Mixture of Fluids involving Entropy Gradients and Acceleration Waves in Interfacial Layers
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n Mixture of Fluids involving Entropy Gradientsand Acceleration Waves in Interfacial Layers
Henri Gouin a , ∗ , Tommaso Ruggeri b a Laboratoire de Mod´elisation en M´ecanique et Thermodynamique E.A. 2596Universit´e Paul C´ezanne-Aix-Marseille III, Case 322, 13397 Marseille Cedex 20France b Department of Mathematics and Research Center of Applied MathematicsC.I.R.A.M. University of Bologna Via Saragozza 8 40123-I Bologna Italy
Abstract
Through an Hamiltonian action we write down the system of equations of motionsfor a mixture of thermocapillary fluids under the assumption that the internal en-ergy is a function not only of the gradient of the densities but also of the gradient ofthe entropies of each component. A Lagrangian associated with the kinetic energyand the internal energy allows to obtain the equations of momentum for each com-ponent and for the barycentric motion of the mixture. We obtain also the balance ofenergy and we prove that the equations are compatible with the second law of ther-modynamics. Though the system is of parabolic type, we prove that there exist twotangential acceleration waves that characterize the interfacial motion. The depen-dence of the internal energy of the entropy gradients is mandatory for the existenceof this kind of waves. The differential system is non-linear but the waves propagatewithout distortion due to the fact that they are linearly degenerate (exceptionalwaves).
Key words:
Fluid mixtures, acceleration waves, interfacial layers
PACS: ∗ Corresponding Author
Email addresses: [email protected] (Henri Gouin), [email protected] (Tommaso Ruggeri).
URL: (Tommaso Ruggeri).
Preprint submitted to European Journal of Mechanics B/Fluids 28 October 2018
Introduction
Liquid-vapor and two-phase interfaces are generally represented by a materialsurface endowed with an energy related to Laplace’s surface tension. In fluidmechanics and thermodynamics, the interface appears as a surface separat-ing two media. This surface has its own characteristic behavior and energyproperties [1]. Theoretical and experimental detailed studies show that, whenworking far from critical conditions, the capillary layer has a thickness equiv-alent to a few molecular beams [2].Molecular models such as those used in kinetic theory of gas lead in interfaciallayers to laws of state associated with non-convex internal energies, e.g., thevan der Waals models [3,4,5]. These models appear advantageous as they pro-vide as even more precise verification of Maxwell’s rule applied to isothermalphase transition [6]. Nonetheless, they present two disadvantages:First, for densities that lie between phase densities, the pressure may becomenegative. Simple physical experiments can be used, however, to cause tractionthat leads to these negative pressure values [7,8].Second, in the field between bulks, internal energy cannot be represented by aconvex surface associated with the variation of densities and entropy. This factseems to contradict the existence of steady equilibrium state of the matter inthis type of region.To overcome these disadvantages, the thermodynamic investigative replacesthe non-convex portion corresponding to internal energy with a plane domain.The fluid can no longer be considered as a continuous medium. The interfaceis represented as a material surface with a null thickness. In this case, theonly possible representation of the dynamic behavior of the interface is one ofa discontinuous surface, and its essential structure remains unknown.In the equilibrium state it is possible to eliminate the above disadvantagesby appropriately modifying the stress tensor of the capillary layer, which isexpressed in an anisotropic form. As a consequence, the energy of the con-tinuous medium must change [3,4,9]. A representation of the energy near thecritical point therefore allows the study of interfaces of non-molecular size.This approach is not new and, in fact, dates back to van der Walls [10] andKorteweg [11]; it corresponds to what is known as the Landau-Ginzburg the-ory [5]. The representation proposed in the present study is based on thenotion of internal energy which is more convenient to use when the tempera-ture is not uniform. One of the problems that complicates this study of phasetransformation dynamics is the apparent contradiction between Korteweg’sclassical stress theory and the Clausius-Duhem inequality [12]. Proposal madeby Eglit [13], Dunn and Serrin [14], Casal and Gouin [15] and others rectifythis apparent anomaly for liquid-vapor interfaces of a pure fluid.To study capillary layers and bulk phases, the simplest model in continuum2echanics considers an internal energy ε as the sum of two terms: a first onecorresponding to a medium with a uniform composition equal to the local oneand a second one associated with the non-uniformity of the fluid [3,10]. Thesecond term is approximated by a gradient expansion, typically truncated tothe second order. In the simplest version of the theory we have ε = ρ α ( ρ, s ) + m ( ∇ ρ ) , where ρ is the matter density, s the specific entropy, α the specificinternal energy of the fluid assumed to be homogeneous and m is a coefficientindependent of s , ∇ ρ and of any higher derivatives. Obviously, the model issimpler than models associated with the renormalization-group theory [16].Nevertheless, it has the advantage of easily extending well-known results forequilibrium cases to the dynamics of interfaces [17,18]. In such a model, s varied with ρ through the interface in the same way as in the bulk, then s would always be that function of ρ which, at given temperature T satisfied ∂α∂s ( ρ, s ) = T. (1)With this assumption, s = s ( ρ ) and ε = ε ( ρ ), so that the points representingsingle-phase states in the ρ, s, ε space lie