Keywords: no. 43.20.Ks, 43.20.Tb, 43.40.Rj

This paper deals with vibroacoustic under heavy fluid-loading conditions. The aim is to show how the high-order perturbation expansion approach described yields a very simple approximation of the spectrum of a heavy fluid-loaded structure. When the fluid-loading is "light" (e.g. metallic structures in contact with air) perturbation methods have been classically used to compute the spectrum of a fluid-loaded structure. Because of the computational efficiency of the expansions involved, it is now necessary to extend this method to heavy loading problems, such as those involving a metallic structure in contact with water. As we will see, although direct high order expansion methods yields unrealistic results, a judicious re-ordering of the various terms leads to surprisingly simple and efficient analytical results. Numerical examples are given in the simple case of a clamped rectangular plate under various fluid-loading conditions.

Introduction

Accurately describing the effects of fluid loading on a vibrating structure leads to an overwhelming decrease in the computational efficiency of the various method available. This is due to the fact that the operator describing the loading is not only frequency dependent but also non local. The numerical methods used to compute the solution in vacuo generally leads to the resolution of linear systems with either small but frequency dependent matrices (as in the case of Boundary Element Methods) or large but sparse and frequency independent matrices (as in that of Finite Element Methods). When the loading is taken into account, the size of the involved matrices increase considerably, becoming full and frequency dependent. Any method reducing these drawbacks is therefore most welcome.

A great amount of effort has gone into developing suitable methods for this purpose. Generally, in each method one tries to remove or to approximate the frequency dependence of the fluid-loading operator. One of the most recent and popular of these is the space state method proposed by J.A. Giordano and G.H. Koopman [6,14]. This method is based on the coupling between the Boundary Element Methods and Finite Element Methods, used to discretize the acoustic and structural domains respectively, in a canonical state space which takes in the form of a standard eigenvalue problem, where the frequency dependence of the loading matrix is removed by using low frequency approximations. While this method can be applied to heavy loading problems (such as those involving a metal structure in water), it is restricted to the low frequency range. An iterative method, also based on the coupling between the Boundary and Finite Element Methods, had been proposed by McCollum and Siders [START_REF] Mccollum | [END_REF] to perform modal analysis on heavily fluid-loaded structures. It is based on the eigen-analysis of the structure in vacuo. As we already noticed, the principal difficulty induced by the fluid-loading is caused by the frequency dependence of the matrix elements describing it. Then at each step, the frequency dependence of these elements is fixed. At the first step, for each in vacuo mode, the fluid-loaded system is solved by using its eigenfrequency to compute the matrix elements describing the loading. This allows to solve a standard eigenvalues problem; which gives a first estimate of the fluid-loaded resonance mode and frequency. The next step uses these values and the procedure is repeated until convergence. But, as noted by [START_REF] Pathak | [END_REF], for a certain range of parameters this iterative scheme can give incorrect results; it is also very time consuming since it must be repeated for each mode under consideration.

Asymptotic analysis can be conduced when the loading is light, in the case of a metal structure in air, for instance. It consists in introducing a parameter ǫ, which is the ratio between the surface mass density and the fluid density, which is low. Using perturbation expansions [START_REF] Filippi | [END_REF]4,[START_REF] Nayfeh | Perturbation Methods[END_REF], one can then construct an approximate solution, based on the in vacuo eigenmodes, which is not only easier to calculate than the solution of the exact problem but also leads to a better understanding of the various phenomenons involved. The main limitation here is that this method has a rather small validity range which it is difficult to estimate.

In this paper, we show how a simple modification of the usual high-order perturbation expansion methods yields a very simple approximation of the eigenmodes and resonance modes of a heavy fluid-loaded structure.

The resonance modes correspond to the free oscillations of the system and provide a natural tool for studying the effects of damping on sound radiation. The concept of eigenmodes and resonance modes for fluid-loading structure had been developed recently [START_REF] Filippi | [END_REF][START_REF] Habault | [END_REF]. In the literature, the resonance modes have been referred to under numerous denominations. The terms most commonly used are "generalized eigenmodes" and "complex modes". The latter denominations reflect the mathematical nature of these modes: they are a generalization of the eigenmodes of a mechanical system in which the frequency dependence of the various operators (such as the stiffness and mass operators) gives rise to complex mode values. The resonance modes and eigenmodes are different when frequency dependent damping is taken into account. This occurs in the case of material damping (in viscoelasticity), structural damping (in boundary dissipation problems) and in acoustic radiation problems.

Because of the frequency dependence, estimating the resonance modes is a much more difficult task than that involving eigenmodes.

In section 2 we present the equations. Section 3 deals with a high-order perturbation expansion of both eigenmodes and resonance modes. It is shown here how by re-arranging the various terms in the eigenvalue expansion, a simple analytical expression for the eigenvalues is obtained. In section 4, some numerical examples are given to show the efficiency of the method proposed. Some comments have been made in the Conclusion

Statement of the problem

In this paper, we deal with harmonic time dependence exp(-ıωt), omitted to simplify the reading.

Let us consider an infinite domain Ω with a finite or infinite boundary ∂Ω that contains a fluid at rest. A thin finite structure with thickness h and density ρ p occupies a finite surface Σ of ∂Ω. M is taken to denote a point on the structure and Q a point in the infinite domain. Let us take by U (M, ω) to denote the normal displacement of the structure. O is the operator, independent of ω, that describes the elastic behavior of the structure, such as the bi-Laplacian in the Kirchhoff plate equation or the Donnell-Mushtari operator in the case of a thin shell. At the boundary ∂Σ of the structure, one adds mechanical boundary conditions. f (M, ω) is the excitation force. A perfect fluid at rest with density ρ f and sound speed c f occupies Ω; the acoustic pressure, which is denoted by P (Q, ω), is governed by the Helmholtz equation. A Neumann condition is imposed on Σ. U (M, ω) and P (Q, ω) satisfy the following system of equations:

OU (M, ω) -ρ p hω 2 U (M, ω) = F (M, ω) -P (M, ω) in Σ (1) ∆P (Q, ω) + ω 2 /c 2 f P (Q, ω) = 0 in ∂Ω (2)
Boundary condition for U (M, ω) on ∂Σ (3)

∂ n P (Q, ω) = 0 on ∂Ω -Σ (4) lim M ∈Ω→Q∈Σ ∂ n P (Q, ω) = -ρ f ω 2 U (M, ω) on Σ.
(5)

Integrodifferential equation for the displacement

This system of equations for U and P can be reduced to an integrodifferential equation for U by using Green's representation for the pressure. Let G(Q -Q ′ , ω) be the Green's function for the domain Ω with a Neumann boundary condition on ∂Ω. This function can be obtained analytically for baffled plates or shells and numerically for other configurations. Using Green's representation for the pressure, it is easy to show that the pressure inside Ω is given by the integral equation

P (Q, ω) = ω 2 ρ f Σ U (M ′ , ω)G(Q -M ′ , ω)dM ′ . (6)
Introducing this result into equation ( 1), one obtains the integrodifferential boundary value problem for the structure

OU (M, ω) -ρ p hω 2 U (M, ω) -ǫ Σ U (M ′ , ω)G(M -M ′ , ω)dM ′ = F (M, ω) (7)
Boundary condition for U (M, ω) on ∂Σ

where ǫ = ρ f /ρ p h is a small parameter in the case of a metal structure in air.

Eigenmodes and resonance modes.

One now introduces [START_REF] Filippi | [END_REF] the weak form of the system given by equations [START_REF] Lancaster | Lambda-matrices and Vibrating Systems[END_REF][START_REF] Leissa | Vibration of Plates[END_REF]. It reads : find U (M, ω)

satisfying the boundary condition such that for each V (M, ω) satisfying the boundary condition, the following equation holds

a(U, V ) -Λ ( U, V -ǫβ ω (U, V )) = F, V (9)
where Λ = ρ p hω 2 . a(U, V ) corresponds to the elastic (or potential) energy of the structure. U, V is

the usual inner product U, V = Σ U (M )V * (M )dM , ρ p hω 2 U, V is the kinetic energy. β ω (U, V ) = Σ Σ U (M )G(M -M ′ , ω)V * (M ′
)dM dM ′ is the radiation impedance. Since in the general case G(M -M ′ , ω) depends on the frequency β ω (U, V ) is obviously frequency dependent. ρ p hω 2 ǫ p β ω (U, V ) is the energy exchanged between the structure and the fluid. Because of the frequency dependence of β ω (U, V ), it is necessary to distinguish the eigenmodes from the resonance modes [START_REF] Filippi | [END_REF][START_REF] Habault | [END_REF]. It is just recall here the corresponding definitions and basic properties of the two kind of modes.

The eigenmodes Ũm (M, ω) and the eigenpulsations ωm (ω) of this problem are the non zero solutions of

a( Ũm , V ) -Λm (ω) Ũm , V -ǫβ ω ( Ũm , V ) = 0, for each V (M ). ( 10
)
The eigenpulsations are given by ω2 m (ω) = Λm (ω)/ρ p h.

The resonance modes Ûm (M ) (or generalized eigenmodes or complex modes) and the resonance pulsations ωm are defined as the nontrivial, frequency independent solutions of

a( Ũm , V ) -ρ p hω 2 m Ûm , V -ǫβ ωm ( Ûm , V ) = 0, for each V (M ). (11)
While the resonance modes are harder to compute than the eigenmodes, they only need to be calculated once. Moreover, there is a simple relation between the eigenmodes and the resonance modes. From the definitions of these two kinds of modes, it is easy to see that one has:

Ûm (M ) = Ũm (M, ωm ), (12)
ω2 m = ω2 m (ω m ). ( 13
)
Equation ( 13) implies that the resonance pulsations are the fixed points of the eigenpulsations. The only difficulty in computing the resonance modes is how to compute the resonance frequencies. In the general case, discretizing of the problem leads to a system of simultaneous equations with a full complex matrix. The complex zeros of the determinant of this matrix are the resonance frequencies, and their estimation is generally a very difficult task because the coefficients of the matrix depend on the frequency, which leads to estimating the zeros of a non-linear non-convex function in the complex plane. Problems of this kind have not yet been given rigorous numerical treatment. Once the resonance frequencies are known, the resonance modes are estimated by performing a simple matrix complex eigenmode search. Nevertheless, in the particular case of small damping problems (internal of by radiation), perturbation expansions lead to a particularly simple expression for the eigenmodes and eigenfrequencies [4]. In the following paragraph, we show how to use a perturbation method in the case of heavy fluid-loading.

