Multi-Gradient Fluids Henri Gouin

Multi-gradient fluids Henri Gouin

To cite this version:

Henri Gouin. Multi-gradient fluids. Ricerche di matematica, Springer Verlag, In press, 10.1007/s11587-018-0397-8. hal-01731201

HAL Id: hal-01731201 https://hal.archives-ouvertes.fr/hal-01731201 Submitted on 13 Mar 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Preprint submitted to Ricerche di Matematica - Springer (2018)

Multi-gradient ﬂuids

Henri Gouin

Received: 25 January 2018

Abstract An internal energy function of the mass density, the volumetric entropy and their gradients at n-order generates the representation of multi- gradient fluids. Thanks to Hamilton’s principle, we obtain a thermodynamical form of the equation of motion which generalizes the case of perfect compressible fluids. First integrals of flows are extended cases of perfect compressible fluids. The equation of motion and the equation of energy are written for dissipative cases, and are compatible with the second law of thermodynamics. Keywords Multi-gradient fluids · Equation of motion · Equation of energy · First integrals · Laws of thermodynamics Mathematics Subject Classification (2000) 76A02 · 76E30 · 76M30

1 Introduction

Phase transitions between liquid and vapour are associated with a bulk internal- energy per unit volume ε0(ρ, η) which is non-convex function of mass density ρ and volumetric entropy η. Consequently, in continuous theories, the simplest model allowing to study inhomogeneous ﬂuids inside interfacial layers considers an internal energy per unit volume ε in the form

1 2 ε = ε0(ρ, η)+ λ |gradρ | , (1) 2

The paper is dedicated to Professor Tommaso Ruggeri. The work was supported by National Group of Mathematical Physics GNFM-INdAM (Italy).

Aix-Marseille Univ, CNRS, Centrale Marseille, M2P2 UMR 7340, Marseille, France. Tel.:+33-491485120 E-mail: [email protected]; [email protected] ORCID iD: 0000-0003-4088-1386 2 Henri Gouin

where the second term is associated with the non-uniformity of mass density ρ and λ is a coefficient independent of η and grad ρ. The energy form in Eq. (1) has been introduced by van der Waals and is widely used in the literature [1,2]. The model describes interfaces as diffuse layers and has many applications for describing micro-droplets, contact-lines, nanofluidics, thin films, fluid mixtures and vegetal biology [3–6]. Thanks to the second-gradient theory, the model was extended in continuum mechanics, to the behaviour of strongly inhomogeneous media [7–9]. Nonetheless, at equilibrium, expression of energy given by Eq. (1) yields an uniform temperature in all the parts of inhomogeneous fluids [10]. Consequently, the volumetric entropy varies with the mass density in the same way as in the bulks; this fact leads to monotonic variations of densities but they are qualitative fea- tures of non-monotonic behaviours in transition layers which require two or more independently varying densities - entropy included - (chapter 8 in [10]); additively, the temperature through liquid-vapour interfaces is not necessary constant [11]. So, with the form of energy given by Eq. (1), the thermodynamics of inhomogeneous fluids is neglected and it is not possible to model flows with strong variations of temperature such as those across non-isothermal interfaces. In fact, it is difficult to take account of the gradient of temperature in the expression of the internal energy, but we can use the gradient of entropy. The simplest model was called thermocapillary fluids [12]. Such a second- gradient behaviour has also been considered by several authors in physical problems when the temperature is not constant in inhomogeneous parts of complex media [13,14]. We improve the model: when strong variations of mass density and entropy occur, a general case can consider fluids when the volume internal energy de- pends on mass density, volumetric entropy and their gradients up to a convenient n-order (with n ∈ N). The Hamilton principle allows to write the equation of conservative motions in an universal thermodynamic form structurally similar to the equation obtained in the case of conservative perfect fluids [15]. Consequently, all perfect fluid properties, as the Kelvin theorems are preserved and isentropic or isothermal motions can be studied. The case of dissipative motions is considered. From the equation of motion, the conservation of mass and the balance of entropy, we deduce the balance of equation of energy. The equation of energy is not in the same form as that of classical fluids. A new vector term – in form of flux of energy – appears, which takes the gradients of mass density and entropy into account. The Clausius- Duhem inequality can be deduced from the viscous dissipation and the Fourier inequality: that proves multi-gradient fluids are compatible with the second law of thermodynamics. Notations: For any vectors a, b, term aT b denotes the scalar product (line vector aT is multiplied by column vector b) and tensor a bT (or a⊗b) denotes the product of column vector a by line vector bT , where superscript T denotes the transposition. Tensor I denotes the identity transformation. Multi-gradient fluids 3

