1993) provide useful summaries of the dynamical stability of perfectly conducting drops.ELECTROHYDRODYNAMICS 49 More extensive analyses of the static behavior of drops disclose behavior consistent with this picture Sherwood (1988, 1991) studied free drops Wohlhuter & Basaran (1992) and Ramos & Castellanos (1994b) analyzed drops pinned to an electrode. According to the various computations, a dielectric drop immersed in another perfect dielectric elongates into an equilibrium shape as the ? eld increases when S S1 . For S S2 S1 the drops become unstable at turning points in the deformation-? eld strength relation.In the range S1 S S2 there is hysteresis drops are stable on the lower and upper branches of the relation and unstable in between. Values of S2 calculated by various methods are close to the value identi? ed as the maximum value for the existence of a cone. Wohlhuter & Basaran (1992) and Ramos & Castellanos (1994b), who studied pendant and sessile drops between plates, delineate other quantitative effects due to contact angle, drop volume, and plate spacing. How do EHD phenomena modify this picture? Interestingly, solutions for a leaky dielectric cone immersed in another leaky dielectric ? id exist for R ? ? / 17. 6, independent of the dielectric constants (Ramos & Castellanos 1994a). Because of tangential stresses, the ? uids are in motion (Hayati 1992). As before the cone angle is less than 60 , and two solutions exist as long as the conductivity ratio is large enough. The balance between electrical stress and interfacial tension determines the cone angle, and the normal componen t of the viscous stress is zero. As required, the tangential electric stress along the periphery of the cone is balanced by viscous stress. A circulation pattern exists inside and outside the cone with ? id moving toward the apex along the interface and away from it along the axis. One might conjecture that a certain level of conductivity is necessary for the formation of a sharp point and the ensuing micro-jet (see below). Allan & Mason (1962) and Torza et al (1971) observed three modes of drop deformation and breakup at high ? eld-strengths (a) water drops in oil deformed symmetrically, and globules pinched off from a liquid thread (b) castor oil drops in silicone oil deformed asymmetrically with a long thread pulled out toward the negative electrode and (c) silicone oil drops in castor oil ? ttened and broken up unevenly. For modes a and b the initial deformation was prolate for mode c it was oblate. The breakup of oblate drops in steady ? elds involved a complex folding motion w ith a doughnut-like shape. In an oscillatory ? eld, small drops were ejected from part of the periphery. Sherwood (1988) dealt with symmetric deformations (mode a) using a boundary integral technique. Perfect conductors or perfect dielectrics deform into prolate shapes in steady ? elds. With perfect conductors, the tips have a small radius of curvature, and Sherwoods algorithm predicts breakup at the tip with critical ? ld-strengths close to those found by Taylor (1964) and Brazier-Smith (1971). Perfect dielectrics display similar overall behavior, and the maximum Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 50 SAVILLE aspect ratio is near that predicted by energy arguments. With the leaky dielectric model, drops elongate and take on a shape with fattened ends connected by a thin neck. Since the calculation is quasi-static, transient behavior can be followed in cases where breakup occurs.Here th e leaky dielectric model depicts drop elongation followed by breakup into individual droplets, behavior consistent with experimental results. In a leaky dielectric, electrohydrodynamic stresses ? atten the almost-conical tips formed in perfect dielectrics or conductors. Sherwood de? nes two sorts of drop breakup the electrostatic mode where a conical tip develops and breakup is via tip-streaming, and the EHD mode following instability of the elongated thread. Because of the numerical algorithms structure it was not possible to study the other mode of breakup identi? d by Torza et al (1971), which remains a subject for future study along with effects of viscosity. Curiously, conical tips of the sort identi? ed by Ramos & Castellanos (1994a) were not found in Sherwoods calculation. Following tip geometry much beyond the point of instability has not been possible although Basaran et al (1995) report detecting embryonic jets. Their computation includes dynamic effects with ? uid inertia balanced against interfacial tension and electrostatic forces. Although the focus is on perfect conductors and inviscid ? uids, small jets were identi? d emanating from the tips. Inasmuch as electrohydrodynamic effects appeared to suppress conical tip formation (Sherwood 1988), much more effort will be required to resolve the issue of jet creation. In calculations to date, perfectly conducting, inviscid drops produce (immature) jets viscous, leaky dielectric drops do not. Predicting the onset and structure of the thin jet emerging from a Taylor cone is dif? cult, but EHD processes are clearly involved. Observations of liquid drops emerging from a small capillary make this conclusion abundantly clear.Drops become smaller as the potential is raised, and when the