La Jolla

La Jolla California

La Jolla is fun, its a bit more uptown than Mission Beach. Here the surf is about perfect for learners, all day long the waves run two to three feet, with a nice break all up and down the beach. A few hundred yards north of here is the Scripps Oceanography Institute. They take in a lot of ocean water all the time for their laboratory experiments and there is an outfall up there that dumps their used water onto the beach, 24 hours a day. If you go up on the cliff and look, you can see the different colored water from the outfall trailing away out to sea, drifting south and west until it gets beyond sight down near LaJolla point.

There are a few interesting stories about how some rivers and streams do not mingle their waters when they meet a tributary, and this is similar to that. I have sometimes wondered if the Scripps water floating on the surface somehow makes the surfing waves more stable and consistent. I suppose that the Scripps water is warmer, from being in their buildings, but maybe it also has a lighter density from less salt, or ionic differences or something.

One thing is clear from looking at the ocean and that is that an offshore kelp bed can make the water more glassy when it gets to breaking on the beach, and maybe the Scripps water helps do this too. Its an interesting question, and as I sometimes say, there might be some small value in looking for the answer, but I am not going to look for it today. But the thing to do if you look at waves is do some Fourier transforms in your head, and try to identify how many wavelengths you can see in the waves. The highest frequency is the ripples caused by the wind, generally when talking about wind you have to include turbulent eddys or "breezes". So there are probably two frequencies of ripples in the water caused by wind.

Then there are the big ripples caused by the surfers or by sea lions, take your pick. These are quite a bit slower than the wind ripples, but are single point sources, and the waves are dampened out by viscosity after only a few inches. Say these are one tenth the frequency of the ripples. Then you have your chop from the ocean, little garbage waves created due to constructive or destructive interference from a multitude of waves. These things must chop along at some natural frequency of the water since by definition they never got dampened. It might be right to assign them a frequency another order of magnitude slower than the sea lion waves. Another order of magnitude gets you to the swell that causes the breaker, and swells come in sets of 4 to 6 waves in between lulls of 4 to 6 non-waves, so sets have a frequency that is another order of magnitude less. Then you have an occasional swell that comes in from another direction. The swell itself has a frequency equal to what the surfers are using, but it disappears for long periods, another order of magnitude slower.

Summing up the Fourier parts, assigning each a frequency that is an order of magnitude different:
ripples caused by breezes,
normal wind ripples,
mammal ripples
chop,
swell,
set,
cross set

The timing between swells is available from the lifeguards, and so from there you can make a pretty nice model of the waves in the ocean. Then you can try to predict what a kelp bed will do, or what different quality water will do, or the Scripps outfall, to make things glassy. If you want glassy. Note that this is not a surf predictor model. There are plenty of those, and they work pretty good, mostly depending on surf measurement and reporting stations in order to keep themselves calibrated.

And I am at the beach now, and as a segue into what I wanted to talk about, let me tell you that when someone says "have a good day" what that means to me is involuntary memories of long summer days in the blue Pacific, whitewater waves falling down, the back pull of the ocean just before a wave picks you up and sends you ashore, boogie board surfing with my friends, and laughter and a little bit of yahoo yelling. I once heard a hardboard stand up surfer describe what it's like to be in a tube of water with the tube closing in, and water spraying all around you and a deafening noise. Have a good day really means something to me. I try not to think about getting tumbled over while out of breath and underwater or getting my skin thrashed against the broken seashells on the beach when a wave threw me up there before I was ready to go.

What I wanted to talk about instead was imperfect gases and something important that a brilliant scientist named Kantrowitz found during the war. It was classified at the time, so it never made headlines, and if you think about it, even Einstein did not really get famous for about 40 years after his two most important papers. I like the other one the most, the one about magnetism.

Anyway Kantrowitz was trying to get the most combustion out of turbines, and what he found was that nitrogen was a little slow in expanding, compared to the rest of the air, the rest is mostly oxygen, but nitrogen was slow. I say nitrogen is "stubborn" and what Kantrowitz measured was that different shaped nozzles helped the nitrogen expand faster or slower which affects the amount of combustion, more or less of it. He was trying to find an extra ten percent of engine power that the math models predicted would be there, but was not actually there when they built the first jet turbines.

