The vortex theorem in thermodynamics

Transcription

1 L théorèm ds tourbillons n Thrmodynamiqu, J. math. purs t appl. 7 (9), Th vortx thorm in thrmodynamics By M. JOUGUET Translatd by D. H. Dlphnich. Considr a continuous fluid mass that is animatd by a motion that dos not altr its continuity. Lt u, v, w dnot th componnts of th vlocity of th molcul along th thr axs; thy ar continuous functions of tim t and th coordinats x, y z of a gomtric point (viz., Eulr variabls). Th molculs that ar situatd on a closd curv C at th tim t form a closd curv C at any instant. On can stat Hlmholtz s thorm by saying that th curvilinar intgral: u dx + v dy + wdz C prsrvs th sam valu at any instant. Th proof of that thorm that is applicabl to fluids that ar dvoid of viscosity is subordinat to th following rstrictions, in addition: a. Th forcs that act upon ach volum lmnt, whthr intrnal or xtrnal, admit a potntial. b. Th prssur is a function of only th dnsity. Duhm has shown ( ) how thrmodynamics prmits on to study th motion of fluids for which ths rstrictions mak no sns. W propos to study what happns to Hlmholtz s thorm in his thory; w continu to suppos that th viscosity is zro. On will asily prciv vrything that is du to Duhm in th lctur of this not. Not only is his mthod imprintd upon it, but indd som of its pags ar dvotd to rproducing his rsults without modification. W pray that on will xcus us for th fact that h has alrady don that himslf. It sms to us that this rproduction is usful in bringing to light th inadquacy of th statmnts (a) and (b) in hydrodynamics and th possibility of rplacing thm with othr ons that ar mor gnral. ( ) L potntial thrmodynamiqu t la prssion hydrostatiqu, Annals scintifiqus d l Écol Normal supériur (3) (893). Traité d Élctricité t d Magnétism, t. II, 89.

2 Jougut Th vortx thorm in thrmodynamics. I.. In th fluids that ar customarily studid in hydrodynamics, th physical and chmical stat at a point is dfind compltly by th dnsity ρ and th absolut tmpratur T. Hr, w shall suppos that in ordr to mak that dfinition, on must add a finit numbr of algbraic paramtrs to ths two variabls on, for xampl, which w dnot by λ and a finit numbr of gomtric paramtrs on, for xampl, such as th vctor M whos projctions onto th thr axs will b A, B, C. Th variabls ρ, λ, A, B, C, T will b supposd to b normal. Thrmodynamics thn will lad on to put th intrnal thrmodynamic potntial of a fluid mass into th form: () F = ϕ( ρ, λ, A, B, C, T ) ρ dτ + Ψ. E In that xprssion, ϕ (ρ, λ, A, B, C, T) ρ dτ is th intrnal thrmodynamic potntial of th volum lmnt dτ ; th sign rprsnts a tripl intgral that is xtndd ovr all of th volum E of th fluid. Ψ is a complmntary trm that dpnds upon th rlativ position of th various fluid lmnts and th variabls that fix th stat of ach of thm xcpt for th tmpratur. Th most natural hypothsis that on can mak on Ψ is that it can b writtn in th form of a sxtupl intgral ( ):, () Ψ = ψ ( ρ, λ, A, B, C, ρ, λ, A, B, C, x, y, z, x, y, z ) ρρ d τ d τ whos fild is th six-dimnsional st that is obtaind by succssivly associating ach point of E with all of th othr points of th sam domain. ρ, λ, A, B, C rfr to th volum lmnt dτ with th coordinats x, y, z; ρ, λ, A, B, C rfr to th lmnt dτ with th coordinats x, y, z. In truth, sinc a fluid is isotropic, ϕ dpnds upon A, B, C only through th intrmdiary of M. Similarly, A, B, C, A, B, C, x, y, z, x, y, z do not ntr into th function ψ in an arbitrary mannr. That function dpnds upon only th magnitud of th vctors M and M, th angl that thy mak btwn thm, th ons that thy mak with th lin that joins th lmnts dτ and dτ, and th lngth of that lin. Howvr, ths rmarks will b of no intrst in th contxt of what w hav in mind. W suppos that th intgral Ψ can b calculatd by two succssiv tripl intgrals, and w st: (3) V (x, y, z, t) = ψρ dτ. E Th point x, y, z can b found insid of E or outsid of it. By hypothsis, th function V xists in both domains. W suppos, in addition, that it is continuous in ach of ths domains and that it will admit first-ordr partial drivativs with rspct to x, y, z that ar givn by th rul for drivation undr th sign. Ths ar th hypothss that wr ( ) DUHEM, L potntial thrmodynamiqu t la prssion hydrostatiqu.

