Numerical analysis of unsteady implicitly constituted incompressible fluids: 3-field formulation. (English) Zbl 1434.76065
Summary: In the classical theory of fluid mechanics a linear relationship between the shear stress and the symmetric velocity gradient tensor is often assumed. Even when a nonlinear relationship is assumed, it is typically formulated in terms of an explicit relation. Implicit constitutive models provide a theoretical framework that generalizes this, allowing for general implicit constitutive relations. Since it is generally not possible to solve explicitly for the shear stress in the constitutive relation, a natural approach is to include the shear stress as a fundamental unknown in the formulation of the problem. In this work we present a mixed formulation with this feature, discuss its solvability and approximation using mixed finite element methods, and explore the convergence of the numerical approximations to a weak solution of the model.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A05 | Non-Newtonian fluids |
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