But the critical result was the simple observation that under his conditions the air was not an ideal gas. The nitrogen behaved differently than the theoretical models and it affected the observable performance of the turbine system he was working on. You would think there is other evidence of observable differences in air that show it is not an ideal gas. One of these instances of non-ideal behavior might occur near Mach 1. Pilots report an increase in buffeting starting around .94 Mach, or within 6 percent of the speed of sound.

This buffeting could very well be the result of something that is common to all gaseous fluids, but that seems unlikely. It seems more likely that the buffeting is caused by the variations in the gases in the air, some effect that is similar to the Kantrowitz observation that nitrogen is stubborn when pushed under pressure through a nozzle.

It seems like Kantrowitz observed an imperfect behavior of expansion, and the test pilots observed an imperfect behavior of incompressibility. Are these the result of the same imperfect properties in the air? In the nitrogen? One approach to that question would be to ask what kind of observable imperfect behaviors can there be? And to look at that you would start to look at phase changes for each gas, and look at parametric charts of the gas phase behavior under different pressures and temperatures. Langmuir had his people doing things like this, notably Kurt Vonnegut's brother, who identified twelve or eighteen different types of ice, with the ninth type being wholly theoretical and unable to create in the laboratory.

Parametric phase charts would be a productive way to look at the problem, but another approach is whether there are some fundamental properties of the individual molecules that makes a gas behave in an imperfect way. This approach appeals to me a great deal, frankly my attitude has a lot to do with why I never finished my engineering degree, but it appeals to me and that is what I want to talk about today.

I've spent the last couple of years thinking about this, after reading Kantrowitz' paper back in about 2000 or 2001. After about two years I wrote a paper on smartgroups.com/hilsch about how the Kantrowitz behavior is probably responsible for devices that create a temperature difference in air. My position is that the temperature difference is created during the time the nitrogen is too stubborn to expand. During this time the oxygen has to do all the expanding, absorbing all the heat, and making its surroundings colder, the same way the coolant in your air conditioner keeps you cool.

Anyway the oxygen gets hot, and so the nitrogen gets cold, and according to Kantrowitz you have up to about 5 milliseconds to separate them and get a cold flow and a hot flow. After 5 ms all things reach equilibrium - the temperature difference is gone. One way to separate them is with a centrifuge, or spinning them inside a tube, and another way seems to be to vibrate them with sound.

Two or three of my contributions to the smartgroups/hilsch forum are worth reading, and there are a number of interesting papers describing the centrifuge technique which has been known for ninety years or longer. I can never keep it straight in my head but it seems to me that the heavier oxygen molecules (the warm ones) get forced to the outer layer, and the lighter nitrogen (the colder ones) stay in the center of the vortex. It could be wrong, maybe the low density hot oxygen molecules stay in the center. Maybe there is a critical speed where the system flip-flops. I don't care today.

The question I want to know is why the nitrogen is stubborn. Why is it that Kantrowitz found that water vapor or dust made the nitrogen more willing to play ball? Which gas causes the buffeting at .94 Mach? Why?

To answer these questions I think you should ask questions like what shape is the molecule, what is the weight, how does the molecule behave under pressure or temperature changes. You need to consider atomic weights and theory of s p f electron shells, think about ionic potential, electric conductivity or resistance, whether there is gravity or weightlessness, and always remember that the magnetic moment of an electron is 1000 times greater than the magnetic moment of a proton. Heavy things have higher magnetic fields. There are other things to think about too, that are probably outside the environment that normal air works in, such as oxidation, or fission, and others I have never dreamed of. But at some point I am going to want to talk about what happens at hypersonic speeds like Mach 5 or 6, which is dissociation, and at about Mach 8 which is the ionization of molecules.