3 Jougut Th vortx thorm in thrmodynamics. 3 mad by Duhm in his mmoir on th thrmodynamic potntial and hydrostatic prssur that w hav alrady citd. In that mmoir, on will find th analytical discussion of thm, which w shall not duplicat. It prmits on to writ: V x and if w st: (4) ψ ρ ψ ψ = ρ dτ + ρ dτ + ρ dτ x x ρ x + A ψ d B ψ d C ψ ρ τ ρ τ ρ dτ x A + x B + x C, ψ ψ ψ X i = ρ dτ, Yi = ρ dτ, Zi = ρ dτ, x y z ψ ψ ψ ai = ρ dτ, bi = ρ dτ, ci = ρ dτ, A B C ψ I = ρ d τ, ρ ψ Li = ρ d τ thn th valu of V / x will bcom: (5) V ρ A B C = X i I Li ai bi ci. x x x x x x On will likwis hav : V ρ A B C = Yi I Li ai bi ci. y y y y y y V ρ A B C = Zi I Li ai bi ci. z z z z z z Th actions that th rst of th masss xrt upon th lmnt dτ ar composd of a X + Y + Z and th influncs ρ dτ I, ρ dτ L i, ρ dτ a i, ρ dτ b i, ρ dτ c i. forc ρ dτ ( i i i ) Equations (5) show that th forc is not drivd from a potntial; that is why th rstriction (a) of paragraph will not b statd hr. On rmarks that th xistnc of th influncs ρ dτ a i, ρ dτ b i, ρ dτ c i supposs that a coupl acts upon th lmnt dτ : Indd, th virtual work that it dos is non-zro whn th lmnt dτ turns around itslf. On knows that similar coupls ar ncountrd in th study of magntic bodis, and if on adopts th idas of Hlmholtz, in that of th intraction btwn two lctric currnts. 3. Th actions that bodis that ar outsid of th fluid xrt upon it ar, in on cas, th prssur P dω that is applid to ach lmnt dω of th xtrior surfac, and in th

4 Jougut Th vortx thorm in thrmodynamics. 4 othr, th forcs ρ ( X i Yi Zi ) + + dτ and th influncs ρ a dτ, ρ b dτ, ρ c dτ that ar applid to ach volum lmnt dτ. On must add th inrtial forcs: ρ ( jx jy jz ) + + dτ to th lattr, in which j is th acclration. Lt us xprss th virtual work that is don by ths actions. Imagin a virtual modification for which ach matrial point is displacd by: δ x + δ y + δ z. Ω Ω Ω Ω Figur Th volum that is occupid by th fluid is dformd. Bfor th modification, it consistd of th part and th infinitly small part, which wr dividd along th portion Ω of th original surfac. Aftr th modification, it will b composd of and th spac that is confind to along th portion Ω of th original surfac. W agr to dnot th variations that th paramtrs ρ, λ, A, B, C, T xprinc at ach gomtric point by th symbol δ and th ons that thy xprinc at ach matrial point by th symbol. Th virtual work of th xtrnal actions and th inrtial forcs is: δt + δj = S [ P cos( P, x) δ x + P cos( P, y) δ y + P cos( P, z) δ z] dω Ω +Ω + + ρ [( X j ) δ x + ( Y j ) δ y + ( Z j ) δ z x y x + L λ + a A + b B + c C] dτ. On obviously has: λ = δλ + δ x + δ y + δ z, and on can writ analogous rlations for A, B, C. Th xprssion for δt + δj can b transformd into:

