Stabilized stress–velocity–pressure finite element formulations of the Navier–Stokes problem for fluids with non-linear viscosity

Abstract

The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability in the calculation of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf–sup conditions, whereas the dominant convection requires a stabilized formulation in any case. This paper proposes two stabilized schemes of Sub-Grid Scale (SGS) type, differing in the definition of the space of the sub-grid scales, and both allowing to use the same interpolation for the variables $\sigma -u-p$ (deviatoric stress, velocity and pressure), even in problems where the convection component is dominant and the velocity–stress gradients are high. Another aspect considered in this work is the non-linearity of the viscosity, modeled with constitutive models of quasi-Newtonian type. This paper includes a description of the proposed methods, some of their implementation issues and a discussion about benefits and drawbacks of a three-field formulation. Several numerical examples serve to justify our claims.