Thermodynamic form of the equation of motion for perfect fluids of grade n
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Thermodynamic form of the equation ofmotion for perfect fluids of grade n Translation of C. R. Acad. Sci. Paris t. 305, II, p. 833-838 (1987)
Henri Gouin
Universit´e d’Aix-Marseille & C.N.R.S. U.M.R. 6181,Case 322, Av. Escadrille Normandie-Niemen, 13397 Marseille Cedex 20 France
Abstract
We propose a thermodynamic form of the equation of motion for perfect fluidsof grade n which generalizes the one given by J. Serrin in the case of perfectlycompressible fluids ([1], p. 171). First integrals and circulation theorems are deducedand a classification of the flows is given. Key words:
Conservative fluid motions ; Thermodynamics ; Fluids of grade n .
PACS:
In continuum mechanics, first gradient media cannot give a model for fluidswith strong variations of density. Material surfaces need their own character-istic behavior and properties of energy [2]. D.J. Korteweg has been the first topoint out the convenience of fluids of grade upper than one and not to modelinterfacial layers by means of discontinuity surfaces in liquid-vapor interfaces[3]. Recently, this approach was used in the case of dynamic changes of phases[4].Till now, the thermodynamics of fluids was neglected and it was not possibleto model flows with strong variations of temperature such as those associatedwith combustion phenomena or non isothermal interfaces. In fact, it is difficultto take the gradients of temperature into account (it is not possible to consider
Email address: [email protected] (Henri Gouin).
Preprint submitted to : June 29, 1987, (in French version). 31 October 2018 virtual displacement of temperature) but we can use the specific entropythrough internal energy density.To this aim, we usefully describe conservative flows of a compressible fluid bythe means of its internal energy depending on both entropy and density. Thiswill only be a limit mathematical model, however the study of mathematicalstructure of the equations of motion is fundamentally necessary. So, to improvethe model for strong variations of entropy and density in interfacial layers weconsider an internal energy function of the two quantities and their spatialgradients up to a n − of grade n [5].The paper aims to show that the equations of motion for perfect fluids of anygrade can be written in an universal thermodynamic form structurally similarto the one given by J. Serrin in the case of conservative perfect fluids [1].When the thermodynamic form is applied to the second gradient fluids [6,7],it leads to three results: first integrals associated with circulation theorems(such as Kelvin theorems), potential equations representing the motion of thefluid [8,9] and a classification of the flows similar to the one of compressibleperfect fluids [10,11]. n . Perfect fluids of grade n ( n is any integer) are continuous media with aninternal energy per unit mass ε which is a function of the specific entropy s and the density ρ in the form : ε = ε ( s, grad s, ..., (grad) n − s, ρ, grad ρ, ..., (grad) n − ρ )where (grad) p , p ∈ { , ..., n − } , denotes the successive gradients in the space D t occupied by the fluid at present time. We easily may consider the caseof an inhomogeneous fluid but, for the sake of simplicity we will not do it.So, the material is supposed to have infinitely short memory and the motionhistory until an arbitrarily chosen past does not affect the determination ofthe stresses at present time. The virtual works principle (or the virtual powers principle) is a convenientway to find the equation of motion. For conservative motions, it writes as theHamilton principle [8].A particle is identified in Lagrange representation by the position X ( X , X , X )2ccupied in the reference space D . At time t, its position is given in D t bythe Eulerian representation x ( x , x , x ) . The variations of particles motion are deduced from the function family : X = ψ ( x , t ; α ) (1)where α denotes the parameter defined in the vicinity of 0 associated with afamily of virtual motions of the fluid. The real motion corresponds with α = 0[1].Virtual displacements associated with any variation of the real motion can bewritten in the form : δ X = ∂ ψ ∂α ( x , t ; α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α =0 . This variation is dual and mathematically equivalent to Serrin’s one ([1], p.145, [8]). Let L be the Lagrangian of the fluid of grade n : L = ρ (cid:18) V ∗ V − ε − Ω (cid:19) where V denotes the velocity of particles, Ω the potential of mass forces definedon D t and ∗ the transposition in D t . Between times t and t , the Hamiltonaction writes [1,8] : a = Z t t Z D t L dv dt. where dv denotes the volume element.The density satisfies the conservation of mass : ρ det F = ρ ( X ) (2)where ρ is defined on D and F is the gradient of deformation. The motion issupposed to be conservative, then the specific entropy is constant along eachtrajectory : s = s ( X ) . (3)Classical calculus of variations yields the variation of Hamilton action :From δa = a ′ ( α ) | α =0 , we deduce, 3 a = Z t t Z D t (cid:20) ( Lρ − ρ ε ′ ρ ) δρ + ρ V i δV i − ρ ε ′ s δs (4) − ρ ( ε, ρ, i δρ, i + ... + ε, ρ, i ...in − δρ, i ...i n − + ε, s, i δs, i + . . . + ε, s, i ...in − δs, i ...i n − ) (cid:21) dx dx dx dt. The definition of dual virtual motions yields : δ (grad) p ρ = (grad) p δρ and δ (grad) p s = (grad) p δs. By using the Stokes formula, let us integrate by parts. Virtual displacementsare supposed to be null in the vicinity of the edge of D t and integrated termsare null on the edge. We deduce : δa = Z t t Z D t (cid:26)(cid:20) ( Lρ − ρ ε ′ ρ − n − X p =1 ( − p ( ρ ε, ρ, i ...ip ) , i ...i p (cid:21) δρ (5) − (cid:20) ρ ε ′ s + n − X p =1 ( − p ( ρ ε, s, i ...ip ) , i ...i p (cid:21) δs − ρ V i δV i (cid:27) dx dx dx dt. With div p denoting the divergence operator iterated p times on the edge of D t , we obtain Rel. (5) in tensorial form : δa = Z t t Z D t Lρ − ρ ε ′ ρ − n − X p =1 ( − p div p ρ ∂ε∂ (grad) p ρ ! δρ − ρ ε ′ s + n − X p =1 ( − p div p ρ ∂ε∂ (grad) p s ! δs − ρ V ∗ δ V dv dt. By taking (2) into account, we obtain : δρ = ρ div δ X + 1det F ∂ρ ∂ X δ X where div denotes the divergence operator relatively to Lagrange variables in D . We also get : δs = ∂s ∂ X δ X . The definition of velocity implies : ∂ X ∂ x ( x , t ) V + ∂ X ∂t ( x , t ) = 0 , ∂δ X ∂ x V + ∂ X ∂ x δ V + ∂δ X ∂t = 0 . Let us consider δ V = − F (cid:5) δ X , where (cid:5) denotes the material derivative. Denoting p = ρ ε ′ ρ + ρ n − X p =1 ( − p div p ρ ∂ε∂ (grad) p ρ ! θ = ε ′ s + 1 ρ n − X p =1 ( − p div p ρ ∂ε∂ (grad) p s ! h = ε + pρ and m = 12 V ∗ V − h − Ω , then, Rel. (5) yields : δa = Z t t Z D t " m δρ − ρ θ δs + ρ (cid:5) ( V ∗ F ) δ X dv dt = Z t t Z D ρ " ( (cid:5) V ∗ F ) − θ grad ∗ s − grad ∗ m δ X dv dt where grad denotes the gradient operator in D . The principle for any displacement δ X null on the edge of D , δa = 0 implies: (cid:5) V ∗ F = θ grad ∗ s + grad ∗ m. (6)Noting that ( Γ ∗ + V ∗ ∂ V ∂ x ) F = (cid:5) V ∗ F , we get : Γ = θ grad s − grad( h + Ω) . (7)Taking Rel. (3) into account, we obtain : (cid:5) s = 0 . Relation (7) is the generalization of Rel. (29.8) in [1]. This is a thermodynamicform of the equation of motion of perfect fluids of grad n .Obviously, term p has the same dimension as pression, θ has the same di-mension as temperature and h has the same dimension as specific enthalpy. It5eems natural to call them pression, temperature and enthalpy of the fluid ofgrad n , respectively. n . Relation (7) leads to the same conclusions as those obtained in [1], [7,8,9,10,11]but for fluids of grad n . Let us remind the most important results.With J denoting the circulation of velocity vector along a closed fluid curve C convected by the flow, dJdt = Z C θ ds. The
