Bignose

Here's the crux of the problem right here. This is a pretty much impossible request. There are popularizations and analogies of the concepts, but they are only the tiniest tip of the iceberg. To really understand the current theories, you have to put in the work. There is no other way. I'm sorry if I am the one who had to break this fact to you, but it needs to be made clear. Before you out-of-hand dismiss the current idea, you need understand what it really is saying. And no popularization or "bullet-point" synopsis is going to provide real understanding. To get real understanding, you will have put in the work. A good book to start with learning a lot of this is Roger Penrose's The Road to Reality. It is a massive book -- 1100 pages. But it will give a pretty good overview of the current state of understanding. It isn't going to be completely state-of-the-art, you'd have to read the physics journals to get that, but it does cover the major aspects of the current theory. And, yes, it is very mathematical in nature because the current theories are mathematical, though this book is pretty good because it introduces a lot of the necessary math in the beginning of the book. It isn't a textbook, and it really is still only a survey of the major points of the theories, but it also goes much more in depth than any of the popularizations out there that gloss over details Penrose's book won't. Discussion about ideas is fine. As has been written several times over the last few days, discussions are very often the seed to growing new theories. But, they are only the seed. To move past the discussion phase, you need to start doing science. That means investigating what the current theory says about your new idea; that means turning your idea into some kind of mathematical statement so that you can make predictions with it; that means taking your predictions and comparing them with known experimental results and/or developing new experiments to test your idea. That is science. Just talking about things isn't really science -- like I suggested it is more philosophy than anything. The science comes from doing the work to take the idea from mere discussion (or storytelling) and turning it into a predictive tool. I don't mean this to come off harsh if that is what it sounds like to you -- I really don't. But, I think it is also important to make it very clear what is and isn't science, as this is a science forum.

Then, quite simply, you aren't doing anything related to science at all. Like I wrote in your other thread, all you are doing then is telling stories. You also aren't going to gain a lot of traction here on a science-based forum. Because every single story you post, the scientific minded members are going to ask you for evidence, going to ask you for quantitative predictions, going to ask you if your idea predicts things as well as the very-well established and substantiated current theories we have today. I'm not trying to tell you to leave this forum, but I am saying that this tactic of throwing out stories and then refusing (either through inability or lack of desire or any other reason) to approach them scientifically isn't going to go over well on a scientific forum. You may enjoy yourself more, and will get a better reception to this kind of tactic on a philosophy forum or something similar.

Well, then don't just throw these statements out there as facts, please. This is a science board, and as such we are going to ask you to justify your statements. Yes, even (and especially) in the speculations section. If you make a statement that something is the "only" way that something can be, there should be a good reason for saying it. If you are just making a joke (that isn't terribly obviously a joke) then you aren't really adding to the scientific discussion.

Not to sound too flip, but this exists already. It's called university. There is a reason the vast majority of scientific papers are co-authored -- because they are bringing together people of different strengths to make the final result stronger all around. This includes "idea" people and "math" people. Most departments (i.e. physics department, chemistry department, mechanical engineering department, etc.) have a whole range of people and most departments intentionally seek out people who are different from each other to (borrow a cliche) "make the whole greater than just the sum of its parts."

You've really missed iNow's point. Temperature is measure of the average speed of the constituent molecules of a substance. That is the definition of temperature. Speed goes up, temperature goes up. Furthermore, the definition of absolute zero is a point where the average speed of all the molecules is equal to zero; that is, no movement of the molecules at all. So, how can something get so hot -- that is specifically that its constituent molecules speed up so much that they stop?

How do you justify this statement? Why is the "giant balloon" the "only way"?

You are neglecting a critical word -- quantifiable. If I use the mathematics of my idea to predict that the particle will be detected 58.9 mm from the detector 13.1 seconds after the experiment starts and it actually isn't detected until 28 seconds after the experiment starts -- then I know I have a problem with my idea. How are you going to make a quantifiable prediction with words? "The particle will be detected after a short amount of time, but it wasn't, it showed up after a moderate amount of time..." What does any of that really mean? Your definition of "short" and "moderate" may be very different than mine. Your definition of 13.3 seconds is exactly the same as mine. Words can create a pretty picture. They can be exceptionally descriptive, they can tell a great story. But, they aren't quantitative. That requires numbers. And, without quantitative predictions, you aren't doing science. You are telling a story. If you want to do science, you need to make quantitative predictions, with in almost every case is going to require math. There is no other way around it. And finally, I think that "but the trade off is with processing speed." is an unfair generalization. There are several people I know who think in math terms faster than they think in "words". If you get very good at the math, you can pull this off too. Math is a perfectly valid thought process in its own right.

You miss the biggest point -- that math is there to make quantifiable and verifiable predictions. Without being able to make predictions that can be verified as right or wrong, all you have is a story. You don't even have a "hypothesis", much less a "theory". To get your idea beyond being just a collection of words, a story, you have to make quantifiable predictions and that requires math. Period.

As I said, Reaper, people are human with all the inherent flaws of humanity. And, as also pointed out, the papers were published, just not in the top journals in Germany. It would end up being the same thing with any new theory today. Unless it is exceptionally well supported by evidence, any brand new alternative theory today would end up being published in smaller low-impact journals. That's where new ideas start, unless the proposer of the idea has a very good reputation. Is this entirely fair? Probably not. People with good reputations propose dumb things all the time, and people without any reputations propose good things all the time. The main point is that people without reputations that can provide evidence to back up their ideas get published. People with reputations who want to publish something still need to provide evidence, too! The common theme here -- they all need evidence. The reason "Null Physics" isn't in any peer-reviewed scientific journal is because the evidence in non-existent or wrong. Is is possible that the ideas could be right? Sure. But, there needs to be evidence backing it up. The valid questions that are raised by the people on the forum, and the many, many other actual scientists need to be answered. If the new idea cannot answer them, then it isn't much of an idea, then is it? So, I guess I should have been a little more explicit. Has there ever been a case of a scientific theory that is well supported by evidence that was refused to be published just because the scientific community didn't want it to get out? Not because of any political or religious or racial overtones. Simply because "the community" refused it? ----- There is any even bigger reason why I don't think that you will see this happen today. Every low-impact journal desperately wants to be a high-impact journal. And what better way to become a high-impact journal than to be the journal that is publishing "the next big thing". But, what happens almost always is that in reaching to get the rights to publish "the next big thing" is that some slightly dodgy papers get through. People read some of these slightly dodgy paper, and the journal get a bad reputation and it stays low-impact. Nevertheless, whether you get your paper into a high-impact or low-impact, the simple truth is that all a paper needs to be published is evidence backing up the idea. If the idea is sound, keep publishing results based on it, and it will get noticed and you will get a good reputation and then you'll get published in the high-impact journals. Like I said, it isn't fair. But, the system does work in that only evidence-based ideas get published. And, the converse is true, too. If it isn't evidence-based, it doesn't get published. So, where are the peer reviewed papers based on "Null Physics"? And, the larger question I asked, where are the ideas that are supported by evidence that were shot down by "the conspiracy"?

Quite. Look at the Ideal gas law: PV=nRT. If the temperature of the gas in the balloon is doubled, T, and the volume, V, of the balloon remains unchanged, then the pressure is also doubled -- no need to add more mass to the balloon to double the pressure at all! Pressure, P, Volume, V, mass (n is moles of the gas), R is the gas constant, and Temperature, T are all linked together. You cannot simply say that a balloon at 2 atm of pressure has more mass than a balloon at 1 atm of pressure, because there are other reasons why it may be at 2 atm than just more stuff inside it. Now, most gases do not obey the Ideal gas law exactly, but follow it close enough to make a very good approximation. And all the other non-ideal gas laws are also dependent on all of pressure, temperature, volume, and moles/mass of the gas.

Since it is known that the Earth is surrounded by the vacuum of space, how is it possible -- given the statement I quoted here -- that an atmosphere remains about the Earth? Shouldn't the gases of the Earth's atmosphere "fly instantly"? They obviosuly don't or we wouldn't be able to live, so why don't they?

