Linear stability analysis in fluid–structure interaction with transpiration. Part I: Formulation and mathematical analysis

Comput. Methods Appl. Mech. Engrg. 192 (2003) 4805–4835

www.elsevier.com/locate/cma

Linear stability analysis in fluid–structureinteraction with transpiration.

Part I: Formulation and mathematical analysis

Miguel �AAngel Fern�aandez a,*, Patrick Le Tallec b

a INRIA, Projet MACS, F-78153 Le Chesnay Cedex, Franceb �EEcole Polytechnique, DGAE, F-91128 Palaiseau Cedex, France

Received 27 September 2002; received in revised form 12 July 2003; accepted 18 July 2003

Abstract

The aim of this work is to provide a new Linearization Principle approach particularly suited for problems in fluid–

structure stability. The complexity here, and the main difference with respect to the classical approach, comes from the

fact that the full non-linear fluid equations are written in a moving (i.e. time dependent) domain. The underlying idea of

our approach uses transpiration techniques [J. Fluid Mech. 4 (1958) 383; G. Mortch�eel�eewicz, Application of linearized

Euler equations to flutter, in: 85th AGARD SMP Meeting, Aalborg, Denmark, 1997; P. Raj, B. Harris, Using surface

transpiration with an Euler method for cost-effective aerodynamic analysis, in: AIAA 24th Applied Aerodynamics

Conference, number 93-3506, Monterey, Canada, 1993; AIAA 27(6) (1989) 777], with the formalization and lineari-

zation recently developed in [R�eev. Europ�eeenne �EEl�eem. Finis, 9(6–7) (2000) 681, A. Dervieux (Ed.), Fluid–Structure

Interaction, Kogan Page Science, London, 2003 (Chapter 3)]. This allows us to obtain a new grid independent coupled

spectral problem involving the linearized Navier–Stokes equations and those of a reduced linear structure. The coupling

is realized through specific transpiration conditions acting on a fixed interface, while keeping a fixed fluid domain. We

provide a rigorous mathematical treatment of this eigenproblem. We prove that the corresponding eigenmodes,

characterizing the free evolution of the system, can be obtained from the characteristic values of a compact operator

acting on a Hilbert space. Moreover, we localize the eigenfrequencies of the system in a parabolic region of the complex

plan centered along the positive real axis.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Fluid–structure interaction; Linear stability; Linearization; Transpiration; Spectral analysis

* Corresponding author. Address: �EEcole Polytechnique F�eed�eerale de Lausanne, IACS, CH-1015 Lausanne, Switzerland.

E-mail address: [email protected] (M.�AA. Fern�aandez).

0045-7825/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.07.001

mail to: [email protected]

4806 M. �AA. Fern�aandez, P. Le Tallec / Comput. Methods Appl. Mech. Engrg. 192 (2003) 4805–4835

1. Introduction

An important question in the research of experimentalists and applied mathematicians, is the stability of

an equilibrium state in a mechanical system. That is, if the equilibrium is slightly disturbed, do the per-

turbations grow or decay? This question has an important role in the design of complex industrial systems.

The linear stability theory deals with perturbations of small size acting at the initial time, by supposing

that the generated fluctuations remain small for all time t > 0. In this case, a Linearization Principle ap-

proach, see for instance [18], reduces the stability problem to the analysis of a specific spectral problem.Namely, neglecting the high order terms in the full non-linear equations, we obtain a linearized problem

governing (at first order) the fluctuations around the equilibrium state. The study of these fluctuations can

be generally obtained from the behavior of the harmonic solutions, called normal modes, which are directly

characterized by the eigenvalues of the associated spectral problem (see Fig. 1).

In this work we focus on the linear stability of a coupled fluid–structure system involving an incom-

pressible Newtonian fluid and a reduced structure. The mathematical and numerical work reported here

and in Part II (see [15]) is intended as the first stage of a rigorous approach to obtain, at low computational

cost, reliable numerical predictions of the physical stability of such systems. The work in this paper aims, onthe one hand, at deriving mathematically a coupled eigenproblem of minimal complexity, overcoming all

difficulties arising when dealing with moving domains, and on the other hand, at providing a rigorous

mathematical and numerical analysis of its solutions.

Indeed, the complexity here, and the main difference with respect to the classical approach [17,18],

comes from the fact that, due in particular to large structural displacements, the full non-linear fluid

equations are written in a moving (i.e. time dependent) domain. This is classically overcome by the in-

troduction of an arbitrary Lagrangian–Eulerian (ALE) formulation (moving grid approach) transporting

these equations to a fixed arbitrary reference configuration [10,20,22]. Although this approach has beensuccessful in certain stability analysis [20,25,26], the direct implementation and linearization of such a

coupled fluid structure problem with ALE formulation requires the modification of the flux vectors and

the introduction of a smooth user dependent auxiliary mapping (and its corresponding grid motion) in the

fluid domain, whose accurate implementation is expensive in real applications. In order to overcome these

drawbacks, and to be able to solve at low cost fluid structure interaction problems at moderate defor-

mation, aeronautical engineers have successfully developed transpiration techniques [21,27,28,33] which

Fig. 1. Linearization principle.

M. �AA. Fern�aandez, P. Le Tallec / Comput. Methods Appl. Mech. Engrg. 192 (2003) 4805–4835 4807

bypass the construction of this map and of flux corrections through modifications of the interfaceboundary conditions. The price to pay numerically is the accurate calculation of an equilibrium reference

solution, which must be done once for all on a rather fine mesh. The resulting unsteady or spectral

problem is then simpler [28], and moreover, as it will be seen in the present paper, they have a rather nice

mathematical structure.

The present work implements this transpiration philosophy within a mathematically based Linearization

Principle requiring, on the one hand, a specific definition of the state fluctuations (taking into account the

fluid domain motion), and on the other hand, a new linearization method (adapted to this fluctuations

definition), leading to a coupled, grid independent, linear problem with minimal complexity. This problemwill involve the linearized incompressible Navier–Stokes equations and those of the reduced linear struc-

ture. The coupling will be realized by means of specific transpiration boundary conditions. The harmonic

solutions of this coupled linear problem are solutions of a new spectral problem for which we are able to

provide a rigorous mathematical analysis.

The outline of this paper is as follows. In Section 2, we introduce the fluid–structure interaction problem

under study and its mathematical description. We use the classical ALE formulation for the fluid. Section 3

provides a generalized Linearization Principle approach particularly suited for problems involving moving

boundaries. This is achieved from the linearization–transpiration method [21,27,28,33], formalized in[12,13]. Finally, in Section 4 we provide a mathematical study of the eigenproblem defined in Section 3. This

is done by proving that the eigenvalues of this spectral problem can be obtained from the characteristic

values of a coupled vibration operator, whose inverse is compact, and whose spectrum is therefore well

characterized. This last result was already announced as a brief note in [16]. In addition, a new result

concerning the localization of the spectrum is provided.

The formalization of the linearization–transpiration approach and the analysis of the vibration operator

will give a new insight on the structure of the coupled problem, which is potentially useful for the mathe-

matical and engineering practice of fluid–structure interaction problems.

2. Mechanical problem

The modeling of fluid–structure interaction systems under large displacements involves, in a general way,

the coupling of two formulations: the solid classically treated in lagrangian formulation, and the fluid

described by an ALE formulation [10,30]. In this section we introduce the mechanical problem under study

and its mathematical description.

2.1. Geometry

We consider a solid located at time tP 0 in a domain XsðtÞ � R3 with boundary cðtÞ. As in many

problems of aeroelasticity at low Mach numbers, such as bridge deck profiles under the action of an ex-

ternal wind [15], we will assume that it is surrounded by a fluid in R3, although the mechanical and nu-

merical formulations to follow can be extended to other geometries such as those described in [31]. We

introduce a control volume X � R3 containing the solid at each time tP 0. The notation oX stands for theboundary of X. Hence, the fluid evolution is restricted to the domain XfðtÞ ¼ X� X

sðtÞ. In the sequel we set

C ¼ oX and

oXfðtÞ ¼ C [ cðtÞ;stands for the fluid domain boundary (see Fig. 2).

We assume the fluid to be Newtonian viscous, homogeneous and incompressible. Its behavior is

described by its velocity and pressure. The elastic solid under large displacements is described by its

Fig. 2. Geometric description.


displacement, velocity and stress tensor. The classical conservation laws of continuum mechanics drive the

evolution of these unknowns.

2.2. The fluid

The fluid satisfies the following incompressible Navier–Stokes equations which take the Eulerian con-

servative form:

ouEot jx

þ divE uE � uE � 1

qE

rðuE; pEÞ� �

¼ 0; in XfðtÞ;

divEuE ¼ 0; in XfðtÞ;ð1Þ

where qE, uE and pE stand, respectively, for the fluid density, velocity and pressure. In addition, the fluid

stress tensor is given by

rðuE; pEÞ ¼ �pEI þ 2leðuEÞ;with l the dynamic viscosity of the fluid and

eðuEÞ ¼1

2½rEuE þ ðrEuEÞT�;

the strain rate tensor.

Remark 1. In the sequelo

otjastands for the time derivative operator keeping the space variable ‘‘a’’ fixed.

In addition, the subindex ‘‘E’’ stands for the Eulerian description or space derivatives.

