Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

Abstract

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity and pressure with non-constant viscosity. The analysis is performed by the classical Babuška-Brezzi theory, and we state that any inf-sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates which are further confirmed through computational examples.

Robust a posteriori error analysis for rotation-based formulations of the elasticity/poroelasticity coupling

“…Rotation-based formulations are found in applications to the modelling of non-polar media and helicoidal motion (see, e.g., [6,20,25] and the references therein). The resulting theory has a similarity with vorticity-based formulations for incompressible flow such as [5,8,12,13,18].…”

Augmented finite element formulation for the Navier--Stokes equations with vorticity and variable viscosity

“…easily split in the usual rotation form (because, in general, − div(ν∇u) = ν curl(curl u) − ν∇(div u)). Such an addition (and addressed for vorticity-based Stokes, Brinkman, and Oseen formulations in [9,11,29]) yields a non-symmetric variational form that can be augmented using residual terms from the constitutive and mass conservation equations. In the present treatment, additional terms appear due to the variable viscosity, which require a regularity assumption on the viscosity gradient.…”

“…In the present treatment, additional terms appear due to the variable viscosity, which require a regularity assumption on the viscosity gradient. In [11] the kinematic viscosity ν is assumed in W 1,∞ (Ω), and in the Darcy-heat system analysed in [18] the viscosity is assumed Lipschitz continuous. For our analysis it suffices to take ν ∈ W 1,r (Ω), with r = 2r r−2 , and r ∈ (2,6] for 3D and r ∈ (2, ∞) for 2D.…”

Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

“…For instance, [27] addresses the well-posedness of the vorticity-velocity formulation of the Stokes problem with varying density and viscosity, and the equivalence of the vorticity-velocity and velocity-pressure formulations in appropriate functional spaces is proved. More recently, in [6] we have taken a different approach and employed an augmented vorticity-velocity-pressure formulation for Brinkman equations with variable viscosity. Here we extend that analysis to the generalised Oseen equations with variable viscosity, and address in particular how to deal with the additional challenges posed by the presence of the convective term that did not appear in the Brinkman momentum equation.…”