Talk:Navier–Stokes equations/Archive 2

Archive 1

Archive 2

Give the formula first

Who is writing this article? Give the formula in the very first section and define the variables in it before going for the derivation and other bs. Someone looking for the formula can never find from this article which one is the Navier–Stokes equation. Damn it. 122.176.58.109 (talk) 14:34, 20 September 2011 (UTC)

There are more than one equations that can be used 98.114.196.197 (talk) 17:10, 11 March 2012 (UTC)

Three-dimensional example

The three dimensional exact solution in this article might not be an incompressible flow. The divergence of the velocity vector is non-zero. — Preceding unsigned comment added by 68taileddragon (talk • contribs) 21:54, 1 April 2012 (UTC)

sans serif for tensors instead of blackboard bold?

There has been recent revival of interest in using sans serif here, I replaced \mathbb with \mathsf for the various tensors in this article since this seems to be what the literature uses (sans serif and not bold)... Let’s see the reaction... Maschen (talk) 10:34, 7 May 2012 (UTC)

Navier–Stokes and Helmholtz decomposition comparison

Dear Participants of discussion!

Here http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Pressure-free_velocity_formulation we can read: “This result follows from the Helmholtz Theorem (also known as the fundamental theorem of vector calculus)”. However well-known http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive/2012/Mar#Helmholtz_decomposition_is_wrong that Helmholtz decomposition is total wrong. Fundamental theorem of vector calculus can be written as follows:

${\vec {F}}=\operatorname {grad} \varphi +\operatorname {rot} \operatorname {rot} {\vec {A}}$. (2*)

This formula completely corresponds to transformed Navier–Stokes equations(NSE) for incompressible fluids ($\operatorname {div} {\dot {\vec {u}}}=0$)

$\rho {\vec {F}}-\operatorname {grad} p+\mu \nabla ^{2}{\dot {\vec {u}}}=\rho {\ddot {\vec {u}}}\Rightarrow \rho ({\vec {F}}-{\ddot {\vec {u}}})=\operatorname {grad} p+\operatorname {rotrot} \mu {\dot {\vec {u}}}.$ (3*)

Here,

${\vec {F}}={\vec {F}}_{1}+{\vec {F}}_{2}+\cdots$ vectors sum of a given, externally applied forces (e.g. gravity ${\vec {F}}_{1}$ , magnetic ${\vec {F}}_{2}$ and other), $p$- pressure (scalar function), ${\dot {\vec {u}}}$- velocity vector, ${\ddot {\vec {u}}}=d{\dot {\vec {u}}}/dt$ - acceleration vector, $\rho$ - density (const), $\mu$ - viscosity (const), $\nabla ^{2}$ - Laplace operator.

Equations (3) and (2) are consistent. Hence there is no reason to say that the theory requiring (2) must be false. As we can see from NSE the sum - $\operatorname {grad} p+\mu \nabla ^{2}{\dot {\vec {u}}}=-(\operatorname {grad} p+\operatorname {rotrot} \mu {\dot {\vec {u}}})$ forms the vector field.

Note that we will receive the formula (2) also after similar transformation of the Navier–Stokes for a compressible fluid and after transformation of the Lame equations for an elastic media.

Therefore let's try to formulate the text for editing of this article.

--Alexandr (talk) 11:28, 17 May 2012 (UTC)

I don't understand Helmholtz decomposition (yet), but know that it exists and people use it so it must work. Its one thing to suspect a physical law is wrong, but mathematics itself is never wrong as its founded on logic and axioms (the only "mathematical mistakes" which can ever occur are due to an individual making those mistakes in some analysis). You have raised this at least twice before: in that above link and here, and each time you have been told Helmholtz decomposition is correct while simaltaneously your calculations are in error (although rather than considering yourself as wrong you consider something else to be wrong). I suggest you stop disrupting talk pages with it, and trying to introduce OR. F = q(E+v×B) ⇄ ∑_ic_i 19:39, 19 May 2012 (UTC)

These OR are comparison of two Wikipedia articles (http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations and http://en.wikipedia.org/wiki/Helmholtz_decomposition ). In Wikipedia all contradictions must be deleted. Otherwise it is Wikipedia disrupting. --Alexandr (talk) 12:06, 22 May 2012 (UTC)

Notation: Tensor vs. component-summation?