In a fluid–structure interaction framework the evolution of the fluid domain XfðtÞ is induced by the

structural deformation through the fluid–structure interface cðtÞ. Indeed, by definition we have XfðtÞ ¼X� X

sðtÞ. This suggests to characterize XfðtÞ through a map acting in a fixed reference domain. This ap-

proach is usually used for the solid domain XsðtÞ, by means of the Lagrangian formulation [2,23]. In

stability analysis, we assume the existence of a steady state in which the fluid flows in a fixed domain Xf ,

with velocity u0 and pressure p0 satisfying

div u0 � u0 �1

qðu0; p0Þ

� �¼ 0; in Xf ;

divu0 ¼ 0; in Xf :

ð2Þ

This equilibrium configuration can be used as a reference configuration for the solid Xs ¼ X� Xfwith

boundary c, and for the description of the fluid domain motion. Then, the present control volume X ¼XfðtÞ [ X

sðtÞ will be described by a smooth and injective map:

x : X� Rþ ! X


ðx0; tÞ 7!x ¼ xðx0; tÞ:We set xf ¼ xjXf and xs ¼ xjXs , such that (see [24]):

• for x0 2 Xs, xsðx0; tÞ represents the position at time tP 0 of the material point which is at x0 in the equi-

librium configuration. This corresponds to the classical lagrangian flow,

• the ALE map xf is defined from xsjc, as an arbitrary extension over domain Xf , which preserves C ¼ oX,and such that xfðx0; tÞ ¼ x0 in the equilibrium configuration.

In short, the ALE map x is given by

xðx0; tÞ ¼ ExtðxsjcÞðx0; tÞ; 8x0 2 Xf ;

xðx0; tÞ ¼ xsðx0; tÞ; 8x0 2 Xs:

Here, ‘‘Ext’’ represents an arbitrary extension operator from c to Xf which is user defined and which

satisfies the external and initial boundary conditions

ExtðxsjcÞjC ¼ IC; ExtðIcÞ ¼ IXf :

This map allows us to transport the fluid equations (1) back to the fixed reference domain Xf , leading to the

classical incompressible Navier–Stokes equations written in ALE conservative formulation [10,22,24],

satisfied by u : Xf � Rþ ! R3 and p : Xf � Rþ ! R,

oJuot jx0

þ div J u� ðu� wÞ � 1

qrðu; pÞ

� �F �T

� �¼ 0; in Xf ;

divðJuF �TÞ ¼ 0; in Xf ;

ð3Þ

where the quantities F , J , w are defined by:

F ¼ rx ¼ oxox0

; J ¼ detðF Þ > 0; w ¼ oxot jx0

;

and where q ¼ qE (constant), and u and p are the ALE velocity and pressure defined by transport as:

uðx0; tÞ ¼ uEðxfðx0; tÞ; tÞ; pðx0; tÞ ¼ pEðxfðx0; tÞ; tÞ:

Remark 2. From the definition of x, wjXs represents the solid velocity, whereas wjXf stands for the fluid

control volume velocity. This ‘‘grid’’ velocity is a user defined auxiliary variable which usually differs from

the fluid velocity u inside Xf . A variant to bypass the ALE formulation (3) is to use space time finite ele-

ments as in [36].

2.3. The structure

The evolution of the structure is characterized by its motion xs. We will generally consider the case (at

least for our mathematical analysis, see also [9,11,34]) where the structural displacement is given, around

the equilibrium configuration Xs, by a linear combination of a finite number of vibration modes

ui : Xs ! R3, 16 i6 vs, in such a way that

xsðx0; tÞ ¼ x0 þXnsi¼1

siðtÞuiðx0Þ; 8x0 2 Xs;


with sðtÞ ¼ fsiðtÞg16 i6 ns 2 Rns . Hence, xs ¼ IXs þ Us in Xs, where U ¼ ½u1ju2j � � � juns � is a 3� ns matrix

standing for the reduced modal basis. In this way, the structural behavior is driven by given possibly non-

linear mass and stiffness operators,M andK respectively. Thus, the equations describing the motion of the

structure, around a known configuration, reduce to

K0 þM€ssþKs ¼ fg; ð4Þwith fg 2 Rns the generalized load vector, given by

½fg�i ¼Zcfc � ui da; 16 i6 ns;

and where fc 2 R3 stands for the surface force density applied by the fluid on the structural boundary in its

reference configuration. The reference configuration Xs being at equilibrium means that we have a struc-tural residual stress

½K0�i ¼ �Zcðrðu0; p0ÞnÞ � ui da; 16 i6 ns; ð5Þ

equilibrating the stress of the reference flow on the interface. Here n stands for the unit normal vector on cpointing inside Xs.

2.4. The coupled problem

The coupling between the solid and the fluid is realized through standard boundary conditions at the

fluid–structure interface c, namely, the kinematic continuity of the velocity and the kinetic continuity of thestress [24]:

u ¼ _xxs; on c;fc ¼ �Jrðu; pÞF �Tn; on c;

ð6Þ

Moreover, we endow the fluid equations with a Dirichlet boundary condition on Cin and an outlet con-

dition on Cout, i.e.

u ¼ uCin; on Cin;

rðu; pÞn ¼ 0; on Cout;

with C ¼ Cin [ Cout and Cin \ Cout ¼ ;.In summary, the coupled problem, with an ALE formulation for the fluid, is given on the fixed reference

configuration Xf [ Xs by

oJuot jx0

þ div J u� ðu� wÞ � 1

qrðu; pÞ

� �F �T

� �¼ 0; in Xf ;

divðJuF �TÞ ¼ 0; in Xf ;

u ¼ uCin; on Cin


u ¼ U_ss; on c;

K0 þM€ssþKs ¼ �ZcJUTrðu; pÞF �Tnda;

xs ¼ IXs þ Us; xf ¼ ExtðxsjcÞ;

ðu; s; _ssÞjt¼0 ¼ ðu0; s0; s1Þ;

ð7Þ


with F , J and w as defined in (3). Here, ðu0; s0; s1Þ, stands for the initial fluid velocity and the initial modal

components of the structural displacement and velocity, respectively.

3. Linear stability

In this section we focus on the the linear stability analysis of systems involving a reduced structure

immersed in an incompressible viscous flow, as described by (7). For this purpose, we subject the steadyequilibrium state ðu0; p0; IXÞ to small perturbations ðdu0; ds0; ds1Þ acting on the initial conditions. This

perturbation generates a coupled unsteady state ðu; p; xÞ which satisfies (7) with the following initial con-

dition

ðu0; s0; s1Þ ¼ ðu0 þ du0; ds0; ds1Þ:

From (7), we can obtain a non-linear coupled problem governing the fluctuations

ðdu; dp; dxÞ ¼ ðu; p; xÞ � ðu0; p0; IXÞ: ð8ÞBy supposing that these generated fluctuations remain small for any time t > 0, we can neglect the high

order terms. Thus, we obtain a linearized problem driving, at first order, the fluctuations of the coupledsystem (Section 3.1). At this point, we usually introduce another definition in the linear stability theory [18],

by supposing that each linear fluctuation ðdu; dp; dxÞ can be obtained by superposition of fluctuations of

type ðv; q; dÞe�kt, called normal modes. By substituting this last expression in the linearized coupled equa-

tions, we obtain that (k; v, q, d) is an eigenpair of the spectral problem associated to the coupled linear

problem (Section 3.2), leading to the following characterization:

Definition 3. The permanent state ðu0; p0; IXÞ is called linearly asymptotically stable, if all the eigenvalues of

the spectral problem have positive real parts. The permanent state is called asymptotically unstable, if thereexists, at least, one eigenvalue with negative real part.

In short, this general approach, known as Linearization Principle in [18], reduces the stability problem to

the calculation of the eigenmodes of a specific spectral problem.

Nevertheless, we must point out that the complexity of the linearized problem strongly depends on the

fluctuation definition. It is straightforward to verify, see [3,20,25,26] for instance, that taking (8) as fluc-

tuation definition we obtain a linearized problem where the fluid equations still depend on the interface

motion. Hence, in this sense, its complexity is similar to the full non-linear problem.In the next paragraphs, we will focus on a new derivation of the Linearization Principle. This will be

based on the linearization–transpiration method formalized in [12, Section 5] (see also [13, Chapter 2]),

particularly suited for problems involving moving domains. Thus, we will be able to recover a linearized

coupled problem, where the linear fluid equations are independent of the fluid–structure interface motion.

Indeed, the coupling with the linearized solid equations is performed via transpiration interface conditions

while keeping a fixed fluid domain. In other words, we finally obtain an eigenproblem of minimal com-

plexity. From a mathematical point of view, this can be viewed as a change of variables.

3.1. Linearization

To be mathematically correct and consistent, we must first write the coupled problem (7) in variational

form. By multiplying them by ðvvf1; vvf2; ttÞ 2 DðXÞ3 �DðXÞ � Rns , by integrating by parts in the fluid equa-

tions and by taking into account the boundary conditions, we obtain the following variational formulation:

find u : Xf � Rþ ! R3, p : Xf � Rþ ! R and ds : Rþ ! Rns such that

Z Z4812 M. �AA. Fern�aandez, P. Le Tallec / Comput. Methods Appl. Mech. Engrg. 192 (2003) 4805–4835

Xf

oJquot

� vvf1 dx�Xf

Jf½qu� ðu� wÞ � rðu; pÞ�F �T : rvvf1 þ ðuF �TÞ � rvvf2gdx

þZcðJrðu; pÞF �TnÞ � ðUtt � vvf1Þdaþ

ZcJu � ðF �TnÞvvf2 daþ ðM€ddsÞ � tt þ ðK0 þKdsÞ � tt ¼ 0;

8ðvvf ; ttÞ 2 DðXÞ4 � Rns ; ð9Þ

provided with the boundary conditions

u ¼ uCin; on Cin;

rðu; pÞn ¼ 0; on Cout;u ¼ U _dds; on c;

ð10Þ

and test functions

vvf ¼ vvf1vvf2

!: X ! R3 � R:

In the same way, the reference steady problem (2) and (5) can be written in variational form as:

�ZXf

f½qu0 � u0 � rðu0; p0Þ� : rvvf1 þ u0 � rvvf2gdxþZcðrðu0; p0ÞnÞ � ðUtt � vvf1ÞdaþK0 � tt ¼ 0;

8ðvvf ; ttÞ 2 DðXÞ4 � Rns : ð11ÞThe following step in the linearization is the definition of the fluctuations ðdu; dp; dxÞ, of the perturbed

state ðu; p; xÞ around the equilibrium ðu0; p0; IXÞ. As in [12], these fluctuations are defined by

x ¼ IX þ dx; in X;uðIXf þ dxÞ ¼ u0 þru0 dxþ du; in Xf ;pðIXf þ dxÞ ¼ p0 þrp0 dxþ dp; in Xf ;

ð12Þ

and where

dxs ¼ Uds; dxf ¼ ExtðxsjcÞ � IXf :

Compared to the traditional definition, used in [3,18,20,25,26], the non-standard definition (12) takesexplicitly into account the transport of the reference state due to the fluid domain motion, and the intrinsic

perturbation ðdu; dpÞ of the flow-field at the new spatial point (x0 þ dx).

