Numerical analysis of implicitly constituted incompressible fluids: mixed formulations

Thesis

We consider the numerical approximation of incompressible non-Newtonian flow by means of the finite element method, where the constitutive law is defined through an implicit relation G(S ,D(u)) = 0. The setting considered in this work captures models widely used in applications, such as the Bingham and the Carreau–Yasuda constitutive relations. Since in general it is not possible to solve for the shear stress S in the constitutive relation, the emphasis is placed on formulations treating the shear stress as a variable.

Under the assumption that the constitutive relation defines a monotone graph with r-growth, and that the finite element spaces satisfy appropriate inf-sup stability conditions, the first part of the thesis extends earlier results in the literature to provide a convergence result that guarantees that a subsequence of the numerical approximations converges weakly to a solution of the system, in the optimal range r > 2d/(d+2), where d is the spatial dimension. The qualitative nature of this convergence result is a consequence of the generality of the framework of implicitly constituted fluids, for which e.g. higher regularity estimates are not available. Computational examples show, nevertheless, that the numerical scheme considered exhibits the expected convergence rates, in the situations where these are available.

In the second part of the thesis we develop an augmented Lagrangian preconditioner for a stress-velocity-pressure formulation of the steady system. The preconditioner involves a specialised multigrid algorithm that makes use of a space decomposition that captures the kernel of the divergence operator, and non-standard intergrid transfer operators. Although the current theory for robust multigrid works only for symmetric and positive-definite systems (and thus does not apply to the systems considered in this thesis), the resulting preconditioner exhibits remarkable robustness properties.

In the final chapter of the thesis, the extension to the anisothermal case is carried out. We employ an implicit constitutive relation that allows for a temperature dependence of rheological parameters such as the viscosity and the yield stress, and provide convergence results for the unsteady forced convection system that takes the viscous dissipation term S : D (u) into account, and for the steady Oberbeck–Boussinesq approximation. For the latter an augmented Lagrangian preconditioner is also introduced; this preconditioner exhibits robust convergence behaviour when applied to the Navier–Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity, and viscous dissipation.

Authors: Gazca Orozco, PA

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Multigrid methods; Finite element method; Non-Newtonian fluids
• Subjects: Numerical Analysis; Partial Differential Equations