Talk:Circulation (physics)

Untitled

what (expletive deleted) are the units/dimentions for circulation???

m^2 s^-1 ???!!! — Preceding unsigned comment added by 84.67.17.150 (talk • contribs) 23:39, 11 May 2006 (UTC)

Yes (velocity times length). Linuxlad 07:47, 12 May 2006 (UTC)

More needs to be said about this.

The formula for lift based on this "circulation" parameter is a hypothesis only, contrived out of dimensional analysis, which is a necessary but not a sufficient condition for acceptability. A proof would require derivation from a known law like F = ma or a formal attempt to make it a law itself. Such attempts are readily discouraged by finding cases where it does not hold. Typically, wing profile designs try to optimize the lift force using it and in going to great lengths to shape the profile in so doing, such as by creating a large hump on its upper surface, one quickly finds that the credibility of this theory is stretched beyond allowable limits.

Yet in lieu of anything else to take its place, this is what is the basis for all modern day practice. For wind energy, in particular, which is still in its formative years, some serious concern exists here. AnthonyChessick 21:01, 30 July 2006 (UTC)

Two Points - this is a page on circulation, which has a clear mathematical derivation, and whose dimensionality is sufficiently stated earlier. Turning to the (Kutta-Joukowski) lift formula, (a bolt-on here, but still), see say Batchelor's Fluid Dynamics, formula 6.4.26. The formula is for a 2-d wing, force per unit span. It follows by integrating (the vector form of) F=integral (P.dA) around the surface of the body (or alternatvely by considering momentum flows). HTH Bob aka --Linuxlad 14:14, 25 July 2006 (UTC)

Under 6.4.26 on page 406 of my edition of Batchelor he goes on to say, "The remarkable side-force or 'lift' on the body, which is the foundation of the theory of the lifting action of aeroplane wings, arises from the combined effect of the forward motion of the body and the circulation round it, and is independent of the size, shape, and orientation of the body."

In giving his expertise some credit, I don't think he means this literally but only that he places all his analysis on the flows and is blind to how the body causes them. Still, in making bold statements such as this, he seems given to going to unusual lengths to support Kutta and Joukowski. This is a dangerous fascination with detailed intricacies that are unproven and whose support by the basic laws of motion is in some question and the fact that they have survived so well is only a testament to how strongly the modern world adheres to what is complex and a challenge to fathom thoroughly. The words themselves give away the tenuousness of this analysis. Derivative material from this that appears in other textbooks never includes the all-important momentum considerations and their authors can not really be faulted on this account.

The "Lift" entry has a cautionary statement that it is under some controversy but I would think that this same controversy extends here as well regarding the appearance of the lift formula using circulation, not to belabor this. AnthonyChessick 21:01, 30 July 2006 (UTC)

More can be said about the concept of the Circulation Parameter as a vehicle by which the Lift Force may be calculated. In the Mechanics branch of Physics, a vector cross product strikingly similar to the one shown in this article as handed down from the literature occurs in describing the force moment supporting a rotating body like a gyroscope while it precesses but does not fall to the ground. Can it be that early aerodynamics work actually conceived of the Lift Force as something coming out of the Coriolis acceleration of air depicted as rotating as described with a Circulation Parameter around the airfoil profile? If this is so and I hope it is not in support of what is in the books, then it is simply a non explanation of the Lift Force, an attempt to make it sound like the gyroscopic effect. Everyone is familiar with a hand held bicycle wheel that can be made to spin and then supported with a finger on one end of its shaft while it precesses and seemingly levitates. If this is so, again, this is not a valid mathematical tool by which Lift may be explained and the popularity with which this continues to be held requires it be clearly and meaningfully overturned. AnthonyChessick 17:22, 1 August 2006 (UTC)

A suggestion here is, in other words, to lay the Circulation Parameter to rest with a comment to this effect in the subject entry. Batchelor, and through him, Prandtl and Kutta and Joukowski, make many suppositions and hypotheses concerning fluid flow, such as use of an arcane velocity potential, three dimensional vector differentials especially those in curvilinear co-ordinate systems, the Kutta Condition, reliance on the inviscid nature of (and thus the near "disappearance of drag" at) high Reynolds Numbers, and many ad hoc parameters in their mathematical descriptions that are not defined in a Glossary. The resultant treatments of Fluid Dynamics appear to most sincere users as a problem of deciphering hieroglyphics and as suitable fodder for only computers with the application there even in doubt. This need not be. In other words, I find the material highly "kinematic", that is, with little attention to providing a guide to engineering realities involving averaged mass flows and momentum rather than spending time on unusual behaviors of fluid flows and unusual ways of presenting descriptions of them. The Lift Force is a prime example of this, not to present an invitation to engage in polemics over this observation. Anthony Chessick 17:50, 7 August 2006 (UTC)