Remark 4. The term K0 appearing in Eq. (11) corresponds to the residual stress introduced in [12, Section

5.1]. It equilibrates the stress vector developed by the steady flow on the interface.

As usual, the linearization can be carried out with respect to the new unknown ðdu; dp; dxÞ, by sub-

tracting the reference problem (11) from the perturbed problem (9), and then by neglecting the high orderterms. This leads us to the following coupled linear problem [12, Section 5.3]:Z

Xf

q _ddu � vvf1 dx�ZXf

½ðqdu� u0 þ qu0 � du� drðdu; dpÞÞ : rvvf1 þ du � rvvf2�dx

þZcðru0 dxs þ duÞ � nvvf2 da�

ZXf

f/ðu0; rðu0; p0ÞÞ½I div dx� ðrdxÞT� þ r/ðu0; rðu0; p0ÞÞdxg : rvvf dx

þZc½ðrrðu0; p0Þdxs þ drðdu; dpÞÞn� rðu0; p0ÞgðdxsÞ� � ðUtt � vvf1Þda

þ ðM €ddsþ K dsÞ � tt ¼ 0; 8ðvvf ; ttÞ 2 DðXÞ4 � Rns : ð13Þ


Here we have introduced the flux vector

/ðu; rÞ ¼ I1ðqu� u� rÞ þ I2 � u; I1 ¼1 0 00 1 00 0 10 0 0

0B@

1CA; I2 ¼

0001

0B@

1CA;

the perturbation stresses and strain rates

drðdu; dpÞ ¼ �dpIþ 2leðduÞ; eðduÞ ¼ 1

2½rduþ ðrduÞT�;

and the linearized mass and stiffness operators M and K around the equilibrium state ds ¼ 0. Finally, the

vector gðdxÞ ¼ �½I div dx� ðrduÞT�n represents, at first order, the variation of the surface vector �ndainduced by the displacement dx. It only depends on the trace of dx on c, see [34].

In addition, the boundary conditions (10), once written at first order in terms of ðdu; dp; dsÞ, reduce to

du ¼ U _dds�ru0Uds; on c;du ¼ 0; on Cin;drðdu; dpÞn ¼ 0; on Cout:

ð14Þ

We recover on the fixed interface c the so called transpiration boundary conditions (see [12,13,21,

28,33,34]). Nevertheless, the third volume integral in (13) contains distributed terms in Xf depending on dx.In other words, the linear fluid equations, as for the perturbed one, still depend on the fluid domain motion.

In contrast with the linearization performed in [3,20,25,26], this difficulty can be overcome here by using a

simplified version of Lemma 1 in [12, Section 5.2].

Lemma 5. For each smooth displacement dx 2 C1ðXfÞ3 and each smooth solution, ðu0; rðu0; p0ÞÞ 2 C1ðXfÞ3 �C1ðXfÞ3�3, of the fluid subproblemZ

Xf

/ðu0; rðu0; p0ÞÞ : rvvfdx ¼ 0; 8vvf 2 DðXfÞ4;

we obtain that

�ZXf

f/ðu0; rðu0; p0ÞÞ½I div dx� ðrdxÞT� þ r/ðu0; rðu0; p0ÞÞdxg : rvvfdx

¼Zc½/ðu0; rðu0; p0ÞÞgðdxÞ � ðr/ðu0; rðu0; p0ÞÞdxÞn� � vvf da; 8vvf 2 DðXÞ4: ð15Þ

Proof. See proof of Lemma 1 in [12, Section 5.2]. h

Subtracting now, from (13), the linearized convected problem (15), we obtain that the fluctuation

ðdu; dp; dsÞ, defined by (12), finally satisfies the following variational linear problem (independent of the

extension map dxf ):ZXf

q _ddu � vvf1 dx�ZXf

½ðqdu� u0 þ qu0 � du� drðdu; dpÞÞ : rvvf1 þ du � rvvf2�dx

þZcf½ðqdu� u0 þ qu0 � du� drðdu; dpÞÞn� � vvf1 þ du � nvvf2gda

þZc½ðrrðu0; p0Þdxs þ drðdu; dpÞÞn� rðu0; p0ÞgðdxsÞ� � ðUttÞda

þ ðM €ddsþ K dsÞ � tt ¼ 0; 8ðvvf ; ttÞ 2 DðXÞ4 � Rns : ð16Þ

completed with the boundary conditions (14).


In summary, as in [12, Section 5.3], the linearization has been carried out by subtracting the steadyreference problem (11), and the linearized convected problem (15), from the perturbed problem (9), and by

neglecting high order terms. This linearization provides a fluid problem written in fixed configuration Xf ,

and totally independent of the extension operator inside the fluid domain Xf .

The variational formulation (16) is equivalent to two subproblems coupled along the fixed interface c,see [12, Section 5.4]. Indeed, by integrating by parts with tt ¼ 0, the linearized fluid subproblem reduces to

qoduot

þ divðqu0 � duþ qdu� u0 � drðdu; dpÞÞ ¼ 0; in Xf ;

div du ¼ 0; in Xf ;

provided with boundary conditions

du ¼ 0; on Cin;drðdu; dpÞn ¼ 0; on Cout;du ¼ U _dds�ru0Uds; on c:

Similarly, by taking vvf ¼ 0, the linearized solid subproblem is given by

ðM €ddsþ KdsÞ � tt ¼ZcUTðrðu0; p0ÞgðdxsÞ � rrðu0; p0Þdxsn� drðdu; dpÞnÞda � tt;

for all tt 2 Rns . In this subproblem, the linearized interface load after transport is made of three contribu-

tions: one induced by interface translation rrðu0; p0Þdxsn, one induced by interface rotation rðu0; p0ÞgðdxsÞand one induced by the state variation drðdu; dpÞn.

After division by q in the fluid linearized problem, by setting m ¼ l=q the kinematic viscosity, by in-

troducing the following real ns � ns matrix B0, defining the sensitivity of the frozen stress interface vector to

unit translation and rotation,

B0ij ¼

Zcðrrðu0; p0Þujn� rðu0; p0ÞgðujÞÞ � ui da; 16 i; j6 ns; ð17Þ

and since du and u0 are divergence free, the coupled problem takes the following more compact form:

oduot

þru0duþrduu0 � 2mdiv eðduÞ þ 1

qrdp ¼ 0; in Xf ;

divdu ¼ 0; in Xf ;

du ¼ 0; on Cin;

drðdu; dpÞn ¼ 0; on Cout;

du ¼ U _dds�ru0Uds; on c;

M €ddsþ ðK þ B0Þds ¼ �ZcUTdrðdu; dpÞnda:

ð18Þ

We recover in (18) the linearized Navier–Stokes equations on a fixed domain provided with a transpi-

ration boundary condition on the fixed interface c. For the structure, the linearization introduces a non-

standard geometric term of ‘‘added stiffness’’ B0 ds. Therefore, the obtained problem allows us to take into

account the motion of the structure, while keeping a fixed fluid domain.

As pointed out in [12], the fundamental idea of this linearization comes from the fluctuation definition

(12), which leads to the transpiration interface condition (18)5, and from the transported problem (15),

which enables us to transport the distributed fluid equation dependencies on dxf to the fixed fluid–structureinterface c.


3.2. Spectral problem definition

Finally, the last step in the Linearization Principle, see Fig. 1, is the analysis of the harmonic solutions of

(18), i.e. solutions in the form of normal modes,

duðx; tÞ ¼ uðxÞe�kt; dpðx; tÞ ¼ pðxÞe�kt; dsðtÞ ¼ se�kt; with k 2 C; u : Xf ! C3;

p : Xf ! C and s 2 Cns : ð19Þ

By transferring the expressions (19) in (18) we obtain the following quadratic spectral problem:

ru0uþruu0 � 2mdiv eðuÞ þ 1

qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on Cin;


u ¼ �kUs�ru0Us; on c;

k2Msþ ðK þ B0Þs ¼ �ZcUTrðu; pÞnda;

ð20Þ

with unknown u, p, s and k, and where

rðu; pÞ ¼ �pI þ 2leðuÞ; eðuÞ ¼ 1

2½ruþ ðruÞT�;

and B0 is given by (17).

Remark 6. We must point out that the linearization method developed in [12] and, in particular, the ap-

proach presented here, does not depend on the structure of the equations. Namely, we could use another

model for the fluid or the structure. Nevertheless, our choice is motivated by the spectral analysis provided

in the next section, and by the numerical experiments performed in Part II (see [15]).

4. Spectral analysis

This section provides a first rigorous mathematical study of the new spectral problem (20) obtained in

the preceding section, characterizing the linear stability of fluid–structure systems coupled via transpiration

boundary conditions on a fixed interface.

4.1. Mathematical framework

We suppose X to be an open bounded subset of R3 (see Fig. 3), with locally Lipschitz continuous

boundary C (see [29]). We assume that Xs has a locally Lipschitz continuous boundary c and that �XXs � X.

Fig. 3. The computational domain, X, defined by the system in its equilibrium configuration.


