A new framework for shallow approximations of incompressible flows

A new framework for the asymptotic analysis of incompressible flows of complex non-Newtonian materials is presented in this paper. It allows both to avoid redundant mathematical hypotheses and to dramatically reduce the amount of tedious formal calculations. The starting point of the proposed framework is a generic equation, easily adaptable to most problems of continuum mechanics, for which a thin-layer approximation is developed. We then show how to treat the so-called Gordon-Schowalter derivative, a general objective time derivative involved in non-Newtonian viscoelastic fluids. As a proof of concept of our framework, we apply it to the Maxwell viscoelastic model.