Computations of Evolutions for Isothermal Viscous and Viscoelastic Flows in Open Domains

Abstract

This paper presents numerical computations of evolutions described by the initial value problems (IVPs) in isothermal incompressible viscous and viscoelastic flows in open domains using a space-time finite element model in hpk framework with space-time variationally consistent (STVC) integral forms. The mathematical models for viscous and viscoelastic liquids include the continuity and momentum equations in terms of the total stress tensor and the velocity vector. The second law of thermodynamics, namely the Clausius-Duhem in-equality, forms the basis for the constitutive equations for all matters including the ones considered here. For the liquids considered here, it becomes necessary to decompose the total stress tensor into equilibrium stress tensor and the deviatoric stress tensor. Determination of equilibrium stress using the Clausius-Duhem inequality with incompressibility constraint yields mechanical pressure as a function of temperature in the case of thermofluids otherwise constant but a function of position coordinates and time. The constitutive equations for the deviatoric stress tensor cannot be determined using the Clausius-Duhem inequality, which only requires that the viscous dissipation due to the deviatoric stress be positive. The theory of generators and invariants is used for this purpose. Due to the decomposition of the stress tensor, mechanical pressure appears in the momentum equations and must be kept as a dependent variable in the case of incompressible liquids. In continuum mechanics, it is well known that mechanical pressure is not deterministic from the deformation field but it influences the deformation field. The investigation considered in this paper focuses on the following four main issues: (1) Since pressure field is not deterministic from the flow field its specification on an open boundary must be such that the pressure field in the whole space-time domain is deterministic regardless of the deformation field. (2) Since the wave speed in incompressible liquids is infinite, does it pose problems in numerical computation of bounded evolutions? (3) The issues related to the need for non-zero initial velocity field. (4) Emphasis on bounded evolutions using hpk approximation spaces and space-time variationally consistent integral forms that ensure unconditionally stable computations and provide mechanism for computing bounded evolutions that are ensured to yield time accurate evolutions upon convergence (h-p-refinements). This feature is essential to ensure that the conclusion drawn from the numerical studies for items (1)–(3) are indeed valid. Newtonian and Maxwell fluids are used as constitutive models to present numerical studies for time dependent flows between parallel plates and the backward facing step.