Principal nomenclatures: ρ, η and s denote the mass density, volumetric entropy and specific entropy. ε denotes the volume internal energy. µ˜ and T˜ denote the extended chemical potential and extended temperature. Dt denotes the space occupied by the fluid at time t. D0 denotes the reference space. F denotes the deformation gradient. u and γ denote the velocity and acceleration of the fluid. Ξ, θ and H denote the generalized chemical potential, generalized temperature and generalized enthalpy.

2 Model of multi-gradient ﬂuids

Perfect ﬂuids of grade n + 1 (n ∈ N) are continuous media with a volume internal energy ε which is a function of ρ and η and their gradients in form ε = ε(ρ, ∇ρ, . . ., ∇pρ, . . ., ∇nρ, η, ∇η, . . ., ∇pη, . . ., ∇nη), (2) p where ∇ , p ∈{1,...,n}, denotes the successive gradients in space Dt occupied by the ﬂuid at time t.

p p p p grad ρ ≡ ∇ ρ = ρ ,xj1 ...,xjp and grad η ≡ ∇ η = η ,xj1 ...,xjp , n o n o where the subscript “comma” indicates the partial derivative and xj1 ...xjp T belong to components of Euler variables x = [x1, x2, x3] ∈ Dt, space of the ﬂuid at time t. We indicate indices in subscript position without taking account of the tensor covariance or contravariance. We deduce ∂ε ∂ε ∂ε . ∂ε . dε = dρ + dη + . d∇ρ + . d∇η ∂ρ ∂η ∂∇ρ ∂∇η ∂ε . ∂ε . + ... + . d∇nρ + . d∇nη . (3) ∂∇nρ ∂∇nη . Sign . means the complete contraction of tensors such that,

∂ε . p p . ∂ε . d∇ ρ ≡ d∇ ρ . = ε,ρ,x ...,x dρ,xj ...,xj , ∂∇pρ ∂∇pρ j1 jp 1 p ∂ε . p p . ∂ε . d∇ η ≡ d∇ η . = ε,η,x ...,x dη,xj ...,xj , ∂∇pη ∂∇pη j1 jp 1 p where repeated subscripts correspond to the summation (1). We call the thermodynamical functions, ∂ε ∂ε µ˜ = and T˜ = , ∂ρ ∂η

1 Due to the fact that ε,ρ,x ...,x dρ,x ...,x = dρ,x ...,x ε, ρ,x ...,x and j1 jp j1 jp j1 jp j1 jp ε,η,x ...,x dη,x ...,x = dη,x ...,x ε, η,x ...,x , we indiﬀerently permute the posi- j1 jp j1 jp j1 jp j1 jp tion of the two terms in the summation. 4 Henri Gouin

the extended chemical potential and the extended temperature, respectively. We may also consider the case of an inhomogeneous configuration in the reference space by taking the reference position of the fluid into account; for the sake of simplicity, we will not do it. So, the fluid is supposed to have infinitely short memory and the motion history until an arbitrarily chosen past does not affect the determination of stresses at present time.

