Pore-scale bending and membrane effects in permeo-elastic media

Pore-scale bending and membrane effects in permeo-elastic media

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Pore-scale bending and membrane effects in permeo-elastic media Claude Boutin, Rodolfo Venegas PII: DOI: Reference:

S0167-6636(19)30933-0 https://doi.org/10.1016/j.mechmat.2020.103362 MECMAT 103362

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Mechanics of Materials

Received date: Revised date: Accepted date:

25 October 2019 25 January 2020 13 February 2020

Please cite this article as: Claude Boutin, Rodolfo Venegas, Pore-scale membrane effects in permeo-elastic media, Mechanics of Materials https://doi.org/10.1016/j.mechmat.2020.103362

bending (2020),

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Highlights • Upscaling of wave equation in permeo-elastic media with bending and membrane effects. • Fluid-film system can behave as an effective visco-elastic fluid. • Inner fluid-film resonances yield atypical transport and acoustic properties. • Effective density is altered by pore-scale fluid-film interaction. • Theory is exemplified numerically.

1

Pore-scale bending and membrane effects in permeo-elastic media Claude Boutin and Rodolfo Venegas Claude Boutin Université de Lyon - Ecole Nationale des Travaux Publics de l’Etat, LGCB / LTDS UMR-CNRS 5513 / CeLyA, Rue Maurice Audin, 69518, Vaulx-en-Velin, France. e-mail: [email protected] Rodolfo Venegas1 University Austral of Chile, Institute of Acoustics, P.O. Box 567, Valdivia, Chile; and Acoustics Research Centre, University of Salford, The Crescent, M5 4WT, Salford, United Kingdom e-mail: [email protected]

1

Corresponding author

Abstract Permeo-elastic media are permeable media whose pore fluid network is connected and their solid frame is made of a stiff skeleton onto which highly flexible thin films are fixed. In this work, we investigate acoustic wave propagation in permeo-elastic media accounting for pore-scale membrane and bending effects in the films. To this end, use is made of the two-scale asymptotic homogenisation method to derive upscaled models, of asymptotic analyses to disclose the different wave propagation regimes, and of the finite element method to solve the local fluid-structure interaction boundary-value problems arising from homogenisation. The study evidences the salient acoustic features of permeo-elastic media and shows that these may significantly depart from those of conventional porous media due to the interplay of the elastic and kinetic energies of the films with the viscous and kinetic energies of the fluid. It is concluded that the control of membrane and bending effects may lead to a useful way of tuning the atypical effective acoustical properties of permeo-elastic media. Keywords: sound propagation; permeo-elastic; heterogeneous materials; homogenization; fluidstructure interaction

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1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

1

Introduction

This paper investigates pore-scale bending and membrane effects in permeo-elastic media. This recently introduced [51] class of materials has a representative elementary volume comprising a connected pore fluid network and a solid frame made of a perfectly rigid and impervious skeleton onto which highly flexible thin solids are fixed. Due to their large deformability, these thin solids, to be referred to as films, strongly interact with the fluid at the pore scale. It will be shown that such interaction leads to atypical long-wavelength behaviour, resulting from the unconventional effective mass transport parameter, namely the effective conductivity or density, in comparison with that of classical rigidframe or elastic porous media [5, 6, 34, 3, 26, 14, 46, 28, 2]. The said atypical behaviour i) is induced by the combined effects of the viscous dissipation, the kinematic energy of the saturating fluid as well as that of the films, and the elastic energy of the films which involves the membrane and/or bending effects at the pore scale; and ii) includes unconventional flow regimes where the fluid–film system behaves as an equivalent visco-elastic fluid and inner fluid–film resonances yielding atypical acoustical properties in given frequency bands. Acoustic wave propagation in permeable materials with inner resonances have been intensively studied since the early 2000s [20, 8, 29, 43, 40, 18, 59, 45, 56, 57, 11, 52, 55, 53, 54]. In these materials one, at least, of its cell constituent, e.g. acoustic resonators or highly resistive porous media, experiences a dynamic state while the remaining elements, e.g. channels or highly permeable porous media, experience a quasi-static state and carry the long-wavelength wave. Often denominated metamaterials [20, 40, 18, 59], these materials possess effective complex-valued compressibility with atypical features such as negative real part in a given frequency band [20, 8, 43] or enhanced real and imaginary parts [45, 56, 57, 52, 55, 53, 54], that respectively lead to sub-wavelength band gaps (i.e. frequency bands where wave propagation is forbidden) or broadband sound attenuation. The physical origin of these phenomena lies in inner mechanisms that induce local mass sources that result in a modification of the macroscopic mass balance and its associated effective parameter, i.e. the effective compressibility. Such inner mechanisms leave unchanged, in contrast with permeo-elastic media, the effective parameter associated with the macroscopic fluid flow constitutive law, i.e. the effective conductivity or density. On the other hand, non-permeable metamaterials comprising arrays of resonating plate- or membrane-type constituents placed in a channel have also received attention (see e.g. [60, 42, 41, 16, 15, 40, 49]). Common to these works it is that the effective models, being usually postulated at the macroscopic scale as opposed to being derived using upscaling techniques, consider that the anomalous acoustic behaviour exhibited in given frequency bands is accounted for by an unconventional effective density (or conductivity) and is due to the interaction between the fluid and the resonating constituents. In comparison with studies dealing with large-scale fluid-structure interaction [24] where usually a single macroscopic element, e.g. a perforated, elastic plate [33, 12, 36], interacts with the fluid, the role of pore-scale fluid-structure interaction in periodic permeable media appears to be less explored, with exceptions being, for example, the works [44, 27, 23, 51] where linear effective models are derived or [32, 13] where non-linear Biot-type poroelastic effective models are investigated. This paper on permeo-elastic media follows the development in [51], where the films were modelled as Love-Kirchhoff plates [39] and, consequently, only the influence of bending effects in the films was investigated. However, membrane effects are expected to dominate over bending effects when the films are extremely thin (e.g. a few, or tens of, microns in thickness) due to their small bending stiffness, while both membrane and bending effects can coexist in stretched films [16]. It is then of interest to derive upscaled models accounting for the pre-stress of the films. The present work derives, through the two-scale asymptotic homogenisation method [47, 4], the linear macroscopic description of wave propagation in permeo-elastic media accounting for membrane and/or bending effects in the films. It will be shown that the control of membrane and bending effects may lead to a useful way of tuning the atypical effective conductivity or density of the said media. Moreover, it is envisaged that this work could find applications in the modelling of foams with thin membranes [22, 7, 17, 48, 30] for noise mitigation. 3

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The paper is organised as follows. The macroscopic description of wave propagation in permeoelastic media with films exhibiting bending effects (i.e. for media with plate-type films) is derived by homogenisation in §2. Section 3 presents the said macroscopic description for permeo-elastic media with films exhibiting membranes effects (i.e. for media with membrane-type films). The results presented in §2 and §3 are then generalised in §4 where an upscaled model for wave propagation in permeo-elastic media accounting for pore-scale bending and membrane effects is introduced. Section 5 analyses the effective properties of the model for generalised permeo-elastic media. In §6, numerical examples that evidence the key acoustical properties of generalised permeo-elastic media are discussed. The main outcomes of this work and perspectives are presented in the conclusion. Before presenting the theoretical developments, let us remark that this paper derives the relationship between the mean flow and pressure gradient ∇P in permeo-elastic media that encompass elastic effects, which are not considered in classical porous media. Hence, to avoid ambiguities and lighten the notation, the relationship will be expressed in three different forms, namely V = −K · ∇P ; H · V = −∇P ; or jωρ · V = −∇P , where V is the local velocity averaged over the pore volume (hence, it is independent of the material porosity). We will refer to K, H = K−1 , and ρ = (jωK)−1 as the conductivity (in m2 .Pa−1 .s−1 ), resistivity (in Pa.s.m−2 ), and effective or apparent density (in kg.m−3 ), respectively. Their link with the usual parameters used in poro-acoustics, such as the intrinsic viscous permeability and the dynamic visco-inertial permeability, resistivity, and tortuosity, will be given along the text.

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2

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2.1

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

Permeo-elastic media – Plates Geometry

82

Figure 1 shows sketches of the geometry of a periodic permeo-elastic medium. Its representative elementary volume (REV) Ω comprises a connected pore fluid network Ωf saturated with a Newtonian fluid, a stiff impervious solid frame Ωs with surface Γs = ∂Ωs ∩ ∂Ωf , and a highly flexible thin flat film Γ whose edges ∂Γ are clamped to the frame. The period size of the material (or pore characteristic size) is ` while the in-plane dimension of the film h is both comparable to `, i.e. h = O(`), and much larger than the film thickness t, i.e. t << h. These assumptions allow estimating the material porosity by φ = Ωf /Ω. The macroscopic characteristic length L is related to the wavelength λ through L = λ/2π and the disparity in the involved length scales provides the small parameter ε = `/L << 1.

83

2.2

74 75 76 77 78 79 80 81

84 85

Local description

We consider long-wavelength propagation of harmonic sound waves (ejωt ) in periodic permeo-elastic media with films made of an isotropic elastic material. The local description corresponds to the set of coupled equations that govern the dynamics of the fluid transport in the pore network Ωf and of the films Γ. The equations formulated in Ωf are the linearised equations of conservation of momentum [Eq. (1a)] and mass [Eq. (1b)], namely div(σ) = jωρ0 v in Ωf , ∇ · v = −jωP C in Ωf ,

86 87 88 89 90

(1a) (1b)

where the fluid stress tensor is σ = 2ηD(v) − P I, the oscillatory velocity and pressure are respectively v and P , and I is the second-rank unitary tensor. The strain rate tensor is defined through 2D(v) = gradv + gradT v. The physical parameters are the dynamic viscosity η, equilibrium density ρ0 , and fluid compressibility C. For the sake of clarity in the presentation, thermal exchanges are not included. However, it will be shown later that including them does not pose complications. The films are modelled, in this section, as Love-Kirchhoff plates [39] with negligible thickness and their behaviour is characterised by their out-of-plane 2D bending. Note that as the plates are represented by a 2D model, all the usual 3D variables are integrated over the plate thickness. Furthermore, 4

Ω

L

`

Γ

(a)

Ω Ωs N

Γs L

Γ

∂Γ Ωf

` n

(b) Figure 1: Sketches of the geometry of a permeo-elastic medium. (a) 3D macroscopic medium (left), 3D Representative Elementary Volume REV (middle), and cutaway view of the 3D REV (right). (b) 2D representations of a macroscopic medium (left) and of the REV (right). The clamped films Γ, solid frame Ωs , and pore fluid network Ωf are shown in grey, blue and white, respectively.

both the in-plane deformation of the films and the effects of gravity are neglected. Consequently, the scalar out-of-plane displacement of the films u is determined by the equations of equilibrium of transverse forces along the out-of-plane direction N (2a), moment balance (2b), and the constitutive law (2c); namely

91 92 93 94 95 96 97 98 99 100 101 102 103 104

e · T = −ρe tω 2 u − [σ · N] · N on Γ, ∇ f T = −div(M) on Γ,   e + ν∇ e · ∇uI e M = EI (1 − ν)e e(∇u)

(2a) (2b) on Γ.

(2c)

In these equations, the tilde on the differential operators indicates that these act on the plane Γ of g g T (·))/2 is its e is the in-plane gradient operator and e the films. For example, ∇ e(·) = (grad(·) + grad symmetric part. T is the out-of-plane shear stress vector integrated over the plate thickness and has the dimension of a force per unit length (i.e. N/m). M is the bending moment of the in-plane stress tensor integrated over the plate thickness and, therefore, has the dimension of a moment per unit length (i.e.Nm/m = N). E = E/(1 − ν 2 ) is the “plate” modulus of the films and depends on the Young’s modulus E and Poisson’s ratio ν. Note that visco-elastic films can be handled by allowing the modulus to be complex. The flexural rigidity (or bending stiffness) is EI, where I = t3 /12 is the moment of inertia of the plate; and ρe t is the surface density of the films. Note that the mean density of the system comprising the fluid and the films reads as % = ρ0 + ρe tΓ/Ωf . On the other hand, the films are loaded by their own inertia and the action of the fluid on their faces, as specified by the right-hand side terms of (2a). The latter consists in the jump across Γ of the normal component of the fluid stress vector σ · N, which is expressed by [σ · N] · N, where [·] represents the ’jump’ across Γ (e.g. [a] = a+ − a− , with the superscript + and − denoting the opposite faces of the film). 5

The local description is completed by the boundary conditions (3a), (3b), and (4). These respectively correspond to the no-slip condition on Γs , the continuity of fluid and film velocities on Γ, and the clamping condition of the films on the edges ∂Γ. The unit normal vectors N and n are depicted in Fig. 1.