5 Jougut Th vortx thorm in thrmodynamics. 5 (6) S δt + δ J = [ P cos( P, x) δ x + P cos( P, y) δ y + P cos( P, z) δ z] dω Ω +Ω + ρ [ L δλ + a δ A + b δ B + c δ C] dτ A B C + ρ X jx + L + a + b + c δ x dτ x x x x A B C + ρ Y jy + L + a + b + c δ y dτ y y y y A B C + ρ Z jz + L + a + b + c δ z dτ. z z z z 4. On will obtain th quations of motion by xprssing th ida that: δf δt δj = for any virtual modification for which th tmpratur of ach molcul rmains constant. In th squl, th variabls A, B, C will play a rol that is idntical to that of λ, as thy hav don up to now. On can simplify th notation gratly by now supposing that for th sak of calculations thy do not xist. It is asy to r-stablish th trms that thy contribut at th nd by analogy with th ons that ar givn for λ. 5. Th calculation of δ ϕ ρ dτ can b prformd by following th mthod that was dvlopd by Duhm in his mmoir Sur l équilibr t l mouvmnt ds fluids mélangs ( ). On sts: H = ϕ + ρ, ρ (7) ϕ ES =. S will b th ntropy of th unit of mass of th fluid, which is rgardd as homognous. Th dsird variation will b: ( ) Travaux t Mémoirs ds Facultés d Lill, t. III, Mémoir no., Chap. VI, pp. 9; 893.

6 Jougut Th vortx thorm in thrmodynamics. 6 (8) δ ϕ ρ dτ = ρ δλ δτ H H + ρ ES x ES y + x x δ + + x x δ H + + ES δ z dτ x x + S [ x cos( n, x) + y cos( n, y) + z cos( n, z)] d, ρ ρ δ δ δ ω Ω +Ω in which n dnots th intrior normal to th fluid. 6. W nonthlss insist upon th calculation of δψ. Ψ varis for two rasons:. Du to th variations dρ, dλ, dρ, dλ at any gomtric point of th fild.. Du to th annihilation of mattr that is containd in and th cration of mattr that is containd in. On will writ: (9) δψ = ( ψ + δψ )( ρ + δρ)( ρ + δρ ) dτ dτ + ( ψ + δψ )( ρ + δρ)( ρ + δρ ) dτ dτ, + ( ψ + δψ )( ρ + δρ)( ρ + δρ ) dτ dτ + + ( ψρρ dτ dτ d d d d ψρρ τ τ ψρρ τ τ, ). Th symbol rprsnts th six-dimnsional st that is obtaind by succssivly associating ach point of with all of th othr points of. Th symbol, rprsnts a st that is composd of two parts, th first of which is obtaind by succssivly associating ach point of with all th points of, and th scond of which, by associating ach point of with all th points of. Th symbols,, and, hav analogous intrprtations. First considr th two intgrals that rlat to th rgion whos diffrnc figurs in (9). By rason of symmtry, that diffrnc will b qual to: d d ψ ψ ρ ψ δρ τ τ + ρ ρ δρ dτ dτ + ρρ δλ dτ dτ ρ or rathr: (9 ) V δρ dτ ρi δρ dτ ρ Li δλ dτ,.