3 Equation of motions. Generalized temperature and generalized chemical potential

3.1 Virtual motions

The virtual power principle is a convenient way to obtain the motion equation. For conservative motions, it writes as the Hamilton principle. A particle is T identiﬁed in Lagrange representation by position X = [X1,X2,X3] ∈ D0, reference space of the ﬂuid. The variations of a motion are deduced from the functional family of virtual motions

X = ψ(x,t; β), where x ∈ Dt, β is a scalar parameter deﬁned in the vicinity of 0, when the real motion corresponds to β = 0. The virtual displacement δ associated with any variation of the real motion is written in the form

∂ψ(x,t; β) δX = . (4) ∂β β=0

The variation is dual and mathematically equivalent to Serrin’s one ([15], p. 145). Additively, we only consider variations with compact support in D0; consequently, δX = 0 on boundary ∂D0 of D0. The mass density satisﬁes the mass conservation law,

ρ det F = ρ0(X), (5)

where ρ0 is defined on D0 and F = ∂x/∂X is the deformation gradient. Equivalently, we can write ∂ρ + div (ρ u)=0, ∂t where u is the fluid velocity. The motion is supposed to be conservative and the specific entropy s = η/ρ is constant along each trajectory (isentropic motion)

s = s0(X), (6) or η det F = η0(X) with η0(X)= ρ0(X) s0(X). Multi-gradient ﬂuids 5

Equivalently to relation (6), we can write ∂η + div (η u)=0. ∂t Lemma: The variations of mass density and volumetric entropy verify

div0 (ρ0 δX) div0 (η0 δX) δρ = and δη = , (7) det F det F where div0 denotes the divergence in reference space D0.

The proof of the lemma comes from the velocity deﬁnition, ∂X(x,t) ∂X(x,t) u + =0, ∂x ∂t which implies ∂X ∂δX ∂δX d (δX) δu + u + =0 or δu = −F , (8) ∂x ∂x ∂t dt where d/dt denotes the material derivative. But,

1 1 ∂x ∂δX F F − = I =⇒ δF = −F δF − F = − . ∂X ∂X From the Jacobi identity and Eq. (5), we get

∂δX ρ0(X) δ det F = − det F tr = − det F div0 δX and ρ = , ∂X det F which implies 1 ∂ρ0 δρ = ρ0 div0 (δX)+ δX det F ∂X and consequently relation (7)1. The same calculus is suitable to volumetric entropy η and we obtain relation (7)2. Due to deﬁnition (4), variation δ and space gradients are independent, so they commute (δ∇pρ = ∇pδρ, δ∇pη = ∇pδη). Expression (3) implies,

∂ε ∂ε ∂ε . ∂ε . δε = δρ + δη + . ∇δρ + . ∇δη + ... ∂ρ ∂η ∂∇ρ ∂∇η ∂ε . ∂ε . + . ∇nδρ + . ∇nδη . (9) ∂∇nρ ∂∇nη We deﬁne operator divp as follows : N divp(bj1...jp )= bj1...jp , p ∈ with xj1 ,...,xjp ∈{x1, x2, x3} . ,xj1 ...,xj1 Classically, term bj1...jp corresponds to the summation on the re- ,xj1 ...,xjp peated indices j1 ...j p of the consecutive derivatives of bj1 ...jp with respect 6 Henri Gouin