We also assume that the equilibrium solutions u0 and p0 are smooth functions, let us say for instanceu0 2 C2ð�XXfÞ3 and p0 2 C1ð�XXfÞ. And on the other hand, the ns deformed modal shapes,

ui : Xs ! R3; i ¼ 1; . . . ; ns;

are supposed to be volume preserving, i.e.,Zcui � nda ¼ 0; i ¼ 1; . . . ; ns: ð21Þ

Here n stands for the unit normal vector on c (pointing inside Xs). In the sequel, the 3� ns matrix,

U ¼ ½u1ju2j � � � juns �;stands for the reduced modal basis. The behavior of the structure is also assumed to be characterized by the

linearized mass and stiffness ns � ns matrices, M and K respectively, the mass matrix being supposed to be

symmetric and positive definite. We finally assume that the external boundary C is sufficiently far away

from the structure, in order to impose a Dirichlet condition on the totality of C (see Fig. 3). In other words,we put Cout ¼ ; in (20). This simplifies the analysis enabling us to apply the classical theory for Stokes

problems [8,19]. In such situations, the fluid pressure is defined up to a constant, which we will fix by

assuming p to be of zero average,RXf pdx ¼ 0.

Remark 7. The regularity imposed on the reference flow ðu0; p0Þ is not optimal. We only have chosen, for

simplicity, spaces where the computations that we will carry out make sense.

In this framework, problem (20) reads: find k 2 C, u : Xf ! C3, p : Xf ! C and s 2 Cns , withRXf pdx ¼ 0

and ðu; p; sÞ 6¼ 0, such that


qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;


k2Msþ ðK þ B0Þs ¼ �ZcUTrðu; pÞnda;

ð22Þ

In order to have a better understanding of (22), let us first consider the particular case where the fluid is at

rest at equilibrium ððu0; p0Þ ¼ 0Þ, and where the structure is rigid in translation, ns ¼ 3 and U ¼ I , withdiagonal matrices M and K, M ¼ mI , K ¼ kI . Here, m, k > 0 are real given data. In this case we have

B0 ¼ 0. Hence, problem (22) yields

�2mdiv eðuÞ þ 1

qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;

u ¼ �ks; on c;

k2msþ ks ¼ �Zcrðu; pÞnda;

ð23Þ

which involves the Stokes equations. This kind of problem has been already proposed and completely

studied in [5], for the characterization of the vibration frequencies of a tube rack immersed in a viscous fluidat rest, see also [6]. Condition (23)4 allows to eliminate the displacement s, leading to a purely fluid problem

with non-local boundary conditions on c. In [5] it is shown that problem (23) has eigenvalues, and those can


be calculated from the characteristic values of a certain compact operator. Explicit estimates on the

eigenvalues are also obtained, proving that the eigenvalues with non-zero imaginary part are in finite number.

Numerical computations on (23) have been carried out in [4], corroborating the theory developed in [5].

Consider now the case where the structure is rigid immersed in a permanent flow ðu0; p0Þ 6¼ 0, with

ns ¼ 3, U ¼ I , M ¼ mI , K ¼ kI and where we neglect B0 and the gradient term ru0 in the transpiration

condition. The quadratic eigenvalue problem (22) becomes


qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;u ¼ �ks; on c;

k2msþ ks ¼ �Zcrðu; pÞnda:

ð24Þ

This problem, involving the linearized Navier–Stokes equations, has been proposed in [6,32] generalizing

(23) to the case of a tube rack placed in a cross-flow. To our knowledge, no rigorous analysis on this

problem has been carried out up to now.The purpose of the following sections (Sections 4.2 and 4.3) is to give a general analysis of (22) covering

then the above particular cases.

4.2. Eigenfrequency characterization

In this part we provide the spectral characteristics of (22). This is the aim of the following theorem:

Theorem 8. The eigenvalues of (22) are, at most, a countable sequence of complex numbers, each with finitemultiplicity, which can only cluster at infinity.

Remark 9. This result agrees with that obtained in [5], for (23), and that conjectured in [6,32], for (24), both

being particular cases of (22).

In the next paragraphs we present the detailed proof of Theorem 8. The main principle consists in

defining a compact operator T which characterizes the solutions of (22), see [5–7]. The classical theory of

Stokes problems [8,19], a specific decomposition of the vibration operator as well as fluid condensationtechniques will be used in this proof.

4.2.1. Linearization and shift: definition of operator TIn order to linearize the quadratic term in (22) we introduce the modal velocity z ¼ �ks 2 Cns , as a new

unknown, so that eigenproblem (22) can be rewritten as: find k 2 C, u : Xf ! C3, p : Xf ! C and s, z 2 Cns ,

withRXf pdx ¼ 0 and ðu; p; s; zÞ 6¼ 0, such that


qrp ¼ ku; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;u ¼ Uz�ru0Us; on c;� z ¼ ks;

M�1 ðK�

þ B0ÞsþZcUTrðu; pÞnda

�¼ kz:

ð25Þ


It is clear that (at least formally) problem (25) can be formulated in the classical form,

Ausz

0@

1A ¼ k

usz

0@

1A:

In fact, we would like to focus on the ‘‘inverse’’ problem, i.e., on finding the eigenvalues of A�1:

A�1

usz

0@

1A ¼ 1

k

usz

0@

1A;

in the hope that A�1 could be a compact operator. But A may have no inverse (k can be zero). In order to

overcome this difficulty, see [35], we introduce the following shift of variable:

k ¼ x� r; ð26Þwith r > 0 a real shift, to be fixed at a sufficiently large value, and x 2 C standing for the new unknown.

For this purpose, we therefore introduce the Hilbert space H ¼ L2ðXfÞ3 � Cns � Cns , and the operator

T : ðf ; h; gÞ 2 H ! T ðf ; g; hÞ ¼ ðu; z; sÞ 2 H 1ðXfÞ3 � Cns � Cns ;

where ðu; p; z; sÞ is defined as the ‘‘solution’’ (for r sufficiently large) of the following coupled problem

(obtained from (25)):


qrp þ ru ¼ f ; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;

u ¼ Uz�ru0Us; on c;

� zþ rs ¼ g;

M�1 ðK�


�þ rz ¼ h;

ð27Þ

with a pressure p of zero average,

p 2 L20ðXfÞ ¼ q 2 L2ðXfÞ

ZXf

qdx

��

¼ 0

�:

Remark 10. The underlying idea in the definition of operator T comes from the fact that if ðk; u; p; z; sÞ is aspectral solution of (25), then x ¼ kþ r satisfies xT ðu; z; sÞ ¼ ðu; z; sÞ, thus x 6¼ 0 and ð1=x; u; z; sÞ is an

eigenpair of T , and conversely, if x 6¼ 0 is an eigenvalue of T then 1=x� r is an eigenvalue in (25). Theproof of this equivalence will be detailed later (Theorem 20). But first, we must show that operator T is well

defined, that is we must prove that problem (27) has one and only one solution.

4.2.2. Vibrational decomposition

The proof of existence and uniqueness for problem (27) will use a decomposition of its solution ðu; pÞ inthe following way:

ðu; pÞ ¼ ðu1; p1Þ þ ðu2; p2Þ þ ðu3; p3Þ; ð28Þeach pressure pi being of zero average, where ðu1; p1Þ is solution of the purely fluid problem


ru0u1 þru1u0 � 2mdiv eðu1Þ þ1

qrp1 þ ru1 ¼ f ; in Xf ;

divu1 ¼ 0; in Xf ;u1 ¼ 0; on C;u1 ¼ �Ug; on c;

ð29Þ

ðu2; p2Þ is solution of


qrp2 þ ru2 ¼ 0; in Xf ;

divu2 ¼ 0; in Xf ;u2 ¼ 0; on C;u2 ¼ �ru0Us; on c;

ð30Þ

and ðu3; p3; sÞ is solution of


qrp3 þ ru3 ¼ 0; in Xf ;

divu3 ¼ 0; in Xf ;

u3 ¼ 0; on C;

u3 ¼ rUs; on c;

ðK þ r2M þ B0ÞsþZcUTrðu1; p1Þndaþ

ZcUTrðu2; p2Þndaþ

ZcUTrðu3; p3Þnda ¼ Mðhþ rgÞ:

ð31Þ

Clearly, if ðu1; p1Þ, ðu2; p2Þ, and ðu3; p3; sÞ are solutions of (29)–(31) respectively, the quadruplet

ðu; p; s; z ¼ rs� gÞ, defined by (28), is solution of (27). But the solution of the above subproblems requires

some preliminary results.

4.2.3. Technical lemmas

In the sequel we will use the following result, proved in [13,14]:

Lemma 11. Let v 2 C1ð�XXfÞ3 to be a smooth vector function which satisfies

divv ¼ 0; in Xf ;v ¼ 0; on c:

and x 2 H12ðcÞ3 be such thatZ

cx � nda ¼ 0: ð32Þ

Then we haveZcðrvxÞ � nda ¼ 0: ð33Þ

In other words, the volume preserving condition (32) is conserved by linear transport of velocity v.Concerning the solution of problems (29) and (30), we have the following result whose proof can be

found in Appendix A:

Theorem 12. Let l 2 L2ðXfÞ3 and n 2 H12ðcÞ3 withZ

cn � nda ¼ 0: ð34Þ


Then for any rP 3keðu0Þk0;1;Xf the following problem:

ru0wþrwu0 � 2mdiv eðwÞ þ 1

qrqþ rw ¼ l; in Xf ;

divw ¼ 0; in Xf ;w ¼ 0; on C;w ¼ n; on c;

ð35Þ

has an unique solution ðw; qÞ in H 1ðXfÞ3 � L20ðXfÞ, provided with the classic estimate

kqk0;Xf 6C1ðklk0;Xf þ jwj1;Xf þ rkwk0;Xf Þ; ð36Þ

where C1 > 0 represents a constant which only depends on q, m, u0 and Xf . Moreover, fixing ~rr > 3ku0k1;1;Xf forany rP ~rr we get the following estimates:

kwk0;Xf 6C2 1

�þ 1ffiffi

rp�

knk12;c

�þ 1

rklk0;Xf

�;

jwj1;Xf 6C3ð1þffiffir

pÞ knk1

2;c

�þ 1

rklk0;Xf

�;

ð37Þ

where C2 and C3 are positive constants which only depend on m, u0, ~rr and Xf .