Kelvin theorems are deduced: the circulation of the velocity vector alonga closed, isentropic (or isothermal) fluid curve is constant.For any motion of fluids of grad n , we can write the velocity field in the form: V = grad ϕ + ψ grad s + τ grad χ, (8)the scalar potentials ϕ, ψ, s, τ and χ verifying : (cid:5) ϕ = 12 V ∗ V − h − Ω , (cid:5) τ = 0 , (cid:5) ψ = θ, (cid:5) χ = 0 , (cid:5) s = 0 . (9)Eqations (8) and (9) induce the same classification as for conservative flowsof compressible perfect fluids [8,9] : Oligotropic motions. -
They are motions for which surfaces of equal entropyare vortex surfaces. The circulation of the velocity vector along a closed, isen-tropic fluid curve is null. Equation (8) of the motion yields : V = grad ϕ + ψ grad s. Homentropic motions . - In the whole fluid s is constant and Eq. (8) yields : V = grad ϕ + τ grad χ . The Cauchy theorem can be easily written : ddt ( rot V ρ ) = ∂ V ∂ x rot V ρ . H = V ∗ V + h + Ω , Eq. (7) yields the Crocco-Vazsonyi equationgeneralized to stationary motions of perfect fluids of grad n :rot V × V = θ grad s − grad H . The laws of conservation expressed by the Kelvin theorems correspond to thegroup of permutations of particles of equal entropy.This group keeps the equations of motion invariant. It is associated to anexpression of Noether’s theorem as in [12]. So, it is natural to conjecture suchresults for a general fluid whose internal energy is a functional of the densityand the entropy.
References [1] J. Serrin, Encyclopedia of Physics, VIII/1, Springer, Berlin, 1959.[2] M. Barrere and R. Prud’homme, Equations fondamentales de l’Aerothermo-chimie, Masson, Paris, 1973.[3] D.J. Korteweg, Archives Neerlandaises, XXVIII, p. 1-24, 1901.[4] J. Serrin, ed., New Perspectives in Thermodynamics, Springer, New York, 1986.[5] C. Truesdell and W. Noll, Encyclopedia of Physics, III/3, Springer, Berlin,1965.[6] P. Germain, J. de M´ecanique, 12, p. 235-274, 1973.[7] P. Casal and H. Gouin, C.R. Acad. Sci. Paris, 300, s´erie II, p. 231-234 et p.301-304, 1985.[8] P. Casal, J. de M´ecanique, 5, p. 149-161, 1966.[9] P. Casal and H. Gouin, Lecture Notes in Physics 344, p. 85-98, 1989, andarXiv:0803.3160.[10] H. Gouin, Research Notes in Mathematics, 46, Pitman, London, p. 128-136,1981.[11] H. Gouin, J. de M´ecanique, 20, p. 273-287, 1981.[12] H. Gouin, Mech. Res. Comm., 3, p. 151-156, 1976.[1] J. Serrin, Encyclopedia of Physics, VIII/1, Springer, Berlin, 1959.[2] M. Barrere and R. Prud’homme, Equations fondamentales de l’Aerothermo-chimie, Masson, Paris, 1973.[3] D.J. Korteweg, Archives Neerlandaises, XXVIII, p. 1-24, 1901.[4] J. Serrin, ed., New Perspectives in Thermodynamics, Springer, New York, 1986.[5] C. Truesdell and W. Noll, Encyclopedia of Physics, III/3, Springer, Berlin,1965.[6] P. Germain, J. de M´ecanique, 12, p. 235-274, 1973.[7] P. Casal and H. Gouin, C.R. Acad. Sci. Paris, 300, s´erie II, p. 231-234 et p.301-304, 1985.[8] P. Casal, J. de M´ecanique, 5, p. 149-161, 1966.[9] P. Casal and H. Gouin, Lecture Notes in Physics 344, p. 85-98, 1989, andarXiv:0803.3160.[10] H. Gouin, Research Notes in Mathematics, 46, Pitman, London, p. 128-136,1981.[11] H. Gouin, J. de M´ecanique, 20, p. 273-287, 1981.[12] H. Gouin, Mech. Res. Comm., 3, p. 151-156, 1976.