For now let's stick to gases behaving in the sonic and transonic regions. What I want to consider first is the bi-atomic molecules N2 and O2. Both of them are molecules made up of two atoms, and the theory says that the molecules are probably symmetrical with respect to electric/magnetic fields, and symmetrical when expressing their observable energy through vibrating or translating. In fact you can work out a table showing the different ways a two atom molecule can translate while vibrating, either along the axis, or against the axis. And curiously this makes you wonder if the two atoms are rotating around each other, and if they are is that something that we can measure.

My thoughts on this are that rotation is something that we can not measure, we can't observe it. I suppose that rotation is not something that is expressed by the PV=kRT equation, and wonder where that takes me. The first thing I do is assume that the molecules all rotate at some equal energy level, and that they also vibrate and translate while the neutrons orbit each other. If you look at the periodic table the oxygen atom is heavier, so the oxygen molecule will spin slower and wider, following Kepler, and the nitrogen molecule will be smaller and spinning faster. The angular momentum is the same, but the kinetic energy of the nitrogen is much greater, proportional to the square of the velocity.

So what happens if you picture a chain-link-fence like lattice of nitrogen and oxygen molecules spinning against one another? The big slow oxygen molecules are like 3 foot windmills getting bombarded by tennis balls. Four tennis balls for every windmill. Pretty soon the oxygen stops spinning. The little nitrogen guys are spinning around like crazy bumping into each other, and every so often clobbering up against some big dumb oxygen molecule. When they hit they might roll off, or they might skid up against it depending on their rotation, or they might hit along their axis and just bump off. But the result after a suitable time delay for equilibrium is a gas where the oxygen gets bumped around all day long by energetically spinning nitrogen molecules, nitrogen molecules spinning more or less energetically depending on their last few collisions with other molecules. Quite an exciting game of molecular billiards.

Does the imagined system match the observed performance? Yes. First, what about the case of rapid expansion in a nozzle? Can this model predict what Kantrowitz observed? Yes. Under expansion our model predicts that the oxygen has a very low angular momentum and can easily change its vibration rate to expand rapidly. The nitrogen has a comparatively large angular momentum and cannot readily change its vibrational energy. Its stubborn.

Kantrowitz observation about water and dust could possibly be due to the irregular shapes of these molecules (dust being a suspended solid molecule). The irregular shapes knock the spin out of the nitrogen so that it is not stubborn.

Second, what about the case of Mach .94 buffeting? Does this model handle that? Yes. It could be that starting at about .94 Mach we have reached the sonic speed of some or all of the nitrogen. It can no longer accept vibration as an input to its molecular energy. It behaves like an incompressible gas while the oxygen is still able to absorb energy. Six percent faster and the oxygen is also no longer able to accept the vibration as an input. This is Mach 1. The nitrogen sonic speed, for the spinning nitrogen molecules, has already been reached. At Mach 1, both gases pass along the momentous push of the aircraft like a liquid would, as sound, a sonic boom.

Now oddly, as a final question. Take the familiar curved shape of a Bernoulli wing as perfected by Wilber and Orville and imagine the airflow across it. This wing compresses and then expands the flow across it, creating a boundary layer and a smooth transition in air velocity until the boundary layer separates above or behind the wing. Behind the wing the fast moving low pressure air joins with the slower moving higher pressure air from below the wing, combining turbulently and creating wake vortices and wingtip vortices whose equations were worked out by von Karman.

What do you see? I see the wing as half a nozzle. If you took another wing, inverted it and placed it above the boundary layer, you would have two wings forming one nozzle. The wing and the nozzle are equivalent. The effect observed by Kantrowitz should be observable in wings. It turns out that it is. While working for Langmuir, Schaefer and Vonnegut tested the wing ice found on airplanes. There are gas molecules embedded in wing ice, proving that the air over the wing was cold, colder than the wing itself. It was the air that froze. The converse question is whether the .94 Mach buffeting of an aircraft is observable in a nozzle.

But you can prove the cooling thing for yourself. Drive down the highway on a warm summmer evening. Put your hand out the window just so! The flow from under the side mirror hits your wrist. The fastest part of the flow is cold, colder than the surrounding air. The blood coursing through your hand will get cool, in a few minutes it will be cooling off your whole body. Turn off your A/C! Save on gas! Breathe in the air.

