Properties of thermocapillary fluids and symmetrization of motion equations
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Properties of thermocapillary fluidsand symmetrization of motion equations
Henri Gouin
Aix-Marseille Univ, CNRS, Centrale Marseille, M2P2 UMR 7340,13451 Marseille, France
Abstract
The equations of fluid motions are considered in the case of internal energy de-pending on mass density, volume entropy and their spatial derivatives. The modelcorresponds to domains with large density gradients in which the temperature isnot necessary uniform. In this new general representation writes in symmetric formwith respect to the mass and entropy densities. For conservative motions of perfect thermocapillary fluids , Kelvin’s circulation theorems are always valid. Dissipativecases are also considered; we obtain the balance of energy and we prove that equa-tions are compatible with the second law of thermodynamics. The internal energyform allows to obtain a Legendre transformation inducing a quasi-linear system ofconservation laws which can be written in a divergence form and the stability nearequilibrium positions can be deduced. The result extends classical hyperbolicitytheory for governing-equations’ systems in hydrodynamics, but symmetric matricesare replaced by Hermitian matrices.
Key words:
Thermocapillarity; second gradient models; fluid interfaces;hyperbolicity
Theoretical and experimental studies show that, when working far from crit-ical conditions, the liquid-vapour capillary layer has a few molecular-beams’thickness [1,2,3]. Consequently, liquid-vapour interfaces are generally repre-sented by material surfaces endowed with surface energy related to Laplace’s
Email address: [email protected];[email protected] (HenriGouin).
Preprint submitted to International Journal of Non-Linear Mechanics 4 September 2018 urface-tension [4]. The surfaces have their own characteristic behaviours andenergy properties [5]. In interfacial layers, molecular models - as used in ki-netic theory of gases - express behaviours associated with non-convex internalenergies [6,7,8,9]. These models appear advantageous as they provide a moreprecise verification of Maxwell’s rule applied to isothermal phase-transitions[8,10]. Nonetheless, they present two disadvantages. First, for densities that liebetween bulk densities, the pressure may become negative. However, simplephysical experiments can be used to cause traction that leads to negative pres-sure values [11]. Second, in the field between bulks, internal energies cannotbe represented by convex surfaces associated with the variation of densities.The fact seems to contradict the existence of equilibrium states. To overcomethe disadvantages, the thermodynamics replaces the non-convex portions cor-responding to internal energies by planar domains [12]; the fluid can no longerbe considered as a continuous medium.At equilibrium, an appropriate modification of the layer stress-tensor, ex-pressed in an anisotropic form, can eliminate the previous disadvantages; then,the continuous-medium energies change [6,9] and near the critical point, al-low to study interfaces of non-molecular size [13,14]. The approach is not newand dates back to van der Walls [15,16] and Korteweg [17]; it corresponds towhat is known as a Landau-Ginzburg theory [7]. The contradiction betweenKorteweg’s classical stress theory and the Clausius-Duhem inequality [18] iscorrected by Eglit [19], Dunn and Serrin [20], Casal and Gouin [21].To study capillary layers and bulks, the second gradient theory [22,23] - concep-tually simpler than Laplace’s theory - led to a capillary model for isothermalliquid-vapour interfaces. Fluids endowed with internal capillarity yield equa-tions of motion and energy including additive terms. The internal energy ofsuch fluids is a function of the entropy, the mass density and the gradient ofmass density [24,25,26]. Gradient theory can be extended to solid mechanics,materials, nanofluidics, fluid mixtures [27,28,29,30,31,32] and developed at n -order ( n >
