Numerical Simulations of Viscoelastic Fluid Flows Past a Transverse Slot Using Least-Squares Finite Element Methods

This paper presents a least-squares (LS) finite element method for linear Phan-Thien–Tanner (PTT) viscoelastic fluid flows. We consider stabilized weights in the LS method for the viscoelastic model and prove that the LS approximation converges to the linearized solutions of the linear PTT model; the convergence is at the optimal rate for the velocity in the H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}$$\end{document}-norm and at suboptimal rates for the stress and pressure in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-norm, respectively. For numerical experiments we first consider the flow through a planar channel to illustrate our theoretical results. The LS method is then applied to a flow through the slot channel with two depth ratios and the effects of physical parameters are discussed. Numerical solutions of the channel problem indicate that flow characteristics of the viscoelastic polymer solution are described by the results obtained using the method. Furthermore, we present the hole pressure for various Weissenberg numbers, and compare with that derived from the Higashitani–Pritchard (HP) theory.