POISSON BRACKET FORMULATION OF VISCOELASTIC FLOW EQUATIONS OF DIFFERENTIAL TYPE - A UNIFIED APPROACH

The Hamiltonian formulation of equations in continuum mechanics through a generalized bracket operation is shown here to reproduce a variety of incompressible viscoelastic fluid models, including the Giesekus model (with particular cases the upper-convected Maxwell and the Oldroyd-B models), the FENE-P dumbbell, the Phan-Thien/Tanner, the Leonov, the Bird/DeAguiar, and the bead-spring chain models. The analysis allows comparison of the differential models on a more fundamental level than previously possible by reformulating the equations in terms of the Hamiltonian (system energy) and the dissipation of the system expressed as functionals involving the velocity vector and structural parameter (s). In fact, all of these models involve only slight variations of the same general Hamiltonian and the dissipation tensor. An advantage of this formulation is the establishment of thermodynamic admissibility criteria which in complex flows can shed light on the range of validity and/or faithfulness of the numerical calculations involving the above models. The usefulness of the generalized bracket formulation lies in the systematic approach that it provides in addressing one of the fundamental problems that the engineer working with complex materials has to deal with: how to transfer information that has been painstakingly provided by the physical chemist, addressing fundamental problems on a molecular level, from the microscopic scale to the macroscopic level where the engineer actually needs the model in dealing with everyday industrial problems. It is hoped that this new formulation can be used in the future to systematically generate continuum constitutive models, which are thermodynamically consistent, and based on microscopic analysis. Thus, it is the purpose here to narrow the gap between detailed (molecular) microscopic descriptions of the motions of polymer chains and (macroscopic phenomenological) continuum approaches. We believe that the generalized bracket formulation, due to its inherent simplicity and symmetry, has the potential to provide an answer to very complex situations, such as multicomponent structured media and coupled transport phenomena. © 1990, The Society of Rheology. All rights reserved.