2) [33].The simplest model in continuum mechanics considers a volume internal en-ergy ε as the sum of two terms: a first one ε corresponding to a medium withuniform composition equal to the local one and a second one associated withthe non-uniformity of the fluid and is approximated by a gradient expansion,typically truncated to the second order [6,15]: ε = ε ( ρ, η ) + 12 m (grad ρ ) , where ρ is the mass density (or volume mass), η the volume entropy, ε thevolume internal-energy of the fluid assumed to be homogeneous and m is acoefficient independent of η , grad ρ and of any higher derivatives [9]. In sucha model, η varies with ρ through isothermal interface in the same way as in2ulks and at given temperature T satisfies ∂ε ∂η ( ρ, η ) = T, (1)so, ε = ε ( ρ, T ). At given temperature T , the points representing phasestates in the ( ρ, η, ε ) space lie on a curve instead of surface ε = ε ( ρ, η ).In fact, the assumption is not exact for realistic potentials; in practice thepotential for the two-density form of van der Waals’ theory is not constructedby prescription (1) but by other means [9] (Ch. 8). Aside from the questionof accuracy, there are qualitative features of some interfaces, especially in sys-tems of more than one component, that require two or more independentlyvarying densities - entropy included - for their description; in fact, when wehave non-monotonic behaviours, one-density models inevitably lead to mono-tonic variations of densities [8]. In our case, the model must be extended bytaking account of not only the strong variations of matter density throughinterfacial layers but also the strong variations of entropy. For this purpose,Rowlinson and Widom in [9] (Ch. 3 and Ch. 9) introduced an energy arisingfrom the mean-field theory and depending on the deviations of densities ρ and η from their values at the critical point and on the gradients of densities. Con-sequently, we can also imagine non-isothermal steady motions in zones withlarge density gradients [34].The paper is presented as follows :In Section 2, we consider different forms of equation of motions in the mostgeneral case. The Hamilton principle yields the equation of conservative mo-tions in a symmetric form with respect to mass and entropy volumes. In [35]we considered thermocapillary fluids as fluids with a specific internal energyin the form α = α ( ρ, s, grad ρ, grad s) where s is the specific entropy. But,it is more convenient to consider the volume entropy in place of the specificentropy to obtain a simpler system of equations.In Section 3, we extend the balance equations to viscous fluids. The equationof energy is completed with a heat flux and a heat supply. We get an addi-tive interstitial-working term similar to a heat-flux vector and the processes’equations are compatible with the second law of thermodynamics.In Section 4, we revisit Kelvin’s circulation-theorem and analyse the surfacetension of planar interfaces at equilibrium. The Maxwell rule is extended forthermocapillary fluids.Section 5 is a completely new study. A Legendre transformation yields a sys-tem of equations in a divergence form when conjugated variables - with re-spect to the mass density, volume entropy and their gradients - are used. Thehyperbolicity of the system of governing equations can be studied. Small per-turbations near an equilibrium position are analysed. Eigenvalues associatedwith Hermitian matrices conclude to the stability of equilibrium positions byextending Godunov and Lax-Friedrichs analyses [36,37].3 conclusion ends the paper.For any vectors a, b we use the notation a ⋆ b for the scalar product (the linevector is multiplied by the column vector) and a b ⋆ for