Is any relation between curl and uniform shear available?

• Paulibus
In summary, the conversation discusses the relation between the vector operator curl and rotation in fluids and vector fields, as well as the uniform shear of a solid described by the strain tensor. There is a mention of an alternative description of shear in terms of an integral of curl along a straight line, but the search for this alternative has not been successful. The conversation also touches on the concept of circulation in fluid mechanics and its role in lift force. The participants then delve into the significance of curl in continuum mechanics and its effect on nearby points and displacement vectors. The connection between shear and rotation is explored, and it is suggested that this connection may explain the propensity of sheared fluids to form rotating structures, such as vortex streets and waterspouts.
Paulibus
The relation between the vector operator curl and rotation in fluids and vector fields is treated thoroughly in many texts. And the uniform (pure or simple) shear of a solid is adequately described by the strain tensor. I'd like to put the two together.

My guess is that an alternative description of shear might exist in terms of some kind of integral of curl along a straight line, but I haven't yet found any mention of such an alternative in texts on elasticity, or in others about vector operators. I'd appreciate being pointed to a web accessible treatment, if it exists --- or at least having it explained why my guess is impossibly wrong. Can anybody help?

The curl is referred to as the 'rotation' or 'vorticity' in fluid mechanics.

You should also look up the 'circulation 'which is the line integral taken around a closed path through the vector field of the fluid. This may be non zero even though the curl or vorticity is zero, and accounts for the lift force in aerodymanics for instance.

Thanks, Studiot. But I think that what I'm asking is how, if at all, may shear be modeled as a superposition of infinitesimal circulations with the help of Stokes' theorem. If someone can tell me I could be on the way to re-inventing the wheel, or at least ball bearings!

OK I think I know what you are seeking and I will write out a page or two later today to help.

The key to understanding the significance of curl in continuum mechanics (elasticity or fluids) is that at any point P the curl doesn't affect P at all!. It affects (nearby) points Q and by association the displacement vector PQ. I say nearby because we can assume linear elastic behaviour and ignore effects of higher order products of small quantities.

For a pure shear, the strain tensor is anti-symmetric so has (in three dimensions) three independent components. Those components can be interpreted as the curl vector.

Thanks very much for both kind replies above, Studiot and Halls of Ivy. So the curl vector is in fact the tensor of rank one whose components are those of the second rank tensor that describes pure shear strain? Marvellous. Wonderful. What a short circuit provided for my wooly thinking!

Let me now reinvent something like the wheel, or bigger. Just suppose this identity between descriptions of shear and of rotation is the reason why sheared fluids seem to have a natural propensity to be unstable to the formation of rotating structures. Then:

Fluids cannot sustain static shear stresses. When sheared, fluids seem often to respond by changing phase, as it were from shearing fluid to substructures of rotating fluid. A few examples: in steady wind, vortex streets trail from stretched wires (ever hear wires humming in a light breeze?) or, in trade winds, clouds arranged in vortex streets have been photographed trailing from the island of Guadaloupe. Waterspouts and dust devils are associated with conditions that promote shear. So, on larger scales, are tornadoes and hurricanes.

Here's a stretch extrapolated from this association: our dynamic and gravitating universe is, it is thought, derived from an earlier fluid phase which may also have been susceptible to the instability of fluid shear to rotation. In the presently observed universe this is manifested on many scales by the prevalence of rotating structures condensed as far as possible by gravity while rotating. Together with such structures seen in the sky with the naked eye on a clear night , and the one we live on, there are lots of others: as shown in http://www.bbc.co.uk/news/science-environment-19728375.

Hey, hang on mate, this is the maths forum!

Please note that 'tensor shear strain' is exactly half the 'engineering shear strain'.

Did you understand my comment about the effect of rotation on P and Q?

Yes, I understand the factor between tensor and engineering shear strains, and I did appreciate that rotations have point centres which, qua points, remain unaffected by the curl vector description and rotation --- I hope this was what you were getting at, re. P and Q.

But I shouldn't have to hang on! This is indeed the maths forum, but it's nevertheless embedded in physics forums. And while not hanging on, let me further trespass by adding that the mathematics of Newton's law of gravitation ensures that any rotating and gravitating non-viscous fluid is sheared as it rotates. It does not rotate in quasi-rigid -body fashion. Hence gravity should automatically induce fluid instability to generate ever smaller condensed and rotating sub-structures. Hence the heterogeneous nature of the solar system. A mix of maths and physics acceptable here?

Last edited:
I should be a little careful when talking about natural vortices.

There is a difference between the circular flow pattern of a rotating body of fluid, where the velocity is proportional to the radius and the curl is non zero, and the so called line vortex where the velocity is proportional to the reciprocal of the radius and the curl is zero.

Many naturally occurring phenomena, such as the ones you mention, can be modeled by a so called Rankine vortex, which has a rotational core, up to a certain radius, and then a long diminishing tail of irrotational fluid, going round with the core. The core and tail velocities match at their boundary.

Thanks. The devil is of course in the details, and I agree that one should be careful. I was not aware of the Rankine vortex-model. In the solar system, where most planets describe nearly circular orbits under the gravitational attraction of the sun, Newtons law mandates a reciprocal variation of orbital speed with the square root of the orbit radius, r. In the fluid disc out of which the planets may have condensed it is this feature of gravity that would have tangentially sheared rotating fluid.

Has modelling with the Rankine vortex been applied to Vera Rubin's measurements of galaxy rotation curves?

I apologise for wandering away from the mathematical context of this forum. Returning, can I ask if the vector operator curl is then an axial vector? I've not seen it so labelled.

Last edited:
In post # 5 HallsofIvy wrote: "For a pure shear, the strain tensor is anti-symmetric and so has (in three dimensions) three independent components. Those components can be interpreted as the curl vector." I'm afraid this is exactly wrong. The strain tensor for pure shear is symmetric., as explained most clearly by J.F.Nye in Physical Properties of Crystals; Their Representation by Tensors and Martices, O.U.P.1957, p.93 et seq. Hallsofivy must have been thinking of simple shear, which is a pure shear plus a rotation described by curl. I missed this slip. You were quite right to warn about this source of confusion, studiot! I've opened a thread "How did gravity build objects that rotate" in the Astrophysics forum for anybody who wants to continue with my crazy stuff.

Nevertheless, thanks to HallsofIvy for her/his post. The attention was appreciated.

Last edited:

1. What is curl and uniform shear?

Curl and uniform shear are both mathematical concepts used to describe the behavior of fluid flow. Curl is a vector operation that measures the rotation of a fluid field, while uniform shear refers to the linear stretching or compressing of a fluid field.

2. Is there any relationship between curl and uniform shear?

Yes, there is a relationship between curl and uniform shear. In fact, uniform shear can be thought of as a special case of curl, where the rotation is zero and only the stretching or compressing effect is present.

3. How are curl and uniform shear calculated?

Curl is calculated using vector calculus, specifically the cross product between the velocity field and the unit normal vector. Uniform shear, on the other hand, can be calculated by taking the gradient of the velocity field and finding the shear components.

4. What are the practical applications of understanding the relationship between curl and uniform shear?

Understanding the relationship between curl and uniform shear can be useful in various fields such as fluid dynamics, meteorology, and oceanography. It can help predict the behavior of fluid flow and can be used in the design of structures that interact with fluids, such as aerodynamic surfaces and ships.

5. Are there any real-life examples of curl and uniform shear?

Yes, there are many real-life examples of curl and uniform shear. For instance, when wind blows over a mountain, it creates curl, resulting in turbulence and eddies. Uniform shear can be observed in the ocean, where currents can stretch or compress the water in a linear fashion.

• Differential Geometry
Replies
2
Views
3K
• Classical Physics
Replies
12
Views
2K
• Astronomy and Astrophysics
Replies
31
Views
6K
• High Energy, Nuclear, Particle Physics
Replies
1
Views
960
• Quantum Interpretations and Foundations
Replies
34
Views
4K
• Beyond the Standard Models
Replies
11
Views
2K
Replies
1
Views
779
• Special and General Relativity
Replies
264
Views
28K
• Special and General Relativity
Replies
54
Views
8K
• Special and General Relativity
Replies
9
Views
2K