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Objective flow classification parameters and their use in general steady flows

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Abstract

Three quantitative flow classification parameters have been studied in the context of Tanner and Huilgol’s suggestion of strong and weak flows. Seen in this context, the different types of streamlines possible for general 3-D flows furnish no indication with respect to the flow strength. This is in total contrast to 2-D flows, where the type of the streamline and the strength of the flow go hand in hand. Astarita’s [J Non-Newton Fluid Mech, 6:69–76, 1979] flow classification parameter takes care of this fact and, if properly generalized, can be applied to more general flows: Two other flow classification parameters also have their basis in homogeneous 2-D flows, but their generalization leads, for general flows, to nonuniqueness and other unacceptable results. For 3-D flows, none of the parameters can quantitatively be used in general, and additional parameters, with their basis outside the 2-D flow regime, seem to be called for.

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Notes

  1. This includes the singular case of straight streamlines.

  2. Frequently, the transpose L of this tensor is used, L=A +.

  3. Some of the results look simpler if we retain the D 1,D 2,D 3 notation. In that case, the restraint K 1=0 is always implied.

  4. Note that Eq. 10 implies \({\mathop {e_{i} }\limits^. } = e_{i} = \cdot \Omega = \omega _{D} \times e_{i}\) with ω D the pseudovector of Ω.

  5. For 3-D flows, two additional invariants are needed besides K 1–K 5.

  6. These are the two parameters we referred to in footnote 5. Viewed as the sum of the symmetric tensor D and the skew tensor Ω, there are exactly seven invariants.

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  1. Universität Erlangen, Lehrstuhl für Strömungsmechanik, Cauerstr. 4, 91058, Erlangen, Germany

    P. O. Brunn

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  1. P. O. Brunn
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Correspondence to P. O. Brunn.

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Brunn, P.O. Objective flow classification parameters and their use in general steady flows. Rheol Acta 46, 171–181 (2006). https://doi.org/10.1007/s00397-006-0103-5

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