A ~ ehu.ic, ~,,,~,,~~, o~ (1-4), pp. ~;o-os U989)

ON THE RELATION BETWEEN THE GYARMATI, PRIGOGINE AND HAMILTON PRINCIPLES*

H.C. Hou

Guan~don 0 Reaearch lnstitute for H~dra@a'c Enoineerino $o~oulin~ Saho, Gu~~zhou, Clu'na

~Received 20 November 1988)

The paper shows that both the Hamilton and the Prigogine principle belong to a respective special forro of Gyarmati's principle. The Prigogine and Hamilton principles are the limit cases of the Gyarmati principle, so that one can justify that the Gyarmati principle should be valid not only for the irreversible process, but also for the nondissipative or mixed process. From the simplified Gyarmati principle the classicaI expressiona of the Hamilton principle can be generalized into a complete set of inequalities. The principle of minimum mechanical energy, the principle of minimum potential energy and the principle of minimum kinetic energy all belong to alternative forros of the Hamilton principle. The dissipative procesa should be a decelerated process, whereas the non-dissipative procesa should be ah accelerated one. A possibility ia auggested to solve the problem of the longitudinal proflle of irreversible process by using ah alternative expression of the Gyarmati principle.

1. Introduction

In recent years, following the formation of the Gyarmati principle (named

~Governing principle" or aOnsager principle" by Gyarmati himself [1,2]), many re- search results concerning the relation between the Gyarmati principle, Prigogine principle (Principle of mŸ entro'py production [3,4]) have been obtained. Be- sides these, derivation of some governing equations (Fick equation, Navier-Stokes equation, etc.) from the Gyarrnati principle have been successfully �91 also.

This �91 was obtained mainly by the effort of the Hungarian School around Prof. Gyarmati.

However, little attention has been paid to the relation of the Gyarmati prin-

ciple to the Harnilton and the Prigogine principles [1]. Gyarmati concluded the similarities and differences of both these Principles into four points. It was nec-

essary to recognize that the Hamilton principle, asa classic result formulated a century ago ancl accepted commonly asa governing principle to be usecl for solv- ing many engineering problems, had not been formulated completely: it should be

reconsidered. This paper deals with the different alternative forros of the Hamilton principle,

generalized into a complete set, together with the Prigogine principle related to the

Gyaxmati principle.

*Dedicated to Prof. I. Gyarmati on his 60th birthday

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2. Gyarmati Principle and Prigogine Principle

In this paper we consider mainly the irreversible process of a closed system under isothermal conditions. This simple case was quasi-valid, for example, for water flow. The energy for water flow was taken from potential energy in gravita- tional ¡ In the water process this energy should be transformed into heat and the latter shoulcl be transferred across its boundary of constant temperature. Due to this effect the temperature on the surface of the system would remain constant; the process is an isothermal one.

As is well-known, the Gyarmati principle can be stated by the following inte- gral:

G = f ( a - ~b) dV = max, (1) v

where ois the entropy production, composed by the parts of internal change of entropy plus the entropy flux from outside, and ~b is the dissipation function.

S = f adV = diS/dt + d~S/dt. (2)

The Prigogine Principle can be expressed by

P = dS/dt = 2~ = min. (s)

Though the relation between Gyarmati's principle and the Prigogine princi- pie was considered thoroughly before (Gyarmati [1] concluded that "the Prigogine principle is nota new and independent principle, but an alternative formulation in the language of the entropy production of the forro of Onsager principle valid for stationary states~), some ambiguous concepts still need to be clari¡

Ir the system was incompressible and the process was isothermal, then the time rate of entropy, inherent in the system, should be equal to zero, diS/dt = 0; and ir the process was stationary• then the entropy flux across the boundary of the system from outside should be a constant value, i.e. its time rate should be some constant. Therefore, for this simple case we have

~r = diS/dt + drS/dt = 0 + const = const (4)

and from Eq. (1)

G = c o n s t - f ~ dV = max, (5) v

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GYARMATI, PRIGOGINE AND HAMILTON

or in Prigogine's expression

= f ~b d V = c o n s t - m a x = P min.

This result of ~min" resulted from geometry in Fig. 1.

61

(6)

%onst ~, which can be shown by elementary

G

time

Therefore, the Prigogine principle can be considered as one of the limit cases of Gyarmati's principle, when a should remain a constant value.

Of course, in common, the Prigogine principle was an independent principle. Gyarmati's contribution was that he correctly predicted that the functional (1) should approach a maximum. However, solely upon Eq. (1) one could not predict: along which route should the process approach the state with this minimum value? For the stationary case, this problem had been solved by Prigogine and Glansdorff [4,5]. They stated that the entropy production would decrease along the time:

d P / d t = d 2 S / d t 2 < O, (7)

in which the ~less than" corresponds to the non-equilibrium case and when the process should approach equilibrium state, then dP/dt would approach zero.