While people are human, with all their flaws, you really have to have a negatively slanted view to think that the entire community behaves as you suggest. It really is a simple hurdle that had to be jumped in order to be accepted -- there has to be evidence to support the idea. That is it. The new idea has to be able to explain all the known phenomena at least as well as the current idea. The new idea has have some experiment, or some math, something that says that, yes, this could be the answer. I've written it before, and I'm sure I'm going to write it again, but the vast majority of scientists I know would salivate at the chance to work on something brand new. To work towards discovering something that has never been discovered before. The chance of discovering something that has never been discovered before is one of the primary motivations people become scientists in the first place! But these same people aren't going to run with something just because it makes for a good story, or because a group of people ardently believe in it! There has to be evidence that backs up the idea. Otherwise it is rejected. This is nothing personal. But, indeed it is biased -- it is biased toward following the ideas that are best supported by the existing evidence. No evidence -- no publication in scientific journals. It is really that simple. It makes me very angry when people write garbage accusing the community of this nonsense. Every single scientist is looking for something new -- that's the whole point of science! But, they only follow things that are new AND are supported by evidence. It isn't like buying a new shirt just because it's new, or getting a new set of golf clubs just because they are new -- new science is only acceptable and only becomes science when it is supported for evidence. It isn't acceptable just because it is new. If there is evidence for the new idea, it will get published. It may not be trumpeted as the next great thing in physics (or engineering to math or whatever) -- but it will be published. If it has evidence behind it, it will be published. If it doesn't have evidence behind it, it will not be published. End of story. ------------ Speaking of which, is there actually any evidence of this supposed bias by the community? Can you cite any theory, ever, that wasn't published solely because of the person who presented it? That is, specifically, can you cite any idea that had significant amounts of corroborating evidence, and no significant evidence that clearly indicated it was wrong, and was a truly testable and falsifiable theory? If you know of one, please enlighten me. Otherwise, this "biased community" hypothesis of yours just bites the dust like so many other unsupported hypotheses. And rightfully so.

I prefer to celebrate Festivus, and so I shall begin with the Airing of Grievances: I got a lot of problems with you people over the last year...

I posted the reference to the journal article, and Klaynos provided a hyperlink to it. You may not have access to it, so you might have to pay for it (to get it online) or you might have to go to a university library and photocopy it. This is just one of many very good papers that present the evidence why the current model is so widely accepted. They didn't just pick this current model because they liked the story behind it the best -- experiment after experiment have shown the existence of the particles in the current model. The onus is on you and your alternative idea to show 1) why all these old experiments were mistaken and 2) why your idea explains the experiments at least as good as but hopefully better than the current model. While you are at the university library, you might want to ask the librarian to help you access a database like Web of Science which will show you every paper that references the Briedenbach one above -- all the work that grown out of that result. Because your current model will have to explain all of those results as well as the one in the Briedenbach paper.

How are you going to explain away some of the classical results that started to show the existence of the quarks as they are defined today? See M. Breidenbach (1969). "Observed Behavior of Highly Inelastic Electron-Proton Scattering". Physical Review Letters 23 (16): 935–939. In this paper there was reported that the author discovered three point-like bodies inside a proton. Why weren't there 725 point-like bodies found? Why have there always been 3 and only 3 point-like bodies found?

My statement about taking the derivative of a constant stems from this statement right here. If [math]\phi[/math] is a constant -- as you wrote in your equation -- then taking a derivative with respect to it really doesn't have any meaning, right? It's be like asking what is [math]\frac{df}{d2}[/math] you see? Now, I think what happened here is that R is the constant, not [math]\phi[/math]. So, this statement above isn't right. But, this is what I was responding to. Taking the derivative with respect to a constant doesn't really have any meaning. There is no reason to be frustrated, I'm just trying to help answer the questions that I see, that I know about. You shouldn't feel frustrated, because you could have gotten no help at all. Your feeling frustrated conveys some sense of entitlement -- like you are frustrated that you aren't getting the help you should be -- when instead on a free forum where people are volunteering their time you should probably be grateful that anyone is trying to help you at all. This is 100% a community of volunteers who spend their time helping others. We are not paid. Part of the frustration is that I don't think that you are being 100% clear in what you are asking. I am trying my best, but if you need different answers, you need to be clearer about exactly what you want. If I can help, I will try to help, but this isn't a guarantee. I may not be able to give you the answers you want, and I may give you wrong answers, because we all make mistakes. I may not be able to give you any answer at all. But to feel frustrated I think it really quite unfair to me and to the entire forum.