The length element in the loop direction should really be dl rather than ds as ds usually denotes a surface element (as it does in the equation relating to vorticity later in the article). This means the vorticity equation is wrong at the bottom of the page as stokes theorem relates a line integral to a surface integral. Thus, I'm changing it to dl. — Preceding unsigned comment added by ‎ Adwol (talk • contribs) 12:39, 11 April 2009 (UTC)

Path independence

The present article seems to tacitly assume, but not state directly, that the line integral defining circulation is independent of the path chosen. Seems to me that this should be stated explicitly - while those of us who remember vector calculus know that if a field has a curl of zero everywhere it can be expressed as the gradient of a scalar field , i.e. it's conservative and hence the line integral is path independent. Assuming I'm remembering this correctly.

For those unfamiliar with vector calculus it's not at all obvious that you will get the same number for the circulation regardless of how you go around the foil so we should probably make that explicit. Which raises the question of whether we should say that it doesn't apply to general airflows but only to potential flow. Personally, I'm not at all sure that circulation is even defined for non-conservative flow fields. I'll wait to hear replies before making any changes. Mr. Swordfish (talk) 17:52, 3 May 2020 (UTC)

Your memory of these matters is sound! The numerical value of the line integral is equal to the net value of the vorticity inside the closed path. Vorticity is a mathematical model of the effects of the viscous forces which are non-conservative and act only within the boundary layers. Providing the line integral is taken around a closed path that is entirely outside the boundary layers it is path-independent. The article should say so explicitly. I may be able to find a published source.

Another way of looking at it is to say the lift acting on unit span of a 2-dimensional wing is directly proportional to the circulation determined using any closed path around the wing; there is only one value for the lift so every closed path around the 2-dimensional wing must yield the same numerical value for the line integral and we can conclude that the circulation is path independent. The Kutta-Joukowski theorem is relevant.

Bear in mind that the two boundary layers (upper and lower surfaces) spin in opposite directions so an airfoil experiencing zero lift will have zero net circulation because total clockwise vorticity will cancel total total anti-clockwise vorticity. It is only when lift is being generated that clockwise and anti-clockwise vorticities are different, and the difference is equal to the numerical value of the line integral around any closed path enclosing the airfoil and its boundary layers.

My comments about the boundary layers may be inaccurate because the BL is a feature of real fluid flows whereas vorticity and circulation are features of potential flows. I don’t recall ever seeing an authoritative source that equates vorticity and viscous forces so there may be a bit of original research there.

I agree that circulation is only defined for potential flows around 2-dimensional wings. It arose when Lanchester, Kutta and Joukowski were attempting to use theoretical methods to investigate fluid flow fields around wings and improve the design of airfoil sections. The Wright brothers built themselves a wind tunnel and made measurements in real fluid flows but the above-mentioned three were mathematicians and so their discoveries were made in the field of potential flows. Dolphin (t) 23:31, 3 May 2020 (UTC)

I have inserted some clarifying comments - see my diff. I will find a suitably reliable source and insert it as an in-line citation. Dolphin (t) 04:40, 4 May 2020 (UTC)

In Fundamentals of Aerodynamics (1984) by John D. Anderson, section 3.16 includes the following: "Let curve A be any curve in the flow enclosing the airfoil. If the airfoil is producing lift, the velocity field around the airfoil will be such that the line integral of velocity around A will be finite [and will be the circulation Γ to be inserted in Equation 3.140.]"

Later in 3.16 Anderson writes: "The important point here is that, in the Kutta-Joukowski theorem, the value of Γ used in Equation 3.140 must be evaluated around a closed curve that encloses the body; the curve can be otherwise arbitrary, but it must have the body inside it."