the tensor product (or a ⊗ b the column vector is multiplied by the line vector), where superscript ⋆ denotes the transposition. Divergence of a linear transformation D is thecovector div D such that, for any constant vector c , (div D ) c = div( D c ).The identical transformation is denoted by I . The volume internal energy of a thermocapillary fluid is represented by adevelopment in gradients with respect to ρ and η : ε = ε ( ρ, η, grad ρ, grad η ) . (2)A particular case of volume internal energy can be ε = ε ( ρ, η ) + 12 (cid:16) C (grad ρ ) + 2 D (grad ρ ) ⋆ grad η + E (grad η ) (cid:17) , (3)where C, D, E are assumed to be constant; in special case D = 0 and E = 0,we get Cahn and Hilliard’s fluids [6]. • Thermodynamical potential ε ( ρ, η ) = ρ α ( ρ, s ) is the volume internalenergy of the fluid bulk with volume mass ρ and volume entropy η (the samepotential expression as for compressible fluids). Consequently, dα ( ρ, s ) = Pρ dρ + T ds , where P is the thermodynamical pressure and T the Kelvin temperature.Then, dε = µ dρ + T dη , where µ = ∂ε ( ρ, η ) ∂ρ is the bulk chemical-potential. We get P = ρ µ + η T − ε and η dT = dP − ρ dµ . For thermocapillary fluids associated with Eq. (2) we denote, dε = µ dρ + T dη + Φ ⋆ d (grad ρ ) + Ψ ⋆ d (grad η )with µ = ∂ε∂ρ , T = ∂ε∂η , Φ ⋆ = ∂ε∂ grad ρ , Ψ ⋆ = ∂ε∂ grad η . (4)We always denote P = ρ µ + η T − ε, where P is called the thermocapillary pressure , µ and T are extended by Eq. (4)as the thermocapillary chemical-potential and the thermocapillary temperature ,respectively.In the particular case of Eq. (3) we obtain, µ = ∂ε ∂ρ , T = ∂ε ∂η , Φ ⋆ = C (grad ρ ) ⋆ + D (grad η ) ⋆ and Ψ ⋆ = D (grad ρ ) ⋆ + E (grad η ) ⋆ , where µ ≡ µ and T ≡ T are also the chemical potential and the temperatureof bulks. The mass conservation writes : ∂ρ∂t + div ρ u = 0 . (5)For isentropic motions, the volume entropy conservation writes : ∂η∂t + div η u = 0 . (6)The Hamilton action between time t and time t is [39,40], S = Z t t Z D t L dv dt with L = 12 ρ u ⋆ u − ε − ρ Ω . where L is the Lagrangian, dv is the volume element of physical space D t at time t , dt is the time differential, u is the fluid velocity-vector and Ω theexternal-force potential. We have the properties associated with the variationsof u , ρ and η δ u = d ζ dt , δρ = − ρ div ζ , δη = − η div ζ , (7)5here ζ = δ x notes the variation of Euler position x as defined by Serrin in[41]. Equation (7 ) corresponds to an isentropic variation when the motion isconservative and isentropic.Thanks to Eqs. (4-7), the variation of Hamilton’s action is [42], δS = Z t t Z D t " ρ u ⋆ d ζ dt + ( ρ µ − ε + η T ) div ζ − δ (grad ρ ) ⋆ Φ − δ (grad η ) ⋆ Ψ − ρ ∂ Ω ∂ x ζ dv dt. Relations :( ρ µ − ε + η T ) div ζ = div[( ρ µ − ε + η T ) ζ ]+ " Φ ⋆ ∂ grad ρ∂ x + Ψ ⋆ ∂ grad η∂ x − ρ ∂µ∂ x − η ∂ T ∂ x , and δ (grad ρ ) ⋆ = ∂δρ∂ x − ∂ρ∂ x ∂ ζ ∂ x and δ (grad η ) ⋆ = ∂δη∂ x − ∂η∂ x ∂ ζ ∂ x , imply − δ (grad ρ ) ⋆ Φ = div " Φ ∂ρ∂ x ζ − Φ δρ + ( ρ div Φ ) ζ + " ∂ ( ρ div Φ ) ∂ x − div Φ ∂ρ∂ x ! ζ . and − δ (grad η ) ⋆ Ψ = div " Ψ ∂η∂ x ζ − Ψ δη + ( η div Ψ ) ζ + " ∂ ( η div Ψ ) ∂ x − div Ψ ∂η∂ x ! ζ . Consequently, ρ u ⋆ d ζ dt = ∂∂t ( ρ u ⋆ ζ ) + div[ ρ ( u ⋆ ζ ) u ] − ρ a ⋆ ζ , a denotes the acceleration vector of the fluid. We get, δS = Z t t Z D t ( − ρ a ⋆ − ρ ∂µ∂ x − η ∂ T ∂ x + Φ ⋆ ∂ grad ρ∂ x + Ψ ⋆ ∂ grad η∂ x + ∂∂ x (cid:16) ρ div Φ + η div Ψ (cid:17) − div Φ ∂ρ∂ x + Ψ ∂η∂ x ! − ρ ∂ Ω ∂ x ) ζ dv dt + Z t t Z D t ( div " Φ ∂ρ∂ x ζ − Φ δρ + ( ρ div Φ ) ζ + Ψ ∂η∂ x ζ − Ψ δη + ( η div Ψ ) ζ +( ρ µ − ε + η T ) ζ (cid:21) + ∂∂t ( ρ u ⋆ ζ ) + div[ ρ ( u ⋆ ζ ) u ] ) dv dt. By integration, the second integral vanishes when the virtual displacement isnull on the boundary of [ t , t ] × D t .From Hamilton principle, ∀ x ∈ D t → ζ ( x ) , with ζ ( x ) null on the boundary of D t , δ S = 0 , we can deduce the motion equation of conservative and