Tensor-nonlinear shear flows: Material functions and the diffusion-vortex solutions

This work deals with tensor-nonlinear constitutive relations connecting the deviators of stress tensor and strain rate tensor in incompressible isotropic media which are called in continuum mechanics as Reiner–Rivlin fluids. The connections of quadratic and cubic invariants of two tensors, where two material functions involve, are presented. The main attention is given to one-dimensional shear flows in various curvilinear coordinate systems. The scheme of obtaining of the material functions for shear on the basis of the steady Poiseuille flow in a plane layer is described. The self-similar solutions corresponding to the generalized diffusion of vortex layer both in plane and axially symmetric cases are derived.