Finite element approximation of steady flows of colloidal solutions

¹ Texas A&M University, Department of Mathematics, 3368 TAMU, College Station, TX 77843-3368, USA
² Sorbonne-Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
³ University of Ottawa, Department of Mathematics and Statistics, 150 Louis-Pasteur Pvt, Ottawa ON K1N 6N5 Canada
⁴ Texas A&M University, Department of Mechanical Engineering, College Station, TX 77843-3123, USA
⁵ Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK

^* Corresponding author: dguignar@uottawa.ca; diane.guignard@gmail.com

Received: 16 February 2021
Accepted: 3 August 2021

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.