Mission Point

Mission Point Beach, California

Mission Beach is a lot of fun especially in August. Here you can laze around all day reading and so on, and the whole place is crowded with athletic young people having a good time exercising and socializing. You can see surfers going into the water and chatting on their way in and out. Waxing their boards or showering off at the public showers, chatting about the waves, and the last time they saw or talked to so-and-so, or what did they do last Friday night.

Wednesdays is an especially good day for watching aspiring volleyball players. Get to the beach early and watch them drill with their coach, hitting volleyballs while un-self-consciously wearing their bikini. At around 10 AM the late comers arrive and pretty soon you can watch some pretty lively volleyball games between some very talented and good looking volleyball players. You can see the men can bash the volleyball around too, if thats the kind of game you want to see.

All of them are pretty smart people, probably attending UC San Diego or one of the other colleges. The especially talented athletes probably get some kind of scholarship, and pro volleyball has become a pretty good career for more than a few young people, mostly the men.

The whole thing is a little like the paradise in one of the old James Bond films where good looking young people cavort in the sun, playing athletic games and admiring each others bodies while socializing and just generally living the kind of life that everyone would lead if this were some kind of James Bond utopia, without anyone like Dr Evil to contend with.

But Mission Beach is fun, and definitely a beautiful side of our society, even if gambling, drugs, cable television, adult movies and liquor is a big part of their lives. It works for them, at least for now!

Anyway I have been coming down here occasionally for about 20 years, and when I do it often seems that I will see a formation flight of F-14's taking off from Miramar and heading out over the ocean. Presumably the flight leader is taking the other crew out to practice carrier landings somewhere out at sea, but you never know, and it doesn't take more than a minute for them to get too small to see, and only a few more seconds for them to be beyond the horizon, which is also out of radar range.

But I don't have any radar and I am left here on the beach thinking about the people I used to work with who were military pilots, including several Vietnam veterans, an A6 pilot, an F-4 Marine pilot who was shot down and evaded capture despite the cracked vertebrae caused by his ejection, and another F-4 pilot who was just on the edge of being crazy, but was employed along with the rest of us as computer engineers and flight planning experts.

The guy who was shot down told me the story about it, he ordered his rear seat radar operator to eject, but there is some stigma about ejections, ejection is just not something any real aviator wants to do. So the RIO, or radar intercept officer, was reluctant to bail out first, which would have been safer, and when the pilot finally bailed them both out, the RIO was somehow killed. This is not that unusual, as it is generally fatal to eject from a 400 or 500 mile per hour or faster airplane, or to eject at higher than about 15,000 feet unless you have bottled oxygen and an electrically heated suit. Francis Gary Powers did not have anything like that, and supposedly above 25,000 feet you are going to black out for a while, just from rapid acceleration. You have to hope you come to before it is time to pull the ripcord. But whatever you do when ejecting the explosive ejection is going to break your back in at least one place, destroy a multi million dollar airplane, and is very likely to kill your RIO if it doesn't kill you both.

But my friend at worked bailed out, and evaded, and his crew man died. He looked me right in the eye when he told me this story, so there is no bullshit here - if they would have stayed they would have died, and it is not like there is any doubt as to whether ejection was the right thing to do.

Anyway the almost crazy F-4 Air Force pilot used to walk around the office talking about different aircraft, and fast cars or fast computers and stuff like that, and he always commented about whether something was a hummer or not. I took it that a hummer was an especially fast plane or car, or girl, and being a hummer seemed to me to mean that the sound barrier was involved. I wondered if maybe things hummed when you broke the sound barrier, or if they put some kind of oscillating device in the F-4 to make it hum, thinking that would make it easier for it to get past the sound barrier.

