In the opening chapters of the book, we saw how a variety of conservative phenomena could be described through the Poisson bracket of classical physics. As already mentioned, the majority of systems with which engineers and physicists must deal reveal dissipative phenomena inherent to their nature. In the last chapter we offered a concise overview of nonequilibrium thermodynamics as traditionally applied to describe, close to equilibrium, dissipative dynamic phenomena. In this chapter, we lay the groundwork for the incorporation of non-conservative effects into an equation of Hamiltonian form, and show several well-known examples in way of proof that such a thesis is, in fact, tenable. The unified formulation of conservative and dissipative processes based on the fundamental equations of motion is not a new idea. Among the most complete treatments is the one offered by the Brussels school of thermodynamics [Prigogine et al., 1973; Prigogine, 1973; Henin, 1974] based on a modified Liouville/von Neumann equation. This is a seminal work where, using the mathematical approach of projection operators and relying only on first principles, it is demonstrated how large isolated dynamic systems may present dissipative properties in some asymptotic limit. A key characteristic of their theory is that dissipative phenomena arise spontaneously without the need of any macroscopic assumptions, including that of local equilibrium [Prigogine et al., 1973, p. 6]. The main value, however, is mostly theoretical, demonstrating the compatibility of dissipative, irreversible, processes with the reversible dynamics of elemental processes through a “symmetry-breaking process.” The value of the theory in applications is limited since it relies on a quantum mechanical equation for the density matrix ρ which is, in general, very difficult to solve except for highly simplified problems [Henin, 1974]. In a nutshell, we hope to offer with the present work a macroscopic equivalent of the Brussels school theory which, at the expense of the introduction of the local equilibrium assumption, attempts to unify the description of dynamic and dissipative phenomena from a continuum, macroscopic viewpoint. The main tool to achieve that goal is the extension of the Poisson bracket formalism, analyzed in chapter 5, to dissipative continua.
Physical and Mathematical Fluid Mechanics