But the algebra wasn't fine. You made a mistake. Check it again. Klaynos showed you the step you made the mistake on.

Dude, I downloaded and looked at the paper, I took the time to explain what I saw. And I don't see any question in the last several posts! I think (and I hope) you didn't meant it to come of that way, but your post sure reads awfully rude to someone who has volunteered a lot of time typing up equations trying to help you as best I can. Now, I'm going to give you a second chance here and ask what question I didn't answer, and I'll do my best to answer it for you. But there aren't any guarantees. I'm not the author of the paper or anything, so I don't know what all the authors did in their paper.

I concur with iNow there. Feel free to keep asking questions on the forum -- that's pretty much the whole point of the forums after all! One other point I meant to write before but forgot to was that if there is a concept in that book that just isn't getting across or you just can't get what the author is trying to say, go find another calc book. Your local library probably has at least 5 or 6 other ones on their shelves. There are other ways of looking at a subject that has been taught and written about as much as calculus, so there is never a reason to sit and struggle with one book. For that matter, if you browse a used book seller website, like abebooks.com which lists the inventories from 100's of used book dealers, you can buy many calculus books for only a dollar or two. Sure, they may have been published 20 or 30 years ago, but the content really hasn't changed that much. And, since you are self-teaching it, it wouldn't hurt to get a few other perspectives on the same topic. Different authors are going to emphasize different things because each will have a different idea on what is most important.

The author of a calculus book can't really get too wrong -- Calculus has been around long enough that the only possible "mistakes" are typos and the like. What this really comes down to is presentation -- whether you are able to grasp what the author is trying to say. A big issue here is that do NOT just read a section and move on thinking you "got it". You have to do the practice problems. That's the only way to know if you "got it" or not. I didn't look, but if that book doesn't have lots of practice problems with solutions, then it isn't worth much. Calculus is one of those things where you just need to 100's and 1000's of problems to really own it. Once you own it, then you can move on. But, if you really want to learn it and own the knowledge when you are done, do many many practice problems from each section. I also hope that you don't take this the wrong way, but if you are just now learning calculus, there is a long way to go. Major stepping stones before this problem can be done in even a simple way include ordinary differential equations, partial differential equations, and computational method for solving ODEs and PDEs. Calculus is normally a 3 semester course. ODEs 1 more semester, PDEs another 1 semester, and computational methods of solving them may be another semester or included in the ODE & PDE courses. At least 5 semesters or 2 1/2 years of info that builds on top of itself. I don't say this to discourage you at all, but to try to give you a perspective of how far you have to go just to learn the math. That doesn't include anything on learning the physics of the situation, that is only to learn the math so that you can understand what the symbols in the physics equations mean. Again, I do hope that you don't take this as discouragement, because that is the last thing I want to do, I just want you to realize the hill you are looking to climb. If this is something that needs to be done in a matter of weeks, I think you had best look elsewhere. Or look to greatly simplify the scope of your project.