Years later I met a guy who flew test flights on the first F-4s for the Air Force, and he told me that he took his plane up to Mach 2.26, and as high as 70,000 feet, where it gets dark, and you can see electrical sprites. Well 2.26 is almost as fast as the Concorde can go, a plane that is a much different design than the F-4. Concorde's supersonic design is to F-4s, what the F-4 is to the Cessna, and what the Cessna is to Orville and WIlburs motorized glider. Its just a different design and the idea that a plane that was designed to be faster than Mach 1 could actually go over twice that fast is pretty amazing.

So it made me wonder a bit about what it meant to go Mach 1, and whether the plane actually started humming, and whether it might help reduce "sound friction" if the plane itself could be made to hum. An extension of that is the idea that maybe making the plane electrically charged might help repel air molecules - or something - and reduce drag in that way.

But I didn't know much about it, and then finally after many years I was semi-retired and reading in the library all the things I wished I had time to read when I was 20. It turned out there were some old ideas about slats and spoilers and suction and so on that related to delaying the separation of the boundary layer over the wing. Suction involved a vacuum pump that drew off the air from the top of the wing, sometimes using it as high pressure intake air for the turbines. There seemed to be a lot of confused ideas about it, except that about that time I was on a L-1011 that had holes in the slats, and holes was one way that the old-timers had tried to use to create suction and relieve the boundary layer.

When I worked with the Vietnam veterans I was so confused that I didn't have any questions, and now I was pretty clear on what it was that I did not know. I guess I had always had the questions, I just didn't know what they were yet, to quote one of my old engineering complex math professors.

What is really going on in the boundary layer that makes it such a good thing?
Would it help get past Mach 1 if you were vibrating?
What kinds of things are useful in delaying boundary layer separation?
What is happening after boundary layer separation and can I minimize it?
What is happening to the boundary layer above Mach 1?

It was not an obvious thing, the fact that at Mach 1 you no longer have any boundary layer. You are not born knowing this. There are some excellent diagrams in a paper by von Karman that show the Mach effect as a perspective of pilot's speed. These diagrams let you picture that flying close to Mach 1 is like flying into a tunnel of incompressible air.

Faster than Mach 1 you are past the air before it has time to compress or expand. There are no aerodynamic forces acting on the aircraft. In fact the aircraft is flying through a fluid that is no longer compressible, as if it were water. There are photographs of aircraft flying into a wall of water, or trailing a wall of water behind them. This is best explained by saying that the plane is moving too fast for the air molecules to move out of the way. They can't move, the only thing they can do is pass a sound, a sonic boom, like clapping your hand onto the water at the pool. When the aircraft moves through the air this fast the air molecules suddenly behave as though they were incompressible, and there are air temperature changes involved with becoming incompressible. If there is water vapor present it will all come out, squeezed out, creating the wall of water effect - the contrail.

Picture the air as a wet chain link fence composed of oxygen and nitrogen. A backstop at the ballpark is a good example. The first time the catcher runs back to field a foul ball he jumps up on the fence and all the water shakes off of it. The next time he runs back there the water is gone, he jumps up on the fence and no water comes off. It is the same with incompressible Mach 1 air. The first plane knocks the water out of the air, when the second plane flies through there is no water left.

There are three things that can happen to the water vapor after the first plane flies into it. It can create a fog that is suspended water molecules, it can create a rain or mist that falls down toward the ground, or it can re-evaporate.

As an aside, and as something to think about, let me point out Langmuir's conjecture that water never condenses unless it condenses on something solid. And when water does condense, it gives up its excess of energy in the form of both heat and kinetic energy, adding both heat and movement to the solid thing it condensed on. If anyone has any thoughts - like about how Mach 1 makes the air seem solid to the water molecule, or how condensation can lead to tornadoes, I would like to hear them.

Savannah

Savannah Georgia

Savannah is a really neat little town. There are two interesting aspects to this town from a city planning point of view, and of course the place has a lot of history. One of the historical aspects to the town is that after Cump Sherman besieged and conquered it at the end of "Sherman's March to the Sea", he gave all the land south of Savannah, all the way to Florida, to the Africans. There is some nice land in there, and a lot of really pretty coastal waters. If I am not mistaken John Kennedy Jr got married at one of those islands, and the great lacrosse and football player Jim Brown was born and raised back there.