potential reaches a certain level, the drops emerge in a pulsating fashion. With further increases in the potential, the drop develops a Taylor cone that has a jet emerging from its tip. According to Hayati et al (1986, 1987a ,b) and Cloupeau & Prunet-Foch (1990), effective atomization is possible only when the liquid conductivity lies in a certain range. To ? rst order, the ? ow pattern inside the cone can be approximated by superimposing ? ow into a conical sink onto the conical ? ow driven by a tangential electric stress varying as r 2 .The tangential stress arises from charge conduction to the interface. Charge accumulation stems from differences between the conductivity of the interior and exterior ? uids. The ? ow pattern has a somewhat counterintuitive structure fluid is supplied to the jet from the surface of the cone, while a recirculating eddy moves ? uid down the axis of the cone toward the supply. Of course this analysis omits details of the jet, whose characteristics are submerged in the property at the cone tip. Accounting for charge convection on the surface of the cone, which clearlyAnnu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/0 7/11. For personal use only. ELECTROHYDRODYNAMICS 51 becomes important near the apex (Ramos & Castellanos 1994b, Fernandez de la Mora & Loscertales 1994), has eluded analysis to date. FLUID CYLINDERS Stability of Charged Cylinders (Free Jets) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Here ? ? 2? a/ , a is the radius, Im ( ) and K m ( ) denote modi? d Bessel functions of order m with the prime sign denoting differentiation, and E 1 is the (radial) ? eld strength at the surface. When the inequality fails, the cylinder oscillates. The quantity on the left of Equation 42 is proportional to the growth rate when the cylinder is unstable and to the oscillation frequency when it is stable. For an uncharged cylinder, instability is indicated when ? 1, that is, when 2? a. Electric charge expands the range of unstable 2 wave numbers and increases growth rates. For ao E 1 / = 1, the range is approximat ely 0 ? 1. 35 ( 1. ? a). Interestingly, charge destabilizes non-axisymmetric deformations that are otherwise stable the relation for these modes may be obtained from Equation 42 by equation the index of the Bessel functions with the mode for the angular deformation, cos(m), and changing 1 ? 2 to 1 ? 2 m 2 . Viscous effects dampen the motion, but their effect is such as to make some non-axisymmetric motions relatively more unstable (Saville 1971a). The theory for charged cylinders is in qualitative accord with Huebners (1969) ? nding of non-axisymmetric modes of breakup with highly charged water jets.Similar behavior exists with highly viscous cylinders, where, in addition to destabilizing non-axisymmetric modes, the presence of charge lowers the wavelength of the most unstable mode. Charge relaxation on an initially uniformly charged jet does not appear to have been studied, although given the importance this process has with axial ? elds, the topic is of considerable interest. T aylor (1964) observed that induced charge has a very powerful effect in preventing the break up of jets into drops under certain circumstances and an equally powerful effect in make violently unsteady movements ultimatelyShortly after Rayleighs pioneering paper (Rayleigh 1882), Bassett (1894) showed how charge destabilizes a cylinder by a mechanism similar to that found earlier with drops. Nevertheless, the process is more complex because a cylinder may be unstable even in the absence of electrical forces. If the wavelength of a corrugation exceeds the circumference, then a unhealthy surface may have a smaller area than a circular cylinder and be unstable because it has a lower (free) energy. By examine the dynamics of a charged, inviscid cylinder, Bassett showed that an axisymmetric disturbance of wavelength will grow if ? 0 2 ? ao E 1 K 0 (? ) ? Io (? 2 1 ? 1+? o 0. (43) Io (? ) K o (? ) 52 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org b y Brown University on 08/07/11. For personal use only. disintegrating the jet into drops in others. Taylor was referring to the effects of a ? eld aligned with the axis of a water jet. Racos (1968) experiments with poorly conducting liquids also show a strong stabilizing effect. These results produce a quandary of sorts. Axial ? elds promote stability with dielectric jets (Nayyar & Murthy 1960) because of the action of the normal component of the electric ? eld on the deformed interface. But the required ? ld strengths are much larger than those encountered by Taylor, and the dual nature of electric forces noted by Taylor does not appear to be consistent with the behavior of perfect dielectrics. The role of electric stress can be appreciated by imagining an axisymmetric deformation of the surface of the form 1 + ? (z, t). The normal component of the electric stress on the interface of a perfect dielectric due to axial ? eld is 2 ao E 1 where the dielectric constants of the cylinder and outer ? uid are denoted by and ? . Thus protrusions are pushed inward and depressions outward irrespective of the wavelength of the disturbance.In contrast, the normal stress on a charged, conducting cylinder is 2 ? ? K 1 (? ) ao E 1 1 (z, t), (45) K o (? ) so this stress resists deformation only when the term in brackets is positive, that is, for ? 