# A theoretical framework for steady-state rheometry in generic flow conditions

## TL;DR

This paper presents a comprehensive theoretical framework for analyzing steady-state rheometry in complex flow conditions by decomposing the stress tensor into components based on local flow characteristics, aiding interpretation and modeling.

## Contribution

It introduces a tensorial basis for stress decomposition in incompressible fluids that generalizes classical viscometric functions to arbitrary steady flows.

## Key findings

- Provides a unified stress decomposition applicable to various flow types.
- Enhances interpretation of rheological data in non-viscometric steady flows.
- Facilitates development of more accurate constitutive models.

## Abstract

We introduce a general decomposition of the stress tensor for incompressible fluids in terms of its components on a tensorial basis adapted to the local flow conditions, which include extensional flows, simple shear flows, and any type of mixed flows. Such a basis is determined solely by the symmetric part of the velocity gradient and allows for a straightforward interpretation of the non-Newtonian response in any local flow conditions. In steady homogeneous flows, the material functions that represent the components of the stress on the adapted basis generalize and complete the classical set of viscometric functions used to characterize the response in simple shear flows. Such a general decomposition of the stress is effective in coherently organizing and interpreting rheological data from laboratory measurements and computational studies in non-viscometric steady flows of great importance for practical applications. The decomposition of the stress in terms with clearly distinct roles is also useful in developing constitutive models.