I will insert section 3.16 as an in-line citation in the article. Dolphin (t) 13:10, 4 May 2020 (UTC)

Thanks for making the changes. My take is that since circulation is undefined unless the line integral is path-independent, that should be noted in the Definition section rather than waiting for the Kutta–Joukowski theorem. It's unclear from the Anderson quotes whether he is restricting his comments to 2D potential flow or whether he's writing about flow more generally. Seems to me that a concise statement of necessary and sufficient conditions for circulation to be well defined would be appropriate as part of the definition. Mr. Swordfish (talk) 22:09, 4 May 2020 (UTC)

I have made a change in order to move that clarification out of the section on the Kutta-Joukowski theorem and into the definition. See my diff.

Earlier I wrote that “circulation is only defined for potential flows around 2-dimensional wings.” I was forgetting that the circulation around the vortex system consisting of the bound vortex and a pair of trailing vortices is the essence of the horseshoe vortex on a 3-dimensional wing. The simple concept of the horseshoe vortex leads directly to the lifting-line theory and this is a method for analysing the spanwise lift distribution on a 3-dimensional wing. Dolphin (t) 12:41, 5 May 2020 (UTC)

I still don't think the article is sufficiently clear that circulation is only defined in limited cases i.e. a conservative vector field ( one that has the curl == 0 throughout). While it would be an overstatement to say that it's only defined for 2-D potential flow, I think we need to say very clearly that it is undefined for "most" real world flows, and only has meaning in "a few" mathematical models. Not sure what the best language is to use. Mr. Swordfish (talk) 19:17, 22 October 2020 (UTC)

The definition begins “If V is the fluid velocity ...” Would it help if the word “fluid” is omitted? When I read the section Definition and all that precedes it, it seems clear that circulation is a mathematical concept. In subsequent sections it is clear that it has an application in the field of fluid-dynamic lift but may also have applications in other fields -perhaps in the theory of magnetism.

I don’t have any objection to an addition saying it is defined only for conservative vector fields. Those words could be linked to conservative vector field. Dolphin (t) 22:44, 23 October 2020 (UTC)

New title

The article now has a new title "Circulation(Physics)"

This was implemented without discussion. Unfortunately, I don't recall the former title and the Wikipedia UI doesn't allow me to see the former title. I think it was "Circulation(Fluid dynamics)".

Is this an improvement?

I'm not entirely sure. My background is mathematics, although I studied E&M extensively as an undergrad. I agree that from a mathematical perspective circulation can be defined for any closed path in any vector field, and that for any conservative vector field the circulation is independent of path. So, I think the current treatment is defensible from a theoretical, mathematical perspective.

However, I'm not sure it accurately represents the literature. While circulation can be defined for arbitrary vector fields and arbitrary closed paths, in practice I have only seen it used in fluid dynamics; I don't recall it every being used in E&M other than to prove curl(v)=0 -> path independence of Integral(v dot s).

I would invite User:Ponor to provide some references to the application of circulation to other branches of physics. Or anyone else to venture their opinion. Mr. Swordfish (talk) 04:55, 3 November 2020 (UTC)

@Mr swordfish: In my experience, discussions on wikipedia rarely lead to any action. I've seen stub articles that had tons of improvements elaborated on their talk pages and none added to the articles themselves; so I choose to WP:BEBOLD. Yes, the former title was Circulation (fluid dynamics), you can see that in article's revision history. And yes, I can provide some references, if Feynman is not enough. Let me first say that circulation is always tied to its curve, that's why we say "The circulation Γ of a vector field V around a closed curve C...". You can't say that circulation is independent of path because circulation is defined on and for a path (same as flux through a surface, never just flux). Conservative fields, which have circulation equal to zero over any loop, are the least exciting example. Circulation is often used in books on electromagnetism, as two Maxwell's laws, in their integral form, are about circulation of the electric and magnetic fields: a changing magnetic field gives rise to a circulating electric field; electric currents, changing electric fields produce circulating magnetic fields. In addition to Feynman (cited in the article), I find it in Purcell (Electricity and magnetism, Berkeley physics course pp 89-102, about 30 mentions), Rothwell and Cloud (Electromagnetics, ISBN 0-8493-1397-X, about 10 mentions), Greiner (Classical Electrodynamics, mentioned as circulation integral and circulation voltage), here [1], and here [2], and here [3], and here [4], and here [5], here [6]. Here too Maxwell's_equations#Circulation_and_curl, and I will be using it in a few other articles soon (that's the reason why I moved the title). I think it's fair to say circulation is used and useful elsewhere in physics, not only in fluid dynamics. Ponor (talk) 07:45, 3 November 2020 (UTC)

