Isotropic tensor-valued polynomial function of second and third-order tensors

Third-order tensor-valued functions appear in the theory of turbulence as the stropholysis tensor that parametrizes the breaking of reflectional symmetry in the spectrum of turbulence and as the correlations of triple fluctuating velocity components whose spatial gradients represent the rate of transport of turbulence by random fluctuations. The formulation of rational models for these quantities involves, in the first instance, the correct representation of a third-order tensor-valued isotropic function as a function of tensors of order two and three. In this paper, we present the derivation of an appropriate representation formula and discuss its potential applications.