I find opaque the tensor notation used throughout the "Derivation and Description" section of the article. It seems like Einstein summation notation would be clearer. For example, the viscous term of $\nabla \cdot {\boldsymbol {\mathsf {T}}}$ is one of the following: $\partial _{i}(\mu _{ij}\partial _{j}v_{k})$, $\partial _{i}(\mu _{jk}\partial _{i}v_{k})$, or $\partial _{i}(\mu _{jk}\partial _{j}v_{i})$. All three give different behavior, but it is not immediately obvious which form is intended. Comparison with the section on "incompressible flow of a Newtonian fluid" makes me think the second is what's intended, but it would be nice to have this made explicit.SMesser (talk) 20:47, 1 June 2012 (UTC)

Reference for the three dimentional example for NS equation solution?

I am curious about how the solution of this three dimentinal NS equation is from. There is reference for the two dimentinal example, but the reference is lacking for this 3D example. Is there any reference or citation? Thank you very much for your kind help. — Preceding unsigned comment added by Wanchung Hu (talk • contribs) 10:11, 27 August 2012 (UTC)

Absolute value of pressure matters for compressible flow

In Section 3.2, Stresses, the following is stated: "Interestingly, only the gradient of pressure matters, not the pressure itself". Maybe it should be stated that only the gradient of pressure affects the stress tensor. Currently, the statement above might lead into an impression that fluid properties are independent of absolute value of the pressure. For compressible flow, an equation of state is needed to describe the fluid. For example, a commonly used equation of state for gases is the ideal gas law, where the absolute value of pressure does matter. Correct me if I am wrong. Papow (talk) 13:02, 10 November 2012 (UTC)

I think you are right. Note that the end of the section on stresses mentions the equation of state. The absolute value of pressure also affects the viscous stresses, through its influence on viscosity. -- Crowsnest (talk) 17:04, 10 November 2012 (UTC)

Compressible flow of Newtonian fluids

I think that the 1/3 term in front of the grad div v term is erroneous. The term should be $\left(\mu +\mu ^{v}\right)\nabla (\nabla \cdot \mathbf {v} )$. I think the error stems from the commonly used Stokes assumption $\mu ^{v}=-{\frac {2}{3}}\mu$. 195.241.117.199 (talk) 16:34, 18 December 2013 (UTC)

I have removed the term $\mu ^{v}$ by invoking the Stokes assumption. Salih (talk) 17:22, 18 December 2013 (UTC)

It is seems this problem covered very well here: http://www.docstoc.com/docs/80173363/Navier-paradox 178.137.134.138 (talk) 13:55, 19 December 2013 (UTC)

The revised form of the equation and the first equation in the article you've pointed out are same. Salih (talk)

Compare the revised form of the equation and equations (1), (6) in the article http://www.docstoc.com/docs/80173363/Navier-paradox  !!!! Equation (6) is obtained after exact transformation of equation (1). 178.137.134.138 (talk) 19:44, 19 December 2013 (UTC)

History

Would be nice to have a history section. This one gives some idea, http://liwei-chen.blogspot.ca/2007/09/brief-history-of-navier-stokes-equation.html (but is short) Stephanwehner (talk) 22:11, 11 January 2014 (UTC)

Most difficult mathematical tasks of the millennium

The article says this problem has not been solved. But it has:

http://www.inform.kz/eng/article/2619922Psyadam (talk) 19:22, 12 January 2014 (UTC)

The paper was released a few days ago, with 20 self-citations, written by a relatively unknown mathematician, published in an obscure journal of which he is an editor..... I would wait before claiming the problem is solved.2001:638:902:2001:214:BFF:FE81:48CE (talk) 11:30, 16 January 2014 (UTC)

Derivation and description

Main article: Derivation of the Navier–Stokes equations

The derivation of the Navier–Stokes equations begins with an application of Newton's second law: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid.

THIS SHOULD BE FIXED! I GET WHAT THEY'RE TRYING TO SAY BUT THE WAY IT'S WORDED MAKES THE AUTHORS SOUND PRETTY DUMB.