I accept that the concept of circulation is applied more widely than fluid dynamics, but my view is that this article is overwhelmingly applied to fluid dynamics. For example, in the lead it says "Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolai Zhukovsky." Michael Faraday died around the time Lanchester and Kutta were born, and yet our article says circulation was first used by these three. The article has a lot to say about circulation in fluid dynamics and only a little to say about circulation in electromagnetic induction. The article has been added to only one category - fluid dynamics.

If the article is to become the one covering the concept of circulation in physics (and mathematics?) there is a lot of work to be done to add an adequate cover of aspects relevant to physics (and math?) An alternative that I find more appealing is this article remains dedicated to the concept in fluid dynamics, and a new article with a new title, perhaps Circulation (electrical) is created. Whether these two articles should subsequently be merged would be something for the Wiki community to decide when the new article is available for examination.

At present it looks a bit like an article that occupies an important place in the coverage of fluid dynamics has been re-named by a User who plans to expand it for other (legitimate) purposes at some time in the future. Dolphin (t) 11:57, 3 November 2020 (UTC)

We could have at least three articles about circulation (of v, E, and B), but they'd all be saying the same, that circulation is the line integral of a vector field - just one way to characterize vector fields. My idea is to have one, which will show little examples from different areas and link to full articles, like this one Kutta–Joukowski theorem, or Maxwell's_equations#Circulation_and_curl. The sentence from the lede ("first used by") is unsourced, and I'm not sure it should be here to begin with; I'd say it's likely that the term was first used in fluid dynamics, Feynman for example draws that analogy (as we do with all other abstract terms). How important is circulation in fluid dynamics? I don't see any mention of it in Fluid dynamics. Even Kutta–Joukowski theorem is only in its "See also" section. (Mathematics? I don't think they care. Ideally, this article would be just Circulation, but we can't have that.) Ponor (talk) 15:23, 3 November 2020 (UTC)

Thanks for your prompt reply. Traditionally, the lift generated by airfoils and wings could be analysed by only two methods - empirically; and theoretically using the Kutta-Joukowski theorem and lifting-line theory. Circulation is at the heart of these two theoretical methods.

I agree that Kutta-Joukowski theorem is only in the See also section of the article Circulation (physics) but that doesn’t surprise me - the K-J theorem is the application of circulation; circulation is not dependent on K-J theorem. Conversely, the lead in the article about the K-J theorem makes multiple mentions of circulation.

Is it your intention to immediately expand this article to give it a much broader focus on other applications within the field of physics, or mainly to link to this article in other articles within the category of physics? (Your earlier post mentions the latter but not the former.) Dolphin (t) 21:32, 3 November 2020 (UTC)

Dolphin51It seemed weird to link to Circulation (fluid dynamics) if the integral was used in other fields. After reading your old discussions here, I decided to do what I did. Can't promise anything, but I may come back and add more stuff. Physics is in Kutta-Joukowski theorem, and circulation is just a tool. So whatever applies to their model and not to circulation should better go there. Ponor (talk) 20:36, 4 November 2020 (UTC)

After reading the above, I now agree that expanding the article to include a more general treatment of circulation is an improvement. I don't think separate articles are called for; the present separate sections under Uses seems sufficient. BTW, it's been a long time since I took E&M and I don't remember coming across the concept of circulation then; dusting off my copy of Loraine & Corson I don't see any treatment of circulation, which is probably why I don't remember it. That said, the Laplacian is the Laplacian, regardless of whether it's E&M or potential flow, so clearly circulation could be employed as a concept, and I'll take the Fenyman ref as sufficient evidence.

One thing that I'd like to see clearly stated is that "Circulation" for a vector field only makes sense if either the path is specified or the vector field is conservative. Consider the following passage from Anderson's Introduction to Flight:

The circulation theory of lift is elegant and well developed; it is also beyond the scope of this book. However, some of its flavor is given as follows...

Equation (5.73) is the Kutta–Joukowsky theorem; it is a pivotal relation in the circulation theory of lift. The object of the theory is to (somehow) calculate Γ for a given V∞ and airfoil shape...