isentropic fluids. From Hamilton principle, we deduce ρ a + ρ grad µ + η grad T − ∂ grad ρ∂ x Φ − ∂ grad η∂ x Ψ (8) − grad ( ρ div Φ + η div Ψ ) + div ⋆ ( Φ grad ⋆ ρ + Ψ grad ⋆ η ) + ρ grad Ω = 0 . From relations ∂ ( ρ div Φ ) ∂ x = (div Φ ) ∂ρ∂ x + ρ ∂ div Φ ∂ x and ∂ ( η div Ψ ) ∂ x = (div Ψ ) ∂η∂ x + η ∂ div Ψ ∂ x , div Φ ∂ρ∂ x ! = (div Φ ) ∂ρ∂ x + Φ ⋆ ∂ grad ρ∂ x anddiv Ψ ∂η∂ x ! = (div Ψ ) ∂η∂ x + Ψ ⋆ ∂ grad η∂ x , we obtain, ρ a + ρ grad( µ − div Φ + Ω) + η grad( T − div Ψ ) = 0 , (9)7r, a + grad( µ − div Φ + Ω) + s grad( T − div Ψ ) = 0 , (10)where s = η/ρ .From Eq. (9) we deduce, a + grad Ξ − θ grad s = 0 (11)with θ = T − div Ψ and Ξ = µ − div Φ + Ω + ( T − div Ψ ) s. In case of internal energy (3) we get θ = T − D ∆ ρ − E ∆ η and Ξ = µ − C ∆ ρ − D ∆ η +Ω+( T − D ∆ ρ − E ∆ η ) s, where ∆ is the Laplace operator. From dµ = d P ρ − s d T , Eq. (9) can be written ρ a + grad P + ρ grad ( Ω − div Φ ) − η grad div Ψ = 0 . (12) If we denote p = P − ρ div Φ − η div Ψ and σ = − p I − Φ ∂ρ∂ x − Ψ ∂η∂ x , equation (8) yields ρ a − div ⋆ σ + ρ grad Ω = 0 . (13)In the case of internal energy (3), we obtain the value of σ , σ = − p I − ( C grad ρ + D grad η ) ∂ρ∂ x − ( D grad ρ + E grad η ) ∂η∂ x . (14) If the total entropy of the fluid in domain D t is constant, its variation is null, δ Z D t η dv ≡ δ Z D t ρs dv = 0 , and it exists a constant Lagrange multiplier T such that the variation ofHamilton’s action δS ≡ Z t t Z D t ρ δ u ⋆ u − ερ − Ω + T s ! dv dt = 08s null, with always δ u = d ζ dt and δρ = − ρ div ζ . From variation field ζ , we get the same equation of motions (Eq. (9)).When ζ = 0, independent variation of η ( δη = ρ δs ) yields δS = Z t t Z D t ( − δε + ρ T δs ) dv dt = 0 . Due to Eq. (4), δε = T δη − Ψ ⋆ δ grad η and ζ = 0 implies δ grad η = ∂δη∂ x ! ⋆ .Consequently, δS = Z t t Z D t [( T − T ) δη − Ψ ⋆ δ grad η ] dv dt = Z t t Z D t ( T − T + div Ψ ) δη dv dt − Z t t Z D t div ( Ψ δη ) dv dt. We consider that δη = 0 on the boundary of D t . By integration on the D t -boundary, the second integral is null and the Hamilton principle yields : T = T + div Ψ , and in the special case of a volume energy in form (3), T = T + D ∆ ρ + E ∆ η . (15)We note that θ = T − div Ψ is constant equal to T which is the temperature inthe homogeneous parts of thermocapillary fluids (corresponding to the bulks). For a viscous fluid, we add a stress tensor in the Newtonian form σ v = τ (tr D ) I + 2 κ D , where D is the velocity deformation tensor; τ , κ are constant. We are in firstgradient model for the viscosity but experiments prove that such a model isalways correct for capillary layers [45]. The Hamilton principle becomes the9rinciple of virtual powers (or virtual works) [10] and Eq. (13) allows to obtain ρ a − div ⋆ ( σ + σ v ) + ρ grad Ω = 0 , where σ verifies Eq. (14). We extend the results proposed in [19,20,21,46]. Let us note M = ρ a − div ⋆ ( σ + σ v ) + ρ grad Ω B = ∂ρ∂t + div ρ u N = ρ ( T − div Ψ ) ˙ s + div q − r − tr ( σ v D ) F = ∂e∂t + div [( e I − σ − σ v ) u ] − div ( ˙ ρ Φ + ˙ η Ψ ) + div q − r − ρ ∂ Ω ∂t (16)where e = 12 ρ u ⋆ u + ε + ρ Ω is the total volume energy of the fluid, q and r arethe heat flux vector and the heat supply, respectively; superscript ˙ denotesthe material derivative and the free enthalpy is h ≡ ε + pρ . We get : Theorem 1
For an internal energy in form (2) and for any motion of ther-mocapillary fluids, F − M ⋆ u − (cid:18) u ⋆ u + h + Ω (cid:19) B − N ≡ . (17)The proof is proposed in Appendix 1. Corollary 2
For any motion of conservative thermocapillary fluids, the con-servation of specific entropy ˙ s = 0 (or ∂η/∂t + div η u = 0 ) is equivalentto ∂e∂t + div [( e I − σ ) u ] − div ( ˙ ρ Φ + ˙ η Ψ ) − ρ ∂ Ω ∂t = 0 . Corollary 3