High-Order Perturbation expansions

Let us expand into a Rayleigh-Schrödinger perturbation series [START_REF] Nayfeh | Perturbation Methods[END_REF] both the eigenmodes Ũm and the eigenvalues Λm in ǫ :

Ũm (M ) = Ũ (0) m (M ) + ǫ Ũ (1) m (M ) + • • • + ǫ s Ũ (s) m (M ) + • • • (14) Λm = Λ(0) m + ǫ Λ(1) m + • • • + ǫ s Λ(s) m + • • • (15)
The Rayleigh-Schrödinger Method is one of the most powerful methods available to dealing with vibroacoustics problems involving low frequency dependent damping such as those arising in viscosity or sound radiation in light fluid. Substituting the two perturbation expansions into the weak formulation given by equation (10), collecting and equating the coefficients of equal power ǫ to zero yields:

ǫ 0 : a( Ũ (0) m , V ) -Λ(0) m Ũ (0) m , V = 0, (16)
. . .

ǫ s : a( Ũ (s) m , V ) -Λ(0) m Ũ (s) m , V = l=s l=1 Λ(l) m Ũ (s-l) m , V - l=s-1 l=0 Λ(l) m β ω ( Ũ (s-l-1) m , V ), (17)
Equation ( 16) is the usual relation giving the eigenmodes of the elastic structure in vacuo. If the eigenmodes Ũ (0) m (M ) of the elastic plate in vacuo are normalized, it is easy to obtain the first order term of the eingenvalue, that is

Λ(1) m Λ(0) m = β mm ω , (18)
in the previous equation, it has been denoted

β mn ω = β ω ( Ũ (0) m , Ũ (0) * n ).
To simplify the reading this notation will be used in the rest of the paper. Ũ (1) m (M ) is expanded into a series of zeroth-order

modes: Ũ (1) m (M ) = n α 1n m Ũ (0) n (M )
where the coefficients α 1n m are easy to calculate [START_REF] Nayfeh | Perturbation Methods[END_REF]. For all s > 1, one has:

Λ(s) m = l=s-1 l=0 Λ(l) m β ω ( Ũ (s-l-1) m , Ũ (0) m ) - l=s-1 l=1 Λ(l) m Ũ (s-l) m , Ũ (0) m . (19)
In a similar manner, each component of the eigenmode's perturbation expansion is developed into a series of zeroth-order modes:

Ũ (s) m (M ) = n α sn m Ũ (0) n (M ). (20)
One obtains, for the eigenvalues:

Λ(2) m Λ(0) m = (β mm ω ) 2 + p =m α 1p m β mp ω , (21)
Λ(3) m Λ(0) m = (β mm ω ) 3 + 2β mm ω p =m α 1p m β mp ω + p =m α 2p m β mp ω , (22)
Λ(4) m Λ(0) m = (β mm ω ) 4 + 3(β mm ω ) 2 p =m α 1p m β mp ω + 2β mm ω p =m α 2p m β mp ω + p =m α 3p m β mp ω +   p =m α 1p m β mp ω   2 -α 2m m p =m α 1p m β mp ω , (23)
Λ(5) m Λ(0) m = (β mm ω ) 5 + 4(β mm ω ) 3 p =m α 1p m β mp ω + 3(β mm ω ) 2 p =m α 2p m β mp ω + 2β mm ω p =m α 3p m β mp ω +3β mm ω   p =m α 1p m β mp ω   2 -2β mm ω α 2m m p =m α 1p m β mp ω + p =m α 4p m β mp ω + 2 p =m α 1p m β mp ω p =m α 2p m β mp ω - p =m α 1p m α 3p m β mp ω -α 2m m p =m α 2p m β mp ω . ( 24
)
Now let us calculate the eigenvalue expansion up to order 5:

Λm Λ(0) m = 1 + s=5 s=1 ǫ s Λ(s) m Λ(0) m + • • • + • • • . ( 25
)
Upon re-ordering the terms, one obtains:

Λm Λ(0) m = 1 + ǫβ mm ω + ǫ 2 (β mm ω ) 2 + ǫ 3 (β mm ω ) 3 + ǫ 4 (β mm ω ) 4 + ǫ 5 (β mm ω ) 5 + • • • +ǫ 2 p =m α 1p m β mp ω 1 + 2ǫβ mm ω + 3ǫ 2 (β mm ω ) 2 + 4ǫ 3 (β mm ω ) 3 + • • • +ǫ 3 p =m α 2p m β mp ω 1 + 2ǫβ mm ω + 3ǫ 2 (β mm ω ) 2 + • • • +ǫ 4      p =m α 3p m β mp ω [1 + 2ǫβ mm ω + • • •] +   p =m α 1p m β mp ω   2 [1 + 3ǫβ mm ω + • • •] -α 2m m p =m α 1p m β mp ω [1 + 2ǫβ mm ω + • • •]    +ǫ 5 • • • (26)
For ǫβ mm ω < 1, one can recognize in each square bracket the expansion of 1/(1 -ǫβ mm ω ) and its successive powers. By identifying the expansion with the original functions, one obtains the modified high order perturbation expansion:

Λm Λ(0) m = 1 (1 -ǫβ mm ω ) + ǫ 2 p =m α 1p m β mp ω (1 -ǫβ mm ω ) 2 + ǫ 3 p =m α 2p m β mp ω (1 -ǫβ mm ω ) 2 +ǫ 4      p =m α 3p m β mp ω (1 -ǫβ mm ω ) 2 + p =m α 1p m β mp ω 2 (1 -ǫβ mm ω ) 3 -α 2m m p =m α 1p m β mp ω (1 -ǫβ mm ω ) 2      +ǫ 5 • • • (27)
In some particular cases (see Appendix), this method gives exact results and can be viewed as an analytic continuation process. If all the terms β mp ω are zero for m = p, the first order of the modified perturbation expansion leads to exact results, even if ǫβ mm ω is greater than 1. This result also shows that the true convergence condition of the various perturbation series is not simply ǫ < 1 but ǫβ mm ω < 1. It is worth noting that this makes the numerical analysis almost impossible because the convergence condition is different for every mode and depends on the frequency.

Numerical Results

In this section, numerical examples are given for baffled fluid-loaded clamped rectangular plates under heavy loading conditions. This plate occupies a domain of the z = 0 plane in IR 3 defined by

x ∈]0, a[×y ∈]0, b[. The same fluid occupies the two half-space x > 0 and x < 0. Let us recall that in this case, the Green's function for the Neumann problem of the Helmholtz equation is given by:

G(M -M ′ , ω) = -1 4π   e ı ω c f d(M,M ′ ) d(M, M ′ ) + e ı ω c f d(M,M ′′ ) d(M, M ′′ )  
where d(M, M ′ ) is the Euclidian distance between M and M ′ , M ′′ is the symmetric of M ′ with respect to the z = 0 plane. Then, the only numerical difficulty arising with this method lies in the computation of the various modal acoustic impedance β mnpq ω given by the integral, with X = ax and Y = by :

β mnpq ω = a 2 b 2 1 0 1 0 1 0 1 0 U (0) mn (x, y)G(x -x ′ , y -y ′ , ω)U (0) pq (x ′ , y ′ )dxdx ′ dydy ′ (28)
The evaluation of this kind of integral is considerably simplified when the eigenmodes are given in separate form U (0) mn (x, x) = X m (x)Y n (y) by a linear transformation in the integral domain given by X = x -x ′ and Y = y -y ′ [START_REF] Mangiarotty | [END_REF]. One thus obtains :

β mnpq ω = 4a 2 b 2 1 0 1 0 A mp (X)B nq (Y )G(X, Y, ω)dXdY, (29)
A mp (X) = 1-X 0 X m (X + x ′ )X p (x ′ )dx ′ , B nq (Y ) = 1-Y 0 Y m (Y + y ′ )Y p (y ′ )dy ′ . ( 30
)
In many cases, such as the one studied here, simple analytic expressions are available for the functions A mp (X) and B nq (Y ), which make it possible to compute the modal impedances by simply using double integrals.

In order to check the validity of the method, the spectrum of the structure is calculated using the original and modified perturbation expansions up to order 3 and compared with exact computations. The exact computations are based on a Chebychev-collocation approximation of the boundary integral equations associated with the problem [10]. Roughly, the two unknowns in the problem, namely the displacement and the acoustic sound pressure jump across the plate, are the solutions of two coupled integral equations. These functions are approximated by series of Chebychev orthogonal polynomials of the first kind. Using a collocation method, the new unknowns (the coefficients of the two series) are the solutions of a linear system of simultaneous equations. The complex zeros of the determinant of the matrix of this system are the resonance frequencies. These frequencies are computed using a Newton-Raphson method.

For each mode, denoted by its index mn, the resonance spectrum is calculated by solving the relation 13: ω2 mn = ω2 mn (ω mn ) = Λmn (ω mn )/ρ p h. For each mode, there are two frequencies that satisfy this relation, namely ωmn and -ω * mn . These frequencies are estimated using the approximate expressions for the eigenvalues Λmn (ω) given by the direct perturbation expansion (26) or by the modified perturbation expansion (27) truncated at orders ranging from 1 to 3. The roots ωmn of the equation ω 2 = ω2 mn (ω) were obtained using the Mathematica [START_REF] Wolfram | The Mathematica Book[END_REF] root finding routine without any particular difficulty.