A major thrust of ideal incompressible flow theory, many times called potential flow theory, is to calculate Γ

Note that Anderson doesn't specify a path, but merely talks about the value of circulation for a given flow field. He mentions potential flow but elides over the fact that the flow has to meet certain requirements for this value to be independent of the path.

Since this is basically the standard into text for aeronautical engineering, my impression is that many readers are left with the impression that "circulation" is a value that one can compute for arbitrary flow fields and that the "circulation theory of lift" is applicable in general. It's not. You either need to be working in a specific model, or the path has to be specified for that number to make sense. I don't think the present article is sufficiently clear on that. I recently inserted language to that effect (since removed):

This number is well-defined for any conservative vector field since it evaluates to the same value regardless of the path taken; it is not well-defined for arbitrary flow fields since number may be path dependent. In potential flow with a region of vorticity, all closed curves that enclose the vorticity have the same numerical value for circulation. The circulation around a region of vorticity is the same for all closed curves that enclose the vorticity. (Anderson, John D. (1984), Fundamentals of Aerodynamics, section 3.16. McGraw-Hill. ISBN 0-07-001656-9)

I'd like to see something along those lines included. Thanks. Mr. Swordfish (talk) 18:44, 4 November 2020 (UTC)

Mr. Swordfish, I understand your concern, but this is not a failure of how circulation is defined and calculated but of how it's used; the model that uses it may be wrong, limited or inappropriate, not the values you get by using different paths. (Anderson: "...but elides over the fact that the flow (!) has to meet certain requirements for this value to be independent of the path") The statement I removed was false. Circulation is defined for all fields and is always defined for a specific path. In conservative fields it evaluates to exactly zero, and as you can see from EM laws, it is well defined and nonzero whenever there are currents present, or changing electric and magnetic fields (thus not only in static/conservative fields). If your model or theory need circulation to evaluate to the same number regardless of path, then your theory is only applicable to such cases. "In potential flow of a fluid with a region of vorticity, all closed curves that enclose the vorticity have the same value for circulation" is true and it's still in the article; as long as vorticity region is inside your loop, all of it gets accounted for by Stokes' theorem. Anything that applies to fluid dynamics models should go into respective articles; I know nothing about that, but would be happy to read. Best, Ponor (talk) 20:36, 4 November 2020 (UTC)

Thanks for your prompt and thoughtful reply.

You say "If your model or theory need circulation to evaluate to the same number regardless of path, then your theory is only applicable to such cases." This is exactly what I would like to see included in the article. The "circulation theory of lift" only makes sense if the circulation is path independent. That is, the Kutta–Joukowski theorem states that the lift is directly proportional to the circulation. If the circulation depends on the path, then it begs the question of "which circulation?".

Now, perhaps this should be included in the Kutta–Joukowski theorem article, and that article does sort of say it in a roundabout way, but I think we would do a service to our readers by including it here. (Note that the K-J article refers here for the details.) We appear to disagree about that, since you have stated "Anything that applies to fluid dynamics models should go into respective articles." The reason I disagree is that in non-technical (and sometimes somewhat technical) discussions of aerodynamic lift, the "theory of circulation" is often cited (see, for instance, the Anderson quote above.) It's one of those buzz-words you encounter a lot. While I don't have anything to back this up, my hunch is that a large portion and perhaps the majority of the traffic to this page is people curious about the "circulation theory of lift". One of the things I try to keep in mind when editing is to think first about the intended or likely audience. Mr. Swordfish (talk) 16:55, 5 November 2020 (UTC)

Hey Mr. Swordfish. The best thing you can do is to find a source that says exactly what you want to put in, that "circulation is only meaningful when it evaluates to the same value regardless of the path" (You won't find it, because the statement is not true. Take for example your "fluid with a region of vorticity": if you integrate through the region of vorticity, you get only a fraction of the value of circulation that you'd get when the whole region is encircled, by Stokes' theorem. So in the case of a vector field with vorticity in some little region, you can choose a path that gives a different value for circulation, which, according to you, makes circulation ill-defined. But why would you calculate circulation "over all closed curves that enclose the vorticity" if circulation itself in this field is not well defined? Why would you kill the messenger (circulation) instead of killing... idk... theory or the velocity field?)

