Abstract
This paper presents formulations, computations and investigations of the solutions of classes C00 and C11 for two dimensional viscoelastic fluid flows in u, v, p, τijp, τijs with Phan-Thien-Tanner (PTT) constitutive model using p-version least squares finite element formulation (LSFEF). The main thrust of the research work presented in the paper is to employ ‘right classes of interpolations’ and the ‘best computational strategy’ 1) to obtain numerical solutions of governing differential equations (GDEs) for increasing Deborah numbers 2) investigate the nature of the computed solutions with the aim of establishing limiting values of the flow parameters beyond which the solutions may be possible to compute, but may not be meaningful.
The investigations presented in this paper reveal the following: a) The manner in which the stresses are non-dimensionalized significantly influences the performance of the iterative procedure of solving non-linear algebraic equations. b) Solutions of the class C00 are always the wrong class of solutions of GDEs in variables u, v, p, τijp and τijs and thus spurious. c) C11 class of solutions are the right class of solutions of the GDEs in variables u, v, p, τijp and τijs. d) In the flow domains, containing sharp gradients of the dependent variables, conservation of mass is difficult to achieve at lower p-levels (worse for coarse meshes). e) An augmented form of GDEs are proposed that always ensure conservation of mass at all p-levels regardless of the mesh and the nature of the solution gradients. f) Stick-slip problem is used as a model problem. We demonstrate that converged solutions are possible to compute for all flow rates reported and that the detailed examination of the solution characteristics reveals them to be in agreement with all the physics of the flow, g) Numerical studies with graded meshes and high p-levels presented in this paper are aimed towards establishing and demonstrating detail behavior of local as well as global nature of the computed solutions, 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 definitions j) A significant aspect and strength of our work is that we utilize straightforward 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.
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