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106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

2.3

Physical analysis

v = 0 on Γs ,

(3a)

v = jωuN on Γ,

(3b)

e · n = 0 on ∂Γ. u = 0 and ∇u

(4)

We aim at obtaining a macroscopic description that captures the contribution of the local physical phenomena to the macroscopic behaviour of permeo-elastic media. Consequently, we focus, within the framework of scale separation, on describing the case for which all the physical mechanisms interact at the local scale. In accordance with the assumption of scale separation, and as in long-wavelength sound propagation in conventional porous materials [3, 4, 28], the pressure varies at the macroscopic scale while the fluid velocity and its rate of deviatoric deformation fluctuate locally. As a consequence, one has that |∇P | = O(P/L) and |div(2ηD(v))| = O(ηv/`2 ), where, for example, v is an estimation of |v|. On the other hand, the divergence of the velocity is assumed to vary with the wavelength and is consequently estimated as |∇·v| = O(v/L). Regarding the relative order of magnitude of the terms in the momentum conservation equation (1a), we consider the situation in which the viscous and inertial terms balance the pressure gradient, i.e. O(ηv/`2 ) = O(ρ0 ωv) = O(P/L). Furthermore, the estimations of the terms of the equation of conservation of mass (1b) satisfy O(v/L) = O(ωP C). The condition of continuity of the film and fluid velocities Eq. (3b) leads to O(v) = O(ωu). Therefore, the velocity of the films varies at the local scale, which is also the case for T and M. The estimations of the terms in the equation of conservation of momentum (2a), allowing to account for visco-elasto-inertial fluid-film interaction, are thus given by [51]      ηv 
EIu ` 2 O = O ρ tω u = O = O . (5) P e `4 ` L

127

Note that due to the local variations of the deviatoric viscous stress, its jump across Γ also fluctuates at the local scale. Hence, one has that O([ηv/`]) = O(ηv/`). On the other hand, since the pressure varies macroscopically, the jump of the pressure across Γ is estimated as [51] O([P ]/`) = O(P/L). In addition, O(ρe t) = O(ρ0 `), which is consistent with the estimates above.

128

2.4

124 125 126

129 130 131 132 133 134 135 136 137 138 139 140 141 142

Homogenisation procedure

The scale separation between the pore characteristic size and the wavelength permits the use of the two-scale asymptotic method of homogenisation [4] to derive the equivalent macroscopic description of acoustic wave propagation in permeo-elastic media. By taking L as the reference length, the following independent space variables x and y = ε−1 x are introduced. These are respectively associated with fluctuations at the macroscopic and local scales. It then follows that i) the unknown variables depend, a priori, on the two space variables, i.e. f = f (x, y) (which are from now on unbolded to lighten the notation), and ii) the usual gradient operator ∇ becomes ∇ = ∇x + ε−1 ∇y . To reflect the physics of the phenomena, the use of two space variables is combined with a rescaling of the usual equations based upon a single space variable. Such a rescaling procedure constrains the magnitude of the (simple or multiple) gradient of a quantity Q to be consistent with its physical estimate. Indeed, the actual physical gradient of a quantity Q(x, y) that varies at the large scale is of the order of ∇x Q, and that is properly expressed by ∇x Q + ε−1 ∇y Q. But when the quantity varies at the local scale, the actual physical gradient is of the order of ∇y Q and has to be expressed as ε(∇x Q + ε−1 ∇y Q), which introduces the rescaling by the scale ratio ε. For instance, div(D(v)) must be rewritten as ε2 div(D(v)) 6

148

to express that the fluid velocity actually varies at the pore scale. The rescaled local description is presented in Section 2.4.1. The next step in the homogenisation procedure consists of expressing the physical in the form of asymptotic expansions in powers of the small parameter ε as P variables i f (i) (x, y) where f = v, p, u. Inserting these series into the rescaled equations and ε f (x, y) = ∞ i=0 identifying the terms with like powers lead to the boundary-value problem discussed in §2.4.2 as well as the macroscopic description presented in §2.4.3.

149

2.4.1

143 144 145 146 147

150 151 152 153

Rescaled local description

Using the estimates identified in the physical analysis and noting that i) the velocity of the elastic films is v = jωu; and ii) the gradient operator is formulated using the introduced spatial variables, i.e. ∇ = ∇x + ε−1 ∇y ; Eqs. (1) and (2) and boundary conditions (3) and (4) are rewritten in rescaled form [51] as follows ε2 div(2ηD(v)) − ∇P = jωρ0 v in Ωf ,

(6a)

∇ · v = −jωP C in Ωf ,

(6b)

v = 0 on Γs ,

(6c)

v = vN on Γ, e ε∇ · T = jωρe tv − [(2ηεD(v) − ε−1 pI) · N] · N on Γ, f T = −εdiv(M) on Γ,   EI 2 e e · ∇vI e M = ε jω (1 − ν)e e(∇v) + ν ∇ on Γ,

(6d)

e · n = 0 on ∂Γ. v = 0 and ε∇v

(6e) (6f) (6g) (6h)

159

The ε2 scaling in Eq. (6a) allows one to account for viscosity effects while the mass balance (6b) remains unscaled to account for the macroscopic compressibility of the effective saturating fluid. On the other hand, the physical analysis in §2.3 revealed that the jump in pressure is an order smaller than the pressure and this leads to the ε−1 scaling of the pressure jump in Eq. (6e). The rescaling of the other terms in Eqs. (6e)–(6h) is because the fluid and film velocities, M, and T fluctuate locally. Therefore, the gradient operator acting on them is rescaled by ε.

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2.4.2

154 155 156 157 158

161 162 163 164

Fluid-film interaction cell problem

Identifying at ε−1 , it follows from the equations of conservation of momentum that ∇y P (0) = 0 in Ωf and [P (0) ] = 0 on Γ. This means that the pressure is a macroscopic variable, i.e. P (0) = P (0) (x). Further identification leads to the following fluid-film interaction cell problem (with σ (1) = 2ηDy (v(0) ) − P (1) I) divy (σ (1) ) = jωρ0 v(0) + ∇x P (0) ∇y ·

v(0)

v(0)

in Ωf ,

= 0 in Ωf ,

= 0 on Γs ,

= on Γ, e y · T(0) = +jωρe tv (0) − [σ (1) · N] · N on Γ, ∇ f y (M(0) ) on Γ, T(0) = −div   e y v (0) ) + ν ∇ ey · ∇ e y v (0) I M(0) = EI (1 − ν)e e ( ∇ on Γ, y jω v(0)

165 166 167

v (0) N

e y v (0) · n = 0 on ∂Γ. v (0) = 0 and ∇

(7a) (7b) (7c) (7d) (7e) (7f) (7g) (7h)

This is a linear problem forced by the macroscopic pressure gradient ∇x P (0) . The unknowns of interest are the velocities of the fluid v(0) and of the films v (0) . The problem is solved through a weak formulation, expressed for convenience in terms of v(0) and v (0) , as follows.

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168 169 170 171 172 173 174 175

Consider the Hilbert space W of complex Ω − periodic velocity fields w defined in Ωf ∪ Γ that fulfil the kinematic restrictions, namely w is divergence free ∇y · w = 0 in Ωf , w = 0 on Γs , w = wN on Γ, e y w · n = 0 on ∂Γ. and w = 0 and ∇ The weak formulation of (7) is derived by multiplying the equation of conservation of momentum in the fluid (7a) by w (i.e. the conjugate of w ∈ W) and integrating it over Ωf ; multiplying the momentum balance of the film (7e) by w and integrating it over Γ; and using the divergence theorem, boundary conditions on Γs and Γ, and periodicity. This leads to (see [51] for the full derivation and appendix A for a similar derivation in the case of combined bending and membrane effects): af (v(0) , w) = −h(σ (1) · N) · wiΓ − ∇x P (0) · hwi,

∀w ∈ W, 176

177

ap (v

(0)

, w) = h(σ

· N) · wNiΓ ,

where h·i and h·iΓ stand for the following operators in Ωf and Γ Z Z 1 1 h·i = · dΩ ; h·iΓ = · dΓ, Ωf Ωf Ωf Γ

ap (v(0) , w) =

ηV(v(0) , w) + jωρ0 If (v(0) , w), EI Ep (v (0) , w) + jωρe tI(v (0) , w), jω

If (v

180

(8b)

(9)

(10a) (10b)

with V(v(0) , w) = h2Dy (v(0) ) : Dy (w)i,

179

(8a)

and af (v(0) , w) =

178

(1)

and

(0)

, w) = hv

I(v

(0)

Ep (v

(0)

(0)

(0)

· wi,

(11a) (11b) (11c)

, w) = hv wiΓ , e (v (0) , w)iΓ , , w) = hN

(11d)

e (v (0) , w) = (1 − ν)e e y v (0) ) : e e y w) + ν ∇ ey · ∇ e y v (0) ∇ ey · ∇ e y w. N ey (∇ ey (∇

(12)

Adding Eqs. (8a)–(8b) and recalling that w = wN on Γ, yields ∀w ∈ W,

af p (v(0) , w) = af (v(0) , w) + ap (v(0) , w) = −∇x P (0) · hwi.

(13)

The form af p in Eq. (13) is sesquilinear and coercive on W, while the right-hand-side term is semilinear. Hence, the Lax–Milgram theorem ensures the existence and uniqueness of the solution v(0) in Ωf and, by continuity, that of v (0) on Γ. Since Eq. (13) is linear and the system is forced by ∇x P (0) , the solution v(0) can be written as a linear combination of the three particular Ω − periodic fields k J corresponding to unit pressure gradient ∇x P (0) = eJ , where the eJ (with J = 1, 2, 3) are the unitary (0) basis vectors, i.e. vi = −kJi · (∇x P (0) )J . This can be rewritten as shown in Eq. (14a) where the tensor k is defined by k = k J ⊗ eJ . Note that the velocity of the films v (0) N is straightforwardly given by Eq. (14b).

v 181 182 183 184 185

v(0) = −k(y, ω, Pf , Pp ) · ∇x P (0)

(0)

N=v

(0)

on Γ.

in Ωf ,

(14a) (14b)

It is worth emphasising that the tensor k(y, ω, Pf , Pp ) includes visco-elasto-inertial effects and depends on the local variable y, the angular frequency ω, the physical properties of the saturating fluid and pore space geometry Pf = {η, ρ0 , Ωf }, and the mechanical and geometrical properties of the elastic films Pp = {EI, ρe t, Γ}. This is in contrast with conventional porous media for which the tensor k depends solely on y, ω, and Pf . 8

186

2.4.3

Macroscopic description

187

Applying the operator h·i to the leading-order mass balance equation leads to ∇x · hv(0) i + jωP (0) C = −h∇y · v(1) i,

188 189

where the term h∇y · v(1) i is null due to v(1) = 0 on Γs , periodicity, and the continuity of the fluid velocity across the interface Γ. The macroscopic description is then given by ∇x · hv(0) i = −jωP (0) C,

(16a)

hv(0) i = −K(ω, Pp , Pf ) · ∇x P (0) ,

190 191 192 193 194 195 196 197 198 199 200 201 202

2

(0)

−ω ρ(ω, Pp , Pf ) · hu 204 205

i = −∇x P

(0)

,

207

(17a) (17b)

where hu(0) i = hv(0) i/jω is the mean fluid displacement. The effective dynamic flow resistivity and apparent dynamic density are related to the visco-elasto-inertial conductivity via jωρ = H = K−1 . Eliminating the velocity in (16) yields the following acoustic wave equation at the leading-order ∇x · (K(ω, Pp , Pf ) · ∇x P (0) ) = jωP (0) C,

206

(16b)

where the effective visco-elasto-inertial conductivity tensor is calculated as K(ω, Pp , Pf ) = hk(y, ω, Pp , Pf )i. Eq. (16a) corresponds to the classical macroscopic mass balance in rigid-frame porous media since the solid frame and the films in bending are incompressible and the thermal exchanges have been neglected for the sake of clarity in the presentation. Accounting for heat exchanges leads to replace [51] the adiabatic compressibility C by the effective compressibility C(ω), which can be calculated from the solution of an oscillatory heat conduction problem formulated in the pore space with boundary conditions of zero temperature variation on Γs ∪ Γ (see, e.g. [28, 2, 4]). Note that the latter condition on Γ may only be reconsidered for films of very small thickness enabling heat transfer across them. In contrast, the fluid flow constitutive law (16b), despite its classical appearance, does not correspond to the usual dynamic Darcy’s law [3] since the elasticity and inertia of the films affect the effective conductivity tensor K(ω, Pp , Pf ). It will be shown later that this brings atypical fluid flow regimes. It is convenient to rewrite the fluid flow constitutive law (16b) in physically equivalent forms expressed in terms of the effective dynamic flow resistivity H(ω, Pp , Pf ) or the apparent dynamic density ρ(ω, Pp , Pf ) tensors, i.e. H(ω, Pp , Pf ) · hv(0) i = −∇x P (0) ,

203

(15)

(18)

from which ones deduces the effective speed of sound tensor C(ω) in permeo-elastic media that is directly linked to the effective conductivity, i.e. r q −1 jωK(ω, Pp , Pf ) C(ω) = = Cρ(ω, Pp , Pf ) . (19) C

208

Hence, the features of the waves are closely related to those of the conductivity or the apparent density.