With this method, analytical or numerical knowledge about the in vacuo eigenmodes is necessary.

In the case of a clamped rectangular plate, there are no exact analytical results available, such as those on a simply supported plate, but the analytical approximation by Warburton [START_REF] Leissa | Vibration of Plates[END_REF] gives a quite perfect shape for the eigenmodes and only slightly overestimates the eigenfrequencies by about 3%.

The first results are given in table 1. Four different configurations were tested in water (in air the differences between the exact and approximate solutions are negligible). The plate is rectangular having length a = 1m, width b = 0.7m and varying thickness h. It is made of steel with Young's modulus E = 200GPa, Poisson coefficient ν = 0.3 and density ρ p = 7800Kg/m 3 . The fluid is water with density ρ f = 1000Kg/m 3 and sound celerity c f = 1500m/s. Table 1 gives the data obtained on steel plates in water, which is the usual heavy loading situation, with thickness of 5mm, 1cm, 3cm and 10cm.

In this table, the first resonance frequencies of the plate under various fluid-loading conditions calculated up to the third order of the perturbation expansion are compared with the exact computations. In the exact values column, the fluid-loaded spectrum is given along with the in vacuo one. In order to clearly show the added mass corresponding to a decrease in the real part and the damping that give rise to a negative imaginary part. The seven remaining columns give the results obtained using the original and modified perturbation expansions. The zeroth order is the in vacuo spectrum (which is about 3% larger than the exact one because of the Warburton approximation).

The order 1 of the perturbation expansion are the data obtained by solving equation ( 26) limited to the first order, while the order 1 modified data are those obtained by solving equation ( 27) to the first order, and so on up to the higher orders. All clearly divergent results (with a positive imaginary part) are given in italics.

The most important finding made in this study is that all the results show that the first order modified approximation gives very accurate results under heavy loading conditions with a small numerical cost. Some resonances are obviously less exactly described, such as mode (m = 1, n = 3) in the case of the thinner plates. But even in these cases, the estimate given by the first order modified approximation is fairly accurate, or at least sufficient for the acoustician.

The next results, presented figures 1 and 2, show the evolution of the resonance frequency for the mode (m = 1, n = 1) in figure 1 and for the mode (m = 1, n = 3) in figure 2 when the thickness of the plate is varying. In both figures, the top curve shows the real part of the resonance and the bottom the imaginary part of it. The smooth curves are the results given by the modified approximation while the curves with the diamonds are the exact results. From these curves, it is obvious that again the modified approximations gives particularly good results.

More interesting is the phenomenon presented for the for the mode (m = 1, n = 3). In figure 2 one observes two disjointed curves. The first one, starting from the origin (when the thickness of the plate tends to zero all resonance frequencies are nil) is called the lower branch and the other the upper branch. In the particular case considered here, except for modes (m = 1, n = 1), (m = 2, n = 1) and (m = 1, n = 2), all the modes showed this phenomenon, which is a typical behavior of the non-linear systems. A priori this must not occurs in the present case since the system that one considers here do not present any non-linear classical feature such as structural non-linearity, non-linear propagation in the fluid or non linear moving boundary condition at the interface between the structure and the fluid [15]. The phenomena observed here has a strong link with the concept of frequency non-linear modes [START_REF] Dazel | Proceedings of the Euromech 457[END_REF] which exhibit a non-linearity that depends on the temporal parameters rather than on the geometrical parameters such as propagation of sound in poroelastic media. The frequency non-linear modes are related to problems expressed in the following form M(ω)u = f where u is the unknown, f the forcing term and M(ω) is a non linear frequency dependent operator such as those that occur in poroelastic media or in fluid-structure interaction for which the loading operator described by equation 6 has a non-linear frequency dependence. In this latter case, when the fluid-loading is light, this non-linear dependence is not significant but when the fluid-loading is strong, this non-linearity can be observed as shown by the results presented here.

To be sure that this effect is not a numerical artifact, we have plotted in the figures

Conclusion

The main finding to emerge from this study was that the efficiency of the first order modified approximation does not depend on the loading, at least in the particular case consider here. The second important point is that the modified approximation can be obtained without any difficulty.

Unlike the usual first order approximations, its estimation does not require any computational effort.

With this method, it seems to be possible to describe all the features of the spectrum of a fluid-loaded structure, such as the high added mass and small damping, observed in water in the case of a the thinnest plate or conversely, the small added mass and high damping observed with the thickest plate. As long as one is able to estimate the eigenmodes of the in vacuo structure, using a finite element model, for example, and the loading given by Green's function for the Neumann problem in the Helmholtz equation, it is easy to compute the spectrum of a heavily fluid-loaded structure at a negligible numerical cost. This method obviously requires the a priori knowledge of this Green's function, that of an infinite plane screen or cylindrical surface for example. In most cases, however this function is impossible to compute analytically. It is therefore necessary to use either a numerical calculation of the Green function or an approximate description of the loading, for example based on the Rayleigh integral but the validity of an approximation of this kind still remains to be confirmed. Some aspects of this method need to be studied in greater depth. The most obvious one is the the nature of this approximation, especially the effects of neglecting the modal coupling given by the terms p =m α 1p m β mp ω . It is well known in damping theory studies that neglecting the modal coupling in non-proportional damped systems yields poor results, even when the structure is only slightly damped [1]. The author has no clear explanation of why the first order modified expansion works so well. More precisely, an expansion which neglects the intermodal impedance ought to work badly.

All previous authors that have worked in this subject have suggested that the intermodal impedance plays a significant role in the loading, and that a method which neglects it must fail. But the results found here seem to invalidate this hypothesis, and it would be interesting to know if the hypothesis is also invalid in other geometries. The next point that needs to be studied is the apparition of a frequency non-linear mode caused by the fluid-loading. It must be deeply studied to understand its emergence in the case considered here and if this occurs for other geometries, boundary conditions or other type of structure (like shells or beams).

Ũm (0, ω) = 0, Ũ ′′ m (0, ω) = 0, Ũm (l x , ω) = 0, Ũ ′′ m (L x , ω) = 0 (34)
Then, it is easy to show that one has:

Ũm (x, ω) = 2 L x sin mπ L x x (35) ωm (ω) = mπ L x 2 D ρ p h 1 1 + ǫ p ı ω (36)
The resonance modes Ûm (x) and resonance pulsations ωm are the non-trivial solution of:

D∆ 2 Ûm (x) -ρ p hω 2 m 1 -ǫ p -ı ωm Ûm (x) = 0 (37) Ûm (0) = 0, Û ′′ m (0) = 0, Ûm (l x ) = 0, Û ′′ m (L x ) = 0 (38)
The resonance modss are linked to the eigenmodes by the relations (12): Ûm (x) = Ũm (x, ωm ) and

= - ıǫ p 2 ± D ρ p h mπ L x 4 - ǫ p 2 2 (40)
This root symmetry with respect to the imaginary axis is the condition [START_REF] Habault | [END_REF] for the resonance modes to ensure causality and provide real solutions with a time dependency of the form exp(-ıωt). Obviously with a time dependency of the form exp(ıωt), the imaginary parts of the roots have to be positive.

A.2 Perturbation expansion for the eigenmodes

The weak form of the equation giving the eigenmodes reads : find Ũm (x, ω) and Λm (ω) satisfying the boundary condition such that for each V (x, ω) satisfying the boundary condition, the following equation holds a( Ũm , V ) -Λm Ũm , V -ǫ p β ω ( Ũm , V ) = 0 (41)

where β ω ( Ũm , V ) = -ı/ω Ũm , V and Λm = ρ p hω 2 m (ω). It can be easily seen that one has β mp ω = -ı/ωδ p m , with δ p m the Kronecker delta.

Then, since the weak form of this equation is similar to that of the fluid-loaded plate 10, all the results previously obtained can be used without making any changes. It is easy to show that the zeroth order terms are, with Λ(0) m = ρ p h(ω (0) m ) 2 , given by:

Ũ (0) m (x) = 2 L x sin mπ L x x, Ũ (

( 3 )

 3 and(4) the amplitude of the displacement of the plate, given by the resolution of the Boundary Integral Equations associated with the exact solution, for frequencies close to the two resonances observed for the thickness corresponding to the inflexion point for the real parts and a crossing of the imaginary parts that is, for the (m = 1, n = 3) mode close to h = 0.074cm. The plate is excited by a unit Dirac delta located close to the middle of the plate at x = 0.6m, y = 0.45m. The excitation frequency f , given in the figure caption, has been chosen close to (less than 0.1%) the resonance one; obviously at the resonance, the exact solution has no solution. In the figure (3), the excitation frequency f = 4827 -ı1025 Hz is close to the resonance frequency of the upper branch (that is f u 13 = 4829.64 -ı1024.4 Hz). In the figure (4), the excitation frequency f = 3717 -ı1053Hz is close to the resonance frequency to the lower branch (that is f u 13 = 3716.9 -ı1052.4 Hz). As shown by these figures, the displacement is very close to that of a m = 1, n = 3 mode. Similar results had been observed for many modes; more precisely, only the (m = 1, n = 1), (m = 1, n = 2) and (m = 2, n = 1) modes vary continuously with the thickness, all the other modes presents such a comportment.

( 13 ) 4 = 0 .

 1340 : ω2 m = ω2 m (ω m ). Since the eigenmodes do not depend on the frequency, the eigenmodes and resonance modes are identical. The resonances frequencies are the solutions of ω2 This quadratic equation ca be easily solved, one obtains the two roots: ωm

=Figure 1 :Figure 2 :Figure 3 :Figure 4 :

 1234 Figure 1: 1m×0.7m steel plate in water: evolution of the real and imaginary parts with the thickness of the plate of the (m = 1, n = 1) resonance frequency. Comparison of the exact computations ( f e 11 )with the modified approximation ( f 1 11 ). Top: real part, bottom: imaginary part

Table 1 :

 1 Comparison of exact and approximate spectra for a steel plate in water (values given in Hz)

	Exact

A High order perturbation expansion of a viscous plate

In this Appendix, some classical results [START_REF] Lancaster | Lambda-matrices and Vibrating Systems[END_REF][START_REF] Woodhouse | [END_REF] on the eigenmodes and resonance modes of a viscously damped plate are used to show that the high perturbation expansion is able to give the exact solution in some particular cases.

