Theory of Radial Momentum

Correspondence Log


The pressure of radially flowing fluid is inverse to the distance from its source.


1997-Feb-25 Letter to Exploratorium

1999-June-11 Letter to Brad Snyder, UNR

To Peter Olsen

1999-June-11 to Rick Purcell



San Francisco Exploratorium

Sign on the Levitator Exhibit:

In this exhibit, air blowing down keeps a plywood disc hovering. To do and notice: Lift plywood disc up under the air outlet until it stays there by itself. If the disc is already hovering, try pulling it down. What's going on? When you lift the plywood disc up towards the air outlet you constrict the space that the air can flow through. In order for the same amount of air to get through the increasingly narrow space, it has to go faster. As air moves faster its pressure drops. As you continue to raise the disc, there is a certain point where the air is going so fast that its pressure drops below the pressure of the normal surrounding air. This fast moving, low pressure air above the disc pulls the board up. This lifting force is counteracted by the air blowing down on the disc and at a certain point the two forces cancel each other out and the disc just hovers.

I wrote to The Exploratorium on 97-Feb-25 and received this prompt reply from Dr. Paul Doherty, (a prominent MIT physicist who looks over the Exploratorium):

Oh I see, you are after a fundamental physics explanation of the Bernoulli effect. I too have looked for a long time, and there isn't one. Very few books give the correct statement. They usually say something like: 'air flowing at high speeds exerts low pressures,' which is often incomplete. A more correct statement of Bernoulli would be 'Under certain conditions, as air speeds up its pressure drops.'




To: Brad Snyder, UNR, 89557 Date: June 11, 1999

From: Ed Seykota


Dear Brad,

  I want to thank you for a number of things:

1.     Thinking crisply and getting through to the fundamental issues.

2.     Pointing out that my reading of Potter and Foss was incorrect.

3.     Suggesting that I fractionate my thesis into smaller, publishable chunks.

4.     Noting that breezy-type news-periodicals generally don't take risks.

5.     Suggesting I take some formal course work.

 Based on your comments, I have revised The Bernoulli Fallacy and hopefully strengthened it considerably. I would like you to apply your keen eye to it for another hour and expose any residual misconceptions.

 I am also working to bring The Levitator Model in line with conventional fluid mechanics parlance, particularly (1) recasting my own home-brew terms such as "push-acceleration" into a conventional momentum flux term and (2) clearly enumerating the mass, momentum and thermal energy balances.

 I am still a bit puzzled about how to run the pressure balance for a control volume with fluid entry and exit. I continue to feel that a numerical solution, based on Euler's method of recursive solution of integral equations for small dt offers the most direct approach. I am not sure, however, how to increment the pressure and temperature from mass and momentum flux along adiabatic lines. 

Per course work, I would like to sign up for something in fluid mechanics, perhaps your course if appropriate and available. I hold two BS degrees, one from MIT in EE, and would not mind continuing where I left off. I like simulations. 

My interim goal is to get something into print, preferably a suite of articles about an alternative way to analyze the levitator and other flow-normal lift devices. Perhaps you can suggest a target publication and a template for bringing my writings into acceptable form. 

Enclosed is [$ x] against the time you claimed for reading. I appreciate your generosity.




Dear Peter Olsen

Yes, that would be Daniel Bernoulli, of rho + dv^2 = k fame, who also said, "It would be better for the true physics if there were no mathematicians on earth." 

(In hopes of) a possibility of advancing a 262 year-old science, I offer the following peek under Bernoulli's skirts. 

Daniel Bernoulli, contemporary of Newton and Euler, developed an interest in blood, and also in other fluids, enumerated pressure, density and velocity as properties, and in 1738 (Hydronamica) established the kinetic theory of gases.  

I claim, however, that his famous pressure/velocity relationship has some fundamental limitations that preclude its application from solving for lift in hydrodynamic situations, ironically, in the essential "Bernoulli Effect" phenomena. 

