Galerkin least-squares solutions for purely viscous flows of shear-thinning fluids and regularized yield stress fluids

Abstract

This paper aims to present Galerkin Least-Squares approximations for flows of Bingham plastic fluids. These fluids are modeled using the Generalized Newtonian Liquid (GNL) constitutive equation. Their viscoplastic behavior is predicted by the viscosity function, which employs the Papanastasiou's regularization in order to predict a highly viscous behavior when the applied stress lies under the material's yield stress. The mechanical modeling for this type of flow is based on the conservation equations of mass and momentum, coupled to the GNL constitutive equation for the extra-stress tensor. The finite element methodology concerned herein, the well-known Galerkin Least-Squares (GLS) method, overcomes the two greatest Galerkin shortcomings for mixed problems. There is no need to satisfy Babuška-Brezzi condition for velocity and pressure subspaces, and spurious numerical oscillations, due to the asymmetric nature of advective operator, are eliminated. Some numerical simulations are presented: first, the lid-driven cavity flow of shear-thinning and shear-thickening fluids, for the purpose of code validation; second, the flow of shear-thinning fluids with no yield stress limit, and finally, Bingham plastic creeping flows through 2:1 planar and axisymmetric expansions, for Bingham numbers between 0.2 and 133. The numerical results illustrate the arising of two distinct unyielded regions: one near the expansion corner and another along the flow centerline. For those regions, velocity and pressure fields are investigated for the various Bingham numbers tested

A Stabilized Finite Element Formulation of Non-Newtonian Fluid Model of Blood Flow in A Bifurcated Channel with Overlapping Stenosis

“…Abdullah et al, [6] with the helps of stabilization parameter which is added to the magnetic source term of the momentum equations are able to solve the numerical instability which arised in their solutions in a form of chaotic velocity vector. On the other hand, Skála et al, [7], Machado et al, [5] and Zinani et al, [8], all employed Galerkin least-squares approach that was good deals with two major sources of problems in classical Galerkin formulation. The first one is the requirement to satisfy the inf-sup condition of Ladyzhenskaya-Babuska-Brezzi (LBB) condition to employ the combinations for the finite element interpolations for velocity and pressure fields [2,3].…”

Tailoring of arteriovenous graft-to-vein anastomosis angle to attenuate pathological flow fields

“…The Bird-Carreau constitutive law was used to represent the non-Newtonian, pseudo-plastic nature of blood, as is appropriate for oscillatory flow with high Womersley number ( , in which represents heart rate, in radians per second, and R is the artery radius) 31 – 33 : in which the strain rate tensor is: and effective viscosity changes with the Deborah number, : where n is a constitutive parameter and . Here, is the characteristic relaxation time of the blood, and the effective shear strain rate is 32 , 34 , in which is the second invariant of . The parameters used are listed in Table 1 .…”

Three-dimensional simulation of underfill process in flip-chip encapsulation

“…However, the classical continuous Galerkin FEM might produce spurious numerical oscillations during the solution due to dominating advection and non-LBB stable finite element spaces, which will lead to a failure result. A number of finite element methods have been developed to address this problem without having these spurious effects, such as the SUPG/PSPG method [17,[22][23][24][25], the Galerkin/Least-squares (GLS) method [25][26][27], the characteristic based split (CBS) method [28][29][30], the algebraic subgrid scale (ASGS) method [31,32], the orthogonal sub-scale stabilization (OSS) method [32,33], the Hughes variational multiscale (HVM) method [34,35], the stabilized by pressure gradient projection (SPGP) method [36][37][38], and the local projection stabilization (LPS) method [39,40]. These methods simultaneously suppress spurious oscillations and allow equal-order velocity pressure approximation.…”