Please let us know what is your proposal to rewrite. Salih (talk) 05:18, 15 October 2013 (UTC)

A very common form of the incompressible equations, the non-dimentionalized form with only the Reynolds number as parameter, ${\frac {\partial v}{\partial t}}+v\cdot \nabla v=-\nabla p+{\frac {1}{Re}}\Delta v$, does not appear in the article. It should, at least for the theoretical reasons that solutions to NS are characterized only on one parameter. Gummif (talk) 01:31, 5 April 2014 (UTC)

Where is THE equation(s)?

I was just looking for a quick info on what is, or are, Navier–Stokes equation(s). After glancing the article I still have no idea. Maybe I am reading too fast (I am...), but anyway, it looks confusing. To the point:

• The first equation we get to see, and the only one in the lede, is explicitly NOT a Navier–Stokes equation, quote: «Navier–Stokes equations [...] cannot be put into the quasilinear homogeneous form: $\mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\mathbf {y} _{x}=0$  »; not much useful starting to show it is not, right? It is not a whole lot of other things...
• Then we get a long(ish) derivation, with many variously colourful boxes: «Cauchy momentum equation», «Linear stress constitutive equation», ... «Navier-Stokes momentum equation», aha! probably this is one, but by now, who knows?

The derivation may be fine, but it needed some introduction. It is kind of if the aeroplane article said something like "An aeroplane is something that flies. Here are instructions on building one: ..." before we knew what it looks like at all. Hopefully someone may improve the lede? Thank you! - Nabla (talk) 19:04, 7 February 2015 (UTC)

The incompressible equations appear in the relevant section of the article, but that is fairly far down the page. What's missing here is not just a statement in the lead (which would be ideal), but a clear first section stating the equations and describing various forms of them. The derivation is of comparatively less importance, and I would like to see that whole section moved further down. At the moment, the article is practically unreadable. Sławomir Biały (talk) 19:22, 8 February 2015 (UTC)

Yes, it needs just that. - Nabla (talk) 10:28, 14 February 2015 (UTC)

Relativistic extension ？

No words talk about relativistic hydrodynamics 111.121.12.141 (talk) 14:36, 11 April 2015 (UTC)

General pressure-velocity formulation derivation

There are still some errors and misconceptions in this section in my opinion.

Maybe the problem is also it lacks of some concepts :

• differences between "total pressure" and "equilibrium pressure".
• correlation and differences between first lamé parameter and bulk viscosity.

I wuold write something like:

The Cauchy stress tensor can be decomposed into its isotropic and deviatoric parts as follow:

${\boldsymbol {\sigma }}=-p{\boldsymbol {I}}+{\boldsymbol {\tau }}$

Where the pressure $p$ is defined as minus the mean normal stress and is an osservable quantity. This quantity can be thought as the sum of the equilibrium pressure $p_{e}$ that is equal to total pressure $p$ in absence of any relative motion of the fluid and an argument that depend on the fluid velocity field gradient, it can be written as:

$p\equiv -{\tfrac {1}{3}}tr({\boldsymbol {\sigma }})=p_{e}-\zeta \nabla \cdot \mathbf {u}$

where

$\zeta$ is the bulk viscosity

The bulk viscosity is related to the first lamé parameter $\lambda$ (another coefficient commonly used to describe the stress tensor):

$\zeta =\lambda +{\tfrac {2}{3}}\mu$

By expanding also the material derivative the Cauchy momentum equation becomes explicitly:

$\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {g}$

Assuming I said true things, there are any ideas to explain this in a better way?

These misconceptions lead to inaccuracies in the following steps so the whole section should be fixed.

For example:

$\nabla \cdot (2\mu {\boldsymbol {\varepsilon }})=\mu \nabla \cdot (2\varepsilon )=\mu \nabla \cdot ((\nabla \mathbf {u} )+(\nabla \mathbf {u} )^{\text{T}})=\mu \nabla ^{2}\mathbf {u}$

Seems algebraically wrong to me, It should be something like:

$\nabla \cdot (2\mu {\boldsymbol {\varepsilon }})=\mu \nabla \cdot (2\varepsilon )=\mu \nabla \cdot ((\nabla \mathbf {u} )+(\nabla \mathbf {u} )^{\text{T}})=\mu (\nabla ^{2}\mathbf {u} +\nabla (\nabla \cdot \mathbf {u} ))$

--Tom.sartor (talk) 17:54, 25 January 2015 (UTC)

I also found the sam mistake. I just added minor correction: in the last red bordered equation I have changed $-2/3\mu$ to 1/3 with a note about a divergence of tensor $\nabla \mathbf {u}$.