209

3

210 211 212 213 214

Permeo-elastic media – Membranes

As shown in the previous section, the loading in plate-type films is carried by bending stresses. In films that support membrane effects, the said loading is balanced by the in-plane pre-stresses in the deformed configuration. Considering that the films i) are isotropically and uniformly pre-stressed by σ T = σT I = (T /t)I, where T is the uniform tension per unit thickness; ii) possess negligible bending stiffness; and iii) are loaded by their own inertia and the stresses exerted by the fluid on their faces; 9

215 216 217

the out-of-plane displacement of the films u is governed by the non-homogeneous membrane equation (20a). The local description is then formulated by replacing Eqs. (2a), (2b) and (2c) by (20a) and the boundary conditions Eq. (4) by Eq. (20b). e · (T ∇u) e = −ω 2 ρe tu − [σ · N] · N on Γ, ∇ u = 0 on ∂Γ.

218 219 220 221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236

It follows that the fluid-film interaction cell problem is given by Eqs. (7a)–(7d), Eq. (21) which replaces Eqs. (7e)–(7g), and v (0) = 0 on ∂Γ which replaces Eq. (7h). To solve the resulting cell problem through a weak formulation, we consider the Hilbert space W of complex Ω − periodic velocity fields w defined in Ωf ∪ Γ that fulfil the kinematic restrictions, namely w is divergence free ∇y · w = 0 in Ωf , w = 0 on Γs , w = wN on Γ, and w = 0 on ∂Γ. The weak formulation is obtained as outlined in §2.4.2 (see also appendix A) and reads as af m (v(0) , w) = af (v(0) , w) + am (v(0) , w) = −∇x P (0) · hwi,

(22)

where am (v(0) , w) =

238

(20b)

Since the differential operator in Eq. (20a) is of second order, the condition on the membrane edges reduces to a vanishing motion, as opposed to the fourth-order operator in the bending case that requires an additional condition of vanishing normal gradient. Note that the introduction of a pre-stress in the fluid-film system still leads to a linear local description. Furthermore, it can be considered that T is analogous to an elastic modulus. Hence, in what follows we will denominate the tension (or membrane) pseudo-elastic effects and the bending elastic effects with the common terminology of elastic effects to facilitate the reading. The physical analysis of permeo-elastic media with membrane-type films is similar to that of the said media with plate-type films; with the difference being that O(Ep Iu/`4 ) is replaced by O(T u/`2 ) e y · T replaced in Eq. (5). The rescaled membrane equation is then given by Eq. (6e) with the term ε∇ 2 e · (T ∇u). e by ε ∇ As in the plate-type films case, this leads to a macroscopic pressure P (0) = P (0) (x). Then, the leading-order equation of conservation of momentum of the membrane reads as   ey · T ∇ e y u(0) = −ω 2 ρe tu(0) − [σ (1) · N] · N on Γ. (21) ∇

∀w ∈ W, 237

(20a)

with

T (0) jω Em (v , w)

+ jωρe tI(v (0) , w),

e y v (0) · ∇ e y wiΓ . Em (v (0) , w) = h∇

(23) (24)

247

Following the same arguments as in §2.4.2, the solution of the linear fluid-film interaction cell problem Eq. (22) can be written as Eq. (14). The macroscopic acoustic description and the effective speed of sound are then respectively given by Eqs. (16) and (19), with the visco-elasto-inertial conductivity tensor being replaced by K(ω, Pf , Pm ) = hk(y, ω, Pf , Pm )i, which highlights that K depends on the membrane parameters Pm = {T , ρe t, Γ}. In summary, the mathematical form of the macroscopic description is preserved, while the physics differs. Indeed, the elastic energy in membrane-type films is determined by a simple derivative, while in plate-type films it results from the second order derivatives (or curvature) as it can be appreciated by comparing the terms Em (v(0) , v(0) ) and Ep (v(0) , v(0) ) in Eqs. (24) and (11d).

248

4

239 240 241 242 243 244 245 246

249 250 251

Bending and membrane effects in permeo-elastic media

With reference to Eqs. (2a) and (20a), the case where bending and membrane effects contribute to e are of the same order of magnitude, i.e. O(|T|) = the momentum balance occurs when T and T ∇u e O(|T ∇u|). With the aid of the estimates of these terms (see §2.3), this condition can be expressed 10

252 253 254 255 256 257 258 259

260 261

as O(T ) = O(Ep I/h2 ). In this situation, for a given deformation, the in-plane (or membrane) and bending stresses cumulate and act in parallel. The out-of-plane displacement of the films u is then e being added to T. The physical analysis of the local description governed by Eq. (2a) with T ∇u closely follows that discussed in §2.3. In particular, the term T u/` is of the same order as the terms e being added to T. in Eq. (5). The rescaled momentum balance is then given by Eq. (7e) with T ε∇u The homogenisation procedure leads again to a macroscopic field of pressure P (0) = P (0) (x) and yields the fluid-film interaction cell problem given by Eqs. (7a)–(7d) with the following equation formulated on Γ e y · (T(0) + T ∇ e y u(0) ) = −ω 2 ρe tu(0) − [σ (1) · N] · N, ∇ (25) together with equations (7f)–(7g), and boundary conditions (7h). The weak formulation of this cell problem, derived in Appendix A, then reads as a(v(0) , w) = −∇x P (0) · hwi,

∀w ∈ W, 262

with a(v(0) , w) = ηV(v(0) , w) + jω%I(v(0) , w) +

263

(26)

K E(v (0) , w) = af (v(0) , w) + apm (v(0) , w), jω

(27)

where the elastic parameter K that cumulates the bending and membrane effects is defined as    EI/Γ + T t t2 K= =O E + σT , Ωf Ωf 12Γ

and we use the notations

E(v (0) , w) =

T EI Ep (v (0) , w) + Em (v (0) , w), K K

I(v(0) , w) = so that apm (v(0) , w) =

ρ0 ρe t (0) If (v(0) , w) + I(v , w), % % K E(v (0) , w) + jωρe tI(v (0) , w). jω

272

Following the same arguments as in §2.4.2, the solution of the linear fluid-film interaction cell problem Eq. (26) can be written as Eq. (14). The macroscopic description is given by Eqs. (16)–(19) in which the visco-elasto-inertial conductivity reads now as K(ω, Pf , Ppm ) = hk(y, ω, Pf , Ppm )i, where Ppm stands for the parameters of the flexible, tensioned films, i.e. Ppm = {EI, T , ρe t, Γ}. Note that the effective model degenerates into the effective models introduced in the previous sections for plate-type and membrane-type films when, respectively, membrane or bending effects are negligible. In the three cases, the mathematical form of the macroscopic description is preserved, while the physics differs. In permeo-elastic media with films exhibiting bending and membrane effects, the effective elasticity of the system is consequently determined by both of them.

273

5

264 265 266 267 268 269 270 271

274 275 276 277 278 279 280

Analysis of the effective properties of permeo-elastic media

The analysis of the effective properties of permeo-elastic media is presented in this section. It is clear from the previous developments that the most general case is that of permeo-elastic media with films exhibiting bending and membrane effects. Hence, the analysis is performed for this case, noting that the same formalism holds for permeo-elastic media with plate-type or membrane-type films. In addition, for simplicity, we will focus on materials exhibiting macroscopic isotropy, i.e. K = KI, H = HI, ρ = ρI. Note that in the anisotropic case the same analysis applies in each of the principal directions of the effective tensors.

11

281 282 283 284 285 286 287

5.1

Expression of the conductivity and resistivity

Since we focus on the leading-order description, from now on we drop without ambiguity (unless otherwise specified) the exponent (0) to alleviate the notations, i.e. v(0) , v (0) , P (0) become v, v, P . Using the weak formulation and taking as test field the solution itself, i.e. w = v, yield a(v, v) = −∇x P · hvi, and recalling that for the isotropic case hvi = −K∇x P and Hhvi = −∇x P one obtains a direct relationship between the flow conductivity K, flow resistivity H, and the frequency-dependent local flow in the period. These read as (with v∗ = v/|hvi|) K=

288 289

a(v, v) = a(k, k) = H−1 |∇x P |2

;

H=

a(v, v) = a(v∗ , v∗ ) = K−1 . |hvi|2

(28)

This means that K and H are determined by flows corresponding to a unit pressure gradient (i.e. |∇x P | = 1) or a unit mean velocity (i.e. |hvi| = 1), respectively. Moreover, the expression a(v, v) =

2ηh|Dy (v)|2 i


+ jω ρ0 h|v|2 i + ρe th|v|2 iΓ   1   e y v)|2 iΓ + νh|∆ e y v|2 iΓ + T h|∇ e y v|2 iΓ , + EI (1 − ν)h|e ey (∇ jω

(29)

299

shows that the effective parameters reflect the dissipated viscous, kinetic and elastic (or pseudo-elastic) energies developed by the flows. Note that, for each one of these contributions, the integrand, i.e. the terms in brackets, takes real and positive values. To shed light on the specificities of the dynamic mass transfer in permeo-elastic media, we first consider the two limiting situations of non-deformable and infinitely deformable films and investigate the induced changes in the flow behaviour in the cases of quasi-rigid or very deformable films, respectively. In both circumstances, it is of interest to explore the different combinations of mechanisms (e.g. visco-elastic, visco-inertial, elasto-inertial, etc.) that determine the fluid-film system behaviour. The results are summarised in Table 1. This is followed by an analysis of the effective properties of permeo-elastic media in more general cases.

300

5.2

301

5.2.1

290 291 292 293 294 295 296 297 298

Rigid or quasi-rigid films Rigid films

The “rigid” limiting case, denoted by the subscript r, corresponds to films with infinite bending stiffness and/or infinitely tensioned, i.e. K → ∞. As the “elastic” parameters of E(v, w) in a(v, w) (see Eq. (27)) are infinite, the condition of finite energy imposes that the films are motionless, i.e. vr = 0. Thus, there is no influence of the inertia of the films on the fluid flow. In turn, the fluid flow is governed by an oscillatory Stokes problem formulated in the pore geometry with perfectly rigid films. Hence, the flow in the pores vr (ω) is of visco-inertial type and vr (ω) = −kr (y, ω, Pf , Ppm ) · ∇x P (0) . The latter leads to the usual dynamic Darcy’s law, given for the case of isotropic materials by hvr i = −Kr ∇x P or Hr hvr i = −∇x P , where the frequency-dependent conductivity Kr (ω) and resistivity Hr (ω) read as Kr =

2ηh|Dy (vr )|2 i − jωρ0 h|vr |2 i a(vr , vr ) = |∇x P |2 |∇x P |2

;

Hr = K−1 r .

