Simultaneous invariants of strain and rotation rate tensors and their admitted region (Q277711)

Summary: The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensor \(\mathbf{S}\) and rotation rate tensor \(\mathbf{\Omega}\) 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 \(\mathbf{S}\) and \(\mathbf{\Omega}\) 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 \(\mathbf{S}\) and \(\mathbf{\Omega}\), in which the scalar coefficients depend on simultaneous invariants of these tensors.