7 Jougut Th vortx thorm in thrmodynamics. 7 Morovr, δρ can b xprssd as a function of δx, δy, δz. Indd: ρ ρ ρ δρ = ρ δ x δ y δ z and δ x δ y δ z ρ = ρ + +. Hnc: δρ = ( ρ δ x) ( ρ δ y) ( ρ δ z) + +. An intgration by parts thn transforms th sum (9 ) into: (9 ) ( V ρi ) ( V ρi ) ( V ρi ) ρ δ x + δ y + δ z dτ ρl i δλ dτ + S ( ρv ρ I )[ δ x cos( n, x) + δ y cos( n, y) + δ z cos( n, z)] dω. Ω +Ω Now, tak th intgrals that rlat to,, and in (9). On can writ thm as: namly: ρ d τ ψ ρ d τ ρ d τ ψ ρ d τ +, V ρ dτ. Similarly, th ons that ar takn ovr,, and giv: V ρ dτ. In rgard to th lattr trm, w rmark that th δρ and th δλ ar not infinitly small in th rgion. Howvr, th intgrals that rlat to thm ar: On calculats thm by giving th valus to ρ, λ, V that thos quantitis hav at th points of that ar infinitly clos to th surfac Ω bfor th modification. Th diffrnc (9 ) V ρ dτ V ρ dτ is writtn: S ρ V [δx cos(n, x) + δy cos(n, y) + δz cos(n, z)] dω. Ω +Ω In this calculation, on supposs that: ρ dτ d ψ ρ τ and + ρ dτ ψ ρ dτ ar qual, up to scond ordr. Indd, that is what will happn, in gnral, with th hypothss that Duhm statd for th function ψ. Howvr, it will b fals if on

8 Jougut Th vortx thorm in thrmodynamics. 8 supposs, for xampl, as on dos in th thory of capillarity, that this function has a maningful valu only whn th points (x, y, z), (x, y, z ) ar vry clos to ach othr. W lav that cas asid, and rmark simply that if on ncountrs it thn th only things that will b modifid will b th doubl intgrals of δψ, but not th tripl intgrals. Now, th lattr suffic for th purpos that w hav in mind. By virtu of (9), (9 ), (9 ), w thn writ: () ( V ρi ) ( V ρi ) ( V ρi ) δψ = ρ x y z d L δ + δ + δ τ ρ iδλ dτ S ρ I [ δ x cos( n, x) + δ y cos( n, y) + δ z cos( n, z)] dω. Ω +Ω 7. Formulas (6), (8), () giv δf δt δj. That xprssion is composd of a sum of volum intgrals and surfac intgrals. Th considration of th volum intgrals will giv th quations of motion for th intrior of a fluid mass. W writ thm by rstablishing th trms that ar du to A, B, C: () () = L + Li, = a + ai, A = b + bi, B = c + ci, C H ( V ρi) A B C + ES + = X jx + L + a + b + c, x x x x x x x H ( V ρi) A B C + ES + = Y jy + L + a + b + c, y y y y y y y H ( V ρi) A B C + ES + = Z jz + L + a + b + c. z z z z z z z Suppos that: X dx + Y dy + Z dz + L dλ + a da + b db + c dc ES dt is th xact diffrntial of a function R(x, y, z, t) with rspct to x, y, z at ach instant. Equations () will b writtn:

9 Jougut Th vortx thorm in thrmodynamics. 9 ( ) ( H + V ρi + R) = jx, x ( H + V ρi + R) = jy, y ( H + V ρi + R) = jz, z which ar formulas that can b infrrd from Hlmholtz s thorm in a known way [s POINCARÉ, Théori ds tourbillons, Chap. I. Equations ( ) hav th form of quations (7), pp. in that book.] Th function R will xist, in particular, whnvr th following conditions ar satisfid: a. Th xtrnal actions admit a potntial; i.., X dx + Y dy + Z dz + L dλ + a da + b db + c dc is th xact diffrntial of a function Ω (x, y, z, λ, A, B, C). b. At any instant, thr xists a rlation: (3) K (S, T) = btwn tmpratur and ntropy pr unit mass in th ntir fluid mass. Th lattr condition will b, in turn, crtainly fulfilld if, upon starting with a stat in which th rlation (3) is tru, ach lmnt of mattr xprincs only transformations for which that rlation dos not cas to b vrifid. In particular, it will b fulfilld if, upon starting at an instant at which all of th mass is homognous, ach matrial lmnt xprincs only isothrmal or adiabatic transformations. 8. A considration of doubl intgrals of δf δt δj will giv th boundary conditions. Thy ar: P cos( P, x) = ρ ρ I cos( n, x), ρ (4) P cos( P, y) = ρ ρ I cos( n, y), ρ P cos( P, z) = ρ ρ I cos( n, z). ρ Ths quations show, first of all, that P has th sam dirction as n (th prssur must thn b normal to th surfac); it thn shows that P must hav th valu ρ ρ ρ I.