The following corollary can then be directly obtained from the preceding theorem:

Corollary 13. Let ~rr > 3ku0k1;1;Xf and s 2 Cns . For rP ~rr problems (29) and (30) have an unique solution inH 1ðXfÞ3 � L2

0ðXfÞ, and we have the following estimates:

ku1k0;Xf 6C4 1

�þ 1ffiffi

rp�ðkgk þ kf k0;Xf Þ;

ju1j1;Xf 6C5ð1þffiffir

pÞðkgk þ kf k0;Xf Þ;

kp1k0;Xf 6C6ð1þffiffir

pþ rÞðkgk þ kf k0;Xf Þ;

ku2k0;Xf 6C4 1

�þ 1ffiffi

rp�ksk;

ju2j1;Xf 6C5ð1þffiffir

pÞksk;

kp2k0;Xf 6C6ð1þffiffir

pþ rÞksk;

ð38Þ

where C4, C5 are C6 are positive constants that only depend on q, m, u0, ~rr, U and Xf .

Proof. Since u1 and u2 are divergence free, we have to check the compatibility condition of the trace on c(34), i.e.Z

cðUgÞ � nda ¼ 0;

Zcðru0UsÞ � nda ¼ 0:

The first identity is obvious because, from (21),ZcUTnda ¼ 0;

and g is a given vector independent of space position. The second one can be obtained from a direct ap-

plication of Lemma 11, with v ¼ u0 and x ¼ Ujcs. The corollary holds after direct application of Theorem 12to problems (29) and (30). h


4.2.4. Load transfer operator

Corollary 13 allows us to completely determine ðu1; p1Þ from the data f and g, and ðu2; p2Þ as a linear

function of s. The existence of ðu1; p1Þ as a linear function of f and g means that we can consider the loadZcUTrðu1; p1Þnda

as a given right-hand side in (31)5. Since the structural displacement is by assumption a superposition of a

finite number of vibration modes, we can condense the fluid problem (30) in the definition of the second

fluid loadZcUTrðu2; p2Þnda ¼ F ðrÞs; ð39Þ

with F ðrÞ a ns � ns real matrix, associated to problem (30) and given by the following expression:

FijðrÞ ¼Zcðrðwj; qjÞnÞ � ui da;

where ðwj; qjÞ is the unique solution (see Corollary 13), of the fluid unit problem

ru0wj þrwju0 � 2mdiv eðwjÞ þ1

qrqj þ rwj ¼ 0; in Xf ;

divwj ¼ 0; in Xf ;

wj ¼ 0; on C;

wj ¼ �ru0uj; on c;

ð40Þ

for j ¼ 1; . . . ; ns.

Remark 14. Expression (39) is crucial in our approach, and explains our assumption of a reduced behavior

of the structure. In a more general case, for instance where the displacement of the structure satisfies the

linear elastodynamics equations, our approach would not be valid.

In the sequel we will need an estimate for the load transfer matrix F ðrÞ in terms of r. This is the object ofthe following lemma:

Lemma 15. Let ~rr > 3ku0k1;1;Xf . For each rP ~rr we have

kF ðrÞk6C7ð1þffiffir

pþ rÞ;

with C7 a positive constant which only depends on q, m, u0, ~rr, U and Xf .

Proof. For s 2 Rns , (u2; p2) is uniquely determined from Corollary 13. From (39) we get, up to some

multiplicative constants independent of r and ðu2; p2Þ, the following estimate:

kF ðrÞsk ¼ZcUTrðu2; p2Þnda

6 krðu2; p2ÞkHðdiv;Xf Þ

6 krðu2; p2Þk0;Xf þ kdivrðu2; p2Þk0;Xf

6 kp2k0;Xf þ ju2j1;Xf þ kru0u2 þru2u0k0;Xf þ rku2k0;Xf

6 kp2k0;Xf þ ju2j1;Xf þ rku2k0;Xf : ð41Þ


Corollary 13 provides estimates for ju2j1;Xf ; ku2k0;Xf and kp2k0;Xf depending on r and s. Thus, from (41)and (38)4;5;6, we get

kF ðrÞsk6 C6ð1�

þffiffir

pþ rÞ þ C5ð1þ

ffiffir

pÞ þ rC4 1

�þ 1ffiffi

rp��

ksk6C7ð1þffiffir

pþ rÞksk;

where C7 is a constant which neither depends on r, u2 nor s. Thus, we complete the proof of the lemma. h

Now let us take again the problem of determining (u2; p2) and (u3; p3; s). From (39) we can eliminate

(u2; p2) in (31)5 and, by multiplying by q Eq. (31)1, the original vibrational problem (27) finally reduces to:

find ðu3; p3; sÞ 2 H 1ðXfÞ3 � L20ðXfÞ � Cns verifying

qðru0u3 þru3u0Þ � 2ldiveðu3Þ þ rp3 þ rqu3 ¼ 0; in Xf ;

div u3 ¼ 0; in Xf ;

u3 ¼ 0; on C;

u3 ¼ rUs; on c;

ðK þ r2M þ B0 þ F ðrÞÞsþZcUTrðu3; p3Þnda ¼ Mðhþ rgÞ �

ZcUTrðu1; p1Þnda:

ð42Þ

Such a problem is well posed as indicated by the next theorem.

Theorem 16. There exists r > 3ku0k1;1;Xf such that the coupled problem (42) has an unique solution inH 1ðXfÞ3 � L2

0ðXfÞ � Cns . In addition, for each r sufficiently large, we have the estimate

ksk þ ju3j1;Xf þ kp3k0;Xf 6C8ðkf k0;Xf þ kgk þ khkÞ; ð43Þ

where C8 > 0 is a constant independent of (u3; p3; s) and (f ; g; h).

Proof. We introduce the following Hilbert space:

V ¼ V

8<: ¼ ðv; tÞ 2 H 1ðXfÞ3 � Cns

divv ¼ 0 in Xf

v ¼ 0 on Cv ¼ rUt on c

��9=;;

provided with the norm

kðv; tÞkV ¼ ðjvj21;Xf þ ktk2Þ12:

Let V ¼ ðu; tÞ 2 V, multiplying Eq. (42)1 by �vv and integrating by parts, we get

qðru0u3 þru3u0;�vvÞ0;Xf þ 2lðeðu3Þ; eð�vvÞÞ0;Xf þ rqðu3;�vvÞ0;Xf � rZcUTrðu3; p3Þnda ��tt ¼ 0:

Now, multiplying Eq. (42)5 by r�tt and by adding these two last expressions, we obtain the following vari-

ational formulation for the coupled problem (42): find (u3; sÞ 2 V such that

qðru0u3 þru3u0;�vvÞ0;Xf þ 2lðeðu3Þ; eð�vvÞÞ0;Xf þ rqðu3;�vvÞ0;Xf þ rðK þ B0Þs ��tt þ r3Ms ��tt þ rF ðrÞs ��tt

¼ rMðrg þ hÞ ��tt � rZcUTrðu1; p1Þnda ��tt; 8 2 ðv; tÞ 2 V: ð44Þ


For its analysis, we introduce the sesquilinear form A : V�V ! C and the antilinear form L : V ! C

defined by

Aððu; sÞ; ðv; tÞÞ ¼ qðru0uþruu0;�vvÞ0;Xf þ 2lðeðuÞ; eð�vvÞÞ0;Xf

þ rqðu;�vvÞ0;Xf þ rðK þ B0Þs ��tt þ r3Ms ��tt þ rF ðrÞs ��tt;

Lððv; tÞÞ ¼ rMðrg þ hÞ ��tt � rZcUTrðu1; p1Þnda ��tt;

for U ¼ ðu; sÞ and V ¼ ðv; tÞ in V. In this way, problem (44) takes the following classical form: find

U ¼ ðu3; sÞ 2 V such that

AðU ; V Þ ¼ LðV Þ; 8V 2 V: ð45ÞIn order to check the V ellipticity ofA, and hence the well posedness of problem (45), we integrate by parts

in Xf , using the fact that u0 is divergence free, and that kvk2u0 � n vanishes on C [ c (since u0 is solution of

the reference problem (2) in Xf ), to get the following identity:

2Reðrvu0;�vvÞ0;Xf ¼ ðrvu0;�vvÞ0;Xf þ ðrvu0;�vvÞ0;Xf ¼ ðrvu0;�vvÞ0;Xf þ ðv;r�vvu0Þ0;Xf

¼ ðrvu0;�vvÞ0;Xf þZXf

ðv� u0Þ : r�vvdx

¼ ðrvu0;�vvÞ0;Xf �ZXf

ðrvu0 þ vdivu0Þ � �vvdxþZC[c

kvk2u0 � nda

¼ ðrvu0;�vvÞ0;Xf � ðrvu0;�vvÞ0;Xf ¼ 0: ð46Þ

Let m ¼ min rðMÞ > 0 the smallest eigenvalue of the mass matrix M . We fix ~rr > 3ku0k1;1;Xf and we takerP ~rr. From (46), we obtain that for each V ¼ ðv; tÞ 2 V

ReAðV ; V Þ ¼ qReðru0v;�vvÞ0;Xf þ 2lkeðvÞk20;Xf þ rqkvk20;Xf þ r3Mt ��tt þ rRe½ðK þ B0Þt ��tt � þ rReðF ðrÞt ��ttÞ

P qðeðu0Þv;�vvÞ0;Xf þ 2lkeðvÞk20;Xf þ rqkvk20;Xf þ r3mktk2 þ rRe½ðK þ B0Þt ��tt � þ rReðF ðrÞt ��ttÞ

P 2lkeðvÞk20;Xf � q3keðu0Þk0;1;Xfkvk20;Xf þ rqkvk20;Xf þ rðr2m� kK þ B0k � kF ðrÞkÞktk2

P 2lkeðvÞk20;Xf þ r r2m� K þ B0 � C7 1þ

ffiffir

pþ r

�� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}aðrÞ

ktk2; ð47Þ

where the last inequality is a consequence of Lemma 15. We take rP ~rr sufficiently large such as aðrÞP 0,

which is possible because the dominating term grows at infinity as r2m, with m > 0. Then, the last inequality

in (47) yields

ReAðV ; V ÞP 2lkeðvÞk20;Xf þ raðrÞktk2 P minf2l; raðrÞgðkeðvÞk20;Xf þ ktk2Þ:

This, with Korn�s inequality, implies that A is V-elliptic, i.e.,

ReAðV ; V ÞP bkðv; tÞk2V; 8V 2 C;

where b > 0 depends on M , K, B0, q, l, u0, r and Xf . Consequently, Lax-Milgram�s theorem gives the

existence and uniqueness of ðu3; sÞ 2 V as solution of (42). The pressure p3 is obtained from Theorem 12.

