POISSON BRACKET FORMULATION OF INCOMPRESSIBLE-FLOW EQUATIONS IN CONTINUUM-MECHANICS

The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets is presented for a number of incompressible fluids, including the Euler inviscid fluid, the Newtonian viscous fluid, a perfectly elastic medium, the upper-convected Maxwell, and the Oldroyd-B viscoelastic fluids. The analysis, expanding previous results reported by Grmela, leads to a generalized Poisson bracket formalism from which all of the above-mentioned cases can be recovered. Furthermore, the Poisson bracket formulation can easily incorporate model changes, as shown in the application of the hydrodynamic interaction correction to the Hookean dumbbell (upper-convected Maxwell) model. The Hamiltonian formulation is fully explained here through a novel interpretation of the functional derivative through which the constraints of the flow are incorporated into the continuum equations. The Poisson bracket formulation, as developed here, can be used for the systematic development of constitutive equations of material behavior. This new approach allows a significant reduction in the number of arbitrary parameters and assumptions, compared to the traditional continuum modeling procedures. Simultaneously, the Hamiltonian formulation provides more structure to the system of equations and an easier interpretation of the various terms present in the constitutive equations. © 1990, The Society of Rheology. All rights reserved.