For any motions of dissipative thermocapillary fluids, equationof energy ∂e∂t + div [( e I − σ − σ v ) u ] − div ( ˙ ρ Φ + ˙ η Ψ ) + div q − r − ρ ∂ Ω ∂t = 0 is equivalent to ”equation of entropy” ρ ( T − div Ψ ) ˙ s + div q − r − tr ( σ v D ) = 0 . (18)10erm ˙ ρ Φ + ˙ η Ψ has the physical dimension of a heat flux vector; it correspondsto the interstitial working term [20] and reveals the existence of an additionalterm to the heat flux even if the motion is conservative. The result extendsthe ones obtained for capillary fluids when terms associated with grad η arenot taken into account. For any motion of thermocapillary fluids, tr ( σ v D ) ≥ Planck’s inequality [47] ρ ( T − div Ψ ) ˙ s + div q − r ≥ . We assume the
Fourier law in the general form, q ⋆ grad θ ≤ , with θ = T − div Ψ and we obtain ρ ˙ s + div (cid:18) q θ (cid:19) − rθ ≥ , which is the extended form for thermocapillary fluids of Clausius-Duhem’sinequality . We note that temperature θ corresponds to the temperature valuein homogeneous parts of thermocapillary fluids. Theorem 4
The velocity circulation on a closed, isentropic fluid-curve is con-stant.
The circulation of velocity vector u on a closed fluid-curve C is J = I C u ⋆ d x .From [41] p. 162, ddt I C u ⋆ d x = I C a ⋆ d x and thanks to Eq. (11), we deduce I C a ⋆ d x = I C grad ⋆ Ξ d x = 0 , which proves the theorem. 11 orollary 5 In a homentropic motion (the entropy is uniform in the fluid),the velocity circulation on a fluid-curve is constant.
Theorem 6
The velocity circulation on a closed fluid-curve such that
T − div Ψ = T is constant. From Eq. (10) we get, a + grad Ξ − ( T − div Ψ ) grad s = 0 . But, a −
12 grad u = ∂ u ∂t + ∂ u ∂ x u − ∂ u ∂ x ! ⋆ u = ∂ u ∂t + rot u × u . For a stationary motion,rot u × u = ( T − div Ψ ) grad s − grad Ξ + u ! . (19)Equation(19) is the generalized Crocco-Vazsonyi relation for thermocapillaryfluids. We consider a planar interface between liquid and vapour bulks of a thermo-capillary fluid. In the interfacial layer, density gradients are important. Withinternal energy (3), the stress tensor is σ = − (cid:18) P − C ⋆ ρ grad ρ − D grad ⋆ ρ grad η − E ⋆ η grad η (cid:19) I − ( C grad ρ + D grad η ) ∂ρ∂ x − ( D grad ρ + E grad η ) ∂η∂ x . When the extraneous force potential is neglected, the equation of the equilib-rium is div σ = 0 . For a flat interface, normal to grad ρ and grad η , the coordinate normal tothe interface being denoted z , the eigenvalues of stress tensor σ are λ = −P + C dρdz ! + D dρdz dηdz + E dηdz ! λ = −P − C dρdz ! − D dρdz dηdz − E dηdz ! (associated with direction normal to the plane of interface).In an orthonormal system with third coordinate z , the stress tensor writes σ = λ λ
00 0 λ . The equation of balance momentum in the planar interface implies λ = − P , where P is the common pressure in the bulks. The force per unit of length onthe edge of the interface is (see Fig. 1) : F = Z z z λ dz = − P ( z − z ) + Z z z C dρdz ! + 2 D dρdz dηdz + E dηdz ! dz, where z − z corresponds to the physical interface thickness. Due to the smallthickness of the interface, − P ( z − z ) is negligible. Let us note H = Z z z C dρdz ! dz, H = Z z z D dρdz dηdz dz, H = Z z z E dηdz ! dz . The line force per unit of length on the interface edge is H = H + H + H , where H represents the surface tension of the planar interface at equilibrium.If we consider the approximation ∂ε ∂η ( ρ, η ) = T , (20)where T is the temperature value in the liquid and vapour bulks, then η isa function of ρ . Due to the variation principle, the surface tension calculatedfor capillary fluids (corresponding to D = 0 and E = 0) with approximation(20) is necessary greater than the surface tension when ∂ε ∂η ( ρ, η ) = T + D ∆ ρ + E ∆ η . ig. 1. Interpretation of the surface tension In fact, experiments prove that the entropy effects are small enough on surfacetension value and when the critical point is approached, the one - and two-density theories become equivalent as a general property of critical point ([9],Ch. 3), [14].