on a curve instead of on a surface ε = ε ( ρ, s ). This was the original assumption of van der Waals which waslater justified by Ornstein in 1909 for a system composed of molecules withlong-ranged but weak attractive force; this assumption is not exact for morerealistic potentials .As coexistence curves these are no way peculiar; the only peculiarity is that thesingle-phase states -in this version of this approximation- have collapsed ontothe coexistence curve instead being represented by the points of an extendedtwo-dimensional region of which the coexistence curve is merely a boundary.There is then no proper two-density description of the one-phase states of aone-component system in the lowest order of the mean-field approximation.There is such a description of the two-states, where not even in mean-field ap-proximation is there any discernible peculiarity; but in practice the potentialfor the two-density form of the van der Waals theory is then not constructedby the prescription in (1) but by other means. For example Rowlinson andWidom introduce in [4], chapter 9, an energy arising from the mean-field the-ory and depending on the deviations of the densities s and ρ , say, from theirvalues at the critical point and the gradients of these densities. It is also seenthat in c -components systems, c + 1 densities -the densities ρ , ..., ρ c of the c components and the entropy density s - may vary independently through theinterface.Aside from the question of accuracy, there are also qualitative features of someinterfaces in physical-chemistry, especially in systems of more than one com-ponent, that require two or more independently varying densities for their de-scription. An example is strong positive or negative adsorption of a component Main sentences in this paragraph and more precisely the comments on staticinterfaces are issued from the book of Rowlinson and Widom [4] and its bibliography. associated with a non-monotonic profile ρ i ( z ) where z is the spatial variable.In the one-density theory based on the approximation ∂α∂s − T = 0 and ∂α∂ρ i = 0for all j = i , the resulting one-density model leads inevitably to a monotonic ρ i ( z ) . In a theory based on two or more densities, by contrast, we may havea realistic trajectory with which is associated non-monotonic behavior of oneor more of the components if we suppose ρ j ( z ) to be monotonic.We must also allow the independent variation of at least two densities, entropyincluded, if we are to account contact angles in three-phase equilibrium : J.Cahn made the remark that we might use a two- (or more-) density van derWaals theory to describe the case of non-spreading in the three phase equilib-rium, ([4], chapter 8). Then, at equilibrium, Rowlinson and Widom pointedout that for single fluids the model must be extended by taking into accountnot only the strong variations of matter density through the interfacial layerbut also the strong variations of entropy.Also in dynamics, for an extended Cahn and Hilliard fluid , the volume inter-nal energy ε is proposed with a gradient expansion depending not only ongrad ρ but also on grad s corresponding to a strong heat supply in the changeof phases: ε = f ( ρ, s, ∇ ρ, ∇ s ) . The medium is then called a thermocapillaryfluid [19,20].To extend the model to fluid mixtures corresponding to c -component systemsand realistic potentials in molecular theory of fluid interfaces, the internal en-ergy is assumed to be a functional of the different densities of the mixture.In all the cases where strong gradient of densities occurs - for example shocksor capillary layers - the internal energy is chosen as a function of successivederivatives of densities of matter and entropies. To be in accordance with thephysical phenomena presented in [4], one will consider the internal energy ofa two-component fluid mixture with an internal energy depending also on thegradients of entropy of each component. The internal energy is a Galilean in-variant, it does not depend on the reference frame; hence the internal energydepends also on the relative velocity between the two components of the mix-ture.The conservative motions of thermocapillary fluid mixture are relevant to theso-called second gradient theory [21] and we obtain a complete set of bal-ance equations for conservative motions; we extend this result to the dissi-pative case. Our goal is simply to verify the consistency of our model withthe second law of thermodynamics. We consider a special case of dissipativethermocapillary mixtures where the introduction of dissipative forces is onlydone in the framework of the first gradient theory; we deduce the Fick lawas a consequence of a friction behavior between the components and from theequations of motion of the components. In such a case, extended thermody-namic principle (as Gibbs identity) provides a set of equations that satisfy theentropy principle, thereby making these irreversible motions compatible withthe second law of thermodynamics. 4he idea of studying interface motions as localized traveling waves in a multi-gradient theory is not new and can be traced throughout many problems ofcondensed matter and phase-transition physics [22]. In Cahn and Hilliard’smodel [3], the direction of solitary waves was along the gradient of density[17,22]. The introduction of the model of thermocapillary fluid mixture pro-vides a better understanding of the behavior of motions in fluid mixture inter-faces: it is possible to obtain the previous solitary waves but also a new kindof adiabatic waves may be forecasted. These waves are associated with thespatial second derivatives of entropy and matter densities. For this new kindof adiabatic waves, the direction of propagation is normal to the gradient ofdensities. In the case of a thick interface, the waves are tangential to the inter-face and the wave velocities depend on the constitutive