2 to xj1 ...xjp ( ). Operator divp is the extended divergence at order p. Operator divp decreases from p the tensor order and term ∇p increases from p the tensor order. We deduce, ∂ε . ∂ε ∂ε . ∇δρ = div δρ − div δρ ∂∇ρ ∂∇ρ ∂∇ρ ∂ε . 2 ∂ε . ∂ε ∂ε . ∇ δρ = div . ∇δρ − div δρ + div2 δρ ∂∇2ρ ∂∇2ρ ∂∇2ρ ∂∇2ρ . . ∂ε . ∂ε . ∇nδρ = divA + (−1)ndiv δρ ∂∇nρ n n ∂∇nρ with Ap, p ∈{1,...,n} veriﬁes ∂ε . ∂ε . A = . ∇p−1δρ − div . ∇p−2δρ + ... p ∂∇pρ ∂∇pρ . q−1 ∂ε . p−q p−1 ∂ε + (−1) div 1 . ∇ δρ + ... + (−1) div 1 δρ. q− ∂∇pρ p− ∂∇pρ We obtain the same relation for the volumetric entropy, ∂ε . ∂ε ∂ε . ∇δη = div δη − div δη ∂∇η ∂∇η ∂∇η ∂ε . 2 ∂ε . ∂ε ∂ε . ∇ δη = div . ∇δη − div δη + div2 δη ∂∇2η ∂∇2η ∂∇2η ∂∇2η . . ∂ε . ∂ε . ∇nδη = divB + (−1)ndiv δη ∂∇nη n n ∂∇nη with Bp, p ∈{1,...,n} veriﬁes ∂ε . ∂ε . B = . ∇p−1δη − div . ∇p−2δη + ... p ∂∇pη ∂∇pη . q−1 ∂ε . p−q p−1 ∂ε + (−1) div 1 . ∇ δη + ... + (−1) div 1 δη. q− ∂∇pη p− ∂∇pη We denote n Ξ =µ ˜ − div Φ1 + div2 Φ2 + ... +(−1) divn Φn, (10)  ˜ n  θ = T − div Ψ 1 + div2 Ψ 2 + ... +(−1) divn Ψ n,

a11, a12, a13 2    For example, when A = a21, a22, a23 , then  a31, a32, a33  divA = a11,x +a21,x +a31,x , a12,x +a22,x +a32,x , a13,x +a23,x +a33,x , and 1 2 3 1 2 3 1 2 3 div2 A = a11,x1,x1 + a21,x2,x1 + a31,x3,x1 + a12,x1,x2 + a22,x2,x2 + a32,x3,x2 + a13,x1,x3 + a23,x2,x3 + a33,x3,x3 . Multi-gradient ﬂuids 7

with

∂ε ∂ε Φp = = ε,ρ,x ...,x , Ψ p = = ε,η,x ...,x , (11) ∂∇pρ j1 jp ∂∇pη j1 jp n o n o where xj1 ,...,xjp ∈ {x1, x2, x3}. We call Ξ and θ the generalized chemical potential and the generalized temperature, respectively.

3.2 The Hamilton action. Generalized enthalpy

Let L be the Lagrangian of the ﬂuid, i.e.

1 2 L = ρ uT u − ε − ρ Ω, with uT u = |u| , 2

where ε is deﬁned by Eq. (2), u denotes the particle velocity and Ω, function of (x,t), denotes the potential of external forces. Between times t1 and t2, the Hamilton action writes t2 a = L dx dt, Zt1 ZDt

where dx dt denotes the volume element in time-space [t1,t2] × Dt. Classical methods of variation calculus yield the variation of Hamilton’s action

′ δa = a (β)|β=0.

From δΩ = 0, we deduce

t2 1 δa = ρ uT δu + uT u − Ω δρ − δε dx dt. 2 Zt1 ZDt From eqs. (9) to (11), we deduce the value of δε and

t2 1 δa = ρ uT δu + uT u − Ξ − Ω δρ − θ δη 2 Zt1 ZDt + div (A1 + ... +An +B1 + ... +Bn) dx dt, o where term div(A1 + ... +An +B1 + ... +Bn) can be integrated on ∂Dt and consequently has a zero contribution. Then,

t2 T 1 T δa = ρ0 u δu + det F u u − Ξ − Ω δρ − (det F ) θ δη dXdt, 2 Zt1 ZD0 8 Henri Gouin