I also don't like the whole structure of the artile. The most general case should come first. Now we have the derivation for incompressible flow as first, :followed with more general where at the end special case of incompressible flow is mentioned. — Preceding unsigned comment added by En odveč (talk • contribs) 17:15, :15 February 2015 (UTC)

Edit: I have deleted my changes. Why? Because compressible flow comes later, so $\nabla \cdot \mathbf {u}$ is zero and therefore divergence of tensor $\nabla \mathbf {u}$ is $\nabla ^{2}\mathbf {u}$ together with deviatoric part of $p$. But obviously article isn't clearly written.

Yes I'm agree with you, and yes I notice that now that formula is correct but at time i wrote this comment the formula was used in the general expression of N-S eq. , anyway I think that also in the Incompressible version should be pointed out that $\nabla \cdot \mathbf {u} =0$ because of Incompressible hypothesis.

Something like:

$\mu \nabla \cdot (2\varepsilon )=\mu \nabla \cdot ((\nabla \mathbf {u} )+(\nabla \mathbf {u} )^{\text{T}})=\mu (\nabla ^{2}\mathbf {u} +\nabla (\nabla \cdot \mathbf {u} ))=\mu \nabla ^{2}\mathbf {u} \quad \quad {\text{Where}}\,\,\nabla \cdot \mathbf {u} =0\,\,{\text{for incompressible hypothesis.}}$

and also:

$\nabla \cdot (2\mu {\boldsymbol {\varepsilon }})=\mu \nabla \cdot (2\varepsilon )\quad \quad {\text{For constant}}\,\mu \,{\text{hypothesis.}}$

But maybe this is too verbose for someone.

--Tom.sartor (talk) 18:45, 15 February 2015 (UTC)

Hey, somebody exchanged my assumptions. I did really not mean assuptions. — Preceding unsigned comment added by 179.67.157.6 (talk) 22:08, 13 September 2015 (UTC)

References

recent carnage

@Sławomir Biały and Ancheta Wis: hhahahaha can't stop laughing at the "physics" (deliberate quotes) guy trying to insert equations from his own work on a mathematical analysis page hahhahaha.

this is probably off topic so i apologise for the ping, but i thought i'd share because i saw the edits last night and i was like "... mmm, nah i'll let someone else do it" hhahahha

• Comment. Wikipedia requires independent secondary sources. It has been asserted in an edit summary that this abstract is a secondary source. That is wrong. It is a primary source written by the originator of the concept. (We will leave aside the question of whether it is in fact a reliable source, although mostly special sessions like these lack meaningful peer review required for reliability in this setting.) A secondary source would be a secondary review article or textbook, written by someone else, that substantiates the result, and evaluates its relative importance in the subject. Generally to include results like this in an encyclopedia article, they should be fairly standard things that can be found in textbooks. Otherwise, putting them in the main article is undue weight, which is one of our editorial policies. Also, I suggest that the editor restoring this content familiarize himself with the editing guideline WP:COI, and I encourage him to disclose any conflicts of interest he may have in adding this material (such as being the author of the linked abstract). Sławomir Biały (talk) 13:51, 22 June 2016 (UTC)

I should add that people do study fluids from a relativistic point of view. Relativistic hydrodynamics is a well-studies subject. Entire monographs have been written on the subject, and I can find no reference to Estakhr's work. So even in the context of relativistic hydrodynamics, this appears to be undue weight/fringe. Sławomir Biały (talk) 15:45, 22 June 2016 (UTC)

of course since this is an encyclopedia, i must limit my conjecture as to how such an article was published in a purportedly prestigious physics journal, but i think it's for the initiated scholars/readers in this field to put it together.

the "sexiness" (appeal) of the topic (navier-stokes), combined with the claim of having a "correction" in an area (physics) where such a topic holds great promise, does raise some eyebrows. oh well. such is the publishing industry today. — Preceding unsigned comment added by 174.3.155.181 (talk) 17:33, 22 June 2016 (UTC)