We consider the case when the volume internal energy is in form (3). In thecase of capillary fluids (corresponding to D = 0 and E = 0), the Maxwell ruleof planar liquid-vapour interface at equilibrium can be written in equivalentform Z ρ l ρ v ( µ − µ ) dρ = 0 , where ρ l and ρ v are the mass density in the liquid and vapour bulks; µ is thecommon value of the chemical potential in the bulks [11]. We denote η l and η v the volume entropies in the liquid and vapour bulks, respectively.Equation of temperature (15) of thermocapillary fluids yields T − T = D d ρdz + E d ηdz . Without body forces, equation of equilibrium (10) of thermocapillary fluidsyields grad( µ − div Φ ) = 014r by integration, µ − µ = C d ρdz + D d ηdz . Consequently, Z ρ l ρ v ( µ − µ ) dρ + Z η l η v ( T − T ) dη = Z z z " C d ρdz dρdz + D d ηdz dρdz + d ρdz dηdz ! + E d ηdz dηdz dz = C dρdz ! + D dρdz ! dηdz ! + E dηdz ! z z ≡ . The generalisation of Maxwell’s rule for thermocapillary fluids writes in theform : Z ρ l ρ v ( µ − µ ) dρ + Z η l η v ( T − T ) dη = 0 . Conservative motions with balance equation of energy lead to an interestingclass of quasilinear systems previously pointed out by Godunov [36], Friedrichsand Lax [37]. In classical mechanics and relativity, many studies on hyperbolicsystems were developed in the literature for hydrodynamics, elasticity andclassical materials [49,50,51,52]. The section extends results presented in [53]for the capillary-fluids’ simplest case. The small motions near an equilibriumposition are studied thanks to a convenient system of governing equationsassociated with a Legendre transformation of the internal energy.
Let us denote β ≡ grad ρ , χ ≡ grad η and j ≡ ρ u . The gradient of themass-conservation balance verifies another conservation equation, ∂ β ∂t + grad div j = 0 , (21)Conversely, if we consider β as an independent vector verifying Eq. (21), andif we add initial condition β | t =0 = grad ρ | t =0 , β ≡ grad ρ becomes a consequence of governing equation (21).Similarly, the gradient of the balance of entropy verifies another conservationequation, ∂ χ ∂t + grad div( η u ) = 0 . (22)In the same way, if we add initial condition χ | t =0 = grad η | t =0 , χ ≡ grad η becomes a consequence of governing equation (22) and we canconsider χ as an independent vector verifying Eq. (22).Without body forces, with the new notations, Eqs. (5, 6, 12, 21, 22) immedi-ately yield the system of governing equations in the form ∂ρ∂t + div j = 0 ∂η∂t + div ηρ j ! = 0 ∂ j ⋆ ∂t + div jj ⋆ ρ + P I ! − ρ grad ⋆ (div Φ ) − η grad ⋆ (div Ψ ) = 0 ∂ β ∂t + grad (div j ) = 0 ∂ χ ∂t + grad div( ηρ j ) = 0 . (23)With the new notations, the total volume energy of the fluid is E = j ⋆ j ρ + ε . We denote q = µ − u ⋆ u d E = q dρ + T dη + u ⋆ d j + Φ ⋆ d β + Ψ ⋆ d χ . The Legendre transform of E with respect to ρ, η, j , β , χ isΠ = ρ q + η T + j ⋆ u + Φ ⋆ β + Ψ ⋆ χ − E . (24)Conjugate variables q, T , u , Φ , Ψ verify ∂ Π ∂q = ρ, ∂ Π ∂ T = η, ∂ Π ∂ u = j ⋆ , ∂ Π ∂ Φ = β ⋆ , ∂ Π ∂ Ψ = χ ⋆ . ∂∂t ∂ Π ∂q ! + div " ∂ (Π u ) ∂q = 0 ∂∂t ∂ Π ∂ T ! + div " ∂ (Π u ) ∂ T = 0 ∂∂t ∂ Π ∂ u ! + div " ∂ (Π u ) ∂ u − ∂ Π ∂q ∂ Φ ∂ x − ∂ Π ∂ T ∂ Ψ ∂ x = 0 ∂∂t ∂ Π ∂ Φ ! + div " ∂ (Π u ) ∂ Φ + ∂ Π ∂q ∂ u ∂ x = 0 ∂∂t ∂ Π ∂ Ψ ! + div " ∂ (Π u ) ∂ Ψ + ∂ Π ∂ T ∂ u ∂ x = 0 . (25)When ε = ε ( ρ, η ), we get the classical gas