0. 6. Moreover, with perfectly conducting ? uids some wavelengths are made more unstable. Although axial ? elds are seen to promote stability with dielectrics, large wavelengths (small ? , s) remain unstable. Since the stresses with perfect conductors or dielectrics are normal to the interface, the situation should be different with leaky dielectric materials due to tangential EHD stresses.The leaky dielectric equations have been solved for a viscous cylinder immersed in another viscous liquid under conditions where the current is continuous at the interface, that is, ignoring charge transport by relaxation, convection, and dilat ion of the surface. In terms of the dimensionless parameters in Equation 220 , ? C ? ? P & ? F . If, for example, we choose the process time to be the hydrodynamic time and identify it as that for a relatively inviscid mate? ? rial, (? a 3 / )1/2 , then o / ? (? a 3 / )1/2 ? 1. For distilled water, the electrical ? elaxation time is less than a millisecond and the hydrodynamic time for a 1-mm water jet is over a second for apolar liquids of the sort mentioned earlier, the relaxation time may be longer ( ? 35 ms). In either case ? C ? ? P , so the approximation is appropriate. With leaky dielectrics the normal stress differs from that noted with perfect dielectrics, and there is also a tangential (1 /)2 ? (z, t), Io (? )K 1 (? ) + / Io (? )K 1 (? ) ? (44) ELECTROHYDRODYNAMICS 53 stress due to induced charge. Before deformation the surface is free of charge since the ? eld is parallel to the surface.Upon deformation

Chile Case Analysis Essay

IntroductionThe Multi-Product chilly case showed exclusively the characteristics of a decision case. At the end of the case, Mr. Thompson, the rude(a) manager, had to reflection with a decision whether or not continuing with the changes started by the friendship. Therefore, we approached at the case with a decisive standpoint.1. What atomic number 18 the specific problems & issues facing the company?Managers scheme structure chart shows that the work distri plainlyion could not be done well. There ar too many managers and this might lead to a genuinely high individual competition. The working structure of the company does not promote teamwork, each sectionalization is working individually and the staff is not working as a team and this decreases the effectiveness of the compensation system, which has also been held as an individual activity. Another major issue is the lack of trust on each other, ca retain principally by cultural aspects that lead the employees to be mor e individual everyone is only interested in themselves, not in the area they work in. Sales representatives are only interested on the product they are specifically selling and that causes differences with what the clients are looking for. Therefore, the company struggles to accept changes happening right now.2. How do you quantify the approach the firm has taken so far?Multi-Products Chile has tried to keep up with the new trends in the market by meeting the new and constantly evolving customers needs. Before Thompsons arrival, the mention Accounts review and Integrated Solutions program were already been launched. The new identification of companys Key Accounts as a strategic relationship with the customers was aimed to ca-ca a long-term competitive advantage for the firm and creating new benefits for the clients. Integrated Solutions meant a completely contrary approach in the relationship with the customers the gross revenue representatives focused their efforts from one pr oduct and many clients to one client and a different set of products. The role of the cut-rate sales representative had to evolve becoming a consultant for the customer.The issue of Chilean cultural barriers to overcome, in order change their sales policy, was not an light-headed one. They are a closed culture and the mentioned lack of trust that emerged between the co-workers represented an issue in the team working approach required by the regeneration to an incorporated sales model. In our view the firm has moved in the right direction. Even though the sales were doing well with no sign of a crisis, the addition margin was decreasing. The customers were asking for a different kind of service and, in order to be competitive in the future and coherent with the mod spirit of the firm, a step forward was necessary. Meet the customer needs is the identify to success and now the big challenge is trying to align the approach with the Chilean culture. Moving from a traditional selli ng model to an integrated also represented an opportunity to align their strategy with all the other Multi-Products branches all over the world.3. What actions should the firm take going forward?From an integrated solutions model standpoint, every single employee should be on the same page in term of team motivation, company coherence, and content on the multi-products structure. This innovative model from the traditional business model has a lack of adaptability in the Chilean culture because the companys implementation isnt in its full effect. Therefore, a change in a more streamlined and lean management structure that coincides with