It is now being claimed that this abstract is independent of Estakhr. Given that Estakhr is listed as an author, this is difficult to believe. But an abstract in a conference proceedings, with no GS cites, in a high citation field, is clearly WP:UNDUE weight, even assuming it was not WP:OR. I would suggest the IP to please stop wasting everyone's time with this shameless and dishonorable self promotion. We aren't going to cite this paper, period. Sławomir Biały (talk) 13:31, 23 June 2016 (UTC)

hahahahha he's still going eh? that's so funny.

i support your proposed course of action @Sławomir Biały:. 174.3.155.181 (talk) 19:16, 23 June 2016 (UTC)

Incompressible Flow

This section of the article contains the statement:

• the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $\mathbf {V}$ is an isotropic tensor[.]

It appears that $\mathbf {V}$ is a mistake, because it doesn't appear anywhere else nearby. Was ${\boldsymbol {\tau }}$ what was meant? BlackGriffen (talk) 23:41, 1 March 2016 (UTC)

I went ahead and changed this. 2001:480:20:114:3615:9EFF:FE26:D05C (talk) 16:57, 3 August 2017 (UTC)

Also why does the first blue box say "(expression used for incompressible elastic solids)" but the second blue box says "(expression used for incompressible viscous fluids)" when the only change was that an expression was plugged in (unless I am mistaken)? Is it really the case that one is for elastic solids and the other for viscous fluids, or are both equations for one of them (or both of them)?

Also the article states "Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure", but in incompressible flows, density is constant, so this seems somewhat misleading in that μ depending on density does not make μ non-constant. (I may be missing something though.) 2001:480:20:114:3615:9EFF:FE26:D05C (talk) 16:58, 3 August 2017 (UTC)

"Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass."

That quote is from the second paragraph of the "Use in Games Sections".

Which source is this from? Does anybody have any examples of such improvements? I'm asking not only because it would be a good addition to the article, and because I'm interested in the field myself. — Preceding unsigned comment added by Skylord a52 (talk • contribs) 17:36, 6 March 2018 (UTC)

Clarity

Does anyone else think the mention of the quasilinear homogeneous form is a bit confusing? Sure we can leave in the statement, but I feel that placing the actual form is a confusing and not something to put in the general overview section (especially when it's an abnegation). Perhaps we should add a quick statement of the actual equations and then put a footnote hyperlink to the quasilinear homogeneous form. — Preceding unsigned comment added by Gharveymn (talk • contribs) 22:19, 22 May 2017 (UTC)

I went ahead and added the non-inertial version of the equation at the top of the page to mitigate exactly that issue. I think it is misleading to have an exclusion case as the first equation on an article when the article is specifically referring to an equation. — Preceding unsigned comment added by DavidJonBloom (talk • contribs) 18:32, 17 April 2018 (UTC)

Removing so-called "tensor notation"

Confusingly, this article currently contains two boxes with the same title "Incompressible Navier–Stokes equations (convective form)". The second one is called "tensor notation" by a short text before the box, but that is confusing, too. There is no one such thing as "tensor notation"; instead, the notation applied apparently is what we call Einstein notation here. Since Einstein notation ins used nowhere else in this article, and since it is a straightforward formal conversion from the first form, it only distracts from the topic of the article. Furthermore, its location between the laminar flow example and the explanation of the terms using nabla is particularly disruptive. I will therefore simply remove it. ◄ Sebastian 08:20, 30 April 2018 (UTC)

Problem with weak form

The weak form as written will not satisfy the boundary conditions as written. The given weak form will satisfy the Neumann boundary condition:

$\mu \nabla \mathbf {u} \cdot \mathbf {n} -p\mathbf {n} =\mathbf {h}$

and not the desired boundary condition, which is

$\mu (\nabla \mathbf {u} +\nabla \mathbf {u} ^{T})\cdot \mathbf {n} -p\mathbf {n} =\mathbf {h} .$

While it is true that:

$\nabla \cdot \mathbf {u} =0$

implies

$\nabla \cdot (\nabla \mathbf {u} )^{T}=\nabla (\nabla \cdot \mathbf {u} )=0$

inside the domain, this does not imply that

$(\nabla \mathbf {u} )^{T}=0$

on the Neumann boundary, as the given manipuation

$\sigma ({\boldsymbol {u}},p){\boldsymbol {\hat {n}}}=(-p{\boldsymbol {I}}+2\mu {\boldsymbol {\varepsilon }}({\boldsymbol {u}})){\boldsymbol {\hat {n}}}=-p{\boldsymbol {\hat {n}}}+\mu {\frac {\partial {\boldsymbol {u}}}{\partial {\boldsymbol {\hat {n}}}}}.$

would seem to suggest. In fact, this quantity is nonzero in general. I am not sure how we should remedy this. Delving into the details is messy, but the bottom line is that, although strong form of the Laplace and stress tensor formulations are equivalent, the same is not true of their respective weak forms (unless it is a fully Dirichlet problem). I suggest that we leave the equations in the stress tensor form when integrating by parts, rather than the Laplace form, or alternatively, present a fully Dirichlet problem from the beginning. — Preceding unsigned comment added by 87.15.139.47 (talk) 12:20, 19 June 2020 (UTC)

Not conservation equations?

The current version of the article states that the Navier-Stokes equations are not conservation equations.

The main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations also factor in the Froude limit (no external field) and are not conservation equations, but rather a dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form:

This seems to be at odds with every reliable source I've read. This statement was added back in January 2015 https://en.wikipedia.org/w/index.php?title=Navier%E2%80%93Stokes_equations&diff=next&oldid=643034185 and does not appear to have generated much notice or controversy.

I suppose that this distinction may be supportable in some narrow technical sense, but I don't see what reliable source supports it, and my take is that it's sufficiently at odds with most reliable sources that it should be treated more cautiously and probably doesn't belong in the opening section. Comments? Mr. Swordfish (talk) 16:22, 26 July 2020 (UTC)

Seeing no objections, no support for the material in question on previous Talk page threads, and no citation from a reliable source, I have recast the intro to include the three conservation laws ( momentum, mass, and energy) and hopefully to make it more readable to those who don't already know the material. Mr. Swordfish (talk) 15:28, 30 July 2020 (UTC)

It’s not fair to say Euler is “simpler.” Euler in fact has much more geometric and algebraic structure, and allows for solutions with much more degenerate behavior. The other important difference is that Navier-Stokes are parabolic and therefore smooth out their solutions very nicely.

Plea to use $P$ for pressure not $p$.

Reasons to use or avoid upper case:

1. Many of the pages about pressure use $P$. Example: https://en.wikipedia.org/wiki/Ideal_gas_law There's a disclaimer that different fields use lower case $p$ and others use upper case $P$. I think MOST audiences are like college students. Most of them will be using $P$. In some grad school classes they use $p$, but it cannot be stressed enough by me that those audiences are smaller.

2. Per https://en.wikipedia.org/wiki/Intensive_and_extensive_properties , there intensive things tend to be lower case. This would be why to stick with lower case for pressure.

3. Momentum has the same letter and is lower case. Momentum and pressure having the same variable is confusing.

2/3 reasons say raise case. Please, let's vote and pick the case most people are really using in real situations for the pressure. If I lose the vote so be it, then let's change all the uppercases to lower cases. To be using both cases is blatant tyranny. Jason Arthur Taylor Jasontaylor7 (talk) 20:28, 29 March 2021 (UTC)

4. Thousands of college students learn fluid mechanics from either Fox and McDonald's famed Introduction to Fluid Mechanics or Munson et al.'s eminent Fundamentals of Fluid Mechanics, both use standard $p$ notation: see https://www.google.com/books/edition/Fox_and_McDonald_s_Introduction_to_Fluid/v4T3DwAAQBAJ?hl=en&gbpv=1&pg=PA161&printsec=frontcover and https://www.google.com/books/edition/Munson_Young_and_Okiishi_s_Fundamentals/UBqRCgAAQBAJ?hl=en&gbpv=1&pg=PA248&printsec=frontcover. White's Fluid mechanics also uses $p$: https://www.google.com/books/edition/EBOOK_Fluid_Mechanics_SI_units/OMsvEAAAQBAJ?hl=en&gbpv=1&dq=white%20fluid%20mechanics&pg=PA211&printsec=frontcover Where did the idea for $P$ even come from? The standard textbooks in the field do not use it.

why include HOPF Fibations?

somebody is "showing off ", confusing mere mortals 150.143.189.203 (talk) 15:06, 16 April 2021 (UTC)