As in conventional porous media, the effective mass transfer parameter can be characterised by means of the low frequency conductivity Kr0 and the high frequency apparent density ρr∞ (which accounts for the inertia of the fluid only). These are given by 2ηh|Dy (vr )|2 i ω→0 |∇x P |2

Kr0 = lim 302 303 304

;

Hr (ω) h|vr |2 i = lim ρ0 . ω→∞ ω→∞ jω |hvr i|2

ρr∞ = lim

As usual in poro-acoustics, a characteristic frequency that determines the transition between viscosity- and inertia-dominated fluid flow can be defined. Equating the low-frequency approximation of the viscous dissipated power and the high-frequency approximation of the kinetic power, this 12

305

characteristic frequency for a material with rigid films is estimated as ωvr =

306 307 308 309 310 311

312 313 314 315 316 317 318 319 320

322 323 324 325

5.2.2

Quasi-rigid films

Let us now focus on films with high bending stiffness and/or under large tension so that the elastic term KE(v, w)/jω dominates over the viscous and inertial terms. Consequently, the local fields are slightly modified in comparison with the local fields of a medium with non-deformable films. Hence, following e + . . . and v = jω [51], the local fields can be looked for in the expanded forms v = vr + jω e + .... Note Kv Kv jω e|  |vr | and | jω that for the latter vr = 0 and the corrector fields satisfy | K v v e |  |v |. The difference r K of order of magnitude between the fluid and films motions lead to solve the variational problem Eq. (26) sequentially in the fluid domain and in the films by splitting a into af and apm . As in Eq. (8), and considering that σ v stands for σ (1) , we have af (v, w) = −h[σ v · N] · wiΓ − ∇x P · hwi, K apm (v, w) = E(v, w) + jωρe tI(v, w) = h[σ v · N] · wNiΓ . jω

af (vr , w) = −∇x P · hwi,

328

329

331 332 333

334 335

(33)

(34)

The corrector field in the films jωe v /K is determined by the variational formulation Eq. (32) in which apm reduces to the dominant elastic term and σ v = σ vr . This results in the following formulation ∀w ∈ W,

330

(32)

and fully determines vr in the fluid domain Ωf . Furthermore, as for non-deformable films, one recovers af (vr , vr ) = −∇x P · hvr i.

327

(31)

Note that, without restricting the generality of the reasoning, it will be considered that ∇x P is real valued (by linearity a phase shift on ∇x P would be directly transferred to v). Equation (31) provides a variational equation for the leading-order field vr . However, since vr = 0, the space W is reduced to the subspace Wr whose fields satisfy the additional kinematic constrain w = wN = 0 on Γ. Hence, the variational formulation at the leading-order reads as ∀w ∈ Wr ,

326

(30)

where Kr0 = ηKr0 and αr∞ = ρr∞ /ρ0 denote respectively the intrinsic permeability (in m2 ) and the high-frequency tortuosity with non-deformable films. Note that the low frequency apparent density [46], i.e. ρr0 = limω→0 (Hr (ω) − Hr0 )/jω, will be used later. Let us also recall that, separating Hr into its real and imaginary parts, i.e. Hr = Hr< + jHr= , it can be proven (cf. [10]) that Hr< and Hr= /ω r< are respectively monotonically decreasing and increasing functions of frequency, i.e. dH dω ≤ 0 and d(Hr= /ω) ≥ 0. dω

∀w ∈ W,

321

1 η Hr0 = = , ρr∞ ρr∞ Kr0 ρ0 αr∞ Kr0

E(e v , w) = h[σ vr · N] · wNiΓ ,

(35)

which expresses that the elastic deformation of the films is forced by the jump of the differences of stress on their opposite faces [σ vr · N] due to the leading-order fluid flow vr under the macroscopic pressure gradient ∇x P . Thus, ve is also linearly dependent on ∇x P . Recalling that [σ vr · N] = O(∇x P `), b related to each component Ωf = O(Γ`) and introducing the dimensionless vector of specific motions u of ∇x P , one has that b (y, ω) · ∇x P. ve = u (36)

On the other hand, the corrector jωe v/K of the fluid flow satisfies the residual parts of the variational equation (31), i.e.   jω e, w = −h[σ jω ve · N] · wiΓ . ∀w ∈ W, af v (37) K K 13

Consequently, taking w = vr (that is null on Γ), one deduces that af (e v, vr ) = 0. Moreover, taking v/K (which is not null on Γ) in the variational equation (31), yields w = jωe

336

337

338

e) = −h[σ vr · N] · v eiΓ − ∇x P · he vi, af (vr , v

e) = af (e and since af (vr , v v, vr ), one derives, using Eq. (35), the following expression

339

(38)

b · ∇x P ). vi = E(b u · ∇x P, u −∇x P · he

(39)

Thus, recalling that ∇x P is real valued and accounting for Eq. (36) one obtains Finally, adding the equations (34) and (39) multiplied by jω/K yields af (vr , vr ) −

340

eiΓ = E(e v , ve). −∇x P · he vi = h[σ vr · N] · v

jω jω b · ∇x P ) = −∇x P · hvr + v ei = K?r (ω)|∇x P |2 . u · ∇x P, u E(b K K

Within the assumption of macroscopic isotropy, it is convenient to introduce the complex frequency dependent dimensionless parameter χr (ω) defined by χr (ω) =

341 342

343 344 345 346

b · ∇x P ) E(b u · ∇x P, u . |∇x P |2

347

(41)

Thus, one obtains the following assessments of the corrected conductivity K?r (ω) and resistivity H?r (ω) for weakly deformable films   jωχr jωχr K?r (ω) ≈ Kr + = Kr (ω) 1 + , (42) K KKr     χr Hr 1 jωχr −1 ? = Hr (ω) 1 − jω . (43) Hr (ω) ≈ + Hr K K Eq. (42)–(43) show how the elasticity of weakly deformable films modifies the conductivity and the resistivity. A classical dynamic Darcy’s law is retrieved when either K → ∞, i.e. for films with very large bending stiffness and/or under extremely large tension, or at sufficiently low frequencies ensuring a weak magnitude of the corrector. This condition is formulated implicitly as ω

348

(40)

K KKr = . Hr |χr | |χr |

(44)

Formulae (42), (43), and (44) are specified below by considering the different flow regimes of the rigid film case and taking advantage of the limiting expressions of Kr , Hr , and χr .

349

354

Remarks: - The velocity corrector jωe v/K in Ωf could be determined by the residual part of the variational equation (31) formulated in a subspace Wre of W whose fields satisfy the kinematic constraint w = jωe v N/K on Γ. - In anisotropic media, χr becomes a second-rank tensor.

355

5.2.2.1

350 351 352 353

Quasi-rigid films – Visco-elastic regime

356

In the flow regime dominated by viscosity, which occurs in the frequency range ω  ωvr , the inertial effects can be neglected. One therefore has that Kr (ω) ∼ = Kr0 = Kr0 /η and χr (ω) ∼ = χr0 = O(1). Consequently, the condition (44) reads explicitly as ω  ωer , where ωer is a visco-elastic characteristic frequency of the fluid-stiff-film system estimated as   KKr0 K K = =O Kr0 . ωer = Hr0 χr0 ηχr0 η 14

Thus, in the regime where ω  ωer and ω  ωvr , one can re-express K?r (ω) as     Kr0 jωχr0 Kr0 1 jωχr0 −1 jω −1 ? Kr (ω) ≈ + = ; µr (ω) = + =η 1+ . η K µr (ω) η KKr0 ωer 357 358 359 360 361 362 363 364

This indicates that the fluid flow in this regime is characterised by the static permeability Kr0 and a saturating fluid that behaves [51] as an effective visco-elastic Maxwell fluid [9, 38] of parameter µr (ω), corresponding to a dashpot η connected in series with an equivalent stiff spring KKr0 /χr0 .
−1 −1 Alternatively, from (43), the effective flow resistivity H?r (ω) ≈ Hr0 + jωχr0 /K shows that the fluid-stiff-film system behaves as a resistivity Hr0 = η/Kr0 connected in series with a stiff spring K/χr0 . The combination in series is consistent with the fact that the film is deformed in response to the stress induced by the fluid. In this regime, the apparent dynamic density can be assessed by   1 η χr0 η 2 ? ∼ ρr (ω) = − , (45) jω Kr0 K Kr0

367

which shows that its imaginary part, associated with the effects of viscosity, is conventional and its real part, associated with the elasticity of the system and determined by bending and membrane effects, is negative.

368

5.2.2.2

365 366

Quasi-rigid films – Visco-elasto-inertial regime

369 370 371 372 373 374 375

376 377 378

As the frequency increases, i.e. ω < ωvr , together with ω  ωer , the low-frequency inertia of the fluid also contributes to the behaviour and the fluid flow is of visco-elasto-inertial nature. In such a case, the resistivity involves a weak inertia effect and becomes classically Hr ≈ η/Kr0 + jωρr0 . Reporting this expression in (43), shows that the inertia correction cumulates with the elastic one. This evidences that the fluid-film system behaves as a parallel (Hr0 –resistivity, ρr0 –mass) system connected in series to a stiff spring K/χr0 . The apparent dynamic density can then be estimated by   1 η η 2 χr0 ? ∼ ρr (ω) = + ρr0 − , (46) jω Kr0 K Kr0 which shows that its imaginary part is conventional and its real part can take negative or positive values depending on whether the low-frequency inertia of the fluid or the bending and/or membrane effects in the films dominate. 5.2.2.3 Quasi-rigid films – Elasto-inertial regime In the flow regime where the effects of viscosity are negligible, i.e. ω  ωvr , the flow resistivity becomes Hr (ω) ∼ = jωρr∞ and χr (ω) ∼ = χr∞ . Note that, in absence of viscosity, vr∞ takes a purely b ∞ , solution of Eq. (35), is real imaginary value, while σ vr∞ reduces to p(1) I and is real valued. Then u and therefore χr∞ is real valued, positive and of order 1. It follows that the condition (44) takes the 2 where the elasto-inertial characteristic frequency of the global fluid-stiff-film explicit form ω 2  ωgr system, ωgr , is defined by   K K 2 =O . ωgr = ρr∞ χr∞ ρr∞

379 380

Consequently, in the frequency range ωvr  ω  ωgr , the flow resistivity and the dynamic apparent density read as    −1 1 jωχr∞ −1 ρr∞ ω2 H?r (ω) ≈ + ≈ jωρr∞ 1 − 2 ; ρ?r (ω) ∼ (47) = 2 . jωρr∞ K ωgr 1 − ωω2 gr

This shows that the fluid-stiff-film system can be represented as a spring K/χr∞ connected in series with the high-frequency effective mass ρr∞ of the fluid saturating the pores with non-deformable 15

Table 1: Effective flow resistivity of permeo-elastic media for different flow regimes for the cases of quasi-rigid and highly deformable films (see the main text for more details). Quasi-rigid χr

films

films

Hr

H?r (ω) = jωρ?r (ω)

ω < ωvr =

q

Hr (ω) ∼ =

K ρr∞ χr∞

η Kr0

ω < ωvs =

+ jωρr0

χr (ω) ∼ = χr0

K Kr0 η χr0

ω > ωvr ω < ωgr =

H?s (ω) = jωρ?s (ω)

η Kr0 ρr∞

ω < ωer =

χs

Highly-deformable

η Ks0 ρs∞

ω < ωes = Kη χs0 Ks0

q   Hr (ω) ∼ = jωρr∞ 1 + M2r ωjωvr χr (ω) ∼ = χr∞

ω > ωvs ω > ωgs =

q

K|χs∞ | ρs∞

Hs

Hs (ω) ∼ =

η Ks0

+ jωρs0

χs (ω) ∼ = χs0 q   Hs (ω) ∼ = jωρs∞ 1 + M2s ωjωvs χs (ω) ∼ = χs∞

films. The effect of elasticity tends to reduce the apparent dynamic density of the fluid. Remark: The weak viscous dissipation can also be considered in the frequency range ωvr < ω  ωgr . Indeed, the inertial resistivity corrected by the viscous skin effect is classically given by   p Hr (ω) ∼ M ω /2jω , where Mr is the form factor of the network of pores with jωρ 1 + = r vr r∞ rigid films. Inserting this expression in (43) shows that the viscous and elastic correction add up and gives the following estimate  −1 s !−1 2 1 jωχ M ω ω r∞ r vr ?  ≈ jωρr∞ 1 − q +  Hr (ω) ≈  − 2 . K 2 jω ωgr jωρ 1 + Mr ωvr r∞

381

5.3

2

jω

Infinitely or highly deformable films

383

Following a similar approach as in the previous section, the behaviour of the fluid-film system with infinitely or highly deformable films is now investigated.