10 Jougut Th vortx thorm in thrmodynamics. Σ E S E S Figur. In ordr to dfin th prssur that is xrtd on an intrior lmnt dσ, pass a surfac Σ through dσ that divids th fluid into two parts E and E. Rmov E without supprssing th action that it xrts upon th volum lmnts of E, and which will hav: ρ dτ ψρ dτ E E for a potntial. In ordr to maintain d Almbrt s fictitious quilibrium in E, on must thn apply an xtrior prssur Π to ach lmnts dσ of Σ. By dfinition, Π will b th prssur insid of th fluid. W shall now calculat it. Th thrmodynamic potntial in E is: ϕρ dτ + E ψρρ dτ dτ. E E By virtu of formulas (8) and (), its variation undr a virtual modification will b: H ρ δλ dτ + ρ ES δ x λ + x x H H + + ES δ y + + ES δ z dτ y y z z S ρ [δx cos (n, x) + δy cos (n, y) + δz cos (n, z)] dω +Σ ρ ( V ρi ) ( V ρi ) ( V ρi ) ρl i δλ dτ + ρ δ x δ y δ z dτ E E + + ρ I [δx cos (n, x) + δy cos (n, y) + δz cos (n, z)] dω. + E E S S S +Σ V, I, L i ar th intgrals V, I, L i, whn thy ar takn in only th rgion E. Th work that is don by th action that is xrtd by E on E, with th opposit sign, has th xprssion: ψ ψ ρ ρ τ λ ρ dτ ρ [ d + E E E

11 Jougut Th vortx thorm in thrmodynamics. namly: or finally: + δx ψ ψ ψ ρ dτ + δλ ρ dτ δ z ρ dτ ρ dτ E + E E, V V V δ x + δ y + δ z ρ dτ ( I δρ + L iδλ) ρ dτ E, E ( V ρ I ) ( V ρ I ) ( V ρ I ) ρ L i δλ dτ + ρ δ x δ y δ z dτ E + + S ρ I [δx cos (n, x) + δy cos (n, y) + δz cos (n, z)] dω S +Σ E V, I, L i ar th intgrals V, I, L i, whn thy ar takn in only th rgion E. Finally, th virtual work that is don by th othr xtrnal actions is, with th opposit sign, th xprssion: ρ ( X jx ) δ x + ( Y jy ) δ y + ( Z jz ) δ z dτ E S P [cos (P, x) δx + cos (P, y) δy + cos (P, z) δz] dω S S Π [cos (Π, x) δx + cos (Π, y) δy + cos (Π, z) δz] dω. Σ Upon quating th sum of th thr xprssions abov to zro, and rmarking that: V = V + V, I = I + I, L i = L i + L i, on will obtain th quations of motion for th mass E in th form (), (), and th valu of Π in th form: (5) Π = ρ ρ I, ρ and on vrifis that Π is normal to dσ. Th rlation (5) shows that th prssur on th lmnt dσ dpnds upon th stat of th total mass by th intrmdiary of I. On ss how difficult that it would b to introduc th rstriction (b) of no. hr. 9. W rcall that an xampl of a fluid to which th prcding considrations will apply is givn by th ons whos lmnts oby th law that was proposd by Fay in his studis on th tails of comts.