1 T where dX denotes the volume element in D0. Let us denote m = u u−Ξ − 2 Ω; taking account of Eqs. (7) and (8), we obtain

t2 T dδX δa = −ρ0 u F + (det F ) m δρ − (det F ) θ δη dXdt = dt Zt1 ZD0 t2 T d u F T T ρ0 − ρ0 ∇0 m + η0 ∇0 θ δX dXdt + dt Zt1 ZD0 (" # ) t2 T d ρ0 u F δX − + div0 (mρ0 δX) − div0 (θ η0 δX) dXdt, dt Zt1 ZD0 " # where ∇0 and div0 denotes the gradient and divergence operators in D0, respectively. The last integral can be integrated on the boundary of [t1,t2] × D0 and consequently is null. Then,

t2 T d u F T T δa = ρ0 − ρ0 ∇0 m + η0 ∇0 θ δX dXdt. dt Zt1 ZD0 (" # ) The fundamental lemma of variation calculus and η0 = ρ0 s0 yield

T d u F T T − ∇0 m + s0 ∇0 θ =0. dt Due to d uT F ∂u = γT + uT F , dt ∂x where γ denotes the acceleration vector, we obtain the equation of conservative motions,

γ+grad(Ξ + Ω)+ s grad θ =0 or γ + grad(H + Ω) − θ grad s =0, (12)

where H = Ξ + sθ denotes the generalized enthalpy. Relation (12) is a thermodynamic form of the equation of isentropic motions for perfect ﬂuids which generalizes relation (29.8) in [15].

3.3 Isothermal motions

We assume that t2 η dx dt = S0, (13) Zt1 ZDt where S0 is a constant. Only mass conservation (5) is considered. The volumetric entropy becomes an independent variable which is subject to constraint (13) and action a can be replaced by

t2 b = (L + θ0 η ) dx dt, Zt1 ZDt Multi-gradient ﬂuids 9 where scalar θ0 is a constant Lagrange multiplier associated with integral constraint (13). For any variation κ of η, with δX = 0, we immediately deduce

t2 det F (θ0 − θ) κ dXdt =0 Zt1 ZD0 and consequently θ = θ0, that corresponds to isothermal motions in the sense of generalized temperature θ. When κ = 0, for any variation of δX, we get the equation of motion in the form γ +grad(Ξ+Ω) = 0. At equilibrium, without body forces,

Ξ = µ0 and θ = θ0 , where µ0 denotes the chemical-potential value in ﬂuid bulks.

3.4 Conservative properties of perfect multi-gradient ﬂuids

The circulation of velocity vector u on a closed fluid-curve C is J = uT dx. C From [15] p. 162, I d uT dx = γT dx dt IC IC Thanks to Eq. (12), we deduce: – The velocity circulation on a closed, isentropic fluid-curve is constant. – In a homentropic motion (the specific entropy is uniform in the fluid), the velocity circulation on a fluid-curve is constant. – The velocity circulation on a closed fluid-curve such that θ = θ0 is constant.

From T 1 T ∂u ∂u ∂u ∂u γ − grad u u = + u − u = + rot u × u . 2 ∂t ∂x ∂x ∂t and γ + grad(H + Ω) − θ grad s =0 , in the case of stationary motions, we obtain 1 rot u × u = θ grad s − grad uT u + H + Ω . (14) 2 Equation (14) is the generalized Crocco-Vazsonyi relation for multi-gradient ﬂuids. 10 Henri Gouin

The Noether theorem proves that any law of conservation can be represented by an invariance group. It has been proved that the conservation laws ex- pressed by Kelvin’s theorems correspond to group of permutations consisting of particles of equal specific entropy [16]. It is clear that this group keeps the equation of motions invariant both for the classical perfect fluids and also for multi-gradient perfect fluids. Consequently, it is natural to surmise that the most general perfect fluids are continuous media whose motions verify Kelvin’s theorems.