OK, I looked at your paper there. 1) K is a viscosity-like parameter. It is the factor in front of the power-law model for the shear stress in the non-Newtonian fluid they used. K has units so that when it is multiplied by the shear rate to a power, the correct units of shear stress come out 2) [math]Re_m[/math] is a modified Reynolds number. Modified in several ways. Firstly, they use the mass flux in place of a characteristic velocity. That's what the [math]\Gamma[/math] stands for. Secondly, because of the power law model for the shear stress, the K part has to be raised to a the correct power so that the Reynolds number becomes dimensionless. Then, using this mass flux, they define a characteristic velocity, [math]u_r[/math] All this is fine with what I said above -- because like I said sometimes it takes some manipulation to get the characteristic variables you want. This one looks a lot different because of the different model for the shear stress. The Navier-Stokes equations are only for Newtonian fluids, and Newtonian fluids obey the shear stress rule: [math]\mathbf{\tau} = -\mu\mathbf{\dot{\gamma}}[/math] where [math]\mathbf{\dot{\gamma}} = \nabla\mathbf{v} + (\nabla\mathbf{v})^T[/math] is the rate of strain tensor. For the power law fluid in your paper, [math]\mathbf{\tau} = -K(\mathbf{\dot{\gamma}})^n[/math] where n is some constant. Using this as the shear-stress in the momentum conservation equations yields equations that are non longer Navier Stokes equations. N-S assumes Newtonian flow. 3) That paper isn't too terribly clear about what coordinates it is using. It is a film running down the inside of a spherical shell, but it looks like the authors are trying to use Cartesian coordinates to describe the flow. This is where the extra terms in the continuity equation come in. Because the film thickness, [math]\delta[/math] changes, there are extra terms that must account for the expansion and contraction of the film. In particular, note how the [math]\eta[/math] is a function of both x & y because of the coordinate system chosen. When you use the chain rule, that's why you get derivatives of U with respect to both the nondimnsionalized x & y -- because the nondimensionalized y, the [math]\eta[/math] is a function of both x & y. It really isn't very clear. Then, on top of it, I am really confused because you have some evaporation but I don't seem a mass-loss term in the boundary conditions. It might be worthwhile looking for one of those author's thesis document or a equivalent where there might be more detail listed. I think that I personally would have tried to use spherical coordinates, but that may have had its own set of issues that complicate things. I do know that if I were refereeing that paper, I would have asked the authors for more detail. 4) Here's my final thoughts. Have you done the same problem, just with a Newtonian fluid? Because, in things like this, you really want to learn to walk before you run. Especially if you aren't experienced with non-Newtonian fluids, is you aren't experienced with CFD, etc. It may even be worthwhile to go all the way back and re-solve a falling film problem of just a Newtonian fluid flowing down an inclined plane. Then do a power-law fluid flowing down an inclined plane. Then do a Newtonian fluid film flowing around a sphere like in the problem, then the power low fluid, etc. Introduce one complexity at a time until you build up to the problem at hand. If you try to put too much into it all at once, you'll never know where your mistake is. Or even worse, you'll make two mistakes, and they'll almost cancel each other out until you find and fix one of the them and then things will get a lot worse and you won't understand because you thought you fixed something. Believe me, I've been there with fluid dynamics and CFD problems before. It really is best to start as simple as possible and slowly add complexity to the problem. Even if you think you have a good handle on what to do with the complex phenomena. This suggestion even includes CFD solution methods. Don't jump into a complex solution method -- start with the really old basics and slowly put in the more complex and accurate ones. Again, if you put in too many changes at once you won't know what change or changes caused the error.

OK, I'm just going to start clean here and show you how I learn how to do this, because I still don't think that you have a good idea. Phrases like "substitute the vectors for scalars" seems quite wrong. I think I know what you mean, but the phrasing of those words is very inexact, because you can't just replace a scalar for a vector -- a quantity is a vector, has to remain a vector, otherwise it isn't the same property. For example, if I say 10 mph due North West, that is a vector because it has a magnitude and a direction. I can't just "substitute" in 10 mph. Because that isn't a vector; it doesn't have a direction. In this case, it is just a speed, a different -- though related -- concept from velocity. I suspect that what you meant was to "solve for the components of the velocity" in that you write out the 2 or 3 equations for the components of the velocity vector. That is, the equation for [math]v_x, v_y, v_z[/math] in Cartesian coordinates, or [math]v_r, v_{\theta}, v_z[/math] in cylindrical coordinates and so on for the other coordinates. It is a small-thing, but the words you use have exact meanings especially in math and physics, so you have to chose them carefully Now, on to nondimensionalizing the Navier-Stokes Equations. Now, this can either be done in their vector form, or in their individual component form -- the results are the same. I'm going to do it in the vector form. As a nomenclature note, unprimed variables have dimensions, primed variables are dimensionless, and I use the symbol [=] to mean "has dimensions of" For example velocity, v [=] L/T which means length per time. The N-S eqns (I'm going to drop the pressure and gravity/body force terms, as they make more dimensionless groups -- ideally, for practice, you should probably put them back in and discover them in the same way I will post below). [math]\rho \frac{\partial{\mathbf{v}}}{\partial{t}} + \rho \mathbf{v}\cdot \nabla \mathbf{v} = \mu \nabla^2 \mathbf{v} [/math] Now, the variables are: density = [math]\rho[/math] [=] [math] M/L^3[/math] velocity = [math]\mathbf{v}[/math] [=] [math]L/T[/math] time = [math]t[/math] [=] [math]T[/math] viscosity = [math]\mu[/math] [=] [math]M/(LT)[/math] on the right side of the [=]'s, the M stands for mass, L for length, T for time -- viscosity has units of mass per length per time. To nondimensionalize this, we need some characteristic values. In most cases, there are going to be some pretty obvious ones. The radius or diameter of the pipe is an obvious characteristic length. The average inlet velocity is a an obvious characteristic velocity. Or, the velocity of a moving boundary is an obvious characteristic velocity. Whatever the are, let's just assume that they exist. So, let [math]l_c[/math] be a characteristic length and let [math]v_c[/math] be a characteristic velocity. Note that with a characteristic velocity, we can make a characteristic time, [math]t_c = \frac{l_c}{v_c}[/math] Furthermore let the dimensionless variables be defined as: [math]\mathbf{v}' = \frac{\mathbf{v}}{v_c}[/math] [math]t' = \frac{t v_c}{l_c}[/math] [math]\nabla' = l_c \nabla[/math] We're going to leave the density and viscosity alone. In all honesty, this is because I know the result we are looking for. But, you could actually nondimensionalize them too, if you found a characteristic mass. This would be important in a compressible flow, a characteristic mass could come from a density as a reference temperature and pressure, for example. It is important to note that there is a nondimensional gradient operator, because the dimensional gradient operator has units of inverse length. I like to re-write those equations above so that they are in a form easy to insert into the equaiton: [math] v_c \mathbf{v}' = \mathbf{v}[/math] [math]\frac{t' l_c}{v_c} = t[/math] [math]\frac{1}{l_c} \nabla' = \nabla[/math] Now, just plug these into the N-S equation above: [math]\rho \frac{\partial{v_c \mathbf{v}'}}{\partial{\frac{t' l_c}{v_c}}} + \rho v_c \mathbf{v}'\cdot \frac{1}{l_c} \nabla' v_c \mathbf{v}' = \mu (\frac{1}{l_c} \nabla')^2 v_c \mathbf{v}' [/math] now, let's collect all the constants together in front of each term: [math]\rho \frac{v_c^2}{l_c} \frac{\partial{\mathbf{v}'}}{\partial{t'}} + \rho \frac{v_c^2}{l_c} \mathbf{v}' \cdot \nabla' \mathbf{v}' = \frac{\mu v_c}{l_c^2} \nabla'^2 \mathbf{v}' [/math] Divide through by [math]\rho \frac{v_c^2}{l_c}[/math] and you get: [math]\frac{\partial{\mathbf{v}'}}{\partial{t'}} + \mathbf{v}' \cdot \nabla' \mathbf{v}' = \frac{\mu}{l_c v_c \rho} \nabla'^2 \mathbf{v}' [/math] And finally, note that term in front of [math]\nabla'^2 \mathbf{v}' [/math].. It is a Reynolds number. A velocity times a length times a density divided by viscosity is a Reynolds number. [math]Re=\frac{l_c v_c \rho}{\mu}[/math] so [math]\frac{\partial{\mathbf{v}'}}{\partial{t'}} + \mathbf{v}' \cdot \nabla' \mathbf{v}' = \frac{1}{Re} \nabla'^2 \mathbf{v}' [/math] <--- the dimensionless Navier-Stokes equation. The Reynolds number comes out completely naturally as a result of the nondimensionalization process, and since I did it with the vector equation, it is completely coordinate system independent. Like I said, you should put the pressure and the gravity terms back into the N-S equations and repeat the above to discover that the Froude number [math]Fr = \frac{v_c^2}{g l_c}[/math] and the Euler number [math]Eu = \frac{\Delta p_c}{\rho v_c^2}[/math] (the [math]\Delta p_c[/math] would be a characteristic pressure drop) comes out just as naturally as the Reynolds number comes out. In the same way, the Nusselt number comes out of the equations of heat transfer, and many of the other famous dimensionless numbers. ----- Now, I mentioned that there is another way to come up with these numbers, and that is the very powerful Buckingham Pi Theorem. http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem With it, you take all the variables in the equation, and you predict all the possible dimensionless numbers that can come out of the problem at hand. It won't show you where the dimensionless numbers arise in the equations, but it will alert you to which ones are relevant to the problem at hand. I hope that this makes it all clearer. I do also hope that you will put in the gravity term and find the Froude number. I also hope that you will follow the same procedure on the continuity equation to find the dimensionless continuity equation. This can also be done with the energy equation -- really any of the equations of fluid mechanics.

