Is any relation between curl and uniform shear available?

[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?

 

[Studiot]

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.

 

[Paulibus]

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!

 

[Studiot]

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

Meanwhile please consider this.

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.

 

[HallsofIvy]

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.

 

[Paulibus]

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.

 

[Studiot]

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?

 

[Paulibus]

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?

 

[Studiot]

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.

 

[Paulibus]

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.

 

[Paulibus]

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.