dynamics equations and the conser-vative form of Godunov [36]. In the simplest special case, when ε = ε ( ρ, η ) + C ρ ) , we obtain the results [53]. The system of governing equations generates dispersive relations with multipleeigenvalues near an equilibrium position. In this subsection we extend theresults presented in [54,55,56]. System (25) yields constant solutions( ρ e , η e , j e , β e = 0 , χ e = 0 ) , where subscript e means at equilibrium. Since the governing equations areinvariant under Galilean transformation, we can assume that u e = 0 whichimplies j e = 0.Near equilibrium, we look for the solutions proportional to e i ( k ⋆ x − λt ) , where k ⋆ = [ k , k , k ] is a constant covector, λ a constant scalar and i = − v = v e i ( k ⋆ x − λt ) with v ⋆ = [ q, T , u , Φ , Ψ ] and v ⋆ = [ q , T , u , Φ , Ψ ] . We obtain ∂∂t ∂ Π ∂ v ! ⋆e ≡ ∂∂ v ∂ Π ∂ v ! ⋆e ∂ v ∂t ≡ − i λ ∂∂ v ∂ Π ∂ v ! ⋆e v e i ( k ⋆ x − λt ) . div ∂ Π u ∂ v ! ⋆ = X j =1 ∂∂x j ∂ Π u j ∂ v ! ⋆ = X j =1 ∂∂ v ∂ Π u j ∂ v ! ⋆ ∂ v ∂x j , x ⋆ = [ x , x , x ] anddiv ∂ Π u ∂ v ! ⋆e = X j =1 i F j k j v e i ( k ⋆ x − λt ) , where F j ≡ ∂∂ v ∂ Π u j ∂ v ! ⋆e ; we denote F ≡ X j =1 F j k j . At equilibrium, • For Eq. (23) (or equivalently Eq. (25) ), we add two additive terms toclassical-fluids’ equations : First term, div ∂ Π ∂q ∂ Φ ∂ x ! = (grad ⋆ ρ ) ∂ Φ ∂ x + ρ div ∂ Φ ∂ x ! . At equilibrium, grad e ρ = 0 . Then, from Φ = Φ e i ( k ⋆ x − λt ) ,div ∂ Π ∂q ∂ Φ ∂ x ! e = ρ e div ∂ Φ ∂ x ! = i ρ e Φ ⋆ k k ⋆ e i ( k ⋆ x − λt ) . Second term, div ∂ Π ∂ T ∂ Ψ ∂ x ! = (grad ⋆ η ) ∂ Ψ ∂ x + η div ∂ Ψ ∂ x ! . At equilibrium, grad e η = 0 . Then, from Ψ = Ψ e i ( k ⋆ x − λt ) ,div ∂ Π ∂ T ∂ Ψ ∂ x ! e = ρ e div ∂ Ψ ∂ x ! = i η e Ψ ⋆ k k ⋆ e i ( k ⋆ x − λt ) . Taking account of u = u e i ( k ⋆ x − λt ) , • For Eq. (25) at equilibrium, we add termdiv " ∂ Π ∂q ∂ u ∂ x e = i ρ e u ⋆ k k ⋆ e i ( k ⋆ x − λt ) . • For Eq. (25) at equilibrium, we add termdiv " ∂ Π ∂ T ∂ u ∂ x e = i η e u ⋆ k k ⋆ e i ( k ⋆ x − λt ) . A , C such that A = ∂∂ v " ∂ Π ∂ v ! ⋆ e , C = − C ⋆ = ⋆ ⋆ ⋆ ⋆ ⋆ ∗ O − ρ e kk ∗ − η e kk ⋆ ρ e kk ⋆ O O η e kk ⋆ O O with O = and ∗ = [0 0 0] . Due to i C ⋆ ≡ i C , where overline denotes the complex conjugation; matrix i C is hermitian.The solutions corresponding to the perturbations of system (25) verify : i [ F + i C − λ A ] v e i ( k ⋆ x − λt ) = 0 , where D = D ⋆ ≡ F + i C is Hermitian matrix and A is symmetric matrix;so, λ are the roots of the characteristic equation :det [ D − λ A ] = 0 , and λ is eigenvalue of D with respect to A and v is its eigenvector. Nearan equilibrium state where the local internal energy is locally convex, A ispositive definite; eigenvalues are real and the small perturbations are stablewith respect to equilibrium positions. For conservative processes associated with system (23), Legendre transforma-tion (24) of the internal energy yields a system of governing equations which19xtends the classical models of hyperbolicity to non-local behaviour. The Lax-Friedrichs method [57] is a numerical method we can consider as an alternativeto Godunov’s scheme [58] in which one avoids solving a Riemann problem ateach cell interface, at the expense of adding artificial viscosity. The stability ofquasi-linear perturbations allows to forecast an extention of the Lax-Friedrichsmethod for thermocapillary fluids.
