Computations of Numerical Solutions of Higher Class in 2D Polymer Flows: Upper Convected Maxwell Model

Abstract

Abstract This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two-dimensional viscoelastic fluid flows in primitive variables u, v, p, τxx, τxy, τyy with Upper Convected Maxwell (UCM) constitutive model using p-version least squares finite element formulation (LSFEF). The main emphasis of the investigations, undertaken in this research work, is to employ the right classes of interpolations and the best computational strategy to address, to illuminate on and perhaps, to answer and resolve whether the “continued obsession of developing newer and newer computational strategies to obtain solutions of Maxwell constitutive equations for ever increasing Deborah number (De) is as meritorious as currently believed. The stick-slip problem is used as a model problem in all investigations. Our investigations reveal the following: a) The manner in which the governing differential equations (GDEs) are non-dimensionalized influences the performance of the iteration procedure for solving nonlinear algebraic equations and thus, computational efficiency achieved. b) C00 class of solutions are always the wrong class of solutions and are spurious. c) In the flow domains, containing sharp gradients of dependant variables, conservation of mass is difficult to achieve specially at lower p-levels. d) C11 solutions are in conformity with the continuity considerations in GDEs. e) An augmented form of the GDEs and associated C00 and C11 formulations are proposed that always ensure conservation of mass regardless of mesh, p-levels and the nature of solution gradients. This approach yields the most desired classes of C11 solutions. f) We demonstrate that UCM constitutive model cannot describe the flow physics at any Deborah number for dilute polymer solutions of constant viscosity due to the fact that I) in UCM model, solvent stresses are a function of Deborah number. Thus, solvent stresses are wrong at any non-zero Deborah number. II) and solvent shear stresses produce elastic normal stresses. Numerical studies presented in the paper support theses findings. UCM model only generates correct flow physics for De = 0 (Newtonian flow). g) Numerical studies, with graded meshes and high p-levels presented in the paper, are aimed towards establishing and demonstrating detailed behavior of local as well as global nature of the computed solutions. Numerical studies are carefully designed and conducted to establish failure of UCM model in describing flow physics as well as failure of computational process. h) various norms are proposed and tested to judge local and global dominance of elasticity or viscous behavior. i) New definitions are proposed for elongational (extensional) viscosity. The proposed definitions are more in conformity and agreement with the flow physics compared to currently used definition. j) A very significant feature of our research work is that we utilize straight forward p-version LSFEF with C00 and C11 type interpolations without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used. Fully developed flow between parallel plates and stick-slip problems are used as model problems. It is concluded that UCM constitutive model always simulates incorrect behavior for dilute polymer solutions of constant viscosity regardless of Deborah number (except De = 0) and thus, development of newer computational schemes to achieve success at higher Deborah number is of no consequence.