Preprint 7-2018

Operator-GENERIC Formulation of Thermodynamics of Irreversible Processes

Author(s) : Arbi Moses Badlyan , Christoph Zimmer

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 7-2018

MSC 2000

35Q35 Other equations arising in fluid mechanics

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws

37L99 None of the above, but in this section

Abstract :
Metriplectic systems are state space formulations that have become well-known under the acronym GENERIC. In this work we present a GENERIC based state space formulation in an operator setting that encodes a weak-formulation of the field equations describing the dynamics of a homogeneous mixture of compressible heat-conducting Newtonian fluids consisting of reactive constituents. We discuss the mathematical model of the fluid mixture formulated in the framework of continuum thermodynamics. The fluid mixture is considered an open thermodynamic system that moves free of external body forces. As closure relations we use the linear constitutive equations of the phenomenological theory known as Thermodynamics of Irreversible Processes (TIP). The phenomenological coefficients of these linear constitutive equations satisfy the Onsager-Casimir reciprocal relations. We present the state space representation of the fluid mixture, formulated in the extended GENERIC framework for open systems, specified by a symmetric, mixture related dissipation bracket and a mixture related Poisson-bracket for which we prove the Jacobi-identity.

Keywords : GENERIC, thermodynamics of irreversible processes, Onsager-Casimir reciprocal relations, operator equation, weak formulation