There are several geometries people use on commercial rheometers, such as rotating parallel plates, cone-plate, and Taylor-Couette co-cylinder. If there is an inherent stress gradient in these geometries, what is the reason? How can the magnitude of the stress gradient be calculated or quantified?
Check this link: http://ebookbrowse.com/an-introduction-to-rheology-h-a-barnes-j-f-hutton-k-walters-pdf-d306358513 in Chapter 2 from page 26-27 to the end of the geometries section... as well as in "Rheology: Principles measurements and applications" (1993) by Christopher W. Macosko, pages 190 & 206, and of course ...read, read, read!!
I am with Thomas - Read. Try Ferry Viscoelastic properties of polymers" Page 101 in the 3 rd edition. It is an older text, but classic both in technical approach and acceptability for use as a text.
Hi Yanfei, so, gradient of stress is a third rank tensor. If by chance you want the divergence of stress then this is simply the force per unit volume at any point. It seems to me that, since grad(T) is a third rank tensor it is only possible to form a scalar invariant by something like (in Cartesian coordinates only): T^{ab}_,cT^{ab}_,c. Another option relating to an imposed strain-rate field G^{ab}=u^a_,b+u^b_,a as your examples suggests, to combine something like: T^{ab}_,c u^G^{ab} to produce a vector invariant and then take a dot product to form a scalar invariant. But there are other combinations, as for example found inthe various Maxwell visco-elastic models. "gradients" of stress do suggest material memory properties.
I figured out that there is stress gradient in cone-plate, and Taylor-Couette co-cylinder, and now I know how to calculate them.
But I do not see why there stress gradient in rotating parallel disk. If there is, how to calculate it?
Do all rotating plate flow have a stress gradient along the vorticity direction?
I can't lay hands on the exact reference that has the flat plate geometry, but try this reference from chapter 5 of Ferry: K. Walters, Proc.7th Intern.Cong. Rheology, Chalmers Inst Tech., Gothenburg, 1977, p 147. The Ferry reference I responded with earlier (ISBN0-471-04894-1) is discussing the cone-plate geometry and thrust force created as a means of determining stress in the cone -plate geometry. At the end of the section, it says, in a kind of off handed way, ' to do this derivation for flat plate', see the Walter's reference. Walters has his own book as well: K. Walters (1975) Rheometry (Chapman & Hall) ISBN 0-412-12090-9.
If I understand properly what you mean by "along the vorticity direction", you will get a stress gradient as long as there is viscosity.
Sorry I can't be of more help. I haven't been active in the field for quite a while. However, there is probably a reason the rheometer manufacturers no longer sell the parallel disk geometry. It has probably been since the 1970s since that geometry has fallen out of favor experimentally.
In the following paper: J. Non-Newtonian Fluid Mech. 126 (2005) 105–114
"Viscometric flow under apparent wall slip in parallel-plate geometry", check the Eq. 20(a), 20(b), 20(c); and once again ...read, dear Yanfei!
The textbooks already cited by Macosko, Walters, Barners and Ferry are all excellent references. Depending on what is available to you, you could also check Bird/Armstrong/Hassager (Vol. 1), Tanner or Morrison. They all discuss fluid mechanics of different rheometric flows in some depth.
It is incorrect that "rheometer manufacturers no longer sell the parallel disk geometry" - far from it! The P-P setup has useful applications e.g. for samples that can not be loaded into other geometries without prior deformation like soft solids, allows to check for wall slip without flow visualisation (by changing the gap size) amongst other features. As for all rheometer geometries, it's a trade-off between different advantages and disadvantages and has to be judged in context.
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