4 Equation of energy and thermodynamical relations

4.1 Equation of balance of energy

Let us consider a dissipative ﬂuid. The equation of motion written in the conservative case can be extended to viscous ﬂuids in the form

ρ γ + ρ grad Ξ + η grad θ − div σv + ρ grad Ω =0,

where σv denotes the viscous stress tensor of the ﬂuid. Due to the relaxation time of viscous stresses, the viscosity does not take into account of any gradient terms. We denote M,B,N and F as,

M = ρ γ + ρ grad Ξ + η grad θ − div σv + ρ grad Ω,

 ∂ρ  B = + div (ρ u),  ∂t    ds   N = ρθ + div Q − r − tr (σv D) ,  dt  ∂ 1 T  F = ρu u + ρΞ + ηθ − Π + ρ Ω +  ∂t 2   1 T ∂Ω  div ρ u u + ρΞ + ηθ + ρ Ω I + σv u + χ + div Q − r − ρ .  2 ∂t    Terms Q and r represent the heat ﬂux vector and the heat supply, respectively. The Legendre transformation of volume internal energy ε with respect to variables ρ, η, ∇ρ, ∇η, . . . , ∇nρ, ∇nη is denoted Π,

. . . n . n Π = ρµ+ηT + Φ1 . ∇ρ + Ψ 1 . ∇η +...+ Φn . ∇ ρ + Ψ n . ∇ η − ε. (15) Multi-gradient ﬂuids 11

Then, Π is a function of µ, T, Φ1, Ψ 1,..., Φn, Ψ n. 1 Term D = ∂u/∂x + (∂u/∂x)T is the velocity deformation tensor and 2 ∂Φ1 ∂Ψ 1 χ = ρ + η ∂t ∂t . ∂Φ2 ∂Φ2 . ∂Ψ 2 ∂Ψ 2 + ∇ρ . − ρ div + ∇η . − η div + ... ∂t ∂t ∂t ∂t . . . ∂Φ . ∂Φ + ∇n−1ρ . n − ∇n−2ρ . div n + ... ∂t ∂t . p−1 n−p . ∂Φn n−1 ∂Φn + (−1) ∇ ρ . div 1 + ... + (−1) ρ div 1 p− ∂t n− ∂t . ∂Ψ . ∂Ψ + ∇n−1η . n − ∇n−2η . div n + ... ∂t ∂t . p−1 n−p . ∂Ψ n n−1 ∂Ψ n + (−1) ∇ η . div 1 + ... + (−1) η div 1 . p− ∂t n− ∂t Let us note that partial derivative ∂/∂t and divp commute. Vector χ is similar to the ﬂux of energy vector obtained in [17].

Theorem: relation 1 F − M T u − uT u + Ξ + s θ + Ω B − N ≡ 0 2 is an algebraic identity. Equation M = 0 represents the equation of motion, B = 0 is the mass conservation and N = 0 is the classical entropy relation, then for dissipative multi-gradient ﬂuids, F = 0 is the equation of energy, ∂ 1 ρuT u + ρΞ + ηθ − Π + ρ Ω + ∂t 2 1 ∂Ω div ρ uT u + ρΞ + ηθ + ρ Ω I + σ u + χ + div Q − r − ρ =0. 2 v ∂t Proof : Firstly, the proof comes from the identities in Ω terms, (div Q − r) terms and σv terms. Secondly, we denote

M 0 = ρ γ + ρ gradΞ + η grad θ,   ∂ρ  B = + div (ρ u) ,  ∂t    ds  N0 = ρθ ,  dt   ∂ 1 T 1 T  F0 = ρu u + ρΞ + ηθ − Π + div ρ u u + ρ Ξ+ ηθ u + χ ,  ∂t 2 2    12 Henri Gouin

then, ∂Ξ ∂θ d 1 dθ ∂Ξ ∂θ ρ γT u + ρ u + η u = ρ uT u + Ξ + η − ρ − η ∂x ∂x dt 2 dt ∂t ∂t ∂ 1 1 ds = ρ uT u + ρΞ + ηθ + div ρ uTu + ρ Ξ+ ηθ u − ρ θ ∂t 2 2 dt ∂Ξ ∂θ 1 ∂ρ −ρ − η − uT u + Ξ + s θ + div (ρ u) . ∂t ∂t 2 ∂t Moreover, we have