384

5.3.1

382

385 386 387 388 389 390 391 392

393 394

Infinitely deformable films

This “soft” limiting case, denoted in what follows by the subscript s, corresponds to films with zero bending stiffness and under no tension. In this case, the “elastic” term KE(v, w)/jω of the bi-linear form a(v, w) vanishes (see Eq. (27)). Then, a(v, w) = af (v, w) and the variational formulation becomes that of a material where the films are “almost” absent. Indeed, on Γ, the normality of the flow is imposed and the film mass is accounted for by the surface density ρe t. Such a physics at the pore scale corresponds to a local visco-inertial flow vs (ω) = −ks (y, ω, Pf , Ppm ) · ∇x P , that is obviously distinct from vr (ω) and leads to a dynamic Darcy’ law hvs i = −Ks ∇x P or Hs hvs i = −∇x P , where the frequency-dependent resistivity Hs (ω) and conductivity Ks (ω) read as
2ηh|Dy (vs )|2 i + jω ρ0 h|vs |2 i + ρe th|vs |2 iΓ ; Ks = H−1 (48) Hs = s . |hvs i|2 As previously, the mass transfer effective parameters are characterised by the low frequency conductivity Ks0 and the high-frequency apparent density ρs∞ for which both the fluid and films contribute, 16

395

namely 2ηh|Dy (vs )|2 i ω→0 |∇x P |2

Ks0 = lim 396 397

398 399 400 401 402 403

404 405 406 407 408 409 410

;

5.3.2

Highly deformable films

Consider now the case of highly deformable films with non zero but weak bending stiffness and tension. Then, the “elastic” term KE(v, w)/jω is much smaller than the viscous and inertial ones, so that the local flow field and film motions are slightly perturbed from vs and vs . Consequently, following [51], K 0 K 0 one can expand the local variables in the form, e.g. v = vs + jω v + . . . , where | jω v |  |vs | and K | jω vs |  |vs |. Inserting these expressions in the variational formulation (26) yields the following variational formulations for the leading-order field and its first corrector ηV(vs , w) + jω%I(vs , w) = −∇x P · hwi, 0

413 414

0

ηV(v , w) + jω%I(v , w) + E(vs , w) = 0.

∀w ∈ W,

412

(49)

and the characteristic frequency ωvs that determines the transition between viscosity- and inertiadominated flow is given by η Hs0 = , (50) ωvs = ρs∞ %αs∞ Ks0 where Ks0 = ηKs0 and αs∞ = ρs∞ /% denote the intrinsic permeability and the high-frequency tortuosity, respectively. Note that the low frequency apparent density [46], i.e. ρs0 = limω→0 (Hs (ω)−Hs0 )/jω, will be used later and let us recall that, as in the classical case [10], one can separate Hs into real and imaginary parts, i.e. Hs = Hs< + jHs= , and prove that the real part Hs< and imaginary part divided by frequency Hs= /ω are, respectively, monotonically decreasing and increasing functions of frequency, d(Hs= /ω) s< i.e. dH ≥ 0. dω ≤ 0 and dω

∀w ∈ W,

411

Hs (ω) ρ0 h|vs |2 i + ρe th|vs |2 iΓ = lim , ω→∞ jω ω→∞ |hvs i|2

ρs∞ = lim

(51) (52)

As previously, we will consider, without restricting the generality of the developments, that ∇x P is real valued. The leading-order formulation is that of a medium with infinitely deformable films and yields hvs i = −Ks ∇x P . Then, taking w = v0 in Eq. (51) and w = vs in Eq. (52), one deduces that ηV(vs , v0 ) + jω%I(vs , v0 ) = −∇x P · hv0 i, 0

0

ηV(v , vs ) + jω%I(v , vs ) = −E(vs , vs ).

(53) (54)

Then, from the symmetry of the sesquilinear forms Vf and I, one obtains ∇x P · hv0 i = E(vs , vs ),

415

and introducing the complex frequency dependent dimensionless parameter χs (ω) defined by χs (ω) = one has that

416 417

E(vs , vs ) , |hvs i|2

∇x P · hv0 i = χs (ω)|hvs i|2 = χs (ω)|Ks |2 |∇x P |2 ,

(55)

from which, with the assumption of macro-isotropy (otherwise χs would be a second rank tensor) one obtains K K h v0 i = χs (ω)|Ks |2 ∇x P. jω jω Summing this corrector with the leading-order field provides the following assessments of the corrected conductivity K?s (ω) and resistivity H?s (ω) for highly deformable films   K K ? 2 Ks (ω) ≈ Ks − χs |Ks | = Ks 1 − χs Ks , (56) jω jω   K Hs K χs H?s (ω) ≈ Hs + χs = Hs 1 + . (57) jω Hs jω Hs 17

418 419 420 421 422

Eqs. (56)-(57) indicate the nature of the correction of the conductivity and the resistivity induced by the elasticity of the films. This tends to vanish and a classical dynamic Darcy’s law is retrieved when either K → 0, which occurs for films with weak bending stiffness and under small tension, or at sufficiently high frequencies ensuring a weak magnitude of the corrector. This condition can be implicitly expressed as K|χs | ω . (58) |Hs |

424

In what follows, the expressions (56), (57), and (58) are specified for the different flow regimes of the infinitely deformable film case.

425

5.3.2.1

423

Highly deformable films – Visco-elastic regime

426

In the regime where the inertial effects are negligible, i.e. ω  ωvs , Ks (ω) and χs (ω) tend to be real with Ks (ω) ∼ = Ks0 = Ks0 /η and χs (ω) ∼ = χs0 = E(vs0 , vs0 )/|hvs0 i|2 [see Eq. (55)]. Note that χs0 is positive and, by construction, χs0 = O(1). Furthermore, the implicit condition Eq. (58) takes the explicit form ω  ωes , where ωes is a visco-elastic characteristic frequency of the fluid-soft-film system defined as   K K ωes = χs0 Ks0 = O Ks0 . η η Consequently, in the regime ωes  ω  ωvs , one can rewrite Eq. (56) as K?s (ω) 427 428 429 430 431 432 433 434 435 436

Ks0 K ≈ − χs0 η jω



Ks0 η

2

Ks0 = µs (ω)

;

  Kχs0 Ks0 ωes µs (ω) = η + =η 1+ . jω jω

This shows that the flow is characterised by (i) the permeability Ks0 of the pore space including infinitely deformable films, which, as argued in [51], can be estimated by the static permeablity of the pore space without films; and (ii) a saturating effective fluid that behaves, in a rheological sense, as a visco-elastic Kelvin-Voigt fluid [9] with parameter µs (ω). The latter corresponds to a dashpot η connected in parallel to a soft spring Kχs0 Ks0 . The combination in parallel is consistent with the fact that the films and the fluid sustain the same motion. In addition, since Hs /Hs ∼ = 1 and considering Eq. (57), one obtains that H?s (ω) ∼ = Hs0 + Kχs0 /jω. Thus, an alternative physical interpretation is that the fluid-film system behaves as a resistivity Hs0 = η/Ks0 connected in parallel to a soft spring Kχs0 that accounts for bending and membrane effects and is of the order of K. In this fluid flow regime, the apparent dynamic density can be estimated by ρ?s (ω) ∼ =

η Kχs0 − 2 , jωKs0 ω

(59)

438

which shows that its real part takes negative values and is inversely proportional to the square of frequency, consistently with the elastic nature of this term.

439

5.3.2.2

437

Highly deformable films – Visco-elasto-inertial regime

440 441 442 443 444 445 446

447 448

For ωes  ω < ωvs , the low frequency inertia arises and the resistivity is therefore approximated by Hs ∼ = Hs0 − jωρs0 , where ρs0 accounts for the fluid and films contribution to the low frequency inertia. Inserting this expression in Eq. (56) shows that the inertial corrector jωρs0 cumulates with the elasticity-induced corrector Kχs0 and yields the effective flow resistivity corresponding to a parallel resistivity-mass-spring system with parameters Hs0 = η/Ks0 , ρs0 , and Kχs0 . The apparent density in this regime reads as η Kχs0 ρ?s (ω) ∼ + ρs0 − 2 . (60) = jωKs ω Consequently, the real part of the apparent dynamic density can be either positive or negative depending on whether the inertial effects dominate over the elastic effects. 18

449

5.3.2.3

Highly deformable films – Elasto-inertial regime

450

In the flow regime where the viscous effects are negligible, i.e. ω  ωvs , one has that Hs (ω) ∼ = jωρs∞ (hence Hs /Hs ∼ = −1) and χs (ω) = χs∞ , which is real and negative. These results rely on the fact that, in absence of viscosity, vs∞ is purely imaginary (i.e. vs∞ = j|vs∞ |) so that, according to Eq. (55), χs∞ = E(vs∞ , vs∞ )/|hvs∞ i|2 = −E(|vs∞ |, |vs∞ |)/|hvs∞ i|2 ≤ 0, and |χs∞ | = O(1). In addition, the 2 , where ω is a elasto-inertial characteristic implicit validity condition (58) reads explicitly as ω 2  ωgs gs frequency of the global fluid-soft-film system given by   K K 2 ωgs = |χs∞ | = O . ρs∞ ρs∞ Therefore, in the high frequency regime where ω  ωvs and ω  ωgs one obtains the following approximations of the dynamic resistivity and apparent dynamic density ! ! 2 2 ω ω Kχ gs gs s∞ H?s (ω) ≈ jωρs∞ − = jωρs∞ 1 − 2 ; ρ?s (ω) ∼ = ρs∞ 1 − 2 . jω ω ω This shows that the fluid-soft film system can be represented as a spring K|χs∞ | connected in parallel with the high-frequency effective mass ρs∞ of the fluid-film system. Remark : The weak viscous dissipation can also be considered in the regime ω > ωvs and ω  ωgs by correcting the inertial resistivity with   the viscous skin effect in a similar manner as p ∼ previously, i.e. Hs (ω) = jωρs∞ 1 + Ms ωvs /2jω , with Ms being the form factor of the network of pores without films. This expression reported in Eq. (57) yields s s ! ! 2 Ms ωvs Kχs∞ Ms ωvs ωgs ? Hs (ω) ≈ jωρs∞ 1 + − = jωρs∞ 1 + + 2 . 2 jω jω 2 jω ω 451

5.4

Synthesis and general case

456

In a general case, a condensed way of expressing the behaviour is out of reach due to the strong coupling among the three mechanisms involved. However, one can take advantage of the limiting cases with weak elastic effects to identify some general features. In particular, several characteristic frequencies have already been identified. The visco-inertial frequencies are classical in porous media, while the visco-elastic and elasto-inertial characteristic frequencies are specific to permeo-elastic media.

457

5.4.1

452 453 454 455

Visco-inertial characteristic frequency

466

The quasi-rigid and highly deformable cases lead to the visco-inertial characteristic frequencies ωvr = η/(ρr∞ Kr0 ) and ωvs = η/(ρs∞ Ks0 ). If the surface of the films spans a significant area of the pores section, the intrinsic permeability of a medium with rigid films can be significantly smaller than that of a medium with extremely soft films. On the other hand, the high frequency density of a medium with rigid films is expected to be larger than that of a medium with soft films, with a possible exception being materials with heavy films. However, the tortuosity ratio is very likely much smaller than the permeability ratio. As a consequence, one may expect that, in most of the cases, ωvr > ωvs , i.e. softer films may result in lower visco-inertial characteristic frequency of the medium, or equivalently, compared to media without films, the films may increase the visco-inertial characteristic frequency.

467

5.4.2

458 459 460 461 462 463 464 465

Visco-elastic characteristic frequency

In the flow regime of negligible inertia, two visco-elastic characteristic frequencies have been identified, namely ωer = KKr0 /ηχr0 and ωes = KKs0 χs0 /η for the quasi-rigid and highly deformable cases,

19

respectively. Since by construction O(χr0 ) = O(χs0 ) = O(1), we have   Kr0 ωer =O < 1. ωes Ks0

480

Apart from the limiting cases K → ∞ and K → 0, no explicit expression can be given for the viscoelastic frequency, which can, however, be estimated, in some cases as will be shown below, from the numerical solution of the fluid-film interaction cell problem. Despite this, the expressions of K?r and K?s show that for ω  ωer the elastic contribution to the conductivity increases linearly while for ω  ωes it decreases as ω −1 . This leads to infer that for a finite value of K, the elastic contribution will reach a maximum at a specific visco-elastic frequency ωe that should lie in between ωer and ωes . In practice, for a given permeo-elastic material, at frequencies much smaller than ωe the films would tend to be motionless and at much higher frequencies (while inertia remains negligible) the films will follow the fluid motion. ve Let us finally mention that, denoting by Hve = Hve < + jH= the complex resistivity in absence of any inertial effects (e.g. % → 0), it can be established from the variational formulation (see Appendix B) dHve d(ωHve ve < = ) that Hve ≤ 0, < and ωH= are monotonically decreasing functions of frequency, i.e. dω ≤ 0 and dω which is consistent with the above analysis.