We also obtain the estimate

kðv; tÞkV 6kLkV0

b:


This last inequality combined with the estimate (36) for p3 and the estimates (38)1;2;3, for u1 and p1, allow us

to obtain estimate (43), which completes the proof. h

4.2.5. Existence of operator TWe are now ready to prove an existence and uniqueness result for the vibrational problem (27). This is

the object of the following theorem:

Theorem 17. There exists r > 3ku0k1;1;Xf such that problem (27) has a unique solution ðu; p; sÞ 2 H 1ðXfÞ3�L20ðXfÞ � Cns . In addition, for each r sufficiently large, we have the estimate

ksk þ juj1;Xf þ kpk0;Xf 6C9ðkf k0;Xf þ kgk þ khkÞ; ð48Þ

where C9 is a positive constant independent of (u; p; s) and (f ; g; h).

Proof. First, the existence is a direct consequence of decomposition (28) and of the above lemmas. Indeed,

Corollary 13 allows to determine first (u1; p1), then we define problem (42). From Lemma 15 and Theorem

16 we can determine (u3; p3; s). Finally, once s known, we can compute (u2; p2) in (30) from Corollary 13, or

simply from the following expression:

ðu2; p2Þ ¼Xnsj¼1

sjðwj; qjÞ;

where (wj; qj) is the solution of (40).

By linearity in problem (27), the uniqueness of solution can be reduced to prove that problem (27)

provided with zero data only admits the trivial solution. In other words, we should prove that if

ðu; p; sÞ 2 H 1ðXfÞ3 � L20ðX

fÞ � Cns satisfies


qrp þ ru ¼ 0; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;

u ¼ rUs�ru0Us; on c;

ðK þ B0ÞsþZcUTrðu; pÞndaþ r2Ms ¼ 0;

ð49Þ

then ðu; p; sÞ ¼ 0.

In this way, let us suppose that ðu; p; sÞ 2 H 1ðXfÞ3 � L20ðX

fÞ � Cns is a solution of (49). Then we define

(~uu; ~pp) as a solution of

ru0~uuþr~uuu0 � 2mdiv eð~uuÞ þ 1

qr~pp þ r~uu ¼ 0; in Xf ;

div ~uu ¼ 0; in Xf ;

~uu ¼ 0; on C;~uu ¼ �ru0Us; on c:

Corollary 13 ensures the existence and uniqueness of (~uu; ~pp). In addition, from (39), we get

F ðrÞs ¼ZcUTrð~uu; ~ppÞnda:

We set ðuu; ppÞ ¼ ðu; pÞ � ð~uu; ~ppÞ. Thus from (49) we obtain that (uu; pp; s) is a solution of the following problem:


ru0uuþruuu0 � 2mdiv eðuuÞ þ 1

qrpp þ ruu ¼ 0; in Xf ;

div uu ¼ 0; in Xf ;

uu ¼ 0; on C;uu ¼ rUs; on c;

ðK þ r2M þ B0 þ F ðrÞÞsþZcUTrðuu; ppÞnda ¼ 0:

This is a coupled problem (42), supplied with zero data. Then, from Theorem 16 we get uu ¼ 0, pp ¼ 0 and

s ¼ 0. In particular, this last identity implies that ~uu ¼ 0 and ~pp ¼ 0, which gives u ¼ 0 and p ¼ 0.

Finally, inequality (48), giving the continuity of operator T , comes from the estimates on (u1; p1) and(u2; p2) (given by Corollary 13) and from inequality (43) in the preceding lemmas. h

Remark 18. The main idea hidden in the above proof lies in the vibrational decomposition and in the load

transfer operator introduced in Sections 4.2.2 and 4.2.4, respectively. The proof is constructive in the sense

that the same argument motivates a modular algorithm for the numerical solution of (27), as reported inPart II (see [15, Section 4.4]).

4.2.6. Compactness of operator T: proof of Theorem 8

From Theorem 17, operator T is well defined, linear and continuous from H ¼ L2ðXfÞ3 � Cns � Cns to

H 1ðXfÞ3 � Cns � Cns . In addition, since H 1ðXfÞ3 � Cns � Cns ,!H is a compact embedding, the continuity of

T leads to the compactness of T as an operator from H to H [7]. The following classical result, concerning

the spectrum of compact operators [1], then applies:

Theorem 19. Let H and T be, respectively, a Hilbert space and a compact operator on H. The non-zeroelements of the spectrum of T are eigenvalues with finite multiplicity, which can only cluster at 0. Theseeigenvalues are, at most, a countable infinite set. Zero always belongs to the spectrum.

The following theorem then characterizes the eigenvalues of (25) from the non-zero eigenvalues of the

operator T :

Theorem 20. For r sufficiently large we have that, if (k; u; p; s; z) is an eigenpair of (25) then1

kþ r; u; s; z

� �;

is an eigenpair of T , with non-zero eigenvalue, and conversely.

Proof. Let us fix r as in Theorem 17. If ðk; u; p; s; zÞ is a solution of (25), we obtain by definition of an

eigenvector that ðu; p; s; zÞ 6¼ 0. This implies that ðu; s; zÞ 6¼ 0. Indeed, otherwise we would have, from (25),

that rp ¼ 0 in Xf , implying p ¼ 0, because the pressure is defined of zero average. We take x ¼ kþ r.From (25) we get


qrp þ ru ¼ xu; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;u ¼ Uz�ru0Us; on c;� zþ rs ¼ xs;

M�1 ðK�


�þ rz ¼ xz:


Hence, from definition of T (27), we get

xT ðu; s; zÞ ¼ ðu; s; zÞ:But, x cannot be zero since we have just proved that ðu; s; zÞ 6¼ 0. Therefore, we can write

T ðu; s; zÞ ¼ 1

xðu; s; zÞ:

Consequently, if ðk; u; p; s; zÞ is an eigensolution of (25), then

1

x¼ 1

kþ r;

is an eigenvalue of T and ðu; z; sÞ an associated eigenvector.Conversely, if x 6¼ 0 is an eigenvalue of T and ðu; s; zÞ an associated eigenvector, namely,

T ðu; s; zÞ ¼ xðu; s; zÞ;then from the definition of T (27), we get that ðxu;xs;xzÞ satisfies

xru0uþ xruu0 � x2mdiv eðuÞ þ 1

qrp þ xru ¼ u; in Xf ;

divxu ¼ 0; in Xf ;

xu ¼ 0; on C;

xu ¼ xUz� xru0Us; on c;

� xzþ xrs ¼ s;

M�1 ðK�

þ B0ÞxsþZcUTrðxu; pÞnda

�þ xrz ¼ z:

After division by x 6¼ 0, and by denoting q ¼ p=x we obtain


qrq ¼ 1

x� r

� �u; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;

u ¼ Uz�ru0Us; on c;

� z ¼ 1

x

�� r�s

M�1 ðK�

þ B0ÞsþZcUTrðu; qÞnda

�¼ 1

x

�� r�z;

which implies that

1

x

�� r; u;

px; s; z

�;

is a solution of (25).

In summary, the eigenvalues of (25) are in the form

k ¼ 1

x� r;

where x 6¼ 0 is an eigenvalue of T . h


Finally, Theorem 8 holds by combining Theorem 20, the compactness of operator T and Theorem 19.

4.3. Eigenfrequency localization

In this part we address the problem of determining a region in the complex plan containing the eigen-

frequencies of the coupled system. This is the aim of the following theorem:

Theorem 21. The eigenvalues k of (22) lie in the parabolic region

ðImkÞ2 6 aRekþ b; ð50Þ

see Fig. 4, where a and b are two positive constants independent of k.