Let U (x, ω) be the normal displacement of a simply supported viscous plate in the harmonic regime with time dependence exp(-ıωt). Let us recall that in the time domain, the simplest form of the viscous operator is given by η p ∂/∂t, where η p is the damping constant. If we introduce the parameter ǫ p = η p /ρ p h, then U (x, ω) satisfies the modified Kirchhoff equation as well as the boundary conditions:

where

) is the bending rigidity of the plate.


S Gavrilyuk

H Gouin

A new form of governing equations of fluids arising from Hamilton's principle

Keywords: Hamilton's principle, Symmetric forms, Dispersion relations

A new form of governing equations is derived from Hamilton's principle of least action for a constrained Lagrangian, depending on conserved quantities and their derivatives with respect to the time-space. This form yields conservation laws both for non-dispersive case (Lagrangian depends only on conserved quantities) and dispersive case (Lagrangian depends also on their derivatives). For non-dispersive case the set of conservation laws allows to rewrite the governing equations in the symmetric form of Godunov-Friedrichs-Lax. The linear stability of equilibrium states for potential motions is also studied. In particular, the dispersion relation is obtained in terms of Hermitian matrices both for non-dispersive and dispersive case. Some new results are extended to the two-fluid non-dispersive case.

Introduction

Hamilton's principle of least action is frequently used in conservative fluid mechanics [START_REF] Sedov | Mathematical methods for constructing new models of continuum media[END_REF][START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF][START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF]. Usually, a given Lagrangian Λ is submitted to constraints representing conservation in the time-space of collinear vectors j k Div j k = 0, k = 0, . . ., m

where Div is the divergence operator in the time-space. Equation (1.1) means the conservation of mass, entropy, concentration, etc.

Lagrangian appears as a function of j k and their derivatives. To calculate the variation of Hamilton's action we don't use Lagrange multipliers to take into account the constraints (1.1). We use the same method as Serrin in [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF] where the variation of the density ρ is expressed directly in terms of the virtual displacement of the medium. This approach yields an antisymmetric form for the governing equations

m k=0 j * k ∂K k ∂z - ∂K k ∂z * = 0 (1.2)
where z is the time-space variable, "star" means the transposition, and K k is the variational derivative of Λ with respect to j k

K * k = δΛ δj k (1.3)
Equations (1.1) -(1.3) admit particular class of solutions called potential flows

j k = a k j 0 , a k = const, k = 1, . . ., m K * 0 =
∂ϕ 0 ∂z where ϕ 0 is a scalar function. We shall study the linear stability of constant solutions for potential flows.

In Section 2, we present the variations of unknown quantities in terms of virtual displacements of the continuum. In Section 3, we obtain the governing system (1.2) using Hamilton's principle of least action. Conservation laws admitted by the system (1.1) -(1.3) are obtained in section 4. For non-dispersive case, we obtain the equations (1.1) - (1.3) in the symmetric form of Godunov-Friedrichs-Lax [START_REF] Godunov | An interesting class of quasilinear systems[END_REF][START_REF] Friedrichs | Systems of conservation laws with a convex extension[END_REF]. Section 5 is devoted to the linear stability of equilibrium states (constant solutions) for potential motions. We obtain dispersion relations in terms of Hermitian matrices both for dispersive and non-dispersive flows and propose simple criteria of stability. In Section 6-7, we generalize our results for two-fluid mixtures in the non-dispersive case. Usually, the theory of mixtures considers two different cases of continuum media. In homogeneous mixtures such as binary gas mixtures, each component occupies the whole volume of a physical space. In heterogeneous mixtures such as a mixture of incompressible liquid containing gas bubbles, each component occupies a part of the volume of a physical space. We do not distinguish the two cases because they have the same form for the governing equations. We obtain a simple stability criterion (criterion of hyperbolicity) for small relative velocity of phases. In Appendix, we prove non-straightforward calculations.

Recall that "star" denotes conjugate (or transpose) mapping or covectors (line vectors). For any vectors a, b we shall use the notation a * b for their scalar product (the line vector is multiplied by the column vector) and ab * for their tensor product (the column vector is multiplied by the line vector). The product of a mapping A by a vector a is denoted by Aa, b * A means covector c * defined by the rule c * = (A * b) * . The divergence of a linear transformation A is the covector Div A such that, for any constant vector a, Div (A)a = Div (Aa)

The identical transformation is denoted by I. For divergence and gradient operators in the time-space we use respectively symbols Div and ∂ ∂z , where z * = (t, x * ), t is the time and x is the space. The gradient line (column) operator in the space is denoted by ∇ (∇ * ), and the divergence operator in the space by div. The elements of the matrix A are denoted by a i j where i means lines and j columns. If f (A) is a scalar function of A, matrix

B * ≡ ∂f ∂A is defined by the formula (B * ) j i = ∂f ∂A j i = ∂f ∂a i j
The repeated latin indices mean summation. Index α = 1, 2 refers to the parameters of components densities ρ α , velocities u α , etc.

Variations of a continuum

Let z = t x be Eulerian coordinates of a particle of a continuum and D(t) a volume of the physical space occupied by a fluid at time t. When t belongs to a finite interval [t 0 , t 1 ], D(t) generates a four-dimensional domain Ω in the time-space. A particle is labelled by its position X in a reference space D 0 . For example, if D(t) contains always the same particles D 0 = D(t 0 ), and we can define the motion of a continuum as a diffeomorphism from D(t 0 ) into D(t) x = χ t (X) (2.1) We generalize (2.1) by defining the motion as the diffeomorsphism from the reference space Ω 0 into the time-space Ω occupied by the medium in the following parametric form

     t = g (λ , X) x = φ (λ , X) (2.2)
where Z = λ X belongs to a reference space Ω 0 . The mappings

     λ = h (t , x) X = ψ(t , x) (2.
         dt = ∂g ∂λ dλ + ∂g ∂X dX dx = ∂φ ∂λ dλ + ∂φ ∂X dX (2.5)
From equation (2.5) we obtain

dx = u dt + F dx
where velocity u and deformation gradient F are defined by

u = ∂φ ∂λ ∂g ∂λ -1 , F = ∂φ ∂X - ∂φ ∂λ ∂g ∂X ∂g ∂λ -1 (2.6) Let      t = G (λ , X , ε) x = Φ (λ , X , ε) (2.7)
be a one-parameter family of virtual motions of the medium such that

G (λ , X , 0) = g(λ , X), Φ (λ , X , 0) = φ (λ , X)
where ε is a scalar defined in the vicinity of zero. We define Eulerian displacement ζ = (τ , ξ) associated with the virtual motion (2.7)

τ = ∂G ∂ε (λ , X , 0), ξ = ∂Φ ∂ε (λ , X , 0) (2.8)
We note that ζ is naturally defined in Lagrangian coordinates. However, we shall suppose that ζ is represented in Eulerian coordinates by means of (2.3).

Let us now consider any tensor quantity represented by f (t, x) in Eulerian coordinates and

• f (λ , X) in Lagrangian coordinates. Definitions (2.2), (2.3) involve • f (λ , X) = f g (λ, X) , φ (λ, X) (2.9)
Conversely,

f (t , x) = • f h (t, x) , ψ (t,
f (λ , X , 0) = • f (λ , X) or equivalently ∧ f (t , x , 0) = f (t , x)). We then obtain ∼ f (λ , X , ε) = ∧ f G (λ , X , ε) , Φ (λ , X , ε) , ε (2.11)
Let us define Eulerian and Lagrangian variations of

f ∧ δ f = ∂ ∧ f ∂ε (t , x , 0) and ∼ δ f = ∂ ∼ f ∂ε (λ , X , 0)
Differentiating relation (2.11) with respect to ε at ε = 0, we get

∧ δ f = ∼ δ f - ∂f ∂z ζ (2.12)
3 Governing equations Consider a four-dimensional vector j 0 satisfying conservation law Div j 0 = 0 (3.1)

Actually, (3.1) represents the mass conservation law, where j 0 = ρv, ρ is the density and v * = (1, u * ) is the four-dimensional velocity vector. Let a k be scalar quantities such as the specific entropy, the number of bubbles per unit mass, the mass concentration, etc., which are conserved along the trajectories. Consequently, if

j k = a k j 0 , Div j k ≡ ∂a k ∂z j 0 = 0, k = 1, . . ., m (3.2)
Hence j k , k = 1, . . . , m form a set of solenoidal vectors collinear to j 0 . Hamilton's principle needs the knowledge of Lagrangian of the medium. We take the Lagrangian in the form

L = Λ j k , ∂j k ∂z , . . . , ∂ n j k ∂z n , z (3.3)
where j k are submitted to the constraints (3.1), (3.2) rewritten as Div j k = 0, k = 0, . . . , m

Let us consider three examples.

a -Gas dynamics [START_REF] Sedov | Mathematical methods for constructing new models of continuum media[END_REF][START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF][START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF] Lagrangian of the fluid is

L = 1 2 ρ |u| 2 -ε (ρ, η) -ρ Π (z)
, where ε is the internal energy per unit volume, η = ρs is the entropy per unit volume, s is the specific entropy and Π is an external potential. Hence, in variables j 0 = ρv, j 1 = ρsv , z, Lagrangian takes the form

L = 1 2 |j 0 | 2 l * j 0 -l * j 0 -ε (l * j 0 , l * j 1 ) -l * j 0 Π(z) = Λ (j 0 , j 1 , z)
where l * = (1, 0, 0, 0).

