Benchmark tests based on the Couette viscometer—I: Laminar flow of incompressible fluids with inertia effects and thermomechanical coupling

The Couette viscometer is a well-known problem of fluid mechanics, well-suited for the verification of numerical methods. The aim of this work is to extend the classical steady state mechanical solution obtained in fluid mechanics, both to strongly-coupled thermomechanical problems in the case of laminar and incompressible fluid flows, and to solid-type nonlinear behaviours. Extended solutions will allow for the verification of new formulations of a mixed P1+/P1 finite element developed both in fluid and solid mechanics, within a temperature/velocity/pressure formulation coupled with an implicit (backward) Euler algorithm in time. In the present Part I, we address the case of the laminar flow of incompressible fluids with inertia effects and thermomechanical coupling. The verification performed on the reference solutions developed clearly evidence the good behaviour of the fluid finite element. The extension to solid-type nonlinear behaviours for strongly-coupled thermomechanical problems will be the subject of Part II.