Proof. As in Section 4.2.1, we introduce in (22) the shifted variable x ¼ kþ r, with r > 0 a real shift to be

fixed at a sufficiently large value. Thus, from (22) we get

qðru0uþruu0Þ � 2ldiv eðuÞ þ rp þ rqu ¼ xqu; in Xf ;

divu ¼ 0; in Xf ;

u ¼ 0; on C;


k2Msþ ðK þ B0Þs ¼ �ZcUTrðu; pÞnda:

ð51Þ

Following the argument introduced in Section 4.2.2, we decompose ðu; pÞ in two vibrational parts:

ðu; pÞ ¼ ðu1; p1Þ þ ðu2; p2Þ;

Fig. 4. A parabolic region in the complex plan containing the eigenfrequencies of the spectral operator (22).


where ðu1; p1Þ is solution of

qðru0u1 þru1u0Þ � 2ldiv eðu2Þ þ rp1 þ rqu1 ¼ 0; in Xf ;divu1 ¼ 0; in Xf ;u1 ¼ 0; on C;u1 ¼ �ru0Us; on c;

ð52Þ

and ðu2; p2; sÞ is solution of

qðru0u2 þru2u0Þ � 2ldiv eðu2Þ þ rp2 þ rqu2 ¼ xqðu2 þ u1Þ; in Xf ;

divu2 ¼ 0; in Xf ;

u2 ¼ 0; on C;

u2 ¼ �kUs; on c;

k2Msþ ðK þ B0ÞsþZcUTrðu1; p1Þnda ¼ �

ZcUTrðu2; p2Þnda:

ð53Þ

We fix now r > 3keðu0Þk0;1;Xf as large as in Corollary 13. Thus, using the load transfer operator in-troduced in Section 4.2.4 for such values of r, we obtainZ

cUTrðu1; p1Þnda ¼ F ðrÞs:

Therefore, from (53) we get

qðru0u2 þru2u0Þ � 2ldiv eðu2Þ þ rp2 þ rqu2 ¼ xqðu1 þ u2Þ; in Xf ;

divu2 ¼ 0; in Xf ;

u2 ¼ 0; on C;u2 ¼ �kUs; on c;

k2Msþ ðK þ B0 þ F ðrÞÞs ¼ �ZcUTrðu2; p2Þnda:

ð54Þ

Without loss of generality, we will suppose that k 6¼ 0 (if k ¼ 0, then it lies in our parabolic region) and

we normalize ðu; sÞ by dividing it by ku2k1;Xf þ ksk (the case s ¼ 0 and u2 ¼ 0 will imply u1 ¼ 0 from (52),

which is impossible because ðu; p; sÞ is an eigenvector). In the sequel ci will denote (i ¼ 1; 2; . . .) a positive

constant that depends, at most, on q, l, u0, r, U, M , K, B0 and Xf . From estimates (38)4;5, provided in

Corollary 13, we get

ku1k1;Xf 6 c1ksk: ð55Þ

In addition, from (54)4, we can easily verify that

ksk6 c2jkj ku2k1;Xf : ð56Þ

Multiplying Eq. (54)1 by �uu2 and integrating by parts, we also get

qðru0u2 þru2u0; �uu2Þ0;Xf þ 2lkeðu2Þk20;Xf þZcUTrðu2; p2Þnda � ð�kk�ssÞ ¼ kqku2k20;Xf þ xqðu1; �uu2Þ0;Xf :

By multiplying Eq. (54)5 by �kk�ss and by substraction, we obtain the following fundamental energy

equality:

2lkeðu2Þk20;Xf ¼ kðqku2k20;Xf þ jkj2Ms � �ssÞþ ðK þ B0 þ F ðrÞÞs � ð�kk�ssÞ þ xqðu1; �uu2Þ0;Xf � qðru0u2 þru2u0; �uu2Þ0;Xf|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Rðu1;u2;s;kÞ

: ð57Þ


From (55) and (56) and since ðu; sÞ has been normalized we get, for jkjP 1,

jRðu1; u2; s; kÞj6 K þ B0 þ F ðrÞ

kskkksk þ qjkþ rjku1k0;Xfku2k0;Xf

þ q3kru0k0;1;Xfku2k20;Xf þ qffiffiffi3

pku0k0;1;Xf ju2j1;Xfku2k0;Xf

6c3jkj ku2k

21;Xf þ c4ku2k0;Xfku2k1;Xf þ c5ku2k20;Xf

6 c6: ð58Þ

Thus, by dividing (57) by jkj, for jkj6 1 we obtain that

qku2k20;Xf þ jkj2Ms � �ss6 1

jkj ð2lkeðu2Þk20;Xf þ jRðu1; u2; s; kÞjÞ6

c7jkj ;

which implies that

ku2k0;Xf 6c8ffiffiffiffiffiffijkj

p ; for jkjP 1: ð59Þ

By using this last estimate in (58), we obtain

jRðu1; u2; s; kÞj6c9ffiffiffiffiffiffijkj

p ; for jkjP 1: ð60Þ

Therefore, there exists a positive constant Kmin P 1 such that

jRðu1; u2; s; kÞj6 lkeðu2Þk20;Xf for jkjPKmin: ð61Þ

On the other hand, from (59) and (56), the Korn�s inequality and since ku2k1;Xf þ ksk ¼ 1 by con-

struction, we also have that

keðu2Þk20;Xf P c10; for jkjPKmin: ð62Þ

The real part of (57) reads now

2lkeðu2Þk20;Xf ¼ Rekðqku2k20;Xf þ jkj2Ms � �ssÞ þRe½Rðu1; u2; s; kÞ�:

Thus, from (61) we obtain

Rek qku2k20;Xf

�þ jkj2Ms � �ss

�P lkeðu2Þk20;Xf :

for jkjPKmin. This, with (62), implies that

Rek > 0;

qku2k20;Xf þ jkj2Ms � �ssP c11Rek

;

(for jkjPKmin: ð63Þ

On the other hand, by taking the imaginary part of (57) we get

Imk qku2k20;Xf

�þ jkj2Ms � �ss

�¼ �Im½Rðu1; u2; s; kÞ�:

Hence, from (63) and (60) and for jkjPKmin we obtain that

jImkj6 jIm½Rðu1; u2; s; kÞ�jqku2k20;Xf þ jkj2Ms � �ss

6jRðu1; u2; s; kÞj

c11Rek6

c9c11

Rekffiffiffiffiffiffijkj

p 6c9c11

ffiffiffiffiffiffiffiffiffiRek

p;

that is

ðImkÞ2 6 c12 Rek; forjkjPKmin:


In summary, the eigenvalues k of (22) with jkjPKmin are located in the interior of a parabola, in thecomplex plan, whose equation is

ðImkÞ2 ¼ c12 Rek: ð64Þ

Hence, by shifting the parabola (64) to the left, we obtain that the eigenvalues k of (22) can be located in

the parabolic region

ðImkÞ2 6 c12 Rekþ c13;

which completes the proof. h

Remark 22. Estimate (50) is a standard result in hydrodynamic stability theory [18,35]. Theorem 21 gen-

eralizes it to the case of linear stability analysis in fluid–structure interaction.

Theorems 8 and 21 guarantee that

(a) the eigenvalues can not cluster at a leftmost eigenvalue. That is, there is at most a finite number of ei-

genvalues with negative real part,(b) the imaginary part of the leftmost eigenvalues remains bounded.

This implies that most of the eigenvalues of (22) have positive real part and only a small number cross

the imaginary axis. Therefore, to detect a stability change, we only need to compute the few eigenvalues

with smallest real part, i.e. the critical eigenfrequencies of the system. This is the aim of the numerical study

reported in Part II [15].

Remark 23. Properties (a) and (b) are crucial for the reliability of the numerical algorithm used in Part IIfor the approximation of the leftmost eigenvalues of (22), see [15, Section 4.2].

5. Conclusion

The new Linearization Principle approach presented above handles the interaction of an incompressible

Newtonian fluid and of a reduced structure, in large displacements. Based on a linearization–transpiration

method developed in [12,13], we have obtained a new spectral problem coupling the linearized Navier–Stokes equations and those of a reduced linear structure through transpiration conditions on a fixed in-

terface. This allows us to keep a fixed fluid domain and, from a numerical point of view, to overcome all

difficulties arising when dealing with moving grids. In short, we have obtained an eigenproblem of minimal

complexity.

We have characterized and localized the eigenvalues of this new eigenproblem. Indeed, we have shown

that the spectrum consists of a discrete set of complex eigenvalues, each of finite multiplicity, which can

cluster only at infinity. This result is obtained by proving that the eigenproblem can be reduced to that of

finding the characteristic values of a compact operator acting in a Hilbert space. On the other hand, it hasbeen proved by a fundamental energy estimate that the eigenvalues lie in a parabolic region of the complex

plan centered along the positive real axis. In particular, there is at most a finite number of eigenvalues with

negative real part (unstable modes).

The real mechanical and numerical issue is now to obtain reliable numerical predictions of the physical

stability of the coupled system, by computing the leftmost eigenvalues of (20), namely, the critical eigen-

frequencies of the system. This is the subject of Part II (see [15]).


Appendix A

We give hereafter the proof of Theorem 12.

Let us consider the following Hilbert space

H ¼ v 2 H 1ðXfÞ3 divv ¼ 0 in Xf

v ¼ 0 on C [ c

��

;

provided with the H 1ðXfÞ3-seminorm (from the Poincar�ee�s inequality [2] this seminorm becomes a norm), a

divergence free continuous linear lift operator (see [19]),

R : H12ðcÞ3 ! v 2 H 1ðXfÞ3 divv ¼ 0 in Xf

v ¼ 0 on C

��

;

defined for any n 2 H12ðcÞ3 such that

Rc n � nda ¼ 0, and the following continuous sesquilinear form:

a : ðw; vÞ 2 H � H ! aðw; vÞ ¼ a0ðw; vÞ þ rðw;�vvÞ0;Xf 2 C;

with

a0ðw; vÞ ¼ ðru0wþrwu0;�vvÞ0;Xf þ 2mðeðwÞ; eð�vvÞÞ0;Xf :

With this notation, problem (35) can be reformulated in the following variational framework: find

w 2 H 1ðXfÞ3 such that

w� RðnÞ 2 H ;

a0ðw; vÞ þ rðw;�vvÞ0;Xf ¼ ðl;�vvÞ0;Xf ; 8v 2 H :ðA:1Þ

Thus, we obtain that ww ¼ w� RðnÞ 2 H solves the following internal variational problem: find ww 2 Hsuch as

a0ðww; vÞ þ rðww; vÞ0;Xf ¼ ðl;�vvÞ0;Xf � a0ðRðnÞ; vÞ � rðRðnÞ;�vvÞ0;Xf ; 8v 2 H : ðA:2Þ