equations. Finally weobserve that, also if the differential system associated with the wave motionsis non-linear, the waves propagate without distortion due to the fact that theyare linearly degenerate (exceptional waves)[23]. To derive the governing equations and boundary conditions in the dissipative-free case, we use the Hamilton principle of least action [24]. In continuummechanics, the principle with a system endowed with an infinite numberof degrees of freedom was initiated by Lin [25], Herivel [26], Serrin [27],Berdichevsky [28] and many others; it was proposed by Gouin for fluid mix-tures [29]. The main idea is to propose a Lagrangian which yields the behaviorof the medium as the difference between a kinetic and a potential energy perunit volume. Then, the variations of the Hamilton action obtained as a lin-ear functional of virtual displacements allow to find the governing equationsand boundary conditions. For real media, the irreversibility is introduced bothin equations of motion and equation of energy by using a classical approachthrough the dissipative function, diffusion and heat fluxes.We study a mixture of two fluids: the motion of a two-fluid continuum can berepresented using two surjective mappings (i=1,2)( t, x ) → X i = Λ i ( t, x )where ( t, x ) belongs to [ t , t ] × D t , a set in the time-space occupied by thefluid between times t and t . Variables X i denote the positions of each com-ponent of the mixture in reference spaces D i . Variations of particle motions5re deduced from virtual motions X i = Ξ i ( t, x , κ i ) , where scalars κ i are defined in a neighborhood of zero; they are associatedwith a two parameter family of virtual motions of the mixture. The real motioncorresponds to κ i = 0, the associated virtual displacements are [29] δ i X i = ∂ Ξ i ∂ κ i | κ i =0 . They generalize what is obtained for a single fluid [30]. To the virtual dis-placements δ i X i , we associated its image ζ i in the physical space D t occupiedby the fluid mixture at time t [24,31], ζ i = − ∂ x ∂ X i δ i X i . Conservation of matter for each component requires that ρ i det ∂ x ∂ X i ! = ρ io ( X i ) , (2)where ρ io is the reference volume mass in D i and det (cid:16) ∂ x ∂ X i (cid:17) the Jacobiandeterminant of the motion of component i . In differentiable cases eqs (2) areequivalent to the equations of balance of matter densities ρ i ∂ρ i ∂t + div( ρ i u i ) = 0 , (3)where u i denotes the velocity vectors of each component i . Now, we assumethat the mixture has an entropy for each component [32]; for conservativemotions, the equations of conservation of specific entropies s i are ∂ρ i s i ∂t + div( ρ i s i u i ) = 0 . (4)Then, relations s i = s io ( X i )define an isentropic motion of the fluid mixture. We deduce the followingrelations of tensorial quantities [24,29] δ i u i = d i ζ i dt − ∂ u i ∂ x ζ i , δ i ρ i = − div ( ρ i ζ i ) , δ i s i = − ∂s i ∂ x ζ i . (5)where d i dt = ∂∂t + u i . ∇ denotes the material derivative relatively to the com-ponent i . We assume that the volume potential energy of the mixture is in theform ε = ǫ ( ρ i , s i , ∇ ρ i , ∇ s i , w ) , w = u − u the relative velocity of the two components of the mixture.This means that the fluid mixture is a function not only of the densities ofmatter ρ i and specific entropies s i but also of the gradients of ρ i and s i . Thefact that ε depends on two entropies is classically adopted in the literature[29,33,34]. Moreover, for a two-velocity medium, there is no coordinate sys-tem within the framework of which any motion could be disregarded. So, thestandard definition of potential energy leads to its dependence on the relativemotion of the components. The dependence of ε with respect to the relativevelocity is analog to take into account the added mass effect in heterogeneoustwo-fluid theory as done by berdichevsky [28] and Geurst [35,36]. Let us notewe can assume also that ε is depending on ( t, x ); by this way, we introducedirectly the extraneous potential of the body forces.The potential ε is related with the volume internal energy ̟ of the mixturethrough the transformation ̟ = ε − ∂ε∂ w w , so that, e = X i =1 ρ i u i + ̟ is the total energy of the system [24,37].The equation of motion of component i is given by a variational methodassociated with a Hamilton action; the vector field x ∈ D t → ζ i is two timecontinuously differentiable. The Lagrangian of the mixture is L = X i =1 ρ i u i − ε, and consequently, the Hamilton action between the times t and t is I = Z t t Z D t L d x dt. From the definition of virtual motions, we obtain immediately two variationsof the action of Hamilton associated with i = 1 , δ i I = Z t t Z D t (cid:18)(cid:18) u i − ǫ ,ρ i (cid:19) δ i ρ i − ǫ ,ρ i,γ δ i ρ i,γ + ρ i K iγ δ i u iγ − ǫ, s i δ i s i − ǫ ,s i,γ δ i s i,γ (cid:17) d x dt , where subscript γ corresponds to spatial derivatives associated with gradientterms; as usually summation is made on repeated subscript γ from 1 to 3; K i = u i + ( − i ρ i ∂ǫ∂ w ! T where index T denotes the transposition. Then, byintegration by part we obtain, 7 i I = Z t t Z D t (cid:18)(cid:18) u i − ǫ ,ρ i + ( ǫ ,ρ i,γ ) ,γ (cid:19) δ i ρ i + ρ i K iγ δ i u iγ − (cid:16) ǫ, s i − ( ǫ ,s i,γ ) ,γ (cid:17) δ i s i − ( ǫ ,ρ i,γ δ i ρ i ) ,γ − ( ǫ ,s i,γ δ i s i ) ,γ (cid:17) d x dt. Let us denote by ρ i θ i ≡ b ∂ǫ b ∂s i and h i ≡ b ∂ǫ b ∂ρ i , (6)where b ∂ is the variational derivative operator. That is to say, ρ i θ i = ǫ, s i − ( ǫ ,s i,γ ) ,γ ≡ ǫ ,s i − div Ψ i and h i = ǫ ,ρ i − ( ǫ ,ρ i,γ ) ,γ ≡ ǫ ,ρ i − div Φ i , with, Ψ i ≡ ∂ǫ∂ ∇ s i and Φ i ≡ ∂ǫ∂ ∇ ρ i . Introducing R i = u i − h i , and taking into account of the expressions for θ i and h i given by eqs (6), we get δ i I = Z t t Z D t { R i δ i ρ i + ρ i K i · δ i u i − ρ i θ i δ i s i − div ( Φ i δ i ρ i + Ψ i δ i s i ) } d x dt, and from relations (5), we obtain δ i I = Z t t Z D t ( − R i div ( ρ i ζ i ) + ρ i K i · d i ζ i dt − ∂ u i ∂ x ζ i ! + ρ i θ i ∂s i ∂ x ζ i − div ( Φ i δ i ρ i + Ψ i δ i s i ) ) d x dt. Consequently, δ i I = Z t t Z D t ( ρ i ∂R i ∂ x + θ i ∂s i ∂ x − d i K Ti dt − K Ti ∂ u i ∂ x ! ζ i + ∂∂t ( ρ i K Ti ζ i ) − div (cid:16) ρ i R i ζ i − ρ i u i K Ti ζ i + Φ i δ i ρ i + Ψ i δ i s i (cid:17) (cid:27) d x dt. The Stokes formula and relation (5) yield δ i I = Z t t Z D t ρ i ∂R i ∂ x + θ i ∂s i ∂ x − d i K Ti dt − K Ti ∂ u i ∂ x ! ζ i d xdt (7)+ Z t t Z ∂D t g ρ i K Ti ζ i − n . ρ i R i ζ i − ρ i u i K Ti ζ i − Φ i ∂ρ i ∂ x ζ i − Ψ i ∂s i ∂ x ζ i − ρ i Φ i div ζ i ! dσ x dt, ∂D t (of mesure dσ x ) is the boundary of D t , n is the unit external normalvector to ∂D t and g is the velocity of ∂D t . If we consider a vector field x ∈ D t → ζ i and its first derivatives vanishing simultaneously on the boundary ∂D t , the Hamilton principle expressed in the form: ∀ ζ i , δ i a = 0 leads to ∀ ζ i , Z t t Z D t ρ i ∂R i ∂ x + θ i ∂s i ∂ x − d i K Ti dt − K Ti ∂ u i ∂ x ! ζ i d x dt = 0and consequently, d i K i dt + ∂ u i ∂ x ! T K i = ∇ R i + θ i ∇ s i . (8)Let us note that the value of the first member of eq. (8) is equal to d i u i dt + ( − i d i dt ρ i ∂ǫ∂ w ! T + ∂ u i ∂ x ! T u i + ( − i ρ i ∂ u i ∂ x ! T ∂ǫ∂ w ! T and eq. (8) yields ρ i d i u i dt + ( − i div u i ∂ǫ∂ w ! T + d i dt ∂ǫ∂ w ! T + ∂ u i ∂ x ! T ∂ǫ∂ w ! T = ρ i θ i ∇ s i − ρ i ∇ h i . Taking into account of eq. (3) of conservation of mass of component i, wededuce ∂ρ i u i ∂t + div( ρ i u i ⊗ u i ) + ( − i div u i ∂ǫ∂ w ! T + ∂∂t ∂ǫ∂ w ! T + ∂∂ x ∂ǫ∂ w ! T u i + ∂ u i ∂ x ! T ∂ǫ∂ w ! T = ρ i θ i ∇ s i − ρ i ∇ h i , and due to the fact thatdiv u i ∂ǫ∂ w ! T + ∂∂ x ∂ǫ∂ w ! T u i = div ∂ǫ∂ w ! T ⊗ u i , we get the equations of motion of the two components in the form9 ρ i u i ∂t + div( ρ i u i ⊗ u i ) + ( − i ∂∂t ∂ǫ∂ w ! T + ∂ u i ∂ x ! T ∂ǫ∂ w ! T + div ∂ǫ∂ w ! T ⊗ u i = ρ i θ i ∇ s i − ρ i ∇ h i . (9)We consider only the isotropic case where the potential energy ε of the mixturecan be written in terms of the isotropic invariants β ij = ∇ ρ i · ∇ ρ j , χ ij = ∇ ρ i · ∇ s j , γ ij = ∇ s i · ∇ s j , ( i, j = 1 ,
2) and ω = 12 w .ε = ε ( ρ i , s i , β ij , χ ij , γ ij , ω ) , Then, the equation of motion of each component of the mixture is ∂ρ i u i ∂t + div( ρ i u i ⊗ u i ) + (10)( − i ∂∂t ( a w ) + a ∂ u i ∂ x ! T w + div( a w ⊗ u i ) = ρ i θ i ∇ s i − ρ i ∇ h i , where a = ∂ε∂ω .In this case, Ψ i and Φ i can be written Ψ i = X j =1 D ij ∇ ρ j + E ij ∇ s j , Φ i = X j =1 C ij ∇ ρ j + D ij ∇ s j , with C ij = (1 + δ ij ) ǫ, β ij , D ij = ǫ, χ ij , E ij = (1 + δ ij ) ǫ, γ ij , where δ ij is theKronecker symbol.The simplest model is when C ij = C ji , D ij , E ij = E ji are constant. Then, ε = e ( ρ i , s i , w ) + X i,j =1 C ij ∇ ρ i · ∇ ρ j + D ij ∇ ρ i · ∇ s j + 12 E ij ∇ s i · ∇ s j (11)where the associated quadratic form with respect to the vectors ∇ ρ i and ∇ s i is in the form P i,j =1 12 C ij ∇ ρ i · ∇ ρ j + D ij ∇ ρ i · ∇ s j + E ij ∇ s i · ∇ s j .This quadratic form is assumed positive such as the effect of gradient termsincreases the value of the internal energy with respect to a mixture in a ho-mogeneous configuration. 10 .2 Equation of total momentum and equation of energy for conservativemotions of thermocapillary mixtures We limit first to the conservative case. We notice that the equation of motionof each component is not in divergence form. Nevertheless, by summing eqs(10) with respect to i , we obtain the balance equation for the total momentumin a divergence form. In fact, eqs (10) imply X i =1 ∂ρ i u i ∂t + div( ρ i u i ⊗ u i ) ! − div( a w ⊗ w ) = a ∂ w ∂ x ! T w + X i =1 ρ i θ i ∇ s i − ρ i ∇ h i ! (12)In coordinates, the second member of eq. (12) is a w ν w γ,ν + X i =1 ǫ ,s i s i,γ − (cid:16) ǫ ,s i,ν (cid:17) ,ν s i,γ − ρ i (cid:16) ǫ ,ρ i (cid:17) ,γ + ρ i (cid:16) ǫ ,ρ i,νν (cid:17) ,γ , where ν is summed from 1 to 3. Noting that ǫ ,γ = X i =1 ǫ ,s i s i,γ + ǫ ,s i,ν s i,νγ + ǫ ,ρ i ρ i,γ + ǫ ,ρ i,ν ρ i,νγ + a w ν w γ,ν , (13)we obtain a w ν w γ,ν + X i =1 ǫ ,s i s i,γ − (cid:16) ǫ ,s i,ν (cid:17) ,ν s i,γ − ρ i (cid:16) ǫ ,ρ i (cid:17) ,γ + ρ i (cid:16) ǫ ,ρ i,νν (cid:17) ,γ = ǫ ,γ − X i =1 ǫ ,s i,ν s i,νγ + ǫ ,ρ i ρ i,γ + ǫ ,ρ i,ν ρ i,νγ + (cid:16) ǫ ,s i,ν (cid:17) ,ν s i,γ + ρ i (cid:16) ǫ ,ρ i (cid:17) ,γ − ρ i (cid:16) ǫ ,ρ i,νν (cid:17) ,γ = ǫ, γ + X i =1 (cid:18) − ρ i ǫ ,ρ i + ρ i (cid:16) ǫ ,ρ i,ν (cid:17) ,ν (cid:19) ,γ − (cid:16) Φ iν ρ i,γ + Ψ iν s i,γ (cid:17) ,ν , and consequently the equation of motion for the total momentum is ∂ρ u ∂t + div X i =1 ( ρ u i ⊗ u i ) − ρa w ⊗ w − σ ! = 0 (14)where ρ = ρ + ρ is the total volume mass, ρ u = ρ u + ρ u is the totalmomentum and σ = σ + σ is the total stress tensor such that σ iνγ = ( − P i + ρ i div Φ i ) δ νγ − Φ iν ρ i,γ − Ψ iν s i,γ , with P i = ρ i ǫ ,ρ i − ρ i ǫρ . ǫ depends also on ( t, x ) corresponding to an external forcepotential, an additive term appears as body force in relation (13) and in eq.(14) the body force appears in the second member. This is not the case ineqs (9) which include the body forces coming from ǫ (depending on ( t, x )) interms h i . For the sake of simplicity, we do not introduce the body force inequation of the total momentum and equation of total energy.The equation of energy of the total mixture is obtained in the divergence form.Let us define M i = ∂ρ i u i ∂t + div( ρ i u i ⊗ u i ) +( − i ∂∂t ( a w ) + a ∂ u i ∂ x ! T w + div ( a w ⊗ u i ) − ρ i θ i ∇ s i + ρ i ∇ h i ,G i = ∂ρ i ∂t + div( ρ i u i ) ,S = X i =1 ∂ρ i s i ∂t + div( ρ i s i u i ) ! θ i ,E = ∂∂t X i =1 ρ i u i ! + ε − a w ! +div X i =1 ρ i (cid:18) K i · u i − u i − σ i (cid:19) u i ! + ε u − U ! , where U = P i =1 d i ρ i dt Φ i + d i s i dt Ψ i ! corresponds to the interstitial working: in the same way as for Cahn and Hilliard fluids , an additional term that hasthe physical dimension of a heat flux must be added to the equation of energy[13,14,15,28].
