By Peter Glansdorff

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Extra resources for Thermodynamics in Contemporary Dynamics: Lectures Held at the Department of Mechanics of Solids July 1971

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18 fluctuations of the mass fraction in this product are forbidden. In this third section, we shall examine this ques- tion by means of a variational method based on the notion of 10cal potential introduced by p. Glansdorff and I. S. Schechter, I. Prigogine and J. Hamm. 2. The steady state. 10) a~· 1 () Ni aNi f) Ni a -=-V. - + 0 7'T+0 1 J eH I i av· =- V. --~ a V· - --~ a~ th:· f):t. ( aXt er 1 Po Op 9~. k P [ aT N (1-N)-] 1 -- + F. ~ 0 ~o 1 a:t.

2) 2 ~) d 4 Z 2 (,}")+'I1(U .. ) i '" dV > 0 ur = OV ~ ) ~,J' The Benard Problem 53 using as weighting functions: =T 2 1: defined in section 8. s. may be interpreted as the increment of t [(~d) the function: F = P 2 -1 (gOt)d 4 " Z (l, . ) + J' After averaging over the :x: - 2 V (v .. ) - 29 IX T'U' ] d V ~ ,; ~ y plane we may wri te (cf. 2) may then be interpreted as a min- imlDll condi ti on for the function

In many cases, only approximate methods of numerical analysis are available. In this respect, the variational techniques present a particular interest whenever the solution of the problem may be derived as the minimum of some potential associated to the basic equations. < 0, occurs in the form of a non exact differential whereas its reduction to an exact differential of a potential is only possible for particular situations. As an example, most of the non-linear problems are devoid of such an associated potentia~, and therefore cannot be solved by the classical variational techniques as the Rayleigh-Ritz methode Nevertheless, as pointed out hereafter, it remains still possible to split the differential expression of the evolu tion criterion into two parts in such a way, that one of these corresponds to the exact differential of a potential ~ (say).

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