Bernoulli's classic derivation assumes a torus of varying width, perhaps conceived after consuming a mal-formed donut, inside which fluid necessarily circulates in a velocity-varying way. Now, from the principle of conservation of energy, kinetic energy must counter-balance pressure energy, so pressure would then be inverse to the square of velocity. 

The often overlooked tiny-footnote conditions to the scheme, namely, that density is simply and everywhere constant, and that friction is zero, preclude an internal flow motivator, so an actual lab device would contain dormant fluid. 

Bernoulli's solution, then, in the lab, turns out to be another excellent example that a lot of things are all exactly equal to each other when all things are exactly equal to zero. 

Having got my toe onto this theoretical tar-baby, I abandon all caution and jump into its gooey core by claiming that flow-induced pressure gradients are, of all things, a variable density phenomenon. 

Bernoulli's empirical garments seem a bit short in places, as you might yourself witness by modestly inspecting, of all things, a garden hose. 

This ubiquitous and flexible conduit does not collapse, when you widen the faucet, a la Bernoulli's predict-a-pressure program, nor, indeed, does a thin paper tube collapse when you blow lots of high-velocity air through it. 

A paper cone, however, curiously, does collapse,  when you blow into the narrow end. 

I have run numerous trials with various lab apparatus mostly fabricated from acrylic plastic, replete with p & v meters that yield empirical data, all of which supports my thesis. 

Classic texts, such as Potter and Foss, have problems with Bernoulli solutions that just don't work. 

These observations, however, cut directly across the grain of some of the fundamental tenants of formal fluidic faith, so my attempts to secure an audience, at least at U-Nev-Reno have generally induced autonomic disbelief and anger, a somewhat disappointing proxy for logical counter-argument. 

I suspect some of the aversion to my approach has to do with my daring to travel, computationally, out past the simplifying horizon of incompressibility. 

My suggestion to use Euler's numerical method for solving simultaneous integral equations as a way to beat the mathematical intractability seem to find little currency among those bred to think along the lines of the conventional derivations of the closed-form Navier-Stokes equation. 

Were you to be interested enough in the above rantings to study my empirical data and theoretical derivations and forward serious, constructive logical reactions  I would be willing to post a URL for your perusal.

 Yours truly,


Ed Seykota



One final thing, the Bernoulli's attached their names to several things.  I assume you're thinking of the principle dealing with velocity and total pressure.    Peter Olsen, PE,  n2ell  P.O. Box 410, Simpsonville, MD 21150-0410 USA, 410-997-8584   "Engineering is the art of applying a professional knowledge of mathematics and the physical sciences to improve the quality of life."


To: Rick Purcell  June 11, 1999

10505 San Fernando Rd.

Reno NV 89506


Rick Purcell <>


Dear Rick -

 Thank you for the feedback.


 Ed- I was laying in bed this morning thinking about levitators (not a good sign) and realized what has been bothering you about the text book problems. You have picked up on something that tens of thousands of physics students and hundreds or thousands of professors have missed. The author gives the wrong answer. Bernoulli is alive and well and can be applied to the situation. The problem is the simplifying assumption both authors give: that the flow can be simulated as an average value or, for problem 2, that the flow velocity is essentially constant. The levitator will not work with constant velocity - that assumption turns it from a circular device to what you call a slot levitator, which also has constant velocity, and you know empirically that they don't work.

I agree, average velocity is a problem. It is a minor problem, however, compared to using Bernoulli's Principle when you have compressible flow. 

In fact, there is a third possibility. If you could build a circular device with flow from the outside toward the center (with center static pressure communicating with the back of the card), it would actively blow the card away.

 I agree.

 So decelerating flow attracts, constant flow velocity is neutral, and accelerating flow is repulsive for this geometry. I should be able to provide a simple closed form solution using Bernoulli. The authors' simplify the problem to focus on the particular effect they are trying to teach. Additionally, at this level, the students calculus skills are still being developed and may not be up to the task. In this case they have simplified themselves out of business; and probably don't realize it (although they may not admit it).

 I doubt this approach will succeed.