Theorem : For all motions of a thermocapillary fluid mixture, the relation E − S + X i =1 (cid:18) u i − h i + θ i s i (cid:19) G i − M i · u i ≡ is identicaly satisfied. The proof comes from the following algebraic calculation: M Ti u i − (cid:18) u i − h i (cid:19) G i + ρ i θ i d i s i dt ≡ ∂∂t ( 12 ρ i u i ) + div(( 12 ρ i u i ) u i ) + h i G i + ρ i θ i ∂s i ∂t + ρ i ∂h i ∂ x u i + ( − i ∂a w ∂t ! T + a w T ∂ u i ∂ x + div (cid:16) a u i w T (cid:17) u i ∂ε∂t + div X i =1 ρ i h i u i ! ≡ a w T ∂ w ∂t + X i =1 ε ,ρ i ∂ρ i ∂t + div ∂ρ i ∂t ∂ε∂ ∇ ρ i ! − ∂ρ i ∂t div ∂ε∂ ∇ ρ i ! + ε ,s i ∂s i ∂t + ∂ε∂ ∇ s i ∇ ∂s i ∂t + h i div ( ρ i u i ) + ρ i ∂h i ∂ x u i , yields X i =1 h i G i ≡ ∂ε∂t − a w T ∂ w ∂t + X i =1 div ( ρ i h i u i ) − div ∂ρ i ∂t ∂ε∂ ∇ ρ i ! − ε ,s i ∂s i ∂t − ∂ε∂ ∇ s i ∇ ∂s i ∂t − ρ i ∂h i ∂ x u i . Consequently, X i =1 (cid:18) M Ti u i − (cid:18) u i − h i + θ i s i (cid:19) G i (cid:19) + S ≡ ∂∂t X i =1 ρ i u i ! + ε ! − a w T ∂ w ∂t + X i =1 div (cid:18) ρ i (cid:18) u i + h i (cid:19) u i (cid:19) − div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i ! + ( − i ∂a w ∂t ! T u i + a w T ∂ u i ∂ x u i + div (cid:16) a u i w T (cid:17) u i ≡ ∂∂t X i =1 ρ i u i ! + ε − a w ! + X i =1 div (cid:18) ρ i (cid:18) K Ti u i − u i + h i (cid:19) u i (cid:19) − div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i ! Taking into account of the relations ∂ρ i ∂t Φ i ≡ d i ρ i dt Φ i − Φ i ∂ ρ i ∂ x u i and ∂s i ∂t Ψ i ≡ d i s i dt Ψ i − Ψ i ∂ s i ∂ x u i and the definition of the total stress tensor, σ ≡ ε − X i =1 (cid:16) ρ i ε ,ρ i − ρ i div Φ i (cid:17) Id − Φ i ∂ ρ i ∂ x − Ψ i ∂ s i ∂ x , where Id is the identity tensor, we deduce immediately the algebraic identity(15). (cid:3) orollary : All conservative motions of a thermocapillary mixture satisfythe equation of energy balance ∂∂t X i =1 ρ i u i ! + ε − a w ! +div X i =1 ρ i (cid:18) K i · u i − u i − σ i (cid:19) u i ! + ε u − U ! = 0 . (16)This result comes from the simultaneity of relations G i = 0 , S i = 0 and M i = 0 . Let us note that ε − a w corresponds to the volume internal energy ̟ of themixture. The conservative fluid mixture model presented in section 2.1 is relevant tothe so-called second gradient theory [21], [38]. In our form of equations of massconservation for each component (2), equation of the total momentum of themixture (14), equation of energy (16), the diffusion term J = ρ ( u − u ) doesnot directly appear but is deduced respectively from the velocities and densi-ties of the components.Our aim is to verify the consistency of our model with the second law ofthermodynamics. The introduction of dissipative forces is simply done in theframework of the first gradient theory [21]: the dissipative forces applied to thecontinuous medium are divided into volume forces f di and surface forces asso-ciated with the Cauchy stress tensor σ di . Then, the virtual work of dissipativeforces δT i applied to the component i is in the form δT i = f di · ζ i − tr σ di ∂ ζ i ∂ x ! , where δT i is a differential form. For such dissipative motions, no productionof masses due to chemical reactions appears.For the same virtual displacement of two components, ζ = ζ i , ( i = 1 ,
2) ,the total virtual work of dissipative forces is δT = X i =1 f di · ζ − tr σ di ∂ ζ ∂ x ! . When ζ is a translation, the work δT is equal to zero and consequently, X i =1 f di = 0 or f d = − f d ≡ f d (17)14e specify later the behavior of forces f di (they will be associated with thediffusion term) and stress tensors σ di . Taking into account of the dissipativeeffects, the equations of motion for each component become ∂ρ i u i ∂t + div( ρ i u i ⊗ u i ) + ( − i ∂∂t ( a w ) + a ∂ u i ∂ x ! T w + div( a w ⊗ u i ) = ρ i θ i ∇ s i − ρ i ∇ h i + div σ di + f di (18)Taking into account of relation (17), the equation of the total momentumwrites ∂ρ u ∂t + div X i =1 ( ρ u i ⊗ u i ) − ρa w ⊗ w − σ − σ d ! = 0 (19)with σ d = σ d + σ d . The introduction of the heat flux vector q and the heat supply r comes fromclassical methods in thermodynamics [32,34,39]. If we write, M di = M i − div σ di − f di ,S d = S − r + div q + X α =1 f di . u i − tr ( σ di ∆ i ) ,E d = E − r + div q − X α =1 div σ di u i , with ∆ i = 12 ∂ u i ∂ x + ∂ u i ∂ x ! T represents the velocity deformation tensor ofeach component, relation (15) writes as E d − S d − X i =1 M di · u i − ( K i · u i − R i − θ i s i ) G i ≡ Gibbs identity .For the components of the mixture, equations of momenta and equations ofmasses are in the form M di = 0 , G i = 0 . (21)The Gibbs identity (20) and eqs (21) imply S d = E d . If we assume that S d = 0,i.e., X i =1 ∂ρ i s i ∂t + div( ρ i s i u i ) ! θ i + f di . u i − tr ( σ di ∆ i ) − r + div q = , (22)it is equivalent to write E d = 0, i.e., 15 ∂t X i =1 ρ i u i ! + ε − a w ! + (23)div X i =1 ρ i (cid:18) K i · u i − u i − σ i − σ di (cid:19) u i ! + ε u − U + q ! − r = 0 . Eq. (22) is the equation of entropy and Eq. (23) is the equation of energy.