481

5.4.3

468 469 470 471 472 473 474 475 476 477 478 479

482 483 484 485 486 487 488 489

Elasto-inertial characteristic frequencies

The elasto-inertial characteristic frequencies identified for quasi-rigid and highly deformable cases are 2 = K/ρ 2 ωgr r∞ χr∞ and ωgs = K|χs∞ |/ρs∞ , respectively. Both are of the order of O(K/%), however we can infer that they are related to distinct phenomena. Indeed, if the approximated expressions 2 )−1 and ρ? ∼ ρ 2 2 ρ?r ∼ = ρr∞ (1 − ω 2 /ωgr s = s∞ (1 − ωgs /ω ) were correct in a wider frequency range than that considered for their derivation, the apparent density would be infinite at ωgr and would vanish at ωgs . This is reminiscent of anti-resonance phenomena within the coupled fluid-film system as now detailed. Considering a finite value of K, and neglecting the viscous effects in the variational formulation (26), gives jω%I(v, v) +

490

491 492

494 495 496 497

= |hvi|2 H(ω),

and recalling that v∗ = v/|hvi| and v ∗ = v/|hvi| with |hvi| = 6 0, we have equivalently   K 1 ∗ ∗ % I(v∗ , v∗ ) − E(v , v ) = ρ(ω). % ω2

(61)

(62)

This shows that the elastic and the combined fluid and films inertial effects can balance each other at the global resonance frequency ωg defined by ωg2

493

K jω E(v, v)

∗ ∗ K E(vωg , vωg ) = =O % I(v∗ωg , v∗ωg )

  K , %

(63)

∼ %I(v∗ , v∗ )(1 − ω 2 /ω 2 ), that is and around ωg we have the approximated expression ρ(ω→ωg ) = ωg ωg g consistent with the expression of ρ?s and to a global resonance phenomena involving the fluid and the films. At the frequency ωg , the apparent density and resistivity vanish. Consequently, the conductivity K → ∞. In physical terms, the mean fluid velocity takes very large values in response to a finite pressure gradient.

498 499 500

On the other hand, the variational formulation (26) without viscous effects can also be written, after dividing by |∇P |2 and assuming macro-isotropy, as 

 % 1 . −ω 2 I(k, k) + E(k, k) K = jωK(ω) = K ρ(ω) 20

(64)

501 502

503 504 505 506 507 508 509 510

511 512 513 514 515 516

Then, the elastic and the combined fluid-film inertial effects can balance each other at a frequency ωa defined by   K E(kω∗ a , kω∗ a ) K , (65) ωa2 = = O ∗ ∗ % I(kωa , kωa ) %

∼ and around ωa , consistently with the expression of ρ?r , one has the approximated expression ρ−1 (ω→ωa ) = ∗ ∗ ∗ 2 2 KE(kωa , kωa )(1 − ω /ωa ). It then follows that at ωa , the conductivity K = hkωa i = 0 and, consequently, the apparent density ρ → ∞ and H → ∞. Physically, in response to a finite pressure gradient the mean fluid velocity tends to zero. This implies that the internal motions of the fluid-film system compensate each other, as in anti-resonance phenomena. Considering that the motions of the films vary in both cases according to their size, one expects that O(E(kω∗ a , kω∗ a )) = O(E(vω∗ g , vω∗ g )). Hence, the ratio of the two elasto-inertial characteristic frequencies is essentially related to the combined fluid-film inertial effects and can be estimated as ! I(v∗ωg , v∗ωg ) E(kω∗ a , kω∗ a ) I(v∗ωg , v∗ωg ) ωa2 ; |hv∗ωg i| = 1 hk∗ωa i = 0. (66) = =O ωg2 I(k∗ωa , k∗ωa ) E(vω∗ g , vω∗ g ) I(k∗ωa , k∗ωa ) It is worth mentioning that the presence of a pole in K(ω) and ρ(ω) drastically modifies the usual features of the dynamic conductivity (or permeability) and apparent density known in conventional porous media. Note, however, that the resonance and anti-resonance frequencies are defined by discarding the viscous dissipation. Accounting for the latter leads to damped resonance and regularises the singularities of K and ρ. On the other hand, increasing the dissipation (e.g. at lower frequencies) will mask the resonance due to overdamping.

517

523

Remarks: - We only consider here the fundamental resonance/anti-resonance of the fluid-film system. There exists also higher resonance or anti-resonance modes that would result in additional characteristic frequencies for K and ρ. - Due to the strong fluid-film coupling, the resonance frequencies cannot, in general, be precisely assessed from the in-vacuum resonance frequencies of the films [51, 21].

524

5.4.4

518 519 520 521 522

525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543

Mapping of permeo-elastic behaviour

In Figure 2 the different possible permeo-elastic behaviour are mapped onto a synthetic (X, Y ) plot drawn in logarithmic axes. Conveniently, we use the “rigid” characteristic frequencies ωer and ωvr as reference values (another choice could have been ωes and ωvs ) and we use the dimensionless coordinates X = log(ω/ωer ) and Y = log(ω/ωvr ). Now, for a given permeo-elastic (PE) medium ωer and ωvr are unambiguously determined by setting K → ∞. Therefore, whatever the frequency is, X − Y = log(ωer /ωvr ). This means that the possible behaviour of a PE medium according to the frequency can only belong to a straight line parallel to the first diagonal of the X-Y map. For instance, a PE medium with ωer  ωvr (in practice, e.g. ωer > 103 ωvr ) will behave as a classical porous media with rigid films, except at high frequencies where inner resonances may appear and provided that the long wavelength condition is still fulfilled. Conversely, a PE medium with ωes  ωvs (e.g. ωes < 10−3 ωvs ) will mostly behave as a classical porous media without films, except at rather low frequencies where elastic effect can manifest by an apparent Kelvin-Voigt fluid. Interestingly, the behaviour of a PE-medium with ωer ≈ ωvr will move from that of a classical porous media with rigid films saturated by an apparent Maxwell fluid at low frequency to a strongly coupled visco-elastoinertial behaviour at medium frequencies with possibly damped resonance to reach the high frequency behaviour of a classical porous media without films. Finally, it is expected that a PE medium with ωer significantly higher than ωvr , e.g. ωer > 10ωvr , will present, at high frequency, a clear effect of weakly damped resonance.

21

log

ω=O

ω ωvr

ωe v /ω

r

Inertial

q  K %

Resonance

r

>

>

v /ω

r

r

1

ωe

<

<

No Resonance

Highly → ∞ deformable

ωe

Viscous

Damped Resonance

ω = O(ωvs )

ω = O(ωvr )

1 Rigid ← quasi-rigid

/ω

r

vr

O(ωer ) ≤ ω ≤ O(ωvr )

=

log

ω ωer

O )

(1

Figure 2: Logarithmic map of permeo-elastic behaviour with respect to the two dimensionless frequencies ω/ωer and ω/ωvr .

544

545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569

6

Examples and further discussion

This section illustrates the specific properties of the effective conductivity or density that determines straightforwardly, see Eq.(19), the features of the effective speed of sound and wave propagation in permeo-elastic media. A permeo-elastic material with the geometry shown in Fig. 3 is considered to exemplify the theory. Note that it can be periodically repeated laterally to get a periodic material in all the three directions. 1 with rigid structures onto which the films are fixed. The The material is made of a square channel unit cell of the material has dimensions w-by-w-by-h and its rigid structure, shown in red in Fig. 3, has a height hs and width ws . The thin elastic film with thickness t has a square shape of side wc . The four thin air gaps formed in between the rigid structure and the very stiff and impervious walls of the channel connect the front and back fluid-saturated parts of the unit cell and each have a width g. Furthermore and with reference to Fig. 3b, one has w = wc + 2ws + 2g. The results presented below have been obtained by numerically solving the fluid-film interaction cell problem Eqs. (7) using the finite element software Comsol Multiphysics. Second-order Lagrangian elements were used to model the velocities while linear elements modelled the pressure. An unstructured tetrahedral mesh, refined close to the boundaries to resolve the boundary layers, was used. The mesh coincided on the periodic boundaries of the unit cell and a mesh refining analysis was conducted to ensure the convergence of the solution. In what follows, the geometrical parameters of the material will be kept constant, unless otherwise explicitly stated, while some of the mechanical parameters of the films will be varied to illustrate their influence on the effective conductivity or density of the material. The geometrical parameters are w = 20 mm, wc = 15 mm, ws = wr = 2 mm, g = 0.5 mm, h = 6 mm, and hs = 2 mm. The parameters of the films are t = 50 µm, ρe = 800 kg m−3 , ν = 0.36, and E = 2 GPa, i.e. EI = 23.93 × 10−6 N m. These realistic values are in the order of those of typical polymethylpentene films. Note that for the considered cell, the bending stiffness of the film divided by its cross-section area is EI/Γ = 0.1 N m−1 , and the mass of the film is 9 × 10−3 g while that of the gas is 2.55 × 10−3 g. 22

h hs

Channel wall y

z x

w

Elastic films Rigid structure

y

z

Air gap

wc

y wr

x

g

ws

x

(a)

(b)

Figure 3: Geometry of a fluid-saturated permeo-elastic material. (a) Square channel (blue) with a rigid structure (red) onto which the elastic films (grey) are fixed. The right-hand side image shows a section cut of the material and the black arrow indicates the direction of propagation. (b) Top: 3D unit cell and a section cut. Bottom: 2D view of the unit cell. The thin air gap formed between the rigid structure and channel wall is shown in white. (Colour online).

570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598

Figure 4 compares the effective conductivity of materials with perfectly rigid and elastic films. The parameters are as quoted above and the tension per unit thickness of the film is T = 1 N m−1 , which leads to K = 0.5218 × 106 N m−4 . It is clear that the effective conductivity of the permeo-elastic medium approaches that of a material with perfectly rigid films at low frequencies, as predicted by Eq. (42). At higher frequencies, the local resonant behaviour of the permeo-elastic medium becomes evident. The absolute value of the effective conductivity is minimum at ωa and maximum at ωg while its phase takes atypical values in between these characteristic frequencies. Let us also mention that at high frequencies, the absolute value of the conductivity of the permeo-elastic medium used here as example looks close to that of a material with perfectly rigid films. However, this is only a particular situation occurring in the considered frequency range shown in the plot. Indeed, for frequencies higher than the elasto-inertial characteristic frequencies the films effectively behave as highly deformable, as discussed in §5.3. It is also worth recalling that we focus on the fundamental resonance/anti-resonance frequencies of the system. At frequencies higher than those, there exists higher modes that result in additional characteristic frequencies. One can therefore not expect, in general, that at high frequencies the rigid film assumption would provide a good approximation of the behaviour of permeo-elastic media. Considering the same permeo-elastic material as in Figure 4, its normalised apparent dynamic density is shown in Figure 5(a) while Figure 5(b) displays the following associated terms as a cf + I, e where I cf = ωρ0 If (v, v)/|∇x P |2 and function of frequency: Eb = KE(v, v)/ω|∇x P |2 ; Ib = I b = ηV(v, v)/|∇x P |2 . Note that K = V b + j(Eb − I). b Figure 5(c) shows the Ie = ωρe tI(v, v)/|∇x P |2 ; and V b b b c b e e V. b In all these elasticity-to-inertia ratios E/I, E/If , and E/I and Figure 5(d) depicts the ratio |Eb − I|/ plots, the vertical dashed lines with markers show the identified visco-inertial characteristic frequency fv = ωv /2π = 76 Hz, elasto-inertial characteristic frequencies fa = 317.5 Hz and fg = 385.5 Hz, and fρ = 298 Hz. The definition of the latter is discussed below. Figure 5(a) shows that Re(ρ(fa ≤ ω ≤ fg )) ≤ 0 as well as ρ(fg ) → 0 and ρ(fa ) → −∞. Evidently, the latter value is not reached due to the regularisation caused by the small but non-negligible viscosity b with Eb and Ie in Figure 5(b)]. Moreover, as predicted by Eqs. (62)–(63) and effects [cf. the term V (64)–(65), the elasticity of the films is balanced by the inertia of the system at the frequencies fa and b Ib = 1 at the said frequencies. It is worth noting fg . This is explicitly shown in Figure 5(c) where E/ 23