b -Thermocapillary fluids [START_REF] Casal | Equations of motions of thermocapillary fluids[END_REF][START_REF] Slemrod | Dynamic phase transitions in a van der Waals fluid[END_REF] Lagrangian of the fluid is L = 1 2 ρ |u| 2ε (ρ , ∇ρ , η , ∇η)ρ Π(z). Since ∇ρ = ∇ (l * j 0 ) and ∇η = ∇ (l * j 1 ), we obtain Lagrangian in the form (3.3)

L = Λ j 0 , ∂j 0 ∂z , j 1 , ∂j 1 ∂z , z c -One-velocity bubbly liquids [8-10] L = 1 2 ρ |u| 2 -W ρ , dρ dt , N , with dρ dt = ∂ρ ∂z v
where ρ is now the average density of the bubbly liquid and N is the number of identical bubbles per unit volume of the mixture. We define again j 0 = ρv and j 1 = Nv. By using

dρ dt = ∂ρ ∂z v = ∂(l * j 0 ) ∂z j 0 l * j 0
and N = l * j 1 , we obtain Lagrangian in the form

L = Λ j 0 , ∂j 0 ∂z , j 1
The Hamilton principle reads: for each field of virtual displacements z ∈ Ω -→ ζ such that ζ and its derivatives are zero on ∂Ω,

δ Ω Λ dΩ = 0 (3.5)
Since variation ∧ δ is independent of domain Ω and measure dΩ, the variation of Hamilton action (3.5) with the zero boundary conditions for j k and its derivatives yields

δ Ω Λ dΩ = Ω m k=0 δΛ δj k ∧ δ j k dΩ = 0 (3.6)
where δ δj k refers to the variational derivative with respect to j k . In particular, if Lagrangian (3.3) is

L = Λ j k , ∂j k ∂z , z
we get

δΛ δj k = ∂Λ ∂j k -Div     ∂Λ ∂ ∂j k ∂z    
We have to emphasize that in (3.6), the variations ∧ δ j k should take into account constraints (3.4). We use the same method as in [2, p. 145] for the variation of density. This method does not use Lagrange multipliers since the constraints (3.4) are satisfied automatically. The calculation of

∧ δ j k , k = 0, . . . , m is performed in two steps. First, in Appendix A we calcu- late Lagrangian variations ∼ δ v, ∼ δ ρ and ∼ δ a k (expressions (A.
2), (A.5) and (A.6), respectively). Second, by using (2.12) we obtain in Appendix B Eulerian variations

∧ δ j k , k = 0, . . . , m (see (B.3)) ∧ δ j k = ∂ζ ∂z -(Div ζ) I j k - ∂j k ∂z ζ
Let us now define the four-dimensional covector

K * k = δΛ δj k (3.7)
Taking into account conditions (3.4) and the fact that ζ and its derivatives are zero on the boundary ∂Ω, equations (3.6), (B.3) yield

δ Ω Λ dΩ = Ω m k=0 K * k ∂ζ ∂z -(Div ζ) I j k - ∂j k ∂z ζ dΩ = Ω m k=0 -Div (j k K * k ) + j * k ∂K k ∂z ζ dΩ = Ω m k=0 j * k ∂K k ∂z - ∂K k ∂z * ζ dΩ = 0
Hamilton's principle yields the governing equations in the form

m k=0 j * k ∂K k ∂z - ∂K k ∂z * = 0 (3.8)
where K k are given by definition (3.7). The system (3.4), (3.8) represents m + d + 1 partial differential equations for m + d unknown functions u, ρ, a k , k = 1, . . . , m, where d is the dimension of the Ω-space. Since the matrix

R k = ∂K k ∂z - ∂K k ∂z * (3.9)
is antisymmetric and all the vectors j k , k = 0, . . . , m are collinear, we obtain that

j * k R k j 0 ≡ 0. Consequently m k=0 j * k R k j 0 ≡ 0 (3.10)
and the overdetermined system (3.4), (3.8) is compatible.

In the case a k = const, k = 1, . . . , m and Λ = Λ j 0 , ∂j 0 ∂z , the system (3.4), (3.8) can be rewritten in a simplified form

       j * 0 ∂K 0 ∂z - ∂K 0 ∂z * = 0 Div j 0 = 0 (3.11)
We call potential motion such solutions of (3.11) that

K * 0 = ∂ϕ 0 ∂z (3.12)
where ϕ 0 is a scalar function. In this case, the governing system for the potential motion is in the form

       δΛ δj 0 = ∂ϕ 0 ∂z Div j 0 = 0 (3.13)
For the gas dynamics model, the system (3.12), (3.13) reads

                   ∂ϕ 0 ∂t = - 1 2 |u| 2 + ∂ ∂ρ e(ρ) + Π , e(ρ) = ε (ρ, ρs e ) ∇ϕ 0 = u * ∂ρ ∂t + div(ρu) = 0
Eliminating the derivative ∂ϕ 0 ∂t , we obtain the classical model for potential

flows [2]                    ∂u * ∂t + ∇ 1 2 |u| 2 + ∂ ∂ρ e(ρ) + Π = 0 u * = ∇ϕ 0 ∂ρ ∂t + div(ρu) = 0 (3.
K * k j k I -j k K * k + A * k ∂j k ∂z -Λ I + ∂Λ ∂z + m k=0 K * k Div j k - m k=0 j * k R k ≡ 0 (4.1)
where the matrices R k are given by (3.9) and

A * k = ∂Λ ∂ ∂j k ∂z .
In particular, we get Theorem 4.2

The governing equations (3.4), (3.8) are equivalent to the system of conservation laws

Div m k=0 K * k j k I -j k K * k + A * k ∂j k ∂z -ΛI + ∂Λ ∂z = 0 Div j k = 0, k = 0, . . . , m
In the special case of potential flows (3.13) the governing equations admit additional conservation laws

Div cK * 0 -(K * 0 c)I = 0 (4.2)
where c is any constant vector. Indeed,

Div (cK * 0 ) = c * ∂K 0 ∂z * , Div (K * 0 c I) = ∂ ∂z (K * 0 c) = c * ∂K 0 ∂z
Since for the potential flows

∂K 0 ∂z = ∂K 0 ∂z *
we obtain (4.2). In particular, if we take c = l for the gas dynamics equations, we get conservation laws (3.14). Multiplying (4.1) by j 0 and taking into account the identity (3.10), we obtain the following theorem Theorem 4.3

The following scalar relation is an algebraic identity

Div m k=0 (K * k j k I -j k K * k + A * k ∂j k ∂z ) -ΛI j 0 + ∂Λ ∂z j 0 + m k=0 (K * k j 0 ) Div j k ≡ 0 (4.3)
This theorem is a general representation of the Gibbs identity expressing that the "energy equation" is a consequence of the conservation of "mass", "momentum" and "entropy". Examples of this identity for thermocapillary fluids and bubbly liquids were obtained previously in [START_REF] Casal | Equations of motions of thermocapillary fluids[END_REF][START_REF] Gavrilyuk | Media with equations of state that depend on derivatives[END_REF].

Identity (4.3) yields an important consequence. Let us recall the Godunov-Friedrichs-Lax method of symmetrisation of quasilinear conservation laws [START_REF] Godunov | An interesting class of quasilinear systems[END_REF][START_REF] Friedrichs | Systems of conservation laws with a convex extension[END_REF] (see also different applications and generalizations in [START_REF] Godunov | Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media[END_REF][START_REF] Boillat | Non-linear hyperbolic fields and waves[END_REF]). We suppose that the system of conservation laws for n variables q has the form

∂f i ∂t + div F i = 0, f i = f i (q), F i = F i (q), i = 1, . . . , n (4.4)
Let us also assume that (4.4) admits an additional "energy" conservation law

∂e ∂t + div E = 0, e = e (q), E = E (q)
which is obtained by multiplying each equation of (4.4) by some functions p i and then by summing over i = 1, . . . , n ∂e ∂t

+ div E ≡ p i ∂f i ∂t + div F i (4.5)
In particular, if we consider e, E and F i as functions of f i , we obtain from (4.5)

∂e ∂f i = p i , ∂E ∂f j = p i ∂F i ∂f j = ∂e ∂f i ∂F i ∂f j (4.6)
Let us introduce functions N and M such that

N = f i p i -e, M = F i p i -E (4.7)
Consequently, from equations (4.6) and (4.7) we find that

∂N ∂p i = f i , ∂M ∂p i = F i (4.8)
Hence, substituting (4.8) into (4.4), we get a symmetric system in the form

∂ 2 N ∂p i ∂p j ∂p j ∂t + tr ∂ 2 M ∂p i ∂p j ∂p j ∂x = 0 (4.9) If matrix N ij = ∂ 2 N ∂p i ∂p j is
(K * k j k I -j k K * k ) -Λ I j 0 + m k=0 (K * k j 0 ) div j k ≡ 0 (4.10)
Identity (4.10) is exactly of the same type as identity (4.5). It means that the system (3.4), (3.8) can be always rewritten in a Godunov-Friedrichs-Lax symmetric form. Actually, if we take

e = m k=0 K * k j k -(j k K * k ) 1 1 -Λ and E = -P j k K * where P =    0 1 0 0 0 0 1 0 0 0 0 1    the conjugate variables p are p =      u - 1 ρ K * k j 0      , k = 0, . . . , m
Consequently, the Gibbs identity (4.10) gives directly a set of conjugate variables. Therefore, we have proved the following theorem Theorem 4.4

If L = Λ(j k ), the system (3.4), (3.8) is always symmetrisable.

The property of convexity, needed for the hyperbolicity of the governing system, should be verified for each particular case. For example, in the gas dynamics, the energy of the system is given by the formula

e = 1 k=0 K * k j k - 1 k=0 j k K * k 1 1 -Λ = ε(ρ, η) + ρ |u| 2 2 Since K * 0 j 0 = ρ |u| 2 2 - ∂ε ∂ρ , K * 1 j 0 = -ρ
∂ε ∂η the conjugate variables are (see also [START_REF] Godunov | An interesting class of quasilinear systems[END_REF])

p =           u ∂ε ∂ρ - |u| 2 2 ∂ε ∂η           =         u µ - |u| 2 2 T        
where µ is the Gibbs potential and T is the temperature. Obviously, if ε(ρ, η) is convex, the total energy e is convex with respect to ρu, ρ and η.