In the conservative case, the system is closed with two different temperatures θ i ( i = 1 , s i which could remplace equations (4). A possibilityis to consider the case when the exchanges of momentum and energy betweenthe two components are rapid enough to have a common temperature (this isnot the case of heterogeneous mixtures where each phase may have differentpressures and temperatures [40]). Then, in dissipative case, if we know all thedissipative functions, the governing system is closed. Note also that it could bepossible to consider a common temperature and entropy both for conservativeand dissipative case. This case is connected with a conservative equation forthe common entropy s (see Appendix). Another possibility is to assume thatthe entropy is transported along the i th component (say, for example i = 1)which was used for quantum fluids by Landau [41] ∂ρ s∂t + div ( ρ s u ) = 0 . In this case, the independent functions are ρ , s, u , ρ , u , where ρ i are sub-mitted to the constraints (3) and the case of Helium superfluid is a specialcase of our study corresponding to s = 0 and s = s . Using of this hypothesisis nevertheless doubtful for classical fluids.Hence, we may suppose a common temperature only for dissipative case. Thismeans that θ = θ = θ (24)This hypothesis closes the system (21-22). Now we focus on the governing equations for each components of the mixture : M di = 0 ( i = 1 , . For slow motions, we rewrite these equations in the following form : M di ≃ ρ i ∇ h i − ρ i θ ∇ s i − div σ di − f di = 0 . If we consider the case when the motion of each component is regular enough,an approximative case is the case of solid displacements for the motion of eachcomponent; then, div σ di ≃ M di ≃ ρ i ∇ µ i − f di = 0 , µ i = h i − θs i is the chemical potential of the component i of the mixtureat the temperature θ .Considering the difference M d − M d and using relation (17), we obtain, f d ρ − f d ρ ≡ ρ f d ρ ρ = ∇ µ, (25)where µ = µ − µ is the chemical potential of the mixture [42,43]. Let usintroduce the diffusion flux J , J ≡ ρ ( u − u ) = ρ ρ ρ w . (26)Equation (25) implies, f d · w = ∇ µ · J . (27)The term ∇ µ · J corresponds to the entropy production due to the diffu-sion process. Expression (27) is the connection between the mechanical dragforce between components of the mixture and the thermodynamical processof diffusion; consequently, Eq. (22) yields X i =1 ∂ρ i s i ∂t + div( ρ i s i u i ) ! θ i − tr ( σ di ∆ i ) − r + div q − ∇ µ · J = 0 . We have obtained the equations of balance of masses, equation of total momen-tum, Eq. (25) between the components and equation of total energy by usingan energetic method. They are the extension of classical mixture equations toequations of mixtures of fluids involving density gradients.
Relation (22) may be rewritten in the form X i =1 ∂ρ i s i ∂t + div( ρ i s i u i ) ! +div q θ − rθ = 1 θ X i =1 tr ( σ di ∆ i ) ! + f d · w − ∇ θ · q θ This last equation yields the entropy production due to the diffusion, viscosityand heat flux processes. Due to θ >
0, if we assume X i =1 tr ( σ di ∆ i ) ! + f d · w − ∇ θ · q θ ≥ , (28)we get the Clausius-Duhem inequality in the form X i =1 ∂ρ i s i ∂t + div( ρ i s i u i ) ! + div q θ − rθ ≥ . tr ( σ di ∆ i ) ≥ , f d · w ≥ ∇ θ · q ≤ σ di , f d and q :The stress tensor σ di is a symmetric isotropic tensor function of ∆ i such that P i =1 tr ( σ di ∆ i ) > σ di = λ i ( tr ∆ i ) Id +2 µ i ∆ i with µ i > λ i + 2 µ i > q = − χ ∇ θ with χ > . The linear approximation known in the literature as the
Stokes drag formula is adopted [41,42] f d = k w , k > , with f d = − f d ≡ f d and w = u − u . Let us note that relations (25) and (26) together with the drag formula yieldthe property of the diffusion flux J = 1 k ∇ µ, (29)which is the general form of the Fick law [39,41]. So, the Fick law is not directlya linear phenomenological law as the Fourier law but a direct consequence ofequations of motion and the Stokes drag force hypothesis which was previouslynoticed by Bowen for a different model [42]. Now, we will consider conservative motions of thermocapillary mixtures only.As it is well known, wave phenomena - in particular discontinuity waves (wavesacross the front of which some derivatives of the field variables have jumps)- are typical of models that are described through hyperbolic differential sys-tems. A classical example of discontinuity waves in continuum mechanics arethe so-called acceleration waves in which, among the other variables, the ac-celeration jumps across the front while the velocity is continuous [44].Dissipative systems and in particular models for diffusive processes have usualdifferential system with a parabolic structure and discontinuity waves are notadmissible. A typical example of non admissibility is the one of Navier-Stokes-Fourier fluids. In this case a possible approach to obtain hyperbolic system isthe method of the Extended Thermodynamic theory [32], valid also for rarefiedgases. 18evertheless in parabolic systems some discontinuity waves may propagate forparticular initial data. The aim of this paper is to prove that for the presentmodel of thermocapillarity fluid binary mixtures there exists the possibility ofpropagation of two tangential acceleration waves, provided that the internalenergy is at least a function of the entropy gradient of one component. For thisaim, we first briefly recall some very well known questions about discontinuitywaves.A wave is a discontinuity wave if the wave front with Cartesian equation φ ( t, x ) = 0 separates the space in two subspaces in which there exists regularsolutions of the differential system but across the normal direction of the frontsome derivatives of the field suffers a jump [23].As usual, we indicate the jump with a square bracket,[ ] = ( ) φ =0 − − ( ) φ =0 + and we introduce the map between ( t, x ) and ( φ, ξ ) , where ξ ≡ ξ ( t, x ) repre-sents the tangential manifold of the wave surface in time-space. Therefore theassumptions for the discontinuity waves are expressed for a generic function f in the form, [ f ] = 0; " ∂ k f∂ξ γ · · · ∂ξ γ k = 0 ∀ k ;where there exists p ≧ " ∂ j f∂φ j = 0 for 1 ≤ j ≤ p − δ k f ≡ " ∂ k f∂φ k = 0 for k ≥ p. (30)Taking into account (30) and the Hadamard lemma [45], we have " ∂ p f∂x γ · · · ∂x γ p = δ p f n i · · · n i p ; " ∂ p f∂t p = ( − λ ) p δ p f where λ and here n ≡ ( n i ) are respectively the normal velocity and the unitnormal vector to the wave front.The advantages of the previous symbols are that there exists a chain rulebetween the field derivative in the differential systems and the correspondingjump relation, ∂ t → − λ δ ; ∂ γ → n γ δ . (31)We apply now this procedure to our differential systems assuming that acrossthe wave front ρ i , s i and its first derivatives are continuous and there are19umps for the second derivative ( p = 2) while the velocity is continuous andsuffer a jump in the first derivative ( p = 1)[ ρ i ] = [ s i ] = [ δρ i ] = [ δs i ] = 0; [ δ ρ i ] = 0; [ δ s i ] = 0; (32)[ u i ] = 0; [ δ u i ] = 0; i = 1 , . From the balance of mass of the two components (3), taking into account ofeqs (32) and the chain rule (31), we obtain that the normal components of thefirst derivative of the velocities are continuous,[ δu in ] = 0 , (33)where u in = n · u i . If we differentiate the entropy balance law of each component (4), with respect x , we obtain for the discontinuities − v i δ s i + ∇ s i · δ u i = 0 , (34)where v i = λ − u in are the relative velocities of the wave front with respect to the fluid compo-nents. From the balance of momentum we obtain − ρ i v i δ u i + ( − i a { v i δ ( u − u ) + ( δ u i · ( u − u )) n }− ρ i { ∇ s i [ θ i ] − [ ∇ h i ] } = 0 . (35)From eq. (6) we obtain B i ≡ − ρ i [ θ i ] = X j =1 (cid:16) D ji δ ρ j + E ji δ s j (cid:17) A i ≡ − [ ∇ h i ] = n X j =1 (cid:16) C ij δ ρ j + D ij δ s j (cid:17) . (36)Then relation (35) becomes − ρ i v i δ u i + ( − i a { v i δ ( u − u ) + ( δ u i · ( u − u )) n } + B i ∇ s i − ρ i A i = 0 . If we multiply by n and we take into account of relation (33), we obtain ρ i A i · n = ( − i a δ u i · ( u − u ) + B i ∇ s i · n , (37)20hen, we get the final jump conditions from the momentum equations ρ i v i δ u i + ( − i av i δ ( u − u ) − B i ∇ t s i = 0 , (38)where ∇ t s i = ∇ s i − ( ∇ s i · n ) n (39)denotes the tangential component of the gradient of entropy of each compo-nent.Therefore we obtain the algebraic system of 8 equations (33), (34) and (38)for the 10 scalar unknowns δ ρ i , δ s i and δ u i ( i = 1 , . Consequently, weneeds two more conditions that are obtained by compatibility conditions com-ing from boundary conditions associated with the equation of motion for eachcomponent. In fact, we notice in Appendix that if ρ i , s i , ∇ ρ i , ∇ s i are con-tinuous through a surface of weak discontinuities, then div Φ i must be alsocontinuous through the surface [div Φ i ] = 0 . We notice additively that these two conditions are compatible with the Rankine-Hugoniot conditions associated to the total momentum (14) and the totalenergy balance law (16): In fact, eq. (14) yields[ σ + σ ] = 0 , while from eq. (16), we get [ σ u n + σ u n ] = 0 . Then, we obtain the two supplementary equations[div Φ i ] = X j =1 (cid:16) C ij δ ρ j + D ij δ s j (cid:17) = 0 (40)and the system for the discontinuities becomes an homogeneous closed systemof 10 equations for 10 scalar unknowns δ ρ i , δ s i and δ u i ( i = 1 ,
2) in the21orm, [ δu in ] = 0 ρ i v i δ u i + ( − i av i δ ( u − u ) − P j =1 (cid:16) D ji δ ρ j + E ji δ s j (cid:17) ∇ t s i = 0 − v i δ s i + ∇ s i · δ u i = 0 P j =1 (cid:16) C ij δ ρ j + D ij δ s j (cid:17) = 0 (41)We observe that the conditions (36), (37) are constraints for the jump of thethird derivatives of densisties and entropies δ ρ j , δ s j .Now we consider the weak discontinuities near the equilibrium of the fluidmixture; then u i = 0 . Consequently v = λ is the velocity of the acceleration wave. Due to the factthat thermocapillary mixtures can be considered as a mathematical model forinterfacial layers between two mixture bulks [29], the gradients of tensorialquantities ρ i and s i are orthogonal to the interfacial layers and consequentlyare collinear, ∇ s = b ∇ s . Then the second and the third equations of system (41) allow to eliminate u i and to get c s (( a − ρ ) δ s − a b δ s ) + X j =1 (cid:16) D j δ ρ j + E j δ s j (cid:17) ( ∇ t s ) = 0and c s (cid:16) − a b δ s + b ( a − ρ ) δ s (cid:17) + X j =1 (cid:16) D j δ ρ j + E j δ s j (cid:17) ( ∇ t s ) = 0where c s = v ( ∇ t s ) . Therefore we obtain a system of compatibility between the variables δ ρ j and22 s j in the form C δ ρ + C δ ρ + D δ s + D δ s = 0 C δ ρ + C δ ρ + D δ s + D δ s = 0 D δ ρ + D δ ρ + ( E + ( a − ρ ) c s ) δ s + ( E − a b c s ) δ s = 0 D δ ρ + D δ ρ + ( E − a b c s ) δ s + ( E + ( a − ρ ) b c s ) δ s = 0 (42)Let us denote C = C T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C C C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D D D D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , E = E T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E E E E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , B = B T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ − a abab ( ρ − a ) b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ρ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ρ δ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , δ s = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ s δ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and system (42) is writting C δ ρ + D δ s = 0 D T δ ρ + ( E − c s B ) δ s = 0 , which implies (cid:16) A − c s B (cid:17) δ s = 0 . (43)where A = E − D T C − D . From eq.(43) it is simple to verify the property,
Theorem : If EC − D is positive definite and if we consider small diffusion,i.e. a < ρ ∗ with ρ ∗ = ρ ρ ρ + ρ hen, A and B are both symmetric and definite positive, all the eigenvalues c s of eq. (43) are positive and two discontinuity waves exist.3.2 Exceptional waves: As in the hyperbolic case, a wave is exceptional or linearly degenerate (see e.g.[23]) if δλ ≡ . (44)In this case the wave behavior is similar to the behavior in linear case and wedo not get any distortion of the wave or shock formation.It is simple matter to prove that both the waves fulfill the exceptionalitycondition. In fact, taking into account of relations (31),(39), we have δ ∇ t s ≡ λ is function of the modulo of the tangentialgradient of entropy λ ≡ λ ( | ∇ t s | )and then (44) holds. In this paper we prove that the model of thermocapillary fluid mixtures withdissipation yields a system of equations of motions compatible with the secondlaw of thermodynamics at least in simple dissipative cases. The equation ofmotion and the equation of energy of the barycentric motion of the mixtureare in a divergence form in conservative cases.Consequently, Hamilton’s principle applied to fluid dynamics is a direct andsystematic method to obtain the equations of conservative motions. This prin-ciple extended to each component of a mixture of conservative fluids is able todeduce the same number of balance equations than unknown functions. Themethod yields a non ambiguous framework for the case of non-conservativemixtures (with viscosity, diffusion and heat transfer). Non additional assump-tion but constitutive behavior compatible with the second law of thermody-namics is necessary. One obtains the dynamic Gibbs relation and Fick’s lawas a consequence of governing equations.We have seen that the dependance of an entropy gradient is necessary for theexistence of isentropic waves of acceleration along the interfaces: the fact that24he internal energy depends not only on the gradient of matter densities butalso on the gradient of entropy, yields a new kind of waves which does not ap-pear in simpler models. They are exceptional waves in the sense of Boillat andLax [23] and they appear only in, at least, systems with two dimensions. Thesesecond order waves are of weak energy and consequently they are not easy toshow up. Recent experiments in space laboratories in micro-gravity conditions,for carbonic dioxide near its critical point, have showed the possibility of suchwaves [46]. The experimental evidence of such adiabatic waves with otherphysical reasons we have presented in the introduction should strengthen thenecessity to take into account of the dependence of entropy gradients togetherwith density gradients in the expression of the internal energy for continuummodels of capillarity and phase transitions.
