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Advances in Mathematical Physics
Volume 2015 (2015), Article ID 147125, 7 pages
Research Article

Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

1Institute of Mechanics, Lomonosov Moscow State University, Michurinsky Avenue 1, Moscow 119192, Russia
2Chair of Fluid Mechanics, Universität Siegen, Paul-Bonatz-Straße 9–11, 57068 Siegen, Germany

Received 11 August 2015; Accepted 29 September 2015

Academic Editor: Christian Engstrom

Copyright © 2015 Igor Vigdorovich and Holger Foysi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensor and rotation rate tensor and calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensors and are studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensors and , in which the scalar coefficients depend on simultaneous invariants of these tensors.