Abstract
Abstract Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\varOmega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully discrete approximation scheme, using a spatial mixed finite element approximation on general shape-regular simplicial meshes combined with backward Euler time-stepping. We consider the case when the velocity field belongs to the space of solenoidal functions contained in $\textrm{L}^\infty (0,T;\textrm{L}^2(\varOmega )^d)\cap \textrm{L}^q(0,T;\textrm{W}^{1,q}_0(\varOmega )^d)$ with $q\in \left (2d/(d+2), \infty \right )$, which is the maximal range of $q$ with respect to existence of weak solutions. In order to facilitate passage to the limit with the discretization parameters for the sub-range $q \in \left (2d/(d+2), (3d+2)/(d+2) \right )$, we introduce a regularization of the momentum equation by means of a penalty term, and first show convergence of a subsequence of approximate solutions to a weak solution of the regularized problem; we then pass to the limit with the regularization parameter. This is achieved by the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness techniques. For $q \geq (3d+2)/(d+2)$ convergence of a subsequence of approximate solutions to a weak solution can be shown directly, without the regularization term.
Highlights
In the mechanics of viscous incompressible fluids typical constitutive relations relate the shear stress tensor to the rate of strain tensor through an explicit functional relationship
Implicit constitutive theory was introduced in order to describe a wide range of non-Newtonian rheology, by admitting implicit and discontinuous constitutive laws, see [33, 34]
The scheme is based on a spatial mixed finite element approximation and a backward Euler discretization with respect to the temporal variable
Summary
In the mechanics of viscous incompressible fluids typical constitutive relations relate the shear stress tensor to the rate of strain tensor through an explicit functional relationship. The existence of weak solutions to mathematical models of this kind was explored, for example, in [10, 11] for steady and unsteady flows, respectively. The aim of the present paper is to construct a fully discrete numerical approximation scheme, in the unsteady case, for a class of such implicitly constituted models, where the shear stress and rate of strain tensors are related through a (possibly discontinuous) maximal monotone graph. The mathematical ideas contained in the paper are motivated by the existence theory formulated, in the unsteady case, in [11], and the convergence theory for finite element approximations of steady implicitly constituted fluid flow models developed in [16]
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