. ∂Φ1 ∂Φ1 ∂ divΦ1 ∇ρ . = div ρ − ρ ∂t ∂t ∂t 2 . ∂Φ2 . ∂Φ2 ∂ div Φ2 ∂ div2 Φ2 ∇ ρ . = div ∇ρ . − ρ + ρ ∂t ∂t ∂t ∂t . . . ∂ Φ . ∂Φ . ∂ div Φ ∇nρ . n = div ∇n−1ρ . n − ∇n−2ρ . n + ... ∂t ∂t ∂t . ∂ div 1 Φ ∂ div 1 Φ + (−1)p−1 ∇n−pρ . p− n + ... +(−1)n−1ρ n− n ∂t ∂t ∂ div Φ + (−1)nρ n n , ∂t and a similar relation for η,

. ∂Ψ 1 ∂Ψ 1 ∂ divΨ 1 ∇η . = div η − η ∂t ∂t ∂t . ∂Ψ 2 . ∂Ψ 2 ∂ div Ψ 2 ∂ div2 Ψ 2 ∇2η . = div ∇η . − η + η ∂t ∂t ∂t ∂t . . . ∂Ψ . ∂Ψ . ∂ div Ψ ∇nη . n = div ∇n−1η . n − ∇n−2η . n + ... ∂t ∂t ∂t . ∂ div 1 Ψ ∂ div 1 Ψ + (−1)p−1 ∇n−pη . p− n + ... + (−1)n−1η n− n ∂t ∂t ∂ div Ψ + (−1)nη n n . ∂t From Eqs. (3) and (15), we deduce

. . n . n . dΠ = ρ dµ+η dT + ∇ρ . dΦ1 + ∇η . dΨ 1 +...+ ∇ ρ . dΦn + ∇ η . dΨ n which implies

∂Π ∂µ ∂T . ∂Φ1 . ∂Ψ 1 . ∂Φ . ∂Ψ = ρ +η + ∇ρ . + ∇η . +...+ ∇nρ . n + ∇nη . n . ∂t ∂t ∂t ∂t ∂t ∂t ∂t Multi-gradient ﬂuids 13

Partial derivative ∂/∂t and divp commute and consequently,

∂Π ∂µ ∂ divΦ1 ∂ div Φ = ρ − ρ + ... + (−1)nρ n n ∂t ∂t ∂x ∂t ∂T ∂ divΨ 1 ∂ div Ψ + η − η + ... + (−1)nη n n ∂t ∂t ∂t ∂Φ1 . ∂Φ . ∂Φn + div ρ + ... + ∇n−1ρ . n − ∇n−2ρ . div + ... ∂t ∂t ∂t . p−1 n−p . ∂Φn n−1 ∂Φn + (−1) ∇ ρ . div 1 + ... + (−1) ρ div 1 p− ∂t n− ∂t ∂Ψ 1 1 . ∂Ψ 2 . ∂Ψ n + η + ... + ∇n− η . n − ∇n− η . div + ... ∂t ∂t ∂t . p−1 n−p . ∂Ψ n n−1 ∂Ψ n + (−1) ∇ η . div 1 + ... + (−1) η div 1 , p− ∂t n− ∂t and from (10), we get ∂Π ∂Ξ ∂θ = ρ + η + div χ. ∂t ∂t ∂t Then,

T ∂ 1 T 1 T M 0 u = ρ u u + ρΞ + ηθ − Π + div ρ u u + ρ Ξ+ ηθ u + χ ∂t 2 2 ds 1 ∂ρ −ρ θ − uT u + Ξ + s θ + div (ρ u) , dt 2 ∂t which implies

T 1 T F0 − M 0 u− u u + Ξ + sθ B − N0 ≡ 0 2 and proves the theorem.

4.2 The Planck and the Clausius-Duhem inequalities

For all dissipative ﬂuid motions,

tr (σvD) ≥ 0.

Then, from relation N = 0, we deduce the Planck inequality as in [18], ds ρθ + div Q − r ≥ 0. dt We consider the Fourier inequality in the general form

QT grad θ ≤ 0 14 Henri Gouin

and we obtain ds Q r ρ + div − ≥ 0, dt θ θ which is the extended form of the Clausius-Duhem inequality. Consequently, multi-gradient ﬂuids are compatible with the ﬁrst and second laws of thermodynamics.

Final remark: In a forthcoming article, we will prove that system of equations of multi-gradient ﬂuids is a quasi-linear hyperbolic-parabolic system of conservation laws which can be written in Hermitian symmetric-form implying the stability of constant solutions.

Acknowledgments: The results contained in the present paper have been partially pre- sented in Wascom 2017.

References

1. Van der Waals, J. D.: The thermodynamic theory of capillarity under the hypothesis of continuous variation of density. Translation by J.S. Rowlinson. J. Stat. Phys. 20, 200–244 (1979) 2. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system, III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959) 3. Gouin, H., Gavrilyuk, S.: Wetting problem for multi-component fluid mixtures. Physica A. 268, 291–308 (1999). https://arXiv:0803.0275 4. G˘ar˘ajeu, M., Gouin, H., Saccomandi, G.: Scaling Navier-Stokes equation in nanotubes. Phys. Fluids. 25, 082003 (2013). https://arXiv:1311.2484 5. Gouin, H.: Liquid nanofilms. A mechanical model for the disjoining pressure. Int. J. Eng. Sci. 47, 691–699 (2009). https://arXiv:0904.1809 6. Gouin, H.: The watering of tall trees - Embolization and recovery. J. Theor. Biol. 369, 42–50 (2015). https://arXiv:1404.4343 7. Germain, P.: The method of the virtual power in continuum mechanics - Part 2: mi- crostructure. SIAM J. Appl. Math. 25, 556–575 (1973) 8. Gouin, H., Ruggeri, T.: Mixtures of fluids involving entropy gradients and acceleration waves in interfacial layers. European Journal of Mechanics B/Fluids. 24, 596-613 (2005) https://arXiv:0801.2096 9. Eremeyev, V.A., Fischer, F.D.: On the phase transitions in deformable solids. Z. Angew. Math. Mech. 90, 535–536 (2010) 10. Rowlinson, J.S., Widom B.: Molecular theory of capillarity. Clarendon Press, Oxford and Google books (2012) 11. Gouin, H., Seppecher, P.: Temperature profile in a liquid-vapour interface near the critical point. Proc. R. Soc. A. 473, 20170229 (2017). https://arXiv:1703.07302 12. Casal, P., Gouin, H.; Equation of motion of thermocapillary fluids. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers. 306, 99–104 (1988) 13. Maitournam, M.H.: Entropy and temperature gradients thermomechanics: Dissipation, heat conduction inequality and heat equation. C. R. Mec. 340, 434–443 (2012) 14. Bertram, A., Forest, S.: The thermodynamics of gradient elastoplasticity. Continuum Mech. Thermodyn. 26, 269–286 (2014) 15. Serrin, J.: Mathematical Principles of Classical Fluid Mechanics. Encyclopedia of Physics VIII/1, Flügge, S. (ed.), pp. 125–263. Springer, Berlin (1960) 16. Gouin, H.: Noether’s theorem in fluid mechanics. Mech. Res. Commun. 3, 151-156 (1976) 17. Casal, P., Gouin, H.: Connexion between the energy equation and the motion equation in Korteweg’s theory of capillarity. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers. 300, 231-234 (1985) 18. Truesdell, C.: Introduction àla mécanique rationnelle des milieux continus, Masson, Paris (1974)