 It would be interesting to send Halliday the solution and get his comments.

 I would like to wait until I have a tighter, more convincing presentation.



 June 9, 1999

 Ed- I've been having trouble finding time to review your two papers, but wanted to get you some interim thoughts even thought they are incomplete. My comments generally follow the flow of the paper. You mentioned publishing your findings; do you have an audience or publication in mind? Most of the technical papers and journals I see are much more bland then your write-up and they tend to take great care defining their terms, equations, statements, and conclusions.  

Newsy journals, as a rule, do not like to take risks. Standard procedure is to work yourself up the journalistic food chain, starting with the fusty-dusty peer-review ones. 

Regarding 'The Bernoulli Fallacy': 1. I don't know why you limit it to closed systems only; I've not seen that restriction eleswhere. But you could include the term lossless, inviscid, or potential in the first sentence.

 Good call.

 2. I think he is exactly, not fundamentally, correct for his stated conditions (steady, inviscid, incompressible flow with no added work).

 Yes. However, I claim incompressibility and lift are fundamentally incompatible.

 3. Explain what the accounting error is.

 I have abandoned and/or revised this argument in favor of the variable density disallowance for BP.

 4. Why doesn't the B. eqn apply to the levitator, with the caveat of lossless flow? It has steady, incompressible flow.

 To the extent the flow is incompressible it is also absent lift.

 5. Bernoulli does not imply that high velocity means low pressure, but it implies lower pressure than if the fluid was flowing a low velocity.

 It works for non-motivated, loss-less flow; in other words, the kind that does not exist in the thick of the lift.

 6. I would redraw the standard diagram with a smoothed transition section; square corners always distract me when dealing with flow.

 Square corners are not the half of it. The real problem is that there is no source or target for the flow. The diagram, in the limit, implies a vacuum pump that can fire molecules out like rifle shots.

 7. In example #1 you state the approach is incorrect; this would be a good place to briefly state why.

 I am developing the argument along these lines.

 8. In most cases, the Greek letter rho is used to represent density. I understand you are duplicating an existing solution, but lower case d is usually reserved for calculus.

 Good call. I plead laziness and lack of knowing how to do a rho on the word processor. I have learned since.

 9. You spring the term 'valve' on the reader, but it is not apparent to what that refers; I'm guessing you are referring to the area where the flow enters the levitator. You should clarify this.

 Good call.

 10. It is not true that the pressure on top of the card and the pressure below the card are not related. Here are three fundamental things we need to discuss that may help: A. We need to agree on the definition of static, dynamic, and total pressure. B. For subsonic flow issuing from a duct (or the outer edge of a levitator), the flow will adjust such that the static pressure at the exit plane will be equal to ambient pressure. This truly relates the pressure above and below the card. This is not an intuitive situation, but is the way I was taught and I believe it to be true. I'll look for references.

 This is strictly true only when exit velocity = 0. Bernoulli's Principle leads irreconcilable problems at both the center and the outside edge of the disk. At the center the pressure does not agree with experiment; at the outside, the velocity doesn't match. 

A somewhat analogous argument might be a battery hooked to a resistor with lossless leads. If voltage is referenced to the negative terminal of the battery, and you hook a voltmeter between there and the exit of the resistor, it will always read zero volts. You can change the battery voltage and value of the resistor and it will effect current, voltage drop, and dissipated power, but you will still read zero volts at that location. The current will adjust such that there is zero potential leaving the resistor. C. From A. and B.

 The voltage is zero at ground. The current (airflow) at ground (outside of the disk) is not.

 I think I can show that Bernoulli's equation with predict lift for a circular levitator and no lift for a channel levitator. I'll try to fax you an explanation of A, B, and C in a few days. 11. The textbook example #2 makes a horrible statement - that the flow velocity is essentially uniform at 200 fps. If the inner radius is 2" and the outer radius is 12", the velocity will vary by a factor of 6! 12.  

The strange math is not the half of it. Strange math typically accompanies Bernoulli Principle problems. When you straighten out the math, Bernoulli's Principle still fails since it assumes incompressible flow. The real motor behind lift (and my central thesis) is that lift is a function of radial decompression. So Bernoulli's Principle does not apply.

 Again, explain what you mean by a valve.

 Good call.

 13. Your explanation of this problem would better fit conventional format, and read a bit more easily, if the equation numbers follow the equations in parentheses. All the variables should defined below the equations or at the end or beginning of the paper; use rho for density.

 Good call.

 14. The r^2 term disappears between eqn 5 and 6. 15.

 Good call.

 The diagram uses V1, not V0.

 Good call. I did not carry it back to the pipe entry velocity.


 I'm getting my office ready to move this weekend, we have an instrument rack for NASA's DC-8 that has to be completed, tested and delivered to Dryden (Edward AFB) a week from Monday, and a large sampler that needs to ship in two weeks so it is not clear how soon I'll have more discussion for you; but I'll keep trying. Rick


Thanks for the feedback.

June 26, 1999

To: Rick Purcell

10505 San Fernando Rd.

Reno NV 89506


Rick Purcell <>

 Dear Rick -

 Thank you for your fax of 99-Jun-28.

It seems well stated and clearly shows your thinking. Your conclusions seem to follow logically from your premises. It seems to be a pretty good example of applying Bernoulli's Principle.

 At this point I think the matter has some definition and hopefully it will be a fairly simple to demonstrate the validity (or invalidity) of applying Bernoulli by further examining your derivation.

 To move forward, I have enumerated some points that I think demonstrate some of the problems with applying Bernoulli to the levitator. It you can defend these, then I would seem to have little further argument. On the other hand, if you cannot, then I might gain fortification to proceed to a more general statement of the inapplicability of Bernoulli's Principle to the physics of lift.

 So I think, at this point, you are in a position to either (1) put my thesis out of its misery or (2) breathe a little more life into it. Either way, I appreciate your interest and willingness to take the time to examine these concerns about your fax:

 1.     Page #1 drawing. Let's say you simplify things by setting t (plate separation) so that the inter-plate cross section at r1 is the same as the cross section of the feed tube - so there is no initial contraction or expansion of the flow path where it turns the corner. That leaves you with a rather sticky problem: namely, the pressure just to the right of r1 is the very minimum pressure in the system while the pressure just to the left of r1 is the pump source pressure and must therefore be the maximum. So you have this discontinuity from the highest to the lowest pressure, with no change in cross section, all, right in the middle of the most important region of the effect. I do not think a nice theory would have such a huge and inexplicable discontinuity right in the middle of the effect.

 2.     Proceeding along these lines from another angle, you do have to feed the real-world device with a real-world feed pump to make it go. The only way that you can get the drawing to work, consistent with all the constraints, is to feed it with a pump that radiates high velocity molecules at a very low pressure - a kind of thermodynamically radiant black hole. I couldn't find such a device (a vacuum cleaner that ejects high-velocity air molecules out the suction tube) at a hardware store so I think I'd have to rule out using one in this case. And, I think that without such a conflicted contraption at the center, the drawing just won't work.

 3.     Page #1, about halfway down: t = plate separation (influences flow rate, not lift). Experimental results show lift is indeed a function of central pressure, flow rate and separation. Furthermore, the levitator suspension plate actually does come to equilibrium at some t that depends on the other variables. The Bernoulli derivation does not account for this phenomenon, much less predict a value for t.

 4.     Page #1 near the bottom "along streamline, Bernoulli says."  Now we have a radially expanding flow so you either have to say (1) there are really no nice parallel streamlines or (2) there are some sort of "radially separating streamlines" and these are going to invalidate the constant density (incompressibility) assumption. So at this point I would question the application of Bernoulli since you cannot claim parallel flow lines and incompressibility.

 5.     Page #2 bottom: the last equation for P(r) can produce P(r) < 0 for large (r2/r). How can you get a negative pressure? Not in the real world. Only with equations that are somehow drifting away from the real world. I think something may be a little off here.

 Yours truly,