The gmaximum" and ~minimum ", appearing in Eqs (1) and (3) are deter- mined by the mathematical sign. From the physical point of view, any physical and mechanical process is influenced by its own constraint conditions. For example, the river process is constrained by the river bed and sediment transport, etc. Supposing that the constraint conditions in some system could be excluded arbitrarily and suc- cessively, then the exclusion of one constraint condition makes the former minimum decrease to a new minimum; and when aU the constraint conditions should be ex- chded entirely, then the state value of minimum would drop to a "minimum among all mŸ ~, i.e. to sero, and this case would correspond to the static equilibrium:

p = ruin = o. (8)

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Summing up the above mentioned, the Prigogine principle can be formulated by the following set of inequalities and equalities:

{ P>O, dP/dt <_ 0, P = min, (9)

P = 0 .

For the isothermal and incompressible case, as indicated before, these expressions can be replaced by the energy and its time rate in the following:

{ dE/dt >_ O, d'�91 2 <_ O, dE/dt = min,

~ E / d t = o.

(lO)

3. Alternative forro of Hamilton Principle

Let us proceed now to the case of reversible process, i.e. to the case when the dissipation of energT in the system can be disregarded. In this case, the motion should follow the Newton Law. Though the reversible process must be distinguished from the irreversible one it al~o exhibits a characteristic trend to extremum state. Mathematical formulation of this characteristic was expressed by the Hamilton prin- ciple.

In any monograph of mechanics the Hamflton principle for a mass particle is expressed simply in this form:

t~ �91

/ Ldt:min, ~/ t i t i

L de = 0, (11)

where Lis the Langrangian function. For Newtonian mechanics, L i s composed by the kinetic energy Es plus the work W, i.e. L -- Es + W. For the time ti and t2 selected arbitrarily Eq. (11} must lea(] to

L = Ek + W = min. (12)

In the gravitational field, the work-down of particle resulted by the potential energy Ep, then

L = Ek + Ep = E,, = ruin, (13)

where E,n (or simply, E) is the mechanical energy. In this case, the Hamilton principle is expressed by the principle of minimum mechanical energy. For the open

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GYARMATI I PRIGOGINE AND HAMILTON 63

channel flow, the governing effect of this principle was quite evidently expressed, for at the point of water-fall the so-called critical depth should be formed (the critical depth should correspond to the minimum mechanical energy). One of the simple cases from Eq. (13) is that when the mass-particle is in quiescent condition, then Ek = 0. In this case, we would have

Ep : min. (14) !

This is the principle of mŸ potentiM energy, which is a governing principle widely used in solid mechanics [7,8].

Another simple case is when the potential energy can be ignored (for example, neutraJ-buoyant and iluctuating fluid partich). In this case,

Ek = min. (15)

This is the principle of mŸ kinetic energy. Perhaps, it has not been proposed yet in the literature, and could actas a powerful tool for solving the problems of turbulance.

Above we were concerned with the extreme state of a system only. A more profound problem is: Along which route should the system (particle) approach this state, when the system is in gabove minimum ~ state? This problem did not attract more attention, and ir was impossible to attain an answer solely from Eq. (12), i.e. solely from the approach of Newtonian mechanics. Ir could be solved using Gyarmati's principle.

4. Genera l i za t ion of t he H s m ] I t o n Pr inc ip le

Above we considered the Prigogine principle as one of the limit cases of Gyarrnati's principle. Another limit case is when the dissipation of the system can be ignored, or in such a case, when the time rate of dissipation should remain constant along the process. In this case, the Gyarmati integral (1) can be simpli¡ to

G = / ~ d V = m � 9 1 or G = / ( ~ - c o n s t ) d V = m a x . (16)

v v

Again considering this simple case: the incompressible system with isothermal sur- roundings, then

G = / �91 dV = d/dt / r dV = d E / d t = max . (17) v v

According to Eq. (I), sirnilarly to that for the irreversible process, for the reversible (ideal) process other inequalities and equalities as criteria must be supplemented;

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it should trend to increase its time rate of energy along the time: dE/d t > 0, d 2 E / d t 2 > 0. These inequalities should be verified by the example in the next paragraph.