OK, but here's the question I have. What velocity are you using in your Reynolds number here? And why wouldn't you use that same velocity that you are using in the Reynolds number to nondimensionalize the velocity? And furthermore, the Reynolds number comes out of the momentum equation -- it isn't the result of using it in the definition of the characteristic velocity -- it arises naturally from the process of nondimensionalization of the Navier Stokes equation. I don't like giving out too much personal information on the Internet, but I will say that I attended two public schools for engineering. This isn't right: [math]\frac{\partial u}{\partial x} = \frac{\partial}{\partial\left(\frac{x}{R}\right)}\left(\frac{UKRe}{\rho R}\right) = \frac{\partial}{\partial\phi}\left(\frac{UKRe}{\rho R}\right)[/math]. Because you are taking a derivative with respect to a constant -- that will always be equal to zero. ---- If you want some help with this, I can show you the steps to generate the Reynolds number. There are actually 2 or 3 different ways to generate it, and it is important to know them all, in my opinion. An excellent resource on nondimensionalizing is in the book Fundamentals of Fluid Mechanics by Munson, Young, and Okiishi.

Well, 1) Firstly, there is a LaTeX error in your post so I am not sure of what you are trying to do but, 2) Secondly, I guess I am also unsure of what you are trying to do in the first post. Usually, the velocity is nondimensionalized by using a characteristic velocity. Such as the average entrance velocity, or the velocity of a moving boundary. Usually the Reynolds number isn't part of the nondimensionalizing of the velocity, because the Reynolds number contains a velocity itself. Sometimes a characteristic velocity is found by taking a characteristic length and dividing it by a characteristic time, like the stroke of a piston divided by the cycle time of the piston. For an incompressible flow, the dimensionless continuity equation should look like dimensional continuity equation. The Navier-Stokes equation will have dimensionless numbers in them that are the result of the nondimensionalization. The Reynolds number will be in front of the viscous diffusion term, the Euler number will be in front of the pressure term, the Froude number will be in front of the gravity terms, etc. I am also confused by your use of combining x and R and calling R a radius. If you are using x & y, and the form of the continuity equation in your first equation, then you are using Cartesian coordinates -- there shouldn't be a "radius" that is proportional to a coordinate direction in this case. If there is a radius proportional to a coordinate direction, then you need to use something like cylindrical or spherical or maybe even generalized coordinates with an axisymmetry -- it depends on the application. Perhaps if you start from the top, I can try to point you in a better direction. But at the moment, I am pretty confused about what you are trying to do.

1) This forum used LaTex, which will make it a little easier to put in your equations. Just use the [_math_] and the [_/math_] tags (remove the underscores _'s). If you quote my reply here, you'll see the math tags below. 2) When you substitute in dimensionless variables, you just use the normal rules of differentiation of constants. I.e. let [math]x'[/math] be the dimensionless x variable. Something like: [math]x' = \frac{x}{L}[/math] where L is some characteristic length. Rearrange that and you get: [math]x = Lx'[/math] Now, when you take a differential of x, dx, you put in the substitution [math] dx = d(Lx') = Ldx' [/math]. The L comes out because it is a constant. And so on. You can do this for any variable you made dimensionless.