0.5 10-2

Rigid -lms Elastic -lms

0.4

Rigid -lms Elastic -lms

Angle(K(!))=:

jK(!)j [m2 =Pa:s]

0.3

10-3

10-4

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

10-5 101

102 Frequency [Hz]

-0.5 101

103

(a)

102 Frequency [Hz]

103

(b)

Figure 4: Absolute value (a) and normalised phase (b) of the effective conductivity for a material with perfectly rigid (dashed grey line) and elastic (black line) films. 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633

that, in this example, at fa the inertia of both the films and fluid contribute to the acoustic behaviour while at fg the inertia of the system is dominated by the inertia of the films. This is respectively shown b I cf [light in Figure 5(c) where comparable (respectively different but still of the same order) values of E/ b e grey line] and E/I [dark grey line] are attained at fa (respectively fg ). This is consistent with the analysis stating that fa corresponds to an anti-resonance (the film and fluid motions are equivalent but opposite) and fg corresponds to a global resonance (where both phases move together) that is driven by the film which is heavier than the gas. On the other hand, the real part of the dynamic apparent density can be written as Re(ρ) = −F(X )/ωRe(K), where F(X ) = X /(1 + X 2 ) and X = Im(K)/Re(K). It is clear that F is minimum when X → 0, which occurs, as can be seen in Figure 5(d), at fa and fg . Conversely, the absolute value of F is maximum when |X | = 1. Hence, one can implicitly define the characteristic frequency b V| b −1 → 1. At fρ , the maximum of the fρ for which |Im(K)||Re(K)|−1 → 1 or, alternatively, |Eb − I|| real part of the apparent density shown in Figure 5(a) occurs. Note that the amplitude of Re(ρ(ωρ )) increases when the viscosity effects are as small as possible. Oppositely, the peak does not appear in overdamped permeo-elastic materials, as will be shown later. Regarding the viscous characteristic frequency, this is, by definition and in absence of elastic effects, the frequency at which |Im(K)||Re(K)|−1 → 1 [see the dashed line with a square marker in Figure 5(d)]. In this case, as can be inferred from Figure 5(c), the inertia of the system is dominated by that of the fluid [cf. grey lines]. Furthermore, for the permeo-elastic material being used as example, a “pure” visco-elastic characteristic frequency cannot be directly identified since the inertia of the system b V b → 1, as can be seen by comparing the grey line at the frequency of crossing dominates when E/ between the light grey and black lines in Figure 5(b), i.e. at 220 Hz. Figure 6 shows the combined influence of bending and membrane effects on the apparent dynamic density of permeo-elastic materials. We varied the reduced tension per unit thickness, defined as T ∗ = T Γ/EI, from 0 to 100. A value of T ∗ = 0 corresponds to a permeo-elastic material in which only bending effects [51] are accounted for. Note that we have kept Γ/EI constant while varying T and that the other parameters are as in Figure 4. The extreme values of T ∗ respectively correspond to values of the elastic parameter K of 0.0502 × 106 and 5.0681 × 106 N m−4 . As predicted by Eqs. (63) and (65), increasing the tension leads to higher elasto-inertial characteristic frequencies. Since the viscous characteristic frequency does not depend on the elasticity of the system, this is the same for all the materials shown in Figure 6 and is equal to fv = 76 Hz. On the other hand, fρ also increases with the tension and, consequently, the difference between fv and the set of frequencies associated to the resonance (fρ , fa , fg ) becomes larger. This, combined with the fact that for f >> fv the viscosity effects decrease even further, explains the larger amplitude of the peak of the real part of the apparent density at fρ of materials with larger tension. These results strongly suggest that pre-stressing the films 24

fρ fa

fv

fg

fρ fa

fv Eb Ib b V

20 15

100

b [m2 =Pa:s] b or V Eb, I,

10

;(!)=;0

5

10-2

0 -5 -10

10-4

-15 -20

Re(;(!)=;0 ) Im(;(!)=;0 )

-25 -30 50 100

200

300 400 Frequency [Hz]

500

Eb

10-6 600

50 100

Ib

200

(a) fρ fa

fv

b I cf E/

b Ie E/

b Ib E= b I cf E= b Ie E=

500

600

fg

101

100

10-1

2 1 0 50 100

600

b !1 b Vj jIm(K)jjRe(K)j!1 = jEb ! Ijj

3

500

102

6

4

fρ fa

fv

7

5

300 400 Frequency [Hz]

b V

(b) fg

8

Elasticity-to-inertia ratio

fg

102

25

200

b Ib E/

300 400 Frequency [Hz]

500

10-2 50 100

600

(c)

200

300 400 Frequency [Hz]

(d)

b inertial Figure 5: (a) Normalised apparent dynamic density of a permeo-elastic medium. (b) Elastic E, b b I, and viscous V parts of the effective conductivity. (c) Elasticity-to-inertia ratios. (d). Ratio between the absolute values of the real and imaginary parts of the effective conductivity. The vertical lines with markers show the viscous characteristic frequency fv (square), the frequency fρ (upward pointing triangle), and the elasto-inertial characteristic frequencies fa (downward pointing triangle) and fg (circle). The parameters of the permeo-elastic medium are as in Figure 4.

25

30

0

25

-0.1

20

-0.2

Angle(;(!)=;0 )=:

Re(;(!)=;0 )

15 10 5 0 -5 -10 -15 -20 -25 -30 50

T$ T$ T$ T$ T$

=0 =10 =25 =50 =100

100

-0.3 -0.4 -0.5 -0.6

T$ T$ T$ T$ T$

-0.7 -0.8 -0.9

250 Frequency [Hz]

500

-1 50

750 1000

=0 =10 =25 =50 =100

100

250 Frequency [Hz]

(a)

500

750 1000

(b)

Figure 6: Influence of the reduced tension T ∗ = T Γ/EI on the real part (a) and normalised phase (b) of the normalised apparent density of permeo-elastic materials with films exhibiting pore-scale bending and membrane effects. 20

0

15

-0.1 -0.2

5 0 -5 -10 -15 -20 50

;e t = 80 ;e t = 60 ;e t = 40 ;e t = 20 ;e t = 10 100

2

g=m g=m2 g=m2 g=m2 g=m2

-0.3 -0.4 -0.5 -0.6 -0.7 -0.8

250

"f = fg ! fa [Hz]

Angle(;(!)=;0 )=:

Re(;(!)=;0 )

10

-0.9

250 Frequency [Hz]

500

-1 50

750 1000

(a)

200 150 100 50 0 10 20

40

60

80

;e t [g=m2 ]

100

250 Frequency [Hz]

500

750 1000

(b)

Figure 7: Real part (a) and normalised phase (b) of the normalised apparent density for permeo-elastic materials with films with different surface density, ρe t, but constant thickness t = 50 µm. The inset plot shows the bandwidth, i.e. ∆f = fg − fa , of the atypical band where Re(ρ/ρ0 ) ≤ 0 as a function of ρe t. 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649

can be exploited as a mechanism for tuning the acoustic behaviour of permeo-elastic media. Similar trends to those shown in Figure 6 are observed when increasing the elastic modulus E while keeping T constant. This is, however, not shown for the sake of brevity. Figure 7 shows the influence of the surface density, ρe t, of the films of a permeo-elastic material on the dynamic apparent density. The films are subjected to a tension per unit thickness T = 1 N m−1 and their thickness has been kept constant at t = 50 µm. The other parameters are as in Figure 4. The plot shows that a decrease in the surface density leads to higher elasto-inertial characteristic frequencies, as predicted by Eqs. (63) and (65). In addition, permeo-elastic materials with lighter films possess a wider atypical band where Re(ρ/ρ0 ) ≤ 0. This is shown in the inset plot where ∆f = fg − fa is plotted as a function of ρe t, and is consistent with the dominant role of the inertia of the system on the ratio between the elasto-inertial characteristic frequencies estimated in Eq. (66). Furthermore, it is worth mentioning that in permeo-elastic materials with lighter films, the contribution of the inertia of the fluid to the global resonance frequency fg becomes comparable to that of the films, with the opposite trend being found for permeo-elastic materials with heavier films. Figure 8 evidences the possibility of tuning the apparent density of permeo-elastic media by modifying, in a simple manner, the pore space geometry. Specifically, the air gap, of width g, formed in 26

35

Re(;(!)=;0 )

20 15

7m 7m 7m 7m 7m

-0.25 -0.5

10 5 0 -5

-0.75 -1

-10 -15 -20 -25 10

25

50 100 200 Frequency [Hz]

400

1000

10

(a)

650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680

1 0 -1 -2 -3 -4 -5 -6 100

Q = Eb Q = Ib b Q=V

log Q

25

0

g = 750 g = 400 g = 250 g = 200 g = 175

Angle(;(!)=;0 )=:

30

200

25

300 400 Frequency [Hz]

50 100 200 Frequency [Hz]

600

400

800

1000

(b)

Figure 8: Influence of the air gap formed in between the rigid structure and channel wall on the real part (a) and normalised phase (b) of the normalised apparent density of permeo-elastic materials. b I, b and V b as a function of frequency for the material with g = 175 µm. The The inset plot shows E, vertical dashed lines with markers represent the visco-elastic characteristic frequency fe (star), the elasto-inertial characteristic frequency fg (downward pointing triangle), and the frequency at which b = Ib < Eb (square). V

between the rigid structure and channel wall [see Figure 3(b)] is varied. Decreasing g reduces the permeability of the medium and increases both the viscous dissipation and fv . The plots show the influence of g on the normalised apparent density. Focusing on the lower frequency range, it is seen that Re(ρ) can take either positive or negative values depending on whether the low-frequency inertia of the fluid or elasticity of the films dominates. For example, for the material with g = 175 µm, the elastic effect dominates over the low-frequency inertia of the fluid (cf. black and dark grey lines in the inset plot). The visco-elastic characteristic frequency is estimated as fe = 164.7 Hz (see the dashed line with a star in the inset plot) and one has that, for f << fe , the real part of the apparent dynamic density takes a constant negative value, as predicted by Eq. (46). The Re(ρ) then crosses the zero axis at fg = 386 Hz. Note that in this example, a pure visco-inertial characteristic frequency, i.e. in absence of elastic effects, cannot be directly identified (cf. vertical dashed line with a square in the b = Ib < E) b and that the overall behaviour of ρ is largely inset plot indicating the frequency at which V determined by viscosity effects. As the gap is increased, the low-frequency inertia of the fluid can either compensate or dominate over the elasticity of the system, leading, respectively, to effectively zero real part of the dynamic density and Re(ρ) > 0 [see Eq. (46)]. For larger gaps, the visco-inertial characteristic frequency takes smaller values which combined with the balance of the elasticity and the (high-frequency) inertia of the system yields the local resonance/anti-resonance exhibited by the material with g = 750 µm. Figure 9 evidences further the use of pre-stress as a means of tuning the acoustic behaviour of permeo-elastic media. For a material with g = 175 µm and with its films being subjected to a unitary tension per unit thickness, the real part of the apparent density is negative, which occurs at low e As the tension is increased, the quoted inequality becomes a equality or frequencies when Ee > I. reverts, which respectively means that the low-frequency inertia of the fluid compensates or dominates over the elasticity of the system; leading to a near zero Re(ρ), exhibited by the material with T = 4 N m−1 in Figure 9(a), or the positive value exhibited by Re(ρ) at low frequencies for the materials with films subjected to larger tension. On the other hand, Figure 9(b) shows the rather atypical behaviour of the absolute value and normalised phase of the effective conductivity. Regarding the latter, for example, in a classical porous medium the phase varies monotonically from 0 down to −π/2 [cf. 4(b)] while in permeo-elastic media the phase takes a zero value at low frequencies, starts increasing at fa to reach π/2 at fg and then decreases down to −π/2 for f > fg . Finally, let us mention that the atypical behaviour observed in the apparent density in the atypical 27

3#10-3

-0.25

0.5

-0.5

0.25

10-3

-0.75 -1 10 1

10

2

10 Frequency [Hz]

10

3

5 0

Angle(K)=:

16 N=m 8 N=m 4 N=m 2 N=m 1 N=m

Angle(;(!))=:

= = = = =

jKj [m2 =Pa:s]

Re(;(!)=;0 )

0

T T T T T

0 -0.25 -0.5 -0.75

10

-4

-5

-1 10 1

10 2 Frequency [Hz]

10 3

-10 -15

10-5

-20 -25 101

102 Frequency [Hz]

5#10-6 101

103

(a)

102 Frequency [Hz]

103

(b)

Figure 9: Real part of the apparent dynamic density (a) and absolute value of the conductivity (b). The inset plots show the normalised phase of the said effective parameters.

