Finite element approximation of steady flows of colloidal solutions

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.

Numerical approximations of thin structure deformations

“…This methodology has already been successfully applied to a wide range of problems, see for instance [27,30,31]. Moreover, the Γ-convergence as the mesh size goes to zero of the discrete energies obtained using LDG is analyzed in [31][32][33] and [30] for the preasymptotic, prestrain, bilayer, and folding case, respectively. Note that other discretizations are also possible.…”

“…in Ω. In the case where ϕ(x) = (x, 0) T and Φ = ∇ϕ, the boundary conditions (32) are referred to as clamped boundary conditions. A mixed boundary condition occurs when only the value of the deformation on Γ M ⊂ ∂Ω is specified, that is…”

“…The Γ-convergence of E pre s,h to E pre s as h → 0 + is proved in [32] when using continuous (rather than discontinuous) piecewise polynomial approximations of the deformations and when Σ = (i.e. without folding).…”

A review of implicit algebraic constitutive relations for describing the response of nonlinear fluids

“…None of the popular models available in the non-Newtonian fluid mechanics literature come close to fitting such data as the derivative of the stress-shear rate curve changes sign more than once. Recently, [61] studied mathematical issues concerning the equations governing the flow of a fluid described by (43). They established the existence of weak solutions and on making additional assumptions were able to establish uniqueness of those solutions.…”