Abstract
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.
Highlights
The classical incompressible Navier–Stokes constitutive equation and its usual generalisations, the constitutive relations for the incompressible Stokesian fluid, are explicit expressions for the Cauchy stress in terms of the symmetric part of the velocity gradient
The constitutive relation that we introduce first in (2.10) and in (3.1) includes (1.10) as a special sub-class
We study two preliminary model problems without the inertial term; the first one reduces to the Stokes system, while the second model problem involves a monotone nonlinearity treated by the Browder–Minty approach
Summary
The classical incompressible Navier–Stokes constitutive equation and its usual generalisations, the constitutive relations for the incompressible Stokesian fluid, are explicit expressions for the Cauchy stress in terms of the symmetric part of the velocity gradient. Purely implicit algebraic relationship between the stress and the symmetric part of the velocity gradient were not considered to describe non-Newtonian fluids until recently Such models are critical if one is interested in describing the response of fluids which do not exhibit viscoelasticity but whose material properties depend on the mean value of the stress and the shear rate, a characteristic exhibited by many fluids and colloids, as borne out by numerous experiments. With both nonlinearities present in the model, proving the existence of a weak solution, for instance, to the best of our knowledge cannot be done by coupling the techniques used for these two problems, namely the Browder–Minty theorem and the Galerkin method combined with Brouwer’s fixed point theorem and a weak compactness argument. The theoretically established convergence of the scheme is confirmed and convergence of both decoupled algorithms is observed
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