5 Stability of equilibrium states for potential flows.

We assume that L is a function of j k and ∂j k ∂z besides it does not depend on z, i.e.

L = Λ j k , ∂j k ∂z
Let us give some definitions. An equilibrium state is a solution of equations (3.4), (3.8) such that j k = j ke = const. Let ν be a real unit vector. Any vector β can be represented in the form

β = ων + β σ , where ν * β σ = 0
The equilibrium state j ke = 0 is linearly stable in the direction ν if and only if all non-trivial solutions of the form J k e iβ * z of the system (3.4), (3.8) linearized at the equilibrium state j ke are such that ω is real for any real βσ.

Non-dispersive case

We note that in a non-dispersive case the stability means hyperbolicity of governing system [START_REF] Ovsyannikov | Lectures on the gas dynamics foundations[END_REF][START_REF] Serre | Systèmes de Lois de Conservation I[END_REF]. We omit the index "0" and rewrite system (3.13) in the form

       K * ≡ ∂Λ ∂j = ∂ϕ ∂z Div j = 0 (5.1)
where Λ = Λ(j). The Legendre transformation of Λ(j) is where index "e" corresponds to the equilibrium state. The matrix ∆ ′′ (K e ) can be easily calculated in terms of Lagrangian Λ(j). Indeed,

∆(K) = K * j -Λ(j) (5.2)
I = ∂j ∂j = ∂j ∂K ∂K ∂j = ∆ ′′ (K) Λ ′′ (j)
It follows that ∆ ′′ (K e ) = Λ ′′ (j e ) -1

(5.6) Relations (5.5) -(5.6) imply the following result.

Theorem 5.1 (criterion of stability for potential non-dispersive motions)

If the symmetric matrix G = Λ ′′ (j e ) has the signature (-, +, +, +), the equilibrium state j e is stable in any direction ν belonging to the intersection of two cones

C 1 = ν | ν * G -1 ν < 0 and C 2 = ν | ν * Gν < 0
Proof. Since C 1 and C 2 contain the eigenvector corresponding to the negative eigenvalue of G, C 1 ∩ C 2 = ∅. We can find orthogonal coordinates (t, x 1 , x 2 , x 3 ) such that the dispersion relation (5.5) takes the form

-t 2 + 3 i=1 λ i x 2 i = 0, with λ i > 0
In coordinates (t, x 1 , x 2 , x 3 ) the cone C 2 is defined by the inequality

-t 2 + 3 i=1 x 2 i λ i < 0
Let us represent β in the form

β =      t x 1 x 2 x 3      = ω      1 n 1 n 2 n 3      +      -n 1 y 1 -n 2 y 2 -n 3 y 3 y 1 y 2 y 3      The dispersion relation β * G -1 β = 0 implies - 3 i=1 n i y i -ω 2 + 3 i=1 λ i (ωn i + y i ) 2 = 0 It involves ω 2 -1 + 3 i=1 λ i n 2 i + 2ω 3 i=1 (1 + λ i ) n i y i + 3 i=1 λ i y 2 i - 3 i=1 n i y i 2 = 0 14
Due to the following inequality,

3 i=1 λ i y 2 i - 3 i=1 n i y i 2 = 3 i=1 λ i y 2 i - 3 i=1 n i √ λ i λ i y i 2 ≥ 3 i=1 λ i y 2 i - 3 i=1 n 2 i λ i 3 i=1 λ i y 2 i = 3 i=1 λ i y 2 i 1 - 3 i=1 n 2 i λ i ω is real if ν belongs simultaneously to C 1 and C 2 .
The theorem is proved.

For the gas dynamics model, if the volume energy is a convex function of ρ and η = ρs, the matrix G satisfies the conditions of Theorem 5.1.

Dispersive case

If L = Λ j, ∂j ∂z , the governing system is

         K * ≡ δΛ δj = ∂Λ ∂j -Div   ∂Λ ∂ ∂j ∂z   = ∂ϕ ∂z
Div j = 0

The linearised system in a coordinate form is Λ ks j s + Γ p ks j s ,p -D mp ks j s ,mp = ϕ, k , and j s ,s = 0 (5.7)

where the comma denotes the derivative with respect to z k while Λ ks , Γ p ks , D mp ks , calculated at point j e , are defined by the relations

Λ ks = ∂ 2 Λ ∂j k ∂j s , Γ p ks = ∂ 2 Λ ∂j k ∂j s ,p - ∂ 2 Λ ∂j s ∂j k ,p , D mp ks = ∂ 2 Λ ∂j k ,m ∂j s ,p (5.8)
The following symmetry relations result from (5.8)

Λ ks = Λ sk , Γ p ks = -Γ p sk , D mp ks = D pm sk (5.9)
For the solution of (5.7) in the form j s = J s e iβpz p , ϕ = i Φ e iβpz p , we get the following dispersion relation

det    G β β * 0    ≡ -β * Adj (G)β = 0 (5.10)
where Adj (G) is the adjoint matrix to G, and the elements G ks of the matrix G are defined by

G ks = Λ ks + i Γ p ks β p + D mp ks β m β p (5.11)
Equations (5.9) -(5.11) imply that G is hermitian matrix. In particular, if det G = 0 then Adj (G) = det(G) G -1 and the dispersion relation (5.10) is equivalent to β * G -1 β = 0, which is a generalization of (5.5) for the dispersive case. We get the following obvious result: Theorem 5.2 (criterion of stability for potential dispersive motions)

If Γ p ks = 0 and the symmetric matrix G defined by (5.8), (5.11) has the signature (-, +, +, +) for β = 0, the equilibrium state j e is stable for small β in any direction ν belonging to the intersection of the cones C i , i = 1, 2 defined in theorem 5.1.

We note that Γ p ks defined by (5.8) are always zero if the expansion of Lagrangian Λ in Taylor series at the vicinity of equilibrium state does not contain linear terms with respect to ∂j k ∂z .

Sometimes, we are able to obtain a "global" stability. For example, let us consider a particular case of bubbly liquids, for N/ρ = const, defined in Section 3

L = 1 2 ρ|u| 2 + a 2 dρ dt 2 -ε(ρ)
where ε(ρ) is a convex function of the density and a is a positive function of ρ. Then,

L = 1 2 |j| 2 l * j -l * j + a 2 ∂ ∂z (l * j) j l * j 2 -ε(l * j)
Since the governing equations are invariant with respect to the Galilean transformation

t ′ = t, x ′ = x + Ut, u ′ = u + U
we can always assume that u e = 0 (it is sufficient to consider the governing system in the reference frame moving with the velocity U = u e ). Hence, j e = ρ e l, where l * = (1, 0, 0, 0). Omitting index "e", we can calculate the matrix G defined by (5.11)

G =      - ∂ 2 ε ∂ρ 2 + aβ 2 1 0 * 0 I ρ      Hence, Adj (G) =      1 ρ 3 0 * 0 aβ 2 1 -∂ 2 ε ∂ρ 2 I ρ 2     
and dispersion relation (5.10) reads

β 2 1 ρ 3 + aβ 2 1 - ∂ 2 ε ∂ρ 2 ρ 2 β 2 2 + β 2 3 + β 2 4
= 0 (5.12)

Let ν be the direction of time in the time-space, then ν = l and (5.12) is equivalent to

ω 2 ∂ 2 ε ∂ρ 2 -aω 2 = ρ |β σ | 2
(5.13)

The graph of the dispersion relation (5.13) for positive values of ω is presented in Figure 1. We have denoted by

ω * = ∂ 2 ε ∂ρ 2
1 a the eigenfrequency of bubbles and by c 0 = ρ ∂ 2 ε ∂ρ 2 the equilibrium sound speed of bubbly liquid (see [START_REF] Van Wijngaarden | On the equations of motions for mixture of liquid and gas bubbles[END_REF]). Let consider homogeneous binary mixtures. The mixture is described by the velocities u α , the average densities ρ α and the specific entropies s α for each component (α = 1, 2). We introduce two reference frames associated with each component in the form

     t = g α (λ α , X α ) x = φ α (λ α , X α ) (6.1)
and the inverse mappings

     λ α = h α (t , x) X α = Ψ α (t , x) (6.2)
The corresponding families of virtual motions generated by (6.1) are defined by

     t = G α (λ α , X α , ε α ) x = Φ α (λ α , X α , ε α with      G α (λ α , X α , 0) = g α (λ α , X α ) Φ α (λ α , X α , 0) = φ α (λ α , X α )
We define the two Eulerian displacement ζ α = (τ α , ξ α ) where

τ α = ∂G α ∂ε α (λ α , X α , 0), ξ α = ∂Φ α ∂ε α (λ α , X α , 0)
As in Section 3 we define tensor quantities associated with the two virtual motions and variations ∧ δ α and ∼ δ α . In the general case, we have two fourdimensional solenoidal vectors j 0(α) = ρ α v α corresponding to the α th component, where ρ α is the density and v * α = (1, u * α ) is the four-dimensional velocity vector. As in Section 3, we introduce additional physical quantities a k(α) associated with the four-dimensional vectors j k(α) = a k(α) j 0(α) , α = 1, 2, k(α) = 1, . . . , m(α) submitted to the constraints Div j k(α) = 0 (6.3) For Hamilton's action in the form

a = Ω Λ j k(α) , ∂j k(α) ∂z , . . . , ∂ n j k(α)
∂z n , z dΩ the Hamilton principle reads: for each field of virtual displacements z ∈ Ω -→ ζ α such that ζ α and its derivatives are zero on ∂Ω,

