In this chapter, we wish to exploit the availability of the bracket formalism in the description of complex, non-conventional transport phenomena. In the first section, §10.1, we analyze relaxational phenomena in heat and mass transfer. The next section, §10.2, includes the description of phase transitions in inhomogeneous media. The last section, §10.3, contains a first effort to describe inertial effects in viscoelasticity. These problems have rarely been considered in the past, and when they have it has always been from a phenomenological perspective. We explore the availability of the bracket formalism here to provide a more systematic basis for these systems than has heretofore been available, and hence we characterize the models in this chapter as semi-phenomenological. The basic approach that we use is to first establish an appropriate internal variable for the system in consideration, and then to divine an appropriate Hamil-tonian which does, in some limits, produce available phenomenological models. (The latter step indicates why we characterize the models deve-loped in this chapter as “semi-phenomenological.”) As we shall see, describing the models on this more fundamental basis clears up a number of inconsistencies, as well as extending their range of validity without unduly sacrificing their simplicity. In most engineering applications of heat and mass transfer, the simple linear constitutive relations of (6.4-12) are adequate in order to describe the respective transport processes. A couple of very simple examples are the heat flux, when the affinity is the temperature gradient (giving Fourier's law of heat conduction), and the mass diffusion flux, when the affinity is the chemical potential (giving Pick's law of mass diffusion). The importance of such relationships in engineering practice cannot be overestimated. The validity of the linearized equations is generally established by steady-state experiments, so the question that naturally arises is whether or not the same constitutive relationship will hold for transient phenomena. This question cannot be answered as long as only steady-state experiments are performed. From physical considerations alone, it is obvious that the linearized constitutive relationships cannot be complete, in and of themselves.
Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis