Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index

Abstract

We consider a system of nonlinear partial differential equations describing the motion of an incompressible chemically reacting generalized Newtonian fluid in three space dimensions. The governing system consists of a steady convection-diffusion equation for the concentration and a generalized steady power-law-type fluid flow model for the velocity and the pressure, where the viscosity depends on both the shear-rate and the concentration through a concentration-dependent power-law index. The aim of the paper is to perform a mathematical analysis of a finite element approximation of this model. We formulate a regularization of the model by introducing an additional term in the conservation-ofmomentum equation and construct a finite element approximation of the regularized system. We show the convergence of the finite element method to a weak solution of the regularized model and prove that weak solutions of the regularized problem converge to a weak solution of the original problem.

Nonlinear iterative approximation of steady incompressible chemically reacting flows

“…In Section 2 we introduce the necessary notations, state some auxiliary results, and define the weak formulation of the problem. Then, in Section 3 we prove the uniqueness of the steady state to an incompressible chemically reacting fluid flow problem under more restrictive assumptions than those under which existence of weak solutions was shown in [26][27][28]30]. Furthermore, it will be shown that the uniqueness result can be applied to a model of the synovial fluid.…”

“…Remark 1. Compared to the works [25][26][27][28]30], we imposed slightly stronger conditions on the stress tensor S. Most notably, by assuming (implicitly) an infinite shear plateau for the viscosity (cf. the paragraph following ( 44)), we circumvent the difficulty of dealing with Lebesgue and Sobolev spaces with variable exponents.…”

“…However, since-at that time-a finite element counterpart of De Giorgi's estimate was not available for three-dimensional space, the authors had to restrict themselves to the two-dimensional case. The finite element analysis in three space-dimensions was, subsequently, carried out in [28]. In order to circumvent the absence of a discrete De Giorgi regularity estimate, the authors exploited a more complicated limiting process invoking a regularised system.…”

Non-isothermal non-Newtonian fluids: the stationary case

“…More recently, in [16] and [43], optimal error estimates for shear thinning non-Newtonian fluids have been addressed, whereas in [61] a finite element method based on a four-field formulation of the nonlinear Stokes equations has been considered. Moreover, without aiming at completeness, we refer to the book [41] and the references therein, and to [29,36,46,47] as recent relevant contributions on the numerical discretization of generalized Stokes problems. We also point out that the approximation of steady isothermal non-Newtonian incompressible Stokes fluid problems on polytopal meshes have been addressed in [19].…”

A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part I: Convergence analysis

“…We refer to [21,22,49,25,26,20] to name a few. In addition, the convergence analysis for the p-Navier-Stokes problem using FE methods can be found in [29,41,39,40,33]. On the other hand, to the best of the authors knowledge, there are no investigations using DG methods for the p-Navier-Stokes problem (1.1).…”