Acknowledgements
This paper was developed during a stay of Tommaso Ruggeri as visiting pro-fessor in L.M.M.T. of the University of Aix-Marseille III and a stay of HenriGouin as visiting professor in C.I.R.A.M. of the University of Bologna witha fellowship of the Italian GNFM-INDAM and was supported in part (T.R.)by MIUR Progetto di interesse Nazionale
Problemi Matematici Non Linearidi Propagazione e Stabilit`a nei Modelli del Continuo
Coordinatore T. Rug-geri, by the GNFM-INDAM, and by the Istituto Nazionale di Fisica Nucleare(INFN).The authors are indebted to the anonymous referees for their valuable criticismduring the review process.
We obtain also the compatibility due to the boundary conditions. For this aimwe rewrite the variations of the Hamilton action when the equations of themotion (8) are verified. Then, from relation (7), we obtain, δ i I = Z t t Z ∂D t g ρ i K Ti ζ i − n . ρ i R i ζ i − ρ i u i K Ti ζ i − Φ i ∂ρ i ∂ x ζ i − Ψ i ∂s i ∂ x ζ i − ρ i Φ i div ζ i ! dσ x dt. x ∈ D t → ζ i vanishing with its first derivatives on theboundary ∂D t , we deduce immediately on a surface of discontinuity Σ t (where ρ i , s i and its first derivatives are continuous and there are jumps for the secondderivative) the value of the variation of the Hamilton action, δ i I = Z t t Z Σ t g h ρ i K Ti ζ i i − n . " ρ i R i ζ i − ρ i u i K Ti ζ i − Φ i ∂ρ i ∂ x ζ i − Ψ i ∂s i ∂ x ζ i − ρ i Φ i div ζ i dσ x dt ≡ − Z t t Z Σ t n . [ ρ i R i ] ζ i dσ x dt. Due to the fact that δ i I = 0 for a vector field x ∈ D t → ζ i , we obtain[ ρ i R i ] = 0 , and deduce the compatibility conditions (40) across the wave front,[div Φ i ] ≡ X j =1 (cid:16) C ij δ ρ j + D ij δ s j (cid:17) = 0 . The equations of balance of matter densities ρ, ρ are in the form ∂ρ∂t + div( ρ u ) = 0 ,∂ρ ∂t + div( ρ u ) = 0 , (45)where ρ and u denote the density and the velocity of the total mixture, ρ and u denote the density and the velocity of one of the two components (itis equivalent to consider the total density ρ and the concentration c = ρ /ρ betwen the two components of the mixture such that eq. (45) is equivalentto ∂ ( ρc ) /∂t + div( ρc u ) = 0). For conservative motions, the equation ofconservation of the total specific entropy s is ∂ρs∂t + div( ρs u ) = 0 . We consider a volume potential energy of the mixture in the form ε = ε ( ρ, ρ , s, s , ∇ ρ, ∇ ρ , ∇ s, ω ) . ω = 12 w , where w = u − u . As in the section 2, it is possible to deducethe equations of motions through the Hamilton action. We denote ρ θ = b ∂ε b ∂s , h = b ∂ε b ∂ρ , h = b ∂ε b ∂ρ and a = ∂ε∂ω . (In fact, the potential energy of the mixture depends only on the insentropicinvariants ( ∇ ρ ) , ( ∇ ρ ) , ∇ ρ · ∇ ρ , ( ∇ s ) ). By analogous calculations asso-ciated with a Lagrangian of the mixture in the form L = 12 ρ u − ε, and virtual motions such as X = Ξ ( t, x , κ ) and X = Ξ ( t, x , κ ) , we obtainas in section 2, the equations of motion ∂ρ u ∂t + div( ρ u ⊗ u ) + (46) − ∂∂t ( a w ) + a ∂ u ∂ x T w + div( a w ⊗ u ) ! = ρ θ ∇ s − ρ ∇ h and ∂∂t ( a w ) + a ∂ u ∂ x T w + div( a w ⊗ u ) ! + ρ ∇ h = 0 (47)with ρ θ = ρǫ ,s − div Ψ , h = ρǫ ,ρ − div Φ , and h = ρ ǫ ,ρ − div Φ , where Ψ = ∂ε∂ ∇ s , Φ = ∂ε∂ ∇ ρ , Φ = ∂ε∂ ∇ ρ . By summing eqs (46,47), we obtain the balance equation for the total momen-tum in a divergence form ∂ρ u ∂t + div ( ρ u ⊗ u − ρa w ⊗ w − σ ) = 0where σ is the stress tensor such that σ νγ = ( − P + ρ div Φ + ρ div Φ ) δ νγ − Φ ν ρ ,γ − Φ ν ρ ,γ − Ψ ν s ,γ , where P = ǫ − ρǫ ,ρ − ρ ǫ ,ρ . The equation of energy of the total mixture is alsoobtained in the divergence form ∂∂t (cid:18) ρ u + ε − a w (cid:19) + div (cid:18)(cid:18) ρ u − a w . u − σ (cid:19) u − U + ε u (cid:19) = 0 . U = dρdt Φ + d ρ dt Φ + dsdt Ψ and it is possible by a similar calculationthan in section 3 to deduce an acceleration wave associated with the entropygradient. References [1] Levitch V., Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs,New Jersey, 1962.[2] Ono S., Kondo S., Molecular theory of surface tension in liquid in ”Structureof liquids”, S. Fl¨ugge (ed.) Encyclopedia of Physics, X, Springer verlag, Berlin,1960.[3] Cahn J.W., Hilliard J.E., Free energy of a non-uniform system III, J. Chem.Phys. 31 (1959) 688-699.[4] Rowlinson J.S., Widom B., Molecular theory of capillarity, Clarendon Press,Oxford, 1984.[5] Hohenberg P. C., Halperin B.I., Theory of dynamic critical phenomena, Reviewsof Modern Physics 49 (1977) 435-480.[6] Gouin H., Dynamics effects in gradient theory for fluid mixtures, The IMAvolumes in Mathematics and its Applications 52 (1993) 111-122.[7] Rocard Y., Thermodynamique, Masson, Paris, 1952.[8] Bruhat G., Cours de Physique G´en´erale, Thermodynamique, Masson, Paris,1968.[9] Bongiorno V., Scriven L.E., Davis H.T., Molecular theory of fluid interfaces, J.Coll. Int. Sci. 57 (1976) 462-475.[10] van der Waals J.D., Thermodynamique de la capillarit´e dans l’hypoth`ese d’unevariation continue de densit´e, Archives N´eerlandaises 28 (1894-1895) 121-209.[11] Korteweg J., Sur la forme que prennent les ´equations du mouvement des fluidessi l’on tient compte des forces capillaires, Archives N´eerlandaises 2, n o
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