Your "circulation theory of lift" can't be $\Gamma =\oint _{C}\mathbf {V} \cdot \mathrm {d} \mathbf {l}$ (that's what this article is about). Your theory must have some assumptions about the velocity field, shape of the wing, temperature, density etc. (pardon my ignorance). If "circulation theory of lift" is Kutta–Joukowski theorem, then that's where these issues should be discussed. If there is some other "circulation theory of lift", there should be a Circulation theory of lift. There must be something in Kutta–Joukowski theorem that tells you which integration path to take. Is it around the wing, is it diagonally, is it lengthwise, is it 3 ft below, 10 ft above the wing? You won't get any lift if you demand your velocity field to be conservative, because there Γ=zero=lift! Ponor (talk) 20:17, 5 November 2020 (UTC)

After some reading, I realized the airfoil in the theorem is a two-dimensional object, the cross-sectional shape of a wing. That, to me, defines the curve over which the circulation is calculated, and it kind of makes sense, because the circulation would be a measure of the average fluid velocity around the airfoil, taken negative on one side, positive on the other. To get the total lift on the wing you sum over many cross sections, airfoils of possibly different size and shape. A few words about this in the Kutta–Joukowski theorem article or a nice picture would definitely help. Ponor (talk) 23:00, 5 November 2020 (UTC)

Just to be clear, it is not my circulation theory. And I'm not really sure what, exactly, the "circulation theory of lift" entails. That's a phrase used by Anderson and many others when discussing aerodynamic lift. I think it means 2-D potential flow and the K-J theorem, but perhaps it may encompass other theories as well. I do think that many people searching for it will end up here.

To answer your question ("There must be something in Kutta–Joukowski theorem that tells you which integration path to take. Is it around the wing, is it diagonally, is it lengthwise, is it 3 ft below, 10 ft above the wing?") 2-D potential flow can be modeled as a circulatory component plus a linear component, with the center of circulation inside the foil. Since the Laplacian is linear, the sum of these components also solves it. And since that center of the circulation is inside the foil, any path around the foil that encloses the one vortex will have the same non-zero value for circulation. That is, K-J merely requires that the path enclose the foil. This is all fairly standard stuff, and you probably know this already, but you asked, so I answered.

I think our basic point of disagreement (to the extent it is a disagreement) is that the definition of "Circulation" to which you are familiar requires that the path be specified and that (at the risk or reading too much into what you've said) it makes no sense to speak of the "circulation of the vector field" without specifying a path, except in the trivial case. I don't disagree with this; although that definition of circulation is new to me it makes perfect mathematical sense. I won't dispute that it's the standard definition and belongs in the article.

What I've been trying to say is that the notion of a vector field having a circulation can have meaning, but only if the calculation of circulation is (mostly) independent of path. My recollection is that presentations of K-J or 2-D potential flow talk about the circulation of the flow field, and state that the lift (per unit span) is directly proportional to this value - this makes sense if there is exactly one vortex and the path encloses that vortex, but for more general flow fields there is no one value for the circulation of the flow field. So the relationship between circulation and lift is in general not so simple since. Not sure what the best language is to convey this, but the impression I get is that there are a lot of folks who think that circulation is a property of the flow field in general.

I'll look into adding something to either the K-J article or the subsection on fluid dynamics on this page. Mr. Swordfish (talk) 17:56, 6 November 2020 (UTC)

To cite specifics, presently the article says:

The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.

Which is correct, assuming that we're talking about 2-D potential flow. But the implication (to me anyway) is that it is true in general. I think we need to add some language indicating the necessary and/or sufficient conditions for this assertion to be true. Mr. Swordfish (talk) 22:41, 6 November 2020 (UTC)

Mr swordfish: The quotation is true for 2-D flow, sometimes visualised as a uniform wing of infinite span. It isn’t true for a 3-D wing because the circulation around the mid-span progressively reduces as the bound vortex weakens and vortex lines turn through 90 degrees and trail behind the wing, forming the trailing vortices. There are some good diagrams at Horseshoe vortex. Remember that Helmholtz’s first theorem says that the strength of a vortex filament remains constant along its length so the spanwise increase in strength of the trailing vortices can only occur if there is a matching spanwise reduction in the bound vortex. Dolphin (t) 09:19, 22 February 2021 (UTC)