On GENERIC formalisms for complex fluids (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

The General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), proposed by Grmela and Öttinger[l, 2], serves as a general framework for the thermodynamically consistent modeling of continua, wherein the conservative and dissipative mechanisms are clearly distinguished. The conservative mechanism is formulated as a Hamiltonian system using the Poisson bracket, whereas the dissipative mechanism is formulated using a dissipative bracket acting on the entropy functional. Barotropic fluids, Korteweg-type fluids, and complex fluids with complicated micro-structures that necessitate introducing additional structural variables, are all formulated within the GENERIC formalism. In the GENERJC formalism for Korteweg-type fluids, the spatial gradient of the mass density is included in the constitutive relation concerning the internal energy from which the Korteweg stress is derived. The interstitial working proposed by Dunn and Serrin[3] appears in the energy equation. Indeed, the GENERIC formalism provides a fairly simple derivation of the interstitial working, which was derived employing a Coleman-Noll type procedure[4] in an elaborate analysis. The GENERJC formalism also shows that the Korteweg stress and interstitial working are isentropic. This is in contrast with Cahn-Hilliard type models, which are intrinsically dissipative in nature as apparent from their bracket formulations[5]. Complex fluids can also be formulated within the GENERIC formalism[2, 6]. An additional structural variable, called the conformation tensor, can be introduced to model viscoelastic microstructures. The conformation tensor is assumed to be contravariant, similar to the left Cauchy-Green tensor of the deformation, and the time evolution along the flow, which is naturally represented by the Lie derivative of the tensor, is prescribed by the Poisson bracket in the GENERJC formalism. Additional terms in the Poisson bracket are purely kinematic, as long as the entropy function does not explicitly depend on the conformation tensor. It is advantageous to adopt the internal energy density as a state variable instead of the entropy density when constructing a dissipative bracket that models dissipation due to the microstructure.