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In the book "Vectors, Tensors, and the Basic Equations of Fluid Mechanics" by Rutherford Aris I read the following:

If the fluid is such that the torques within it arise only as the moments of direct forces we shall call it nonpolar. A polar fluid is one that is capable of transmitting stress couples and being subjected to body torques, as in polyatomic and certain non-Newtonian fluids.

Can someone help me understand this? In particular, it would be helpful if someone could give me another definition of polar and nonpolar fluids.

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I would say that the use of "polar" in this book is not common to most scientists today, and that under this definition you may safely assume that in almost all circumstances you are dealing with a 'NONpolar' fluid. – Greg P Dec 30 '10 at 3:31
This use of "polar" is not "uncommon", its simply scandalous! Misusing a established wording is a sign of narrow-mindedness of the writer. – Georg May 23 '11 at 8:51

A polar fluid is just a fluid where the constituent molecules have a polarization -- it could be a fluid of molecules that have a magnetic spin moment, or something like H2O where each individual molecule has a nonzero electric dipole -- and at the macroscopic level, as you average over all of the microscopic moments, you get a net polarization for the whole fluid.

I'm no expert on fluid mechanics, but I imagine the polarization somehow couples to the stress tensor in a way that generates torques in whatever equations of motion the author is interested in.

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This is certainly what most people understand by polar vs. nonpolar fluid (i.e. water is polar an oil is non-polar). But it is not clear to me that the author of the book is talking about the same thing. – Greg P Dec 30 '10 at 2:34
In fact, from looking at this book on Google Books, I am fairly certain that the author is not talking about the usual distinction between polar and non-polar fluids familiar, say, to chemists. On page 123 he says that polar fluids have an asymmetric stress tensor. I don't know of an example of this, but I don't think it applies to a classic polar fluid such as water! – Greg P Dec 30 '10 at 3:22
I haven't got to pg.123 yet, but one gets the feeling that these are standard terms from the way the author uses them. I also don't think that water should be considered polar, so this is not the usual distinction between polar and non-polar fluids that chemists use. – becko Dec 30 '10 at 5:47
On the other hand, I think it has to do with the fact that the molecules can store angular momentum internally that doesn't show up macroscopically. But I'm not sure. A confirmation would be helpful. – becko Dec 30 '10 at 5:49
I think the problem with water is that while water molecules are polar, a macroscopic amount of water will not have a spontaneous polarization, since all of the dipoles are randomly oriented. It's possible what the author means is something like a ferromagnetic or ferroelectric fluid, that is spontaneously polarized. – wsc Dec 30 '10 at 6:12

It is about the stress tensor; it is almost always assumed that it is symmetric to satisfy angular momentum conservation. Yet, there are some fluids capable of creating rotation from squeezing (like those spintops with pistons) and thus have some antisymmetric part in their stress tensors.
Aris just calls those fluids polar, what is pretty correct but makes confusion with electromagnetic properties -- I believe that "fluid with non-symmetric stress" or "couple stress" are better keywords.

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Can you elaborate on "those spintops with pistons"? Are they a real thing or a weird model? ;-) – Greg P Dec 30 '10 at 15:18
@GregP After consulting Wikipedia I think I should have written spintops with augers. But this is a real toy. – mbq Dec 30 '10 at 18:01
what page in wikipedia? can you give a link? – becko Dec 30 '10 at 18:10 – mbq Dec 30 '10 at 18:29
I always though that having antisymmetric tensor for a fluid means the fluid have some inner (molecular) angular momentum. That is though the liquid doesn't move it can have an angular momentum. Is this right? BTW having an antisymmetric tensor and linear material equations leads to the notion of the third viscosity. – Yrogirg Aug 15 '12 at 13:24

This is an old Post, but no, water is not a polar fluid. What the author means is that the fluid stress tensor is anisotropic, as was pointed out earlier.

Take for example the classical description of a fluid, under any shear stress the material must continually flow or deform. A polar fluid does not behave this way, it can withstand shear stress. A number of fluids do behave this way, viscoelastic and bingham plastic fluids can withstand shear stress. Cement paste, for example, which is typically a Bingham plastic must first reach a yield stress before it will flow.

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Aris' use of polar is broader than the chemical/electromagnetic usage cited above, but his use may, strictly speaking, include that case as well. The distinction is mainly one of scale. In a polar fluid such as water, the polar nature comes from the molecules themselves, as opposed to, say, long polymer chains or a magnetizable suspended particles. (To cite another, very different example, a suspension of tiny pear-shaped particles, or even micro-organisms, in a gravitational field would also be polar.) If the polar nature is confined to the molecular level, as for water, then the effects at the meso/macro-scopic scales are mainly confined to that substances chemical and physical properties (viscosity, thermal conductivity, melting/boiling point, and the nature of the liquid state itself) and are not apparent as non-symmetric stresses. On the other hand, if the polar nature of the fluid is due to larger particles (or long chain molecules) then there can be non-symmetric stresses and, because the particles can transmit these stresses to the surrounding carrier fluid, the flows of this fluid will exhibit non-Newtonian flow characteristics. So: I don't think Aris had (molecularly) polar fluids in mind, but I do think he had in mind certain suspensions (i.e. ferrofluids and MR fluids) and polymer fluids. To me what is interesting is whether a polar suspension could be used effectively as a "model system" to examine the nature of molecular polar fluids, such as water.

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In Aris' book, non-polar fluids have a symmetric stress tensor ( named T ), and are NOT subject to body torque forces, though they may still have so-called "external" angular momentum ( and "rigid" body rotation ), in the form of a moment of linear momentum, as shown in equation (5.11.4), p.100. Non-polar fluids may or may not be irrotational ( the two terms are NOT synonymous ): meaning their vorticity may or may not be zero. Note: In the derivation of the rate of change of kinetic energy for non-polar fluids in section 6.14: "Dissipation of Energy by Viscous Forces", p.117, the reason the velocity gradient tensor is replaced at one point with the deformation tensor, is because the double dot product ( aka. dyadic product ) of a symmetric tensor ( here: T ) and an anti-symmetric one ( here: Omega, the anti-symmetric part of the velocity gradient tensor ) is always zero ( which does not necessarily mean that Omega was null, which would otherwise be interpreted as a condition of irrotationality, since vorticity vector w = curl( velocity v ) = -2*vec( Omega ) = 2 * angular velocity ). In the book, non-polar fluids refer to Stokesian as well as Newtonian fluids. Note: Newtonian fluids are basically a type of Stokesian fluids in which the eigenvalues of the viscosity tensor are related to those of the deformation tensor via a linear equation rather than via a quadratic, as evidenced p.110.

Polar fluids, on the other hand, have a non-symmetric stress tensor, and ARE subject to body torque, as stated p. 102. They may or may not have rotationality ( vorticity ), depending on the case. Polar fluids are only briefly mentioned in the book. He mentions two kinds in passing: "poly-atomic fluids and certain non-Newtonian fluids", p.102. Note: non-polar fluids can be thought of as a special case of polar fluids, in which the vector of the stress tensor ( Tx = 2 * vec(T) ) is null ( which is another way of saying that the stress tensor of non-polar fluids is symmetric, since its anti-symmetric part would be null ). The "general" equation for the conservation ( aka. "balance" ) of TOTAL ( internal + external ) angular momentum for polar fluids is given in (5.13.10), p.104. Just set Tx = 0 in equation (5.13.10), and you'll get back equation (5.11.4), p.100, which is the version of conservation of -- external only -- angular momentum for non-polar fluids. Aris thus explicitly states, p.123: "For certain class of fluids however ( here called polar fluids ), the stress tensor is NOT symmetric and there may be an internal [ aka. "intrinsic" ] angular momentum [ caused by body torque + stress couple ], as well as an external moment of [ linear ] momentum [ aka. external angular momentum, caused by body force + normal stress ]." Together, these form the "total angular momentum", whose rate of change is given in "general" equation (5.13.6), p.103.

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Welcome to Physics! Note that this site has MathJax enabled, which means you can use Latex-like syntax to add in equations for readability. – Kyle Kanos Mar 19 at 11:06

A water molecule is like a magnet. Bring magnets together they react because of magnetic lines of force. It is said to be "polar". A piece of plastic is not magnetic, putting pieces of plastic together will not have any effect. No magnetic poles ~ "non polar".

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I'm afraid that in the case of molecules, the most common meaning of 'polar' would be to indicate an electric dipole moment. Nothing to do with magnets. And, in the text in question the use of 'polar' is, well, odd... – Jon Custer Mar 1 at 3:24
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. - From Review – Norbert Schuch Mar 1 at 18:10

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