12 Jougut Th vortx thorm in thrmodynamics. II.. Th hypothss that wr mad in no. on th complmntary trm Ψ ar not ncssary. Wakly-magntic prfct fluids offr an xampl in which thy ar not tru. Th action of a magntic mass on an intrior lmnt is not dfind. That implis th impossibility of putting Ψ into th form (). That also implis th impossibility of stating th rstriction (a) of no. for ths bodis. Howvr, if on adopts th idas of Duhm ( ), on can writ th intrnal thrmodynamic potntial of a systm that is composd of prmannt and immobil magnt P and a wakly-magntic prfct fluid E as follows: P+ E. P+ E (6) F = ϕ( ρ, M, T ) ρ dτ ( Aα + Bβ + Cγ ) dτ P Figur 3. Hr, M = A + B + C is th magntic momnt of th lmnt dτ. As for α, β, γ, thy ar th componnts of th magntic fild that is insid th magntic mass along th thr axs. W assum, morovr, that th virtual work that is don by th inrtial forcs and xtrnal actions and act upon th fluid has th form: (7) S δt + δ J = [ P cos( P, x) δ x + P cos( P, y) δ y + P cos( P, z) δ z] dω Ω +Ω + ρ[( X ) ( ) ( ) ], jx δ x + Y jy δ y + Z jz δ z dτ which is only a particular cas of th form (6). Th variation of ϕρ dτ can b calculatd by formula (8): It will b: ( ) Traité d Élctricité t d Magntism, t. II, pp. 59 and 45.

13 Jougut Th vortx thorm in thrmodynamics. 3 (8) ρ δ ϕρ dτ = ( A δ A + B δ B + C δc) dτ M M H + ρ ES x + δ x x H H + + ES δ y + + ES δ z dτ y y z z + S [ x cos( n, x) + y cos( n, y) + z cos( n, z)] d, Ω +Ω ρ ρ δ δ δ ω in which H and S ar dfind by th qualitis (7), as always. Howvr, th variation of Ψ = ( Aα + Bβ + Cγ ) dτ cannot b obtaind by P+ E applying th prcding rsults. It can nonthlss b calculatd by mthod that is compltly analogous to th on that w discussd abov. It was Liénard who first gav th xact valu ( ). W contnt ourslvs by pointing out th rsult of his calculations, whil rturning to his work in th work that follows. (9) δψ = ( α δ A + β δ B + γ δc) dτ + S [ α A + β B + γ C + π M cos ( M, n)] Ω +Ω [ δ x cos( n, x) + δ y cos( n, y) + z cos( n, z)] dω.. Th xprssion δf δt δj is a sum of tripl and doubl intgrals. A considration of th formr will giv th quations of motion insid of th fluid: () M A = α, ρ M M B = β, ρ M M C = γ, ρ M ( ) Prssions à l intériur ds aimants t ds diélctriqus, Lumièr élctriqu, t. II, pp. 7; 984.

14 Jougut Th vortx thorm in thrmodynamics. 4 () H x H y H z + ES = X jx, x + ES = Y jy, y + ES = Z jz. z Imagin that X dx + Y dy + Z ES dt is an xact diffrntial with rspct to x, y, z of a function R (x, y, z, t) at any instant. Equations () thn tak th form: ( ) ( H + R) = jx, x ( H + R) = jy, y ( H + R) = jz, z which lads to Hlmholtz s thorm. Th function R will xist, in particular, whnvr conditions (a ) and (b ) of no. 7 ar vrifid.. Th boundary conditions ar givn by considring th doubl intgrals of δf δt δj. Upon taking () into account in ordr to transform th trm αa + βb + γ C, and upon stting: () Π = ρ + Mr + πm cos (M, n), ρ M on will hav: P cos( P, x) = Π cos( n, x), (3) P cos( P, y) = Π cos( n, y), P cos( P, z) = Π cos( n, z). As in no. 8, on confirms that th prssur on an lmnt dσ that is intrior to th fluid is prcisly Π, and that th prssur is normal to dσ, but it dpnds upon its orintation ( ), vn though th prssur at a point undfind, and th rstriction (b) of no. will mak no mor sns hr than th rstriction (a). 3. In truth, whn a magntic fluid is in motion, ithr conduction currnts or displacmnt currnts will b producd in it. Th study of that motion thus blongs to ( ) LIÉNARD, loc. cit.