the integrated sales model, which could be tweaked according to the Chilean culture, could be the way forward, especially when the profit marginis decreasing where the times of culture adaptability change are needed. The feature that the Chileans are workaholics, and they are dedicated, in turning their dedication to a team oriented strategy could s uffice for the one voice, one face, one company, as component of their integrated solutions model.The incentive structure could be tweaked as well for sales representatives, where a higher commission and bonus model for higher sales in effect could boost the companys performance. If this is streamlined, this could be the factor increase in the companys market share from a low profit margin. Due to the fact that Chileans are naturally competitive work oriented mess, an innovative competition structure that is internal and fun could be established whereby, workers inspire themselves to beat each other, which could overall beat the benchmark targets for the sales and marketing segment in the integrated solutions model. In going forward, merging the integrated solutions model, key accounts and the Chilean culture norms, mores and values of their work standards, where the team as whole could leverage their core strengths in competition, hard work with their sales representatives, coul d be their main(prenominal) factor advantage in the market.4. What are the key supply chain(s) links in this case? How might the supply/value chain be used as an analytical framework for understanding the organizational and managerial challenges facing the company?This case shows different key actors of the supply chain. Starting with the initial ancestry with its way of procurement. Then, they explain the way they deliver their products, which is via truck shipment mostly. The new sale solution they are implementing affects the way they will manage their orders. Adding more go in the process, as customer will need advices. Its overall way of retail is challenged in this case, aside from facing an overhaul in the sales division its supply chain has to be adapted to answer the new needs. The company will need to pivot from a push production to a dedicate production. Hence, most of the supply chain is affected. This is the main organizational challenge, because every aspect of the ir current supply chain will live with to change. They want to reach the future(a) level, going from a simple offering of products to a retailer of solutions, as mentioned earlier. For the supply chain, that means, more reactivity and flexibility.Several layer tools needs to be implemented. They should apply a just in time production, meaning that they will have, at least, to use the lean manufacturing to eliminate the wastes in their production. They will work with smaller trucks to sale more often. A reduction of their waste in the manufacturing process, and the pace they can change their production should be as low as it can. Its sales people need to learn how to work together, but the top management needs to be on the same page regarding this project, everything will start with their support. And its a real mistake that they ease have people openly doubting the overall at this point of the process.The challenge would be to rally those executives. Then create an atmosphere whe re collaboration is valued and useful. The entire success of this overall relies on the way people can adapt to it. And it wont be easy regarding the social value that Chile people developed. The organizational side is important, but should not be the main focus, as it is motioned in the case that they are still delivering in time their products. But keeping these changes in mind is important for the long term, which is also a switch in the companys culture.5. How have STEEPLE factors impacted the company?The social aspect of Chile has a direct impact on our company, as people in Chile are workaholics where they are more focused on individual goals than the companys goal. The use of incentives to individual sale representatives will help increase the sales but this will again drive them towards their own individual goals. Technological innovation helps us in deriving 30 percent of sales. There is potential in the market but the economy is increasing slowly as compared to other Latin American countries. Being political stable since last 10 years, thither are legal issues regarding regulation and standards while dealing with other countries.ConclusionTo conclude, the wisest option for Thompson is to continue the overhaul of the companys management and organizational systems. Because Multi-product Chile is at a critical point where it has to adapt in order to staycompetitive. Even if the project faces difficulties, it has to be achieved. The main factor for success, aside from the technical foul parts, would be to rally the employee towards this project. Starting with the executive, this is inconceivable that they are still facing trust issue from the managers at this point of the project. Regarding the technical part, in order to be able to deliver according to the new standards, they will have to switch their production from a push production to a pull production. We would recommend outset with the basic tools of the lean manufacturing.