Fluid structure interaction problem with changing thickness beam and slightly compressible fluid

In this