683

band, determined by the elasto-inertial characteristic frequencies fa and fg and induced by the local resonance, is translated on to the effective speed of sound. Hence, a quasi-bandgap with a very slow effective speed of sound and an increase of the attenuation coefficient in the said atypical band occur.

684

7

681 682

685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713

Conclusions

This study on permeo-elastic media reveals in which conditions a phenomenon of fluid-film interaction can arise at the scale of the pores and influence the macroscopic observable acoustic behaviour of the said media. It also evidences that the resulting effects are encapsulated in the effective conductivity parameter, which is straightforwardly related to the acoustic features of such media. It is noticeable that a common general framework has been derived for describing the long-wavelength acoustic behaviour of permeo-elastic media in which the films deformation occur by bending and/or membrane effects. However, unsurprisingly, each type of films leads to a specific conductivity. A key feature of permeo-elastic media, compared to conventional porous media, lies in their significantly different behaviour, resulting from the interplay of the elastic and kinetic energies of the films with the viscous and kinetic energies of the fluid. In particular, the theory shows that situations of inner resonance and anti-resonance can be reached for specific frequencies and induce a frequency band of “anomalous” effective speed of sound. Furthermore, the said frequency band can be tuned through a judicious design of the films. It is worth mentioning that experiments on a prototype material in [51] have confirmed these phenomena and the validity of the description for bending films. Further experiments are planed to investigate the case of tensioned films. To ease the developments, this work has been restricted to the study of films clamped on the rigid frame. However, other boundary conditions, such as clamped/free, could also be investigated by closely following the developments presented in this work and would lead to a similar macroscopic description, belonging to the same framework. Also, viscoelastic films could be handled straightforwardly by considering complex moduli. For polymeric films, one can also introduce the variation of their mechanical properties with the temperature that would lead to a dependence of the atypical acoustic behaviour of the permeo-elastic medium on the ambient temperature. To conclude, the results of this work could find applications in the the design and modelling of thin-walled semi-closed foams for noise mitigation [17, 7, 22, 48] and of silencers [50] that incorporate plate- or membrane-type elements [25, 58, 31, 37]. Extensions of this work may include research on wave propagation in permeo-elastic media accounting for the elasticity of the solid frame onto which the flexible films are fixed; the hybridisation of permeo-elastic media and both porous materials and media made of acoustic resonant building-block elements [1, 35]; and the acoustics of thin-walled closed-cell foams [19]. 28

714

718

Acknowledgements This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology). This work was also supported by CeLyA of Université de Lyon operated by ANR (ANR 10 Labex 0060 and ANR 11 IDEX - 0007).

719

A

715 716 717

720

721 722 723

724 725 726 727 728 729

Variational formulation of the fluid-film interaction cell problem with bending and membrane effects

In this appendix, it is derived the variational formulation (26) associated with the fluid-film interaction cell problem given by i) Eqs.(7a)–(7d); ii) the leading-order balance equation (25) of the films, exhibiting bending and membrane effects, expressed in terms of velocity, namely (with σ (1) = 2ηDy (v(0) )−P (1) I) e y · (T(0) + T ∇ e y v (0) ) = jωρe tv (0) − [σ (1) · N] · N on ∇ jω

Ωf

Ωf

Integrating by parts the left-hand side integral and, since ∇y · w = 0, one obtains Z Z (0) (1) divy (2ηDy (v ) − p I) · wdΩ = − 2ηDy (v(0) ) : Dy (w)dΩ Ωf

+

Z

Ωf

(0)

(2ηDy (v

∂Ωf

731 732 733

736 737 738

739

∂Ωf

Then, Eq. (A.2) becomes Z Z − ∇x p(0) · wdΩ = η Ωf

735

)−p

(1)

(A.3)

I) · n · wdΓ.

The surface integral vanishes on Γs due to the no-slip condition Eq. (7c) and on the periodic boundaries of the fluid network due to the periodicity of the flow but not on the opposite sides of the film, and then simplifies into Z Z Z (2ηDy (v(0) ) − p(1) I) · n · wdΓ = σ (1) · n · wdΓ = − [σ (1) · N] · wNdΓ. (A.4) ∂Ωf

734

(A.1)

iii) Eqs. (7f)–(7g); and iv) boundary conditions (7h). Consistently with the kinematic constraints fulfilled by v(0) and v (0) , we shall consider the space W of complex Ω-periodic vectors w defined in Ωf ∪ Γ that are divergence free ∇ · w = 0 in Ωf and e y w · n = 0 and w = 0 on ∂Γ. takes the values w = 0 on Γs , w = wN on Γ, and ∇ Let us focus first on the fluid domain. The balance equation (7a) multiplied by any w ∈ W and integrated over Ωf gives Z Z Z (0) (1) (0) divy (2ηDy (v ) − p I) · wdΩ = jωρ0 v · wdΩ + ∇x p(0) · wdΩ. (A.2) Ωf

730

Γ;

Γ

2Dy (v(0) ) : Dy (w)dΩ + jωρ0

Ωf

Z

Ωf

v(0) · wdΩ +

Z

Γ

[σ (1) · N] · wNdΓ.

(A.5) This equation is the classical variational formulation of the harmonic laminar viscous flow in porous media, except that the surface integral accounts for the fluid-film interaction. Let us focus now on the films and integrate over Γ the momentum balance of the film (A.1) multiplied by w. Thus, Z Z Z e y · (T(0) + T ∇ e y v (0) )wdΓ + jωρe t v (0) wdΓ. [σ (1) · N] · wNdΓ = − ∇ (A.6) jω Γ Γ Γ

Integrating by parts and applying the divergence theorem yield Z Z Z T e (0) e T e (0) e y · (T(0) + T ∇ e y v (0) )wdΓ = − ∇ (T(0) + ∇ v ) · ∇ wdΓ − (T(0) + ∇ y y y v ) · wnds. jω jω jω Γ Γ ∂Γ (A.7) 29

740 741

The integral on the line ∂Γ where the films are clamped vanishes since w = 0 on ∂Γ. Furthermore, from the moment balance Eq. (7f), one has that Z Z f y (M(0) ) · ∇ e y wdΓ = − div e y wdΓ. T(0) · ∇ (A.8) Γ

Γ

742

Integrating by parts the latter integral and considering the symmetry of M(0) , one obtains Z Z Z f y (M(0) ) · ∇ e y w)dΓ − e y w) · nds. e y wdΓ = − div M(0) : e e(∇ (M(0) · ∇

(A.9)

e is defined in Eq. (12). Consequently, Eq.(A.6) is rewritten as where N Z Z Z Z EI T (1) (0) (0) e e e [σ · N] · wNdΓ = N (v , w)dΓ + ∇y v · ∇y wdΓ + jωρe t v (0) wdΓ. jω Γ jω Γ Γe Γ

(A.11)

Γ

Γ

743 744 745

746

∂Γ

e y w · n = 0 on ∂Γ and the integral on the line ∂Γ vanishes. In Since the films are clamped, then ∇ addition, from the bending constitutive law of the films Eq. (7g) expressed in terms of velocity one has Z Z EI (0) e e (v (0) , w)dΓ, e M : e(∇y w)dΓ = N (A.10) jω Γ Γ

751

Finally, the variational formulation (26)–(27) is obtained by replacing Eq. (A.11) into Eq. (A.5). Note that for no tension, i.e. T = 0, one recovers the variational formulation for permeo-elastic materials with plate-type films, while for films with zero bending stiffness, i.e. EI = 0, one recovers the variational formulation for permeo-elastic materials with membrane-type films (expressed in the space W).

752

B

747 748 749 750

Variation of real and imaginary part of Hve (ω) versus frequency

To study the frequency variations of the effective resistivity in visco-elastic regime, denoted by Hve (ω) (assumed to be isotropic, or in given principal direction), let us calculate the virtual crossed energies of a particular velocity field (solution at a given frequency) under a particular velocity field (solution at another frequency) and reciprocally. Consider for this purpose, the velocity fields (va∗ , va∗ ) and (vb∗ , vb∗ ) corresponding to a unit real mean velocity at the frequency ωa (respectively ωb ), i.e. hva∗ i = hvb∗ i = e, 753 754 755

|e| = 1,

arg(e) = 0.

ve The associated forcing pressure gradients are respectively −Hve (ωa )e = −Hve a e and −H (ωb )e = ve ∗ ∗ −Hb e, and the corresponding variational formulation (26) for va (and similarly for vb , ωb , Hve b ) reads as

∀w ∈ W,

ηV(va∗ , w) +

K E(va∗ , w) = Hve a e · hwi. jωa

(B.1)

∗ Choosing now in the ωa -variational formulation (respectively ωb ), the field test w = Hve b vb (respectively ve ∗ w = Ha va ), one obtains

ηV(va∗ , vb∗ )Hve b +

K ve ve E(va∗ , vb∗ )Hve b = Ha Hb , jωa

ηV(vb∗ , va∗ )Hve a +

K ve ve E(vb∗ , va∗ )Hve a = Hb Ha . jωb

Subtracting these two expressions yields the following identity, valid for any frequencies ωa and ωb ,  ve  Hb K Hve a ve ∗ ∗ ηV(va∗ , vb∗ )(Hve − H ) + E(v , v ) − =0 a a b b j ωa ωb 30

756

Taking ωb close to ωa , one deduces that at any frequency (hence the indices are dropped) ηV(v∗ , v∗ )

757

K dHve d(ωHve ) + 2 E(v ∗ , v ∗ ) = 0. dω jω dω

(B.2)

However, choosing w = v∗ in the ω-variational formulation gives ηV(v∗ , v∗ ) +

K E(v ∗ , v ∗ ) = Hve . jω

(B.3)

Then, eliminating ηV(v∗ , v∗ ) from Eqs. (B.2) and (B.3) provides   ve d(ωHve ) K d(Hve ) ve d(H ) ∗ ∗ v ) = −H E(v , − ω , jω 2 dω dω dω so that

dHve K = − 2 E(v ∗ , v ∗ ) dω jω

Using the subscripts

<

and

=

and then

d(ωHve ) = ηV(v∗ , v∗ ). dω

to denote the real and imaginary parts, one has that

dHve 2K ∗ ∗ < = 2 E(v< , jv= ) dω jω

d(ωHve ∗ ∗ =) = 2jηV(v< , jv= ), dω

;

∗ as field test in the In order to establish the sign of these derivative, we use w = (v∗ − v∗ )/2 = jv= ∗ natural and conjugate ω-variational formulation. Since hjv= i = 0, one obtains ∗ )+ ηV(v∗ , jv=

K ∗ E(v ∗ , jv= )=0 jω

and

∗ ηV(v∗ , jv= )−

K ∗ E(v ∗ , jv= ) = 0, jω

that by summing and subtracting yields ∗ ∗ ) = −2 , jv= 2ηV(v<

K K ∗ ∗ ∗ ∗ ) = −2 E(v= ), , jv= , v= E(jv= jω jω

2K ∗ ∗ ∗ ∗ ∗ ∗ ). , v= ) = −2ηV(v= ) = −2ηV(jv= , jv= , jv= E(v< jω 758

Consequently, one finally has that dHve η ∗ ∗ < = −2 V(v= , v= )≤0 dω ω

d(ωHve K ∗ ∗ =) = −2 E(v= , v= ) ≤ 0. dω ω

;

(B.4)

760

ve These expressions prove that Hve < and ωH= monotonically decrease with the frequency, and 10 exemplifies these results.

761

Declaration of competing interests

759

763

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

764

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#105

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Im(!H) [Pa:m!2 ]

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Credit Author Statement

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893 894 895 896

Claude Boutin: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Writing - Original Draft, Writing - Review & Editing, Funding acquisition R. Venegas: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization, Funding acquisition

35