Structure of variational principles in nonequilibrium thermodynamics

Some variational principles describing the behavior of nonstationary dissipative processes are presented. These criteria appear as an extension of Hamilton's principle to dissipative systems. Firstly, processes with vanishing barycentric velocity, i.e., of purely dissipative nature, are considered. The principle proposed is a slight modification of a criterion derived previously by the authors: The central quantity in the expression of the principle is however no longer the Legendre transform of the entropy per unit mass but is here the Legendre transform of the internal energy per unit volume. As an illustrative example, thermodiffusion is considered. Afterwards, the principle is generalized to describe the flow of a compressible, Newtonian fluid in which temperature effects are not neglected. After having shown that the Euler‐Lagrange equations are equivalent to the conservation laws, some particular expressions of the principle, corresponding to special situations like the stationary case and fixed boundary conditions, are given.

Generalization of Hamilton's principle to continuous dissipative systems

In this paper a survey is given concerning to the stochastic modelling approaches in transport processes with a special emphasis on application possibilities for simultaneous heat and mass transfer in drying. First, the mostly used classical modelling methods for drying are discussed which lead to a linear parabolic type of PDE systems supposing constant (state-independent) conductivity coefficients. Powerful discretisation methods are shown for their solution. Basic principles of variational calculus are discussed then with an attention on direct methods. As a simple application a first-order approximation example is formed, and the solution of the system equation is presented. It is also shown, that the thermodynamical state-dependence of the conductivity coefficients has a crucial influence on the flow pattern of the coupled heat and mass transfer, which is particularly obvious in the cases, when the so-called percolative phase transitions take place. It effects a discrete change of the conduc...

Mathematical and physical foundations of drying theories

Hydrodynamic stability by variational methods

The inertial relaxation terms and the variational principles of least action type for non-stationary energy and mass diffusion

A general variational principle concerning non-stationary purely dissipative phenomena, obeying ONSAGER's reciprocal relations, is presented. The principle expresses that the variation of the sum of the total entropy production and the time derivative of the MASSIEU function is equal to zero. It is shown that the EULER-LAGRANGE equations corresponding to arbitrary variations of the state variables yield the conservation laws. Other general criteria for non-stationary processes have been given by GLANSDORFF-PRIGOGINE and GYARMATI. They are compared with our principle.



Two illustrative examples are considered: coupled heat and electrical conduction in an isotropic medium and chemical reactions coupled with diffusion of matter and heat conduction in a multi-component fluid.

A Rather General Variational Principle for Purely Dissipative Non‐Stationary Processes

A variational criterion describing the nonstationary flow of a mixture of chemically active compressible fluid components is presented. The problem considered constitutes a generalization of previous works [1, 2] of the authors.



It is shown that the Euler-Lagrange equations corresponding to arbitrary variations of the chemical potentials, the velocity components, and the temperature are identical to the laws of conservation of mass of each component, of momentum, and of entropy, respectively.



Ein allgemeines Variations-Kriterium fur chemisch aktive kontinuierliche Medien



Es wird ein Variations-Kriterium zur Beschreibung der nichtstationaren Stromung eines Gemisches von chemisch aktiven, kompressiblen Flussigkeiten angegeben. Das betrachtete Problem ist eine Verallgemeinerung der fruheren Arbeiten [1, 2] der Autoren.



Die Euler-Lagrangeschen Gleichungen, welche beliebigen Variationen der chemischen Potentiale, der Geschwindigkeitskomponenten sowie der Temperatur entsprechen, erweisen sich als identische mit den Gesetzen der Erhaltung der Masse einer jeden Komponente, der Momente und der Entropie.

J. H. Lambermont

Papers

A Rather General Variational Principle for Purely Dissipative Non‐Stationary Processes

A General Variational Criterion for Chemically Active Continuous Media