Therefore, the Hamilton principle can be described by the set of inequali- ties, which together with the Prigogine principle are summed up in Table I for comparison:

'rabie I

State Prigogine Principle Hamilton Entropy Energy Principle

NonequUibrium P >_ 0 dE/dt >_ 0 dE/dt >_ 0 (18) dP/dt <_ 0 d2E/dt 2 <_ 0 d2E/dt 2 > 0 (19)

Equilibrium P :ruin. dE/dt:min dE/dt:max (20) (Dynamic) P -- 0 dE/dt = 0 dE/dt : co

Equilibrium E =ruin. (21) (static)

In the last column, E = min was taken from Eq. (14). Its "E ~ has to be distinguished from "E " in other inequalities; this "E " expresses the latent energy in the given system, whereas ~E " in other columns means those energies which should be ~dissipated" (for the irreversible process) or lost (for the reversible process). The difference of both Prigogine and Hamilton principles can be expressed geometrically in Fig. 2.

Table I shows that, in general, the irreversible process belongs to a decelerated process, whereas the reversible process belongs to ah accelerated one. It reveals that a ~reversible ~ process could really not be reversed.

5. E x a m p l e for t he H a m i l t o n P r inc ip l e

Consider the free-fall of a body in the gravitational fie]d a s a simple exampIe from elementary physics. In this case the energy of the body should inc]ude only the potential energy Ep. Then the set of inequa]ities becomes

{ dEp/dt >_ O,

d 2 E p / d t ~ > O, {22) dEp/ dt = max,

Ep = ruin.

The ffee-fall of a body is an acceleration phenomenon and, as ir is well known, this accelerating process belongs to a linear one. The fall velocity and the vertical distance y can be expressed as

{ v = Vo + gr, (23) y --- Pot + (I/2) gr 2,

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GYARMATI, PRIGOGINE AND HAMILTON 65

T /

/ /

/ /

/ /

/ /

/

(3)J /

/ /

q

II

t

F/r ~. Geometry of different processes a: irreversible process; b: reversible process (nonlinear case); c: reversible process (linear case}

where V0 is the velocity of the body at t = O, g is the gravitational acceleration. The potential energy of the body (per unit volume) should be

Ep = ~/y = ~(Vot + ~ 2 / 2 ) , (24)

where "y is the specific weight. The time rate of energy should be

Ep = dEp /d t = "y(V0 + gr). (25)

Ir occurs indeed that Ep should be greater than zero, when t > 0. Furthermore, this increasing process should obey a linear law. When t should approach to a maximum vahe, then Ep would tend to a maximum according to

d E p / d t [~=t m~x = max, and d E p / d t l t = ~ = oo,

as predicted by Eqs (20). The second derivation of E from (24) gives

d2En/dt 2 = "Tg = pg2 > 0, (26)

where p is the density of the body. This value must be positive in any case. There- fore, the validity of inequality (19) has been verified.

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Imagine that in the gravitational ¡ there exists no =g~, the free fall of a body should follow a linear law, then the two-fold derivation of Ep would not assure that its value be greater than zero in Whatever case. Therefore, the existence of g in gravitation was sdapted to this demancl.

6. Energy-var ianee a n d t ime-var ianee

Both systems (I0) above for the irreversible process and system (18)-(21) for the reversible process can be used not only for predicting the overall characteristic of the process, but also for solving some engineering problems. Here we should present only some skill to solve the actual mechsnical problems 5y using a dynamic extreme- approsching characteristic. Ii is well known that the static extreme-approaching characteristic, i.e. the principle of mŸ potential energy, Eq. (14), is widely being used in structure and elasticity theory [7,8].

Fora reversible process, following the law of conservation of energy, supplied to the given system, some constant value Eo : E -- Eo should remain. Then from Eq. (20), in the finite difference expression

dEId t = dEoldt ~ AEo/At = max,

or in this forro

Al ~ Eo/m�91 (27)

As E0 is a constant vahe, its ¡ difference would be a constant vahe also. Once AE0 should be ¡ or be given a constant vahe, then Eq. (27) would lead to minimizing the time-interval At:

At = min (28)

or, again in �91 integral expression

X2

6/ T d z = O, (2o)

where T is a functional of time. This constraint condition has been usecl for deter- mining the trajectory of a small ball on a flexible wire mentioned in any textbook of classical mechanics.

The physical meaning of Eq. (29) reveals a dynamic trend of reversible process to maximally accelerate this process.

Perhaps, this transformation can be used for solving some simple irreversible processes also. Once again, ir the medium was incompressible, the system was surrounded by the boundary with constant temperature and the process belonged

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GYARMATI, PRIGOGINE AND HAMILTON (}7

to a stationary one. Ir the process was stationary, then the energy, transferred into the system, should remain a constant value, E0 =const. But from Eq. (10) we have

A E o / A t = min, (30) then we have

X2 q

A t = max, 6 / T dx = O. (31) X I

The physical meaning of Eq. (31) is that among all the evolution trajectories of the given process only one should really exist, by which the time, exhausted in the whole process, would be a maximum value. In another words, the most delayed process would be the actual process. It reveals the existence of a "dynamical" trend to maximally delay this process.