From a city planning point of view, the interesting thing is that the town was never levelled as part of the 1960's "Urban Renewal" program. All of the houses and businesses are in old buildings dating back to the 1840's. The layout of larger homes include slave quarters, and the town itself was laid out with a large number of parks - every two blocks there is a park. This let in fresh air and sunlight back in the day when everyone burned wood for heat and cooking.

The other interesting thing is that the freeways end here - as originally envisioned by Eisenhower all freeways ended at city limits. They weren't supposed to go through, and one result of the change was that lots of inner city neighborhoods were gutted when freeways were built. Since the government wanted to save money freeways were generally built on inner city land that was the cheapest - the black neighborhoods. So the real result was that up until the 1960s there were a lot of long-established family owned businesses in central cities, and the freeways put an end to that.

When I was in school the example that was used was Knoxville, but today you can go to a lot of cities and find a re-development district near a freeway, and you can bet it used to be a street full of black owned businesses. If you cross under the freeway you will likely find a neighborhood that has always been populated by blacks. So much for the legacy of Cump Sherman!

I have been talking to a lot of people around here about heat, and what it means and how PV=kRT is just the observable thing about heat. And it is not wholly accurate, or else people wouldn't try to invent "wind chill" or "real feal" numbers by including other variables such as humidity and so on.

But one interesting thing came up when I was reading a book by Tim Kantor. Kantor's father was on B-17's and one observation that everyone made on B-17s was that smells propagated faster at altitude. Now once I sat down and computed that if air molecules were vibrating at (as I recall) 4x10^9 vibrations per second, and if the molecules were (I forget) about 400 Angstroms, then the speed of propagation of larger molecules, such as scent bearing oils, would be about 20 meters per second, or 44 miles per hour.

People tell you its a sixth sense when the pretty girl looks up just as you start to gaze at her, but it is not. Its pheromones, and it takes about half a second for your skin temperature to rise, releasing pheromones at 45 MPH, so that she notices you looking almost right away. Now the thing to do to test this is to try staring through windows, or try staring at a cute girl who is 50 meters away. I think you will find I am right about the 20 m/s.

But it is faster at altitude, and I am open to anyones suggestions about why. Speaking of altitude, in Savannah they have a great Army Air Force Museum which includes a display that talks about the missions that Joe Kennedy Jr flew and was killed in. The heroism of this family is a big contrast with the President's family. The Bush family includes Prescott Bush, famous for writing a wholly fictitious account of his heroism for the hometown paper, and George Sr, who is famous for jumping out of perfectly good airplanes, and George Jr who is famous for being absent.

Senator Kerry of course is famous for actually being in battle, and actually taking lead, and if you believe the papers, that makes him a coward. What's up is down. Can you tell a green field from a cold steel rail?

Everglades

Mitchell's Landing Florida

Mitchell's landing is a narrow spot at the end of the road where you can put an airboat or a canoe in. You can also register for a permit to camp in some cleared patches along the road. They have trash cans and two portable toilets, and a dozen circles of broken limestone if you want to pick one out for a fire pit. Its drained swamp and hammocks, with a canal running along the main road, full of fish. You ought to see the heron over in Mitchell's Landing.

I learned just a little bit about teaching once, some concept called building bridges and digging tunnels. Its all about building new knowledge on top of old and showing how things interconnect. I don't know much about lecturing, except that when I was in school there were some really bad examples, and one really good one - my electronics teacher gave a lecture that was absorbing and breathtaking, and you couldn't believe how he could write equations for 47 minutes without looking at notes.

I can't lecture like that but what I am good at is showing new ideas to one person, and telling if they are lost or not. The people I was working with asked me what I do when the person gets a glazed look in their eye and I said - "review" Then take a break. So here is a review of fluids, in about 200 words.

You already know what fluids are, things that flow. It can be a solid or a gas. Hundreds of years ago some good ideas about fluids were written down, someone named Dalton and someone named Boyle. There is a concept of perfect fluids - fluids where every atom behaves like every other one. And there is a concept of partial pressures, where in a mixed gas if the whole volume has a pressure, then each component gas, such as nitrogen or oxygen, also has that pressure. The last thing we know for sure about fluids is that temperature is a function of pressure and volume. PV=kRT where R is Boltzmann's constant.

Then quite a bit of time passed and Navier - the great mathematician, and Stokes - who explained fluorescence, independently came up with some calculus equations that described the state of a gas and how it changed. Navier Stokes is a big deal, and as far as math describing a system its hard to find anything more elegant, but I'm no expert in it. The short answer is Navier Stokes works and can make you a lot of money if you need something like it.

The long answer is different. The long answer says - well what is pressure, really - when we measure temperature what is it really we are measuring - what is involved in being a 'mixed gas' - how do partial pressures get to be the same - in what way is my gas different from ideal. The long answer is we don't really know, pressure and volume and temperature are the most easily observed things. We can observe the gas to have a value for those variables, we can observe how the gas behaves according to Navier-Stokes if we heat it up, and we can observe that for common air it is really hard to tell the difference between our gas and a 'perfect gas'

The variables of pressure volume and temperature are observable and they are also controllable. We can press the gas into a smaller space, or heat it up, or add more gas to change the variables. P V and T are at the same time observable and controllable. So it is safe to use the equations without worrying about the molecular shape of the nitrogen molecule, or the fact that water vapor has an ionic value - it's an electrical dipole, or worrying about the fact that oxygen is likely to combust with other molecules in an oxidation-reduction reaction. We don't need to worry about the fact that 'temperature' is mostly just a measurement of atomic vibrations, and we don't need to worry about the spinning energy of rotation of molecules because we can't observe that easily and we can't control it. We don't need to know. That's eight weeks worth of fluids lectures, without the homework.

Fluid dynamics is a little different. This is when you put fluids in motion and few hundred years ago some ideas were written down by Magnus and one of the Bernoulli's. These ideas are generalizations of more complicated things just like the Boyle Dalton and Navier-Stokes ideas. The generalizations are useful, and they usually work, but moving fluids are more complicated. In my opinion, Magnus and Bernoulli have led to generations of confusion in systems that involve dynamic fluid flow.

What Magnus said is that when a fluid passes over a rotating object it creates lift. What Bernoulli said is that when a fluid speeds up the pressure decreases, useful for creating lift on the top side of a wing. Both are right, and both are generalizations of what is really going on with the molecules in the fluid. In addition I think that both ideas are specific instances of the other man's ideas.

For Magnus the rotating object creates a relative difference in air speeds. One side of the disk speeds up the air, the other side slows it down. Thus you can apply Bernoulli to the relative changes in air speed in the Magnus rotating object. A Bernoulli wing is just a Magnus disk cut in half, and with an angular speed of zero. It might be nice to have a deformable rotating object so that we could spin it and still have it be flat on the bottom, and see what happens. That would be a good test of what Magnus says.

I think that what is really happening with 'lift' is something funny in the air that is not observable. Bernoulli says lift is related to velocity, and Prandtl wrote the equations that determine lift over a two dimensional line along the wing. But velocity is a vector, it has a direction and a magnitude, so it could be that Bernoulli should apply to air that changes direction but does not change speed. That would be a good test of what Bernoulli says.

There are two other basic ideas in fluid dynamics. One is the vortex, or the circulation of fluid that is created by a rotating cylinder. The other is the wake, or the chaotic flow that occurs downstream from an obstruction in the fluid flow. All the other ideas, such as boundary layer, laminar flow, wake vortex, von Karman Streets, modelling wings as if they were a pair or vortexes, Kutta-Joukouski transformations, streamlines, potentials and potential flows, all these ideas come from the basic ideas. The basic ideas are that there is a bow shock, there is a wake flow, vortexes occur naturally in the presence of magnetic fields, and air changes pressure when it changes speed or direction.

That's the review, and I'm glad I got it out of the way!