At this point, we introduce a continuous antilinear form

L : v 2 H ! LðvÞ ¼ ðl;�vvÞ0;Xf � a0ðRðnÞ; vÞ � rðRðnÞ;�vvÞ0;Xf ;

so that problem (A.2) takes the following classic formalism: find ww 2 H such as

aðww; vÞ ¼ LðvÞ; 8v 2 H : ðA:3ÞBy construction we have

Reaðv; vÞ ¼ 2mkeðvÞk20;Xf þReðru0v;�vvÞ þReðrvu0;�vvÞ þ rkvk20;Xf : ðA:4Þ

By using the fact that u0 is divergence free and that v cancels on C [ c, we obtain, on the one hand,

2Reðrvu0;�vvÞ0;Xf ¼ ðrvu0;�vvÞ0;Xf þ ðruv0;�vvÞ0;Xf ¼ ðrvu0;�vvÞ0;Xf þ ðv;r�vvu0Þ0;Xf

¼ ðrvu0;�vvÞ0;Xf þZXf

ðv� u0Þ : r�vvdx

¼ ðrvu0;�vvÞ0;Xf �ZXf

ðrvu0 þ vdivu0Þ � �vvdxþZC[c

kvk2u0 � nda

¼ ðrvu0;�vvÞ0;Xf � ðrvu0;�vvÞ0;Xf ¼ 0; ðA:5Þ


and on the other hand,

2Reðru0v;�vvÞ0;Xf ¼ 1

2ðru0v;�vvÞ0;Xf þ ðru0v;�vvÞ0;Xf

h i¼ 1

2ðru0v;�vvÞ0;Xf þ ðv;ru0�vvÞ0;Xf

h i¼ 1

2ðru0v;�vvÞ0;Xf þ ððru0ÞTv;�vvÞ0;Xf

h i¼ ðeðu0Þv;�vvÞ0;Xf : ðA:6Þ

Hence, from (A.4)–(A.6), we get

Reaðv; vÞP 2mkeðvÞk20;Xf � 3keðu0Þk0;1;Xfkvk20;Xf þ rkvk20;Xf :

Therefore, by taking rP 3keðu0Þk0;1;Xf and from Korn�s inequality [2], we deduce that the sesquilinear

form a is H -elliptic, i.e.,

Reaðv; vÞP 2mkeðvÞk20;Xf P ajvj21;Xf ; 8v 2 H ;

where a > 0 stands for a constant which only depends on m and Xf . Lax-Milgram�s theorem [19] ensures the

existence and uniqueness of ww as solution of (A.3). We also obtain that w ¼ wwþ RðnÞ is a solution of (A.1).

The uniqueness of w comes from (A.1) and from Lax-Milgram�s theorem. In addition, since v cancels on

C [ c, integrating by parts in (A.1)2 yields

ðru0wþrwu0 � 2mdiv eðwÞ þ rw� l;�vvÞ0;Xf ¼ 0; 8v 2 H :

Consequently, there exists (see [8, Propositions 1 and 2 of Chapter 9]) an unique distribution q 2 L20ðXÞ

such that

ru0wþrwu0 � 2mdiv eðwÞ þ rw� l ¼ � 1

qrq; in D0ðXfÞ3;

kqk0;Xf 6 ckru0wþrwu0 � 2mdiv eðwÞ þ rw� lk�1;Xf ;

ðA:7Þ

where c represents a positive constant which only depends on q and Xf . On the one hand, equality (A.7)1allows us to complete the proof of existence and uniqueness of solution for (35) and, on the other hand,

from (A.7)2 we directly obtain the estimate (36). Obtaining the inequalities (37)1;2 requests now a finer

analysis.

By considering the real part of the expression (A.2) with v ¼ ww, and from (A.5) and (A.6) we obtain

ðeðu0Þww; wwÞ0;Xf þ 2mkeðwwÞk20;Xf þ rkwwk20;Xf ¼ Reððl; wwÞ0;Xf � a0ðRðnÞ; wwÞ � rðRðnÞ; wwÞ0;Xf Þ

which implies

� 3keðu0Þk0;1;Xfkwwk20;Xf þ 2mkeðwwÞk20;Xf þ rkwwk20;Xf

6 jðl; wwÞ0;Xf � a0ðRðnÞ; wwÞ � rðRðnÞ; wwÞ0;Xf j6 ðklk0;Xf þ rkRðnÞk0;Xf þ kru0RðnÞk0;Xf þ krRðnÞu0k0;Xf Þkwwk0;Xf þ 2mkeðRðnÞÞk0;XfkeðwwÞk0;Xf

6 klk0;Xf

�þ rkRðnÞk0;Xf þ 3kru0k0;1;XfkRðnÞk0;Xf þ

ffiffiffi3

pku0k0;1;XfkrRðnÞk0;Xf

�kwwk0;Xf

þ 2mkRðnÞk1;XfkeðwwÞk0;Xf

6 ½klk0;Xf þ rkRðnÞk1;Xf þ ð3þffiffiffi3

pÞku0k1;1;XfkRðnÞk1;Xf �kwwk0;Xf þ 2mkRðnÞk1;XfkeðwwÞk0;Xf : ðA:8Þ

Thus, from the continuity of R, from H12ðcÞ3 in H 1ðXfÞ3 and by supposing that rP ku0k1;1;Xf the last

inequality of (A.8) yields


� 3keðu0Þk0;1;Xf|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}r0

kwwk20;Xf þ 2mkeðwwÞk20;Xf þ rkwwk20;Xf

6 2mc1knk12;ckeðwwÞk0;Xf þ c2ðklk0;Xf þ rknk1

2;cÞkwwk0;Xf

¼ 2m c1knk12;c|fflfflfflffl{zfflfflfflffl}

2a

keðwwÞk0;Xf|fflfflfflfflfflffl{zfflfflfflfflfflffl}x

þr c21

rklk0;Xf þ knk1

2;c

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2b

kwwk0;Xf|fflfflffl{zfflfflffl}y

; ðA:9Þ

where c1 and c2 are positive constants that only depend on u0 and Xf .

By taking into account this new notation, and from (A.9), we obtain the following inequality

�r0y2 þ 2mx2 þ ry2 6 ð2mÞð2aÞxþ r2by:

We can complete the squares here, and then we get

2mðx� aÞ2 þ rðy � bÞ2 6 2ma2 þ rb2 þ r0y2: ðA:10ÞIn the sequel we will take r > 3ku0k1;1;Xf which implies r > r0. On the one hand, from inequality (A.10),

we get

rðy � bÞ2 6 2ma2 þ rb2 þ r0y2;

i.e., after division by r

ðy � bÞ2 6 2mra2 þ b2 þ r0

ry2:

By taking the square root, and since a2 þ b2 þ 12 6 ðaþ bþ 1Þ2 for each a; b, 1P 0, we deduce

jy � bj6ffiffiffiffiffi2m

p ffiffir

p aþ bþffiffiffiffir0r

ry;

which implies

y6

ffiffiffiffiffi2m

p ffiffir

p aþ 2bþffiffiffiffir0r

ry;

that is

y 1

��

ffiffiffiffir0r

r �6

ffiffiffiffiffi2m

p ffiffir

p aþ 2b;

i.e., finally

y61

1�ffiffiffiffir0r

r ffiffiffiffiffi2m

p ffiffir

p a

þ 2b

!: ðA:11Þ

Moreover, since

r 2 ðr0;þ1Þ 7!1�ffiffiffiffiffiffir0r;

ris an increasing function, for each fixed ~rr > 3ku0k1;1;Xf P r0 we have

1�ffiffiffiffir0r

rP 1�

ffiffiffiffiffiffir0~rr;

r8rP ~rr:


From (A.11) and the preceding bound we arrive to the following estimate

y6 c31ffiffir

p a�

þ b�; 8rP ~rr: ðA:12Þ

where c3 > 0 is a constant which only depends on r0, ~rr and m.In the same way, from (A.10), we have

2mðx� aÞ2 6 2ma2 þ rb2 þ r0y2;

that is

ðx� aÞ2 6 a2 þ r2m

b2 þ r02m

y2:

By taking the square root, we obtain

jx� aj6 aþffiffir

pffiffiffiffiffi2m

p bþffiffiffiffiffir02m

ry;

which yields

x6 2aþffiffir

pffiffiffiffiffi2m

p bþffiffiffiffiffir02m

ry:

Thus, with (A.12) and the preceding inequality we obtain that for rP ~rr

x6 2aþffiffir

pffiffiffiffiffi2m

p bþ c3

ffiffiffiffiffir02m

r1ffiffir

p a�

þ b�;

from where we deduce

x6 2

�þ c3

ffiffiffiffiffir02m

r1ffiffir

p�aþ

ffiffir

pffiffiffiffiffi2m

p�

þ c3

ffiffiffiffiffir02m

r �b:

This can be written as

x6 c4½aþ ðffiffir

pþ 1Þb�; 8rP ~rr; ðA:13Þ

where c4 > 0 represents a constant which only depends on r0, ~rr and m.Therefore, from (A.12) and (A.13) and the considered notation, we have for each rP ~rr

keðwwÞk0;Xf 6 c4c12knk1

2; c

�þ ð

ffiffir

pþ 1Þ c2

2

1

rklk0;Xf

�þ knk1

2;c

��6 c5ð1þ

ffiffir

pÞ knk1

2;c

�þ 1

rklk0;Xf

�;

kwwk0;Xf 6 c31ffiffir

p c12knk1

2;c

�þ c2

2

1

rklk0;Xf

�þ knk1

2;c

��6 c6 1

�þ 1ffiffi

rp�

knk12;c

�þ 1

rklk0;Xf

�;

ðA:14Þwith c5 and c6 positive constants which only depend on m, u0, ~rr and Xf . The estimates (37)1;2 are obtained

from the fact that w ¼ wwþ RðnÞ and using Korn�s inequality in (A.14), which completes the proof of the

theorem.

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Linear stability analysis in fluid–structure interaction with transpiration. Part I: Formulation and mathematical analysis