Similarly, regarding Eq. (31) a s a mathematical constraint, one would ob- tain a ~trajectorff of evolution of irreversible process also. Recently, though many achievements have been obtained for the irreversible process theory, still most of them were concerned only with the alateral profile" of process, in other words, only the structure or behaviour of the state at or close to equilibrium was being estab- lished. Unfortunately, not so much attention has been payed to the "longitudinal profile ~ of the process, i.e. to the ~process" itself. The constraint (31) may be helpful for research into this problem.

7. Coex i s t ence o f processes a n d the G y a r m a t i P r inc ip l e

In the above paragraphs the c~nnection between the irreversible process and the reversible process has been constructed by the aid of Gyarmati 's principle, so that these processes could be unified and should follow a general principle. How- ever, these processes have been considered separately, as ir they were conflicting with one another. In fact, sometimes two diiferent processes can coexist in a sys- teta. Ir is implicated that not only ditferent irreversible processes (heat conduction, molecular ditfusion, etc.) would be a mixed process which was considered in many monographs, but also that both reversible and irreversible processes would co exist in one compound process. Such phenomena can really exist: water flow with free sufface in open channel in gravitational field should follow the principle of mŸ mum mechanical energy on the one hand, on the other hand, as water is a viscous medinm, as it flows ir would be damped by internal friction, through which some part of the supplied mechanical energy would be transformed into heat.

This ~bi-process" ought to follow that principle which could combine both principles into one unified principle (Ubi-principle'). Perhaps, the Gyarmati prin- ciple was such a principle, though a great deal of research has remained to be done thereafter. A s a first attack, the velocity distribution both for laminar and tur- bulent flows along the whole depth of river has been considered directly through Eq. (1). (See [9]). Ir the energy transferred into the laminar sublayer was assumed unchanged, then the velocity distribution in this layer could be derived also [10].

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8. C o n c l u s i o n a n d d i s c u s s i o n

Gyarrn�91 principle is �91 general principle. One special case was the pure dissi- pative process under stationary and isothermal condition, in this case the Gyarmati principle should be developed into the Prigogine principle (in an integral forro). An- other special case was the non-dissipative or constant-dissipative process. In this case the Gyaxmati principle should be developed into the Hamilton principle. Hence the Gyarrnati principle should be valid not only for the dissipative process, but also for the non-dissipative or the mixed dissipative/non-dissipative process. Further- more, for the classical formulation of the Hamilton principle it should reflect only its static behaviour, but not refer to its dynamic trends towards this state. With the �91 of Gyaxmati's principle, the Hamilton principle could be generalized into a complete set; ir indicated along which route this process should proceed. In the irreversible process theory most attention has been payed to the "lateral profile ~ of the process, nearly not concerned with the process itself (from start to ¡ This was one of the most diflicult tasks, hopefully, more �91 will be focused on it thereafter.

Because the processes in nature sometimes belong to bi-processes, the clif- ferent principles would give diITerent extrema (maximum or minimum), and the physical picture occurred profound enough. Due to these causes a long-term argu- ment axose about the question, whether the water process would approach a state with a minimum or maximum rate of energy dissipation, the analysis of the relation between these principles, could perhaps lead to cease it ¡

References

I. I. Gyarmati, Non-equilibrium Thermodynamics, Springer, Berlin-Heidelberg-New York, 1970. 2. I. Gyarmati, On the Governing Principle of Dissipative Processes and its Extension to Non-

linear Problerns, Ann. Phys., 23, 78, 1968. 3. I. Gyarmati, Acta Chito. Hung., 43, 353, 1965. 4. I. Prigogine, Introduction to Therrnodynarnics of Irreversible Process, Interscience, New York,

1969. 5. S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam,

1969. 6. P. Glansdor/f and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctua~

tions, Wiley-Interscience, New York, 1971. 7. K. WMhi~.u, VariationM Methods in Elesticity and Plasticity, Pergarnon Press, Oxford, 1968. 8. H. Lippmann, Extrernum and Variational Principles in Mechanics, Springer-Verlag, New York,

1972. 9. H.C. Hou, J. Hydr.Eng., ASCE, 113, 5, 1987.

10. H.C.Hou, Mechanics of Drag Reduction, Sci. Press, Beijing, 1987.

Acta Phymica Huncavica 66, 1989