15 Jougut Th vortx thorm in thrmodynamics. 5 lctrodynamics and gos byond th usual mthods of thrmodynamics (Duhm has insistd upon this point); Hlmholtz s mthods will prmit on to addrss it in all of its complxity. Th cas that w hav tratd is an idal cas, namly, th cas of a fictitious fluid that is nithr a conductor nor on that is suscptibl to dilctric polarization. It thus prsnts vry littl of intrst in th thory of lctricity. Hr, w shall assum th viwpoint of nrgtics, and w would simply lik to show by that xampl that is is possibl for matrial systms to xist for which th form () of th trm Ψ is no longr xact, and for which th notion of intrnal forc, which is paramount in mchanics, fails compltly, but can b ffctivly rplacd by that of nrgy. III. 4. W hav thus givn a nw statmnt (a ), (b ) to th rstrictions (a), (b) to which th proof of Hlmholtz s thorm is subordinat that has th advantag of applying to th cass in which th usual statmnts hav no maning. Th conditions (a ), (b ) prsnt th gratst analogy with th ons that Duhm found whil sking th cas in which th quations of motion admit a vis viva intgral ( ) for systms that dpnd upon a finit numbr of variabls. Th following proof of Hlmholtz s thorm will prhaps bring to light th rasons for that paralllism bttr than th prcding dvlopmnts did. 4 3 n Figur 4 In ordr to show that u dx + v dy + w dz is constant, it is ncssary and sufficint to show that j dx x + j y dy + j z dz is zro (s POINCARÉ, Théori ds tourbillons, pp. ). Considr a fluid ring along C with an infinitly small sction that w divid by sctions that ar normal to C into lmnts,,, n that ar infinitly small in thir thr dimnsions and that all contain th sam mass dm. Sinc a matrial systm is dfind by normal variabls, on knows that undr an arbitrary virtual displacmnt (isothrmal or not) th variation of th intrnal thrmodynamic potntial minus th trms that ar du to th variation of th tmpratur will b qual to th forc that is don by th xtrnal actions plus th forcs of inrtia. W writ down that quality for th ring C, ( ) L intégral ds forcs vivs n Thrmodynamiqu, Journal d Mathématiqus purs t appliqués (5) 4 (898), 5.

16 Jougut Th vortx thorm in thrmodynamics. 6 and for th modification of th location and physical and chmical stat that ach lmnt must xprinc, will rplac,, n will rplac. Th thrmodynamic potntial of th ring has th form: n ϕ (ρ, λ, T) dm + ζ, in which ζ dos not dpnd upon th tmpratur of th lmnts,,, n, and will onc mor tak th sam valu whn th total mass of th fluid rturns to th sam stat. Th lft-hand sid of our quality will thn b: Th xtrnal actions consist of: n dt dm. Firstly, th ons that ar du to bodis that ar forign to th fluid. Th virtual work that thy do is, by hypothsis, of th form: n X δ x + Yδ y + Zδ z + L δ x + δ y + δ z dm. On thn has th actions that ar xrtd on th ring by th rst of th fluid. Th work that thy do δη will b zro, sinc all of th mass will rturn to th sam stat. Finally, on has th prssurs that ar applid to th surfac of th ring. Th work that thy do is zro, sinc th fluid is inviscid, so thy will b normal to th lmnts thy prss upon; i.., to th path that is travrsd. W must thn writ: n dt dm = n X δ x + Yδ y + Zδ z + L δ x + δ y + δ z dm n ( j x dx + j y dy + j z dz) dm. In ordr to calculat ach of th trms in that quation, on can obviously no longr imagin ach lmnt substituting for th on that follows, but only that on of thm, for xampl maks a complt circuit around C. On will thn gt: dt = ( X ) dx + Y dy + Zdz + Ldλ ( jxdx + jydy + jzdz).

17 Jougut Th vortx thorm in thrmodynamics. 7 Upon xprssing th ida that ( jxdx + jydy + jzdz) is zro, on will rcovr th sufficint conditions (a ) and (b ) of nos. 7 and. If thy ar satisfid thn on can say that thr xists a vis viva intgral undr th virtual displacmnt that w hav considrd for th lmnt. 5. That proof of Hlmholtz s thorm is only a gnralization of a mthod that was pointd out by Lagrang for finding th quilibrium conditions for liquids ( ). It prsnts two inconvnint aspcts. It lavs th virtual modification that th ring C xprincs somwhat obscur; it is only by having a vry confusd notion of th natur of a fluid that on can glimps th possibility that it will xist. Morovr, it supposs (with no xplanation) that th prssur is normal to th lmnt that it prsss upon. From ths viwpoints, it thrfor cannot rplac th dvlopmnts of nos. through. Thy hav shown us that th normal dirction of th prssur will rsult from th fact that th thrmodynamic potntial ϕ ρ dτ dpnds upon th distribution of mattr around th point x, y, z by th intrmdiary of only its dnsity. Howvr, by contrast, it has th advantag of bing indpndnt of th form of th trm Ψ in no.. In fact, th hypothss that wr mad on δζ and δη (which rplac δψ, hr) follow dirctly from th principls of thrmodynamics, and it sms difficult to ignor that fact. IV. 6. In th motion of a fluid mixtur, Duhm has shown that Hlmholtz s thorm applis to any fluid, in particular, whn th xtrnal actions admit a potntial, and if th tmpratur of th mass is uniform at any instant ( ). Its proof supposs that th various lmnts of th fluid masss xrt no action upon ach othr. It is asy to rmov that rstriction. For xampl, tak two fluids that w distinguish by th indics and. Lt ρ b th dnsity of th mixtur, lt ρ, ρ b th partial dnsitis (ρ = ρ + ρ ), lt λ, λ b two arbitrary paramtrs, and lt T b th tmpratur. W writ th thrmodynamic potntial: F = ϕ (ρ, ρ, λ, λ, T) ρ dτ (ρ, ρ, λ, λ, + ψ ρ, ρ, λ, λ, x, y, z, x, y, z ) ρρ dτ dτ. On supposs that th virtual work that is don by th xtrnal actions has th form: δt = ( X δ x + Y δy + Z δz + L λ ) ρ dτ + ( X δ x + Y δy + Z δz + L λ ) ρ dτ + S P [cos(n, x) δx + cos(n, y) δy + cos(n, z) δz] dω, ( ) Mécaniqu analytiqu, Part I, sction VII, art. 7. ( ) Équilibr t mouvmnt ds fluids mélangs, pp..

18 Jougut Th vortx thorm in thrmodynamics. 8 δx, δy, δz rfr to th fluid, whil δx, δy, δz rfr to fluid. On th surfac, th xprssion: cos (n, x) δx + cos (n, y) δy + cos (n, z) δz will b th sam for both fluids ( ). W st: H = ϕ + ρ, ρ H = ϕ + ρ, ρ ψ I = ρ d τ, ρ ψ I = ρ d τ, ρ ψ L i = ρ d τ, ψ L i = ρ d τ, Th quations of motion ar obtaind by a mthod that is compltly analogous to th on that w applid abov. On will thus arriv at th following formulas for fluid : = L + L i, H ω ( V ρi) + + = X + L x T x x x j x, H ω ( V ρi) + + = Y + L y T y y y j y, H ω ( V ρi) + + = Z + L z T z z z j z, and som analogous formulas for fluid. ω and ω dnot som functions that vrify th quality ω ω = and that thrmodynamics is powrlss to dtrmin. If th xtrnal actions admit a potntial, and if th tmpratur is uniform in th mass ϕ ω at any instant thn Hlmholtz s thorm will b tru. It is furthr tru that + is a ρ function of only T. Unfortunatly, it is difficult to s what sort of physical rality would corrspond to that statmnt. In particular, it is difficult to find th cas hr that ( ) DUHEM, Équilibr t mouvmnt ds fluids mélangs, pp. 37.

19 Jougut Th vortx thorm in thrmodynamics. 9 corrsponds to th adiabatic motions of isolatd fluids. Thr is, morovr, no rason to mphasiz that th quantity of hat that is rlasd by a fluid lmnt that is mixd with anothr fluid will not b a wll-dfind quantity.