By using System (16) in the first member of Eq. (17), dissipative terms q , r, σ v can be algebraically simplified. Also are terms associated with inertia and Ω.The remaining terms are M = − div ⋆ ( σ ) B = ∂ρ∂t + div ρ u N = ρ ( T − div Ψ ) ˙ sF = ∂ε∂t + div [( ε I − σ ) u ] − div ( ˙ ρ Φ + ˙ η Ψ ) , and we have to prove F − M ⋆ u − h B − N ≡ . (26)From ∂ε∂t + div( ε u ) − (div σ ) u − div ( ˙ ρ Φ + ˙ η Ψ ) = ε + pρ B + P ˙ ρ + ρ T ˙ s + Φ ⋆ d grad ρd t + Ψ ⋆ d grad ηdt − p ˙ ρρ + Φ ⋆ ∂ u ∂ x ! ⋆ grad ρ + Ψ ⋆ ∂ u ∂ x ! ⋆ grad η − div ( ˙ ρ Φ + ˙ η Ψ ) = ε + pρ B + ρ ( T − div Ψ ) ˙ s + Φ ⋆ grad ∂ρ∂t + Φ ⋆ ∂ grad ρ∂ x u + Ψ ⋆ grad ∂η∂t + Ψ ⋆ ∂ grad η∂ x u + Φ ⋆ ∂ u ∂ x ! ⋆ grad ρ + Ψ ⋆ ∂ u ∂ x ! ⋆ grad η − grad ⋆ ∂ρ∂t ! Φ − u ⋆ ∂ grad ρ∂ x Φ − (grad ⋆ ρ ) ∂ u ∂ x Φ − grad ⋆ ∂η∂t ! Ψ − u ⋆ ∂ grad η∂ x Ψ − (grad ⋆ η ) ∂ u ∂ x Ψ , ∂ grad ρ∂ x = ∂ grad ρ∂ x ! ⋆ and ∂ grad η∂ x = ∂ grad η∂ x ! ⋆ we get, ∂ε∂t + div( ε u ) − (div σ ) u − div ( ˙ ρ Φ + ˙ η Ψ ) = ε + pρ B + ρ ( T − div Ψ ) ˙ s. Relation div( σ u ) = (div σ ) u + tr σ ∂ u ∂ x ! yields relation (26). (cid:3) Relations ∂ Π ∂q = ρ and ∂ (Π u ) ∂q = ρ u imply Eq. (25 ).Relations ∂ Π ∂ T = η and ∂ (Π u ) ∂ T = η u imply Eq. (25 ).From relation ∂ Π ∂ u = j ⋆ = ⇒ ∂ (Π u ) ∂ u = uj ⋆ + Π I , we obtain,div ∂ (Π u ) ∂ u − ∂ Π ∂q ∂ Φ ∂ x − ∂ Π ∂ T ∂ Ψ ∂ x ! = div ( ρ u u ⋆ ) + ∂ Π ∂ x − div ρ ∂ Φ ∂ x + η ∂ Ψ ∂ x ! = div ( ρ u u ⋆ ) + ρ ∂ µ ∂ x + η ∂ T ∂ x − ρ ∂ div Φ ∂ x − η ∂ div Ψ ∂ x and consequently, the motion equation writes on form (25 ).From relation ∂ Π ∂ Φ = β ⋆ and ∂ (Π u ) ∂ Φ + ∂ Π ∂q ∂ u ∂ x = u β ⋆ + ρ ∂ u ∂ x = ∂ ( ρ u ) ∂ x , we deduce Eq. (25 ).From relation ∂ Π ∂ Ψ = χ ⋆ and ∂ (Π u ) ∂ Ψ + ∂ Π ∂ T ∂ u ∂ x = u χ ⋆ + η ∂ u ∂ x = ∂ ( η u ) ∂ x ,
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