Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids

•Shear stress cannot be is some fluids interpreted as a function of shear rate.•Constitutive models for such fluids can be built using the implicit type approach.•One dimensional implicit type models are capable of fitting experimental data.•Construction of three dimensional models from one dimensional data is ambiguous.•Tensorial implicit models provide a new tool for modelling normal stress differences.

The standard assumption in the phenomenological theory of constitutive relations for non-Newtonian fluids is that the Cauchy stress tensor is a function of the symmetric part of the velocity gradient. By discussing experimental data available in the literature we show that the classical framework is overly restrictive. A simple framework that goes beyond the standard approach is the novel concept of implicit constitutive relations. Here, the basic assumption is that the relation between the stress and the symmetric part of the velocity gradient is given by an implicit tensorial equation. We demonstrate that the implicit type constitutive relations are adequate for fitting the one dimensional (shear stress versus shear rate) experimental data, and we speculate about the possible form of the corresponding three dimensional (Cauchy stress tensor versus symmetric part of the velocity gradient) implicit constitutive relations. Using the representation theorem for isotropic tensorial functions we conjecture that the implicit constitutive relations could lead to novel models capable to describe nonzero normal stress differences. Finally, we provide an example of a nontrivial thermodynamically and dynamically admissible implicit type tensorial constitutive relation. The simple model does predict nonzero normal stress difference, and shows that the conjecture is correct.