δ α Ω Λ dΩ = 0
Here δ α a is the derivative of the Hamilton action with respect to ε α , and ζ α are the virtual displacements expressed in Eulerian coordinates z by means of equations (6.2). The method developed in Section 3 yields the equations of motion in the form

m(α) k(α)=0 j * k(α) ∂K k(α) ∂z - ∂K k(α) ∂z * = 0, K * k(α) = δΛ δj k(α) (6.4)
Div j k(α) = 0, α = 1, 2 and k(α) = 0, . . . , m(α)

As in Theorem 4.1, for the case Λ = Λ j k(α) , ∂j k(α) ∂z , z we can obtain the following identity

Div   2 α=1 m(α) k(α)=0 K * k(α) j k(α) I -j k(α) K * k(α) + A * k(α) ∂j ∂z -Λ I   + ∂Λ ∂z + 2 α=1 m(α) k(α)=0 K * k(α) Div j k(α) - 2 α=1 m(α) k(α)=0 j * k(α) R k(α) ≡ 0
where

A * k(α) = ∂Λ ∂ ∂j k(α) ∂z and R k(α) = ∂K k(α) ∂z - ∂K k(α) ∂z *
Hence, the governing equations (6.4) admit the conservation laws

Div   2 α=1 m(α) k(α)=0 K * k(α) j k(α) I -j k(α) K * k(α) + A * k(α) ∂j k(α) ∂z -Λ I   + ∂Λ ∂z = 0 Div j k(α) = 0, α = 1, 2 and k(α) = 0, . . . , m (α)
We notice that for a one-velocity model, the number of scalar conservation laws is m + d + 1 where d is the dimension of the time-space. The number of unknown variables is m + d. Due to Theorem 4.3, the m + d + 1 conservation laws are connected by the "Gibbs identity". In the case of mixtures, we obtain α m(α) + d + 1 conservation laws for α m(α) + 2d unknown variables. In the general case, the classical approach based on conservation laws, does not allow to obtain Rankine-Hugoniot conditions for this system. Nevertheless, in [START_REF] Gavrilyuk | A variational principle for two-fluid models[END_REF][START_REF] Gavrilyuk | Hyperbolic models of twofluid mixtures[END_REF][START_REF] Gouin | Hamilton's principle and Rankine-Hugoniot conditions for general motions of mixtures[END_REF] we have obtained the jump conditions. The Hamilton principle provides these conditions without any ambiguity.

7 Linear stability of mixtures.

We consider the Lagrangian in the form [START_REF] Gavrilyuk | A variational principle for two-fluid models[END_REF][START_REF] Gavrilyuk | Hyperbolic models of twofluid mixtures[END_REF][START_REF] Gouin | Hamilton's principle and Rankine-Hugoniot conditions for general motions of mixtures[END_REF][START_REF] Geurst | Virtual mass in two-phase bubbly flow[END_REF][START_REF] Geurst | Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures[END_REF]

L = 1 2 ρ 1 |u 1 | 2 + 1 2 ρ 2 |u 2 | 2 -W (ρ 1 , ρ 2 , η 1 , η 2 , w) (7.1)
where w = u 2u 1 and η α is the entropy per unit volume of the α th component. A generalisation of (7.1) for thermocapillary fluids was also proposed in [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]. If W is an isotropic function of w, Lagrangian (7.1) can be rewritten as follows

Λ = 1 2 2 α = 1 |j 0(α) | 2 l * j 0(α) -l * j 0(α)
-W l * j 0(1) , l * j 0(2) , l * j 1(1) , l * j 1(2) , µ

where µ = 1 2 j l * j 0(2) -j 0(1) l * j 0(1) 2 (7.3) Here we shall also restrict our study to the case s α = const. To avoid double indices, we will denote j 0(1) , j 0(2) by j 1 , j 2 . Hence, (7.2) and (7.3) involve

Λ(j 1 , j 2 ) = 1 2 2 α=1 |j α | 2 l * j α -l * j α - ∼ W (l * j 1 , l * j 2 , µ) (7.4)
where

∼ W (ρ 1 , ρ 2 , µ) = W (ρ 1 , ρ 2 , ρ 1 s 1 , ρ 2 s 2 , µ).
Omitting the tilde symbol, we consider the potential motions

       K * α ≡ ∂Λ ∂j α = ∂ϕ α ∂z Div j α = 0, α = 1, 2 (7.5)
Let us consider the Legendre transformation of Λ(j 1 , j 2 ) which is the generalisation of the dispersion relation (5.5). However, to calculate ∆ (γα) in terms of Λ (γα) defined by (7.7), (7.9)), we have to solve the following system of matrix equations

∆(K 1 , K 2 ) =
2 γ=1 ∆ (γα) Λ (βγ) = I δ αβ , δ αβ = 1, α = β 0, α = β
A simpler method is to consider the linearised system generated from (7.5)

2 γ=1 Λ (γα) j γ - ∂ϕ α ∂z * = 0, Div j α = 0
where Λ (γα) are taken at the equilibrium state j ke . Substituting j γ by J γ e iβ * Equation (7.11) is presented in terms of the matrices Λ (γα) calculated directly from the Lagrangian. Let us consider Lagrangian (7.4). Suppose that the velocities of each component are the same at equilibrium (u 1e = u 2e ). Due to the invariance of the governing equations with respect to the Galilean transformation we assume, without loss of the generality, that u 1e = u 2e = 0. Suppressing the index "e" to avoid double indices, we get

                     Λ (11) = ρ 1 - ∂W ∂µ I -ll * ρ 2 1 - ∂ 2 W ∂ρ 2 1 ll * Λ (21) = Λ (12) = 1 ρ 1 ρ 2 ∂W ∂µ (I -ll * ) - ∂ 2 W ∂ρ 1 ∂ρ 2 ll * Λ (22) = ρ 2 - ∂W ∂µ I -ll * ρ 2 2 - ∂ 2 W ∂ρ 2 2 ll * (7.12) by c = 4 3 π R 3 N
where R is the average radius of the bubbles, N is the number of bubbles per unit volume of the mixture. The internal energy per unit mass ε 20 of the gas phase and the virtual mass coefficient m are known functions of ρ 20 and c. By introducing the average density of the liquid phase ρ 1 = ρ 10 (1c) with ρ 10 = const we can rewrite the potential (7.17) as

W = ρ 2 ε 20 ρ 2 ρ 10 ρ 10 -ρ 1 - ρ 10 2 m ρ 10 -ρ 1 ρ 10 |w| 2
The increase of the volume fraction c producing the interaction between gas bubbles changes not only the coefficient m(c) but also the interfacial energy ε in (c) so that potential W can be generalized to the form

W = ρ 2 ε 20 (ρ 20 ) - ρ 10 2 m(c)|w| 2 + ε in (c) (7.18)
Quantity ε in (c) produces an additional pressure term due to the interfacial effect [START_REF] Haley | A characteristic analysis of void waves using two-fluid models[END_REF]. Therefore, Lagrangian (7.1) describes not only binary molecular mixtures but also partial cases of suspensions. Case (7.17) is actually degenerated: w 11 w 22w 2 12 ≡ 0. The introduction of the interfacial term in (7.18) involves the convexity condition w 11 w 22w 2 12 > 0 provided

∂ 2 ε in ∂c 2 > 0.
A APPENDIX

For any function

• f (λ , X) (see definitions (2.9)-(2.10)) and its image f (t , X) in the (t , X)-coordinates, we obtain the following relations

∂ • f ∂λ ∂g ∂λ -1 = ∂f ∂t , ∂ • f ∂X - ∂g ∂X ∂ • f ∂λ ∂g ∂λ -1 = ∂f ∂X (A.1)

A.1 Variation of the velocity

The definition (2.6) of the velocity u yields

• u = ∂φ ∂λ ∂g ∂λ -1
In Lagrangian coordinates, perturbation of u is represented by the formula

∼ u (λ , X , ε) = ∂Φ ∂λ (λ , X , ε) ∂G ∂λ (λ , X , ε) -1
Taking the derivative of ∼ u with respect to ε at ε = 0 and using (A.1), we get

∼ δ u = ∂ • ξ ∂λ ∂g ∂λ -1 - • u ∂ • τ ∂λ ∂g ∂λ -1 = ∂ξ ∂t -u ∂τ ∂t = dξ dt -u dτ dt where d dt = ∂ ∂t + u * ▽ * is the material derivative. We obtain then ∼ δ v = (I -vl * ) ∂ζ ∂z v (A.2)
where v * = (1, u * ), I is the identity tensor and l * = (1, 0, 0, 0).

A.2 Variation of the deformation gradient

In Lagrangian coordinates (λ, X), the perturbation of the deformation gradient is given by (2.

  The second equation (5.1) and (5.4) involve tr ∆ ′′ (K) ϕ ′′ (z) = 0 If we replace ϕ by i Φe iβ * z , we get immediately the dispersion relation β * ∆ ′′ (K e )β = 0 (5.5)

Fig. 1 .

 1 Fig. 1. Dispersion relation for bubbly liquids

  state ∆ (γα) = const. Substituting ϕ γ by iΦ γ e iβ * z in (7.10), we obtain the dispersion relation in the symmetric form det β * ∆ (11) β β * ∆ (21) β β * ∆ (12) β β * ∆ (22) β = 0

z

  and ϕ α by iΦ α e iβ * z , we obtain the dispersion relation in the symmetric form det

FA. 3 F 4 )

 34 with respect to ε at ε = 0 and relation (A.1) gives∼ Variation of the densityThe mass conservation is represented by the formulaρ det F = ρ 0 (X)The perturbation of ρ is in the form ∼ ρ (λ , X , ε) det ∼ (λ , X , ε) = ρ 0 (By using the Euler-Jacobi identity,δ (det F ) = det F tr (F -1 ∼ δ F ) relations (A.3) and (A.4) lead to ∼ δ ρ =tr ρ(Ivl * ) ∂ζ ∂z (A.5)A.4 Variation of specific quantitiesAlong each trajectory the specific quantities a k are constant. Consequently,a k = a 0k (X). The perturbation of a k is such that ∼ a k (λ , X , ε) = a 0k (X).Hence, .2) and (A.5) yield directly the variation of j 0 ∼ δ j 0 = tr (vl * -I) ∂ζ ∂z I -(vl * -I) ∂ζ ∂z j 0 = ∂ζ ∂z -(Div ζ) I j 0 (B.1) Let us consider the variations ∼ δ j k , k = 1, . . . , m with j k = a k j 0 where the scalar fields a k are such that a k = a k0 (X). Due to (A.6) ∼ δ a k = 0 and ∼ δ j k = a k ∼ δ j 0 . Consequently ∼ δ j k = ∂ζ ∂z -(Div ζ) I j k , k = 0, . . . , m (B.2) Hence, (2.12) involves ∧ δ j k = ∂ζ ∂z -(Div ζ) I j k -∂j k ∂z ζ, k = 0, . . . , m (B.3) C APPENDIX Let A , B be two linear transformations depending on z. Let us define the linear form tr A ∂B ∂z such that for any constant vector field a, B a) = Div (A) B a + tr A ∂Ba ∂z = (Div A) B + tr A ∂B ∂z a Let us also recall the following useful formulae for any vector fields b(z), c(z) Div (b c * ) = c * Div b + b * ∂c ∂z * (C.2) Div (b * c I) = ∂ ∂z (b * c) = b * ∂c ∂z + c * ∂b ∂z (C.3) By using formulae (C.1) -(C.3), we have the following identities Div (K * k j k I) ≡ K * k K * k ) ≡ -K * k Div j kj * k proves Theorem 4.1.

  positive definite then the symmetric system (4.9) is t-hyperbolic symmetric in the sense of Friedrichs. Obviously, matrix ∂ 2 N ∂p i ∂p j is positive definite if and only if matrix e ij = ∂ 2 e ∂f i ∂f j is positive definite, since e ij N jp = δ i p , where δ i p is the Kronecker symbol.

	Now, let us rewrite identity (4.3) in the non-dispersive case
	m
	Div
	k=0

All matrices Λ (αβ) are diagonal. Now we are able to calculate the dispersion relation (7.11)

. The dispersion relation (7.15) takes the form

The following result is proved in [START_REF] Gavrilyuk | Variational approach for the construction of hyperbolic two-fluid models[END_REF]:

Theorem 7.1

If potential W (ρ 1 , ρ 2 , µ) satisfies conditions (7.14), all the roots λ of the polynome (7.16) are real.

Conditions (7.14) mean that internal energy

, where w = |w|, is convex [START_REF] Gavrilyuk | A variational principle for two-fluid models[END_REF][START_REF] Gavrilyuk | Hyperbolic models of twofluid mixtures[END_REF][START_REF] Gavrilyuk | Variational approach for the construction of hyperbolic two-fluid models[END_REF]. Hence, if the internal energy is a convex function and the relative velocity w is small enough, the equilibrium state is stable in time direction in the time-space.

It is interesting to note that the Lagrangian of a two-fluid bubbly liquid with incompressible liquid phase (heterogeneous case) has the same form as (7.1). Indeed, in [START_REF] Geurst | Virtual mass in two-phase bubbly flow[END_REF][START_REF] Geurst | Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures[END_REF] the potential W for a bubbly liquid is proposed in the form

where ρ 2 = c ρ 20 , c is the volume concentration of gas phase and the index "0" means the real density of gas and liquid phase. Concentration c is expressed


Sheena Bassett

Leon Wessels

Steven Krauwer

Bente Maegaard

Hella Hollander

Femmy Admiraal

Laurent Romary

Frank Uiterwaal

Connecting the Humanities through Research Infrastructures

Keywords: cross-discipline, policies, language resources, Research Infrastructures, standards

Introduction

Several Research Infrastructures (RIs) exist in the Humanities and Social Sciences, some -such as CLARIN, DARIAH and CESSDA -which address specific areas of interest, i.e. linguistic studies, digital humanities and social science data archives. RIs are also unique in their scope and application, largely tailored to their specific community needs. However, commonalities do exist and it is recognised that benefits are to be gained from these such as efficient use of resources, enabling multi-disciplinary research and sharing good practices. As such, a bridging project PARTHENOS has worked closely with CLARIN and DARIAH as well as ARIADNE (archaeology), CENDARI (history), EHRI (holocaust studies) and E-RIHS (heritage science) to identify, develop and promote these commonalities. In this paper, we present some specific examples of cross-discipline and trans-border applications arising from joint RI collaboration, allowing for entirely new avenues of research.

CLARIN: Language is not just for linguists

Linguists study language, but language is not just studied by linguists. As the European Research Infrastructure Consortium for language resources and technology, CLARIN offers digital language data and tools to scholars, researchers, students and citizen-scientists from all disciplines [START_REF] Hinrichs | The CLARIN Research Infrastructure: Resources and Tools for eHumanities Scholars[END_REF]. CLARIN's vision is to make all digital language data collections and tools from all over Europe and beyond accessible through a single sign-on online environment [START_REF] De | CLARIN: Towards FAIR and Responsible Data Science Using Language Resources[END_REF]. In this section we will highlight two initiatives particularly aimed at stimulating cross-border and cross-disciplinary collaboration using language resources and tools.

Resource Families

To provide user-friendly overviews of available corpora (i.e. collections of digital language data), including information on language, size, annotation, availability and licence, CLARIN has introduced the Resource Families initiative (https://www.clarin.eu/resource-families). Resource Families are data collections that display a high degree of maturity and homogeneity, that are available for most EU languages and are a rich source of social and cultural data [START_REF] Fišer | CLARIN's Key Resource Families[END_REF]. Currently CLARIN offers overviews of seven Resource Families: language data collected from social media, historical corpora, second-language learner corpora, newspaper corpora, parallel corpora, parliamentary corpora and spoken corpora. Resource Families are relevant for multiple scientific disciplines in the digital humanities and social sciences. Parliamentary data for instance, is used by political scientists, historians, and linguists to study complex phenomena such as Euroscepticism and gender stereotypes [START_REF]More information can be found on the PARTHENOS training module Boosting Digital Humanities Research with Parliamentary Data[END_REF].

Language Resource Switchboard

Equally important to language resources are the means to process them. The CLARIN Language Resource Switchboard (https://switchboard.clarin.eu) helps researchers to connect resources to the tools to process them [START_REF] Zinn | The CLARIN Language Resource Switchboard[END_REF]. It provides tools for different types of parsing and annotation, analysis, distant reading, etc. By providing uniform, manually-annotated overviews of corpora and a range of advanced, robust tools for processing and analysing these corpora, CLARIN enables researchers from various disciplines to find cross-national and cross-language patterns in language data far more quickly and easily, on a much larger scale than before.

DARIAH: The Standardization Survival Kit

Whereas some scholarly groups have been very active in defining and using standards in their practices, the project observed that there is a general lack of precise knowledge in the standardization landscape, especially from newcomers to digital methods [START_REF] Klaus | How to work together successfully with e-Humanities and e-Heritage Research Infrastructures[END_REF]. A solution to this information gap is the Standardization Survival Kit (SSK). which assists scholars in finding information on which standards may play a role in their specific scholarly activities and is designed around research scenarios in line with the requirements gathered earlier in the project [7]. The current scenarios illustrate both the importance of standards in research processes, as well as the usefulness of designing an online environment which allows researchers to browse through scenarios and access relevant reference material [START_REF] Romary | Standardization survival kit (Draft)[END_REF]. Additionally, the project developed material (a leaflet and several videos) to make scholars aware of standards. The resulting digital environment was designed in close collaboration with digital specialists from the various communities and RIs represented in the PARTHENOS project with the support of DARIAH who initiated the standards work. The current scenarios cover all types of scholarly domains and methodologies: managing field surveys, creating a textual or a musical corpus, managing archival information as well as using laser techniques for conservation practice in heritage science and includes four specific linguistic scenarios. An important expected future feature will be the possibility to couple the maintenance of validated scenarios with the freedom for scholars to publish their own scenarios through the platform.

PARTHENOS: The need for common policies and implementation strategies

As a cluster project, it has been PARTHENOS' aim to make optimal use of experiences gained throughout all disciplines. Shared best practices and formal policies make it possible to answer research questions in a more overarching way. Not only does this enable researchers from different universities and geographic areas to refer to the same policies in a shared language and mutual understanding. Also, researchers can more easily reuse each other's results, allowing them to build on each other's observations and knowledge. It is PARTHENOS' aim to work towards the integration of research data policies across the disciplines in the humanities, as well as to increase their interdisciplinary exchanges. This is a broad mission, for which one generic solution could never suffice, also due to the current, distributed and previously unmapped landscape of research policies. However, the FAIR Principles have been applied as an underlying framework and three important forms of project output provide solutions to this situation, helping all stake holders in making more research data more findable, accessible, interoperable and re-usable [START_REF] Mark | The FAIR principles provide an important frame of reference for the output PARTHENOS generates on the topic of research data management[END_REF]:

1.) The PARTHENOS Policy Wizard. The Policy Wizard is a tool which helps researchers to discover which data policy applies best to their particular data. It allows the user to select an area of interest, for example History, and find a range of policies suggesting how data should be collected, processed, stored and shared with other researchers.

2.) Data Management Plan template. The PARTHENOS DMP template builds on the Horizon2020 version and has been tailored for the Humanities using expert input. It provides built-in guidance with some pre-filled options for completing a DMP.

3.) PARTHENOS Guidelines. Twenty high-level guidelines are offered as common recommendations to build bridges between different, although tightly interrelated, fields and stakeholders (researchers and data repositories) within the Humanities by the harmonisation of policy definition and their implementation.

Conclusions

Collaboration between RIs for the purpose of developing synergy on common aspects of research has helped initiate harmonisation and led to the development of useful tools whilst also reducing barriers for cross-discipline and cross-border research. Other collaborative examples in PARTHENOS are shared international contacts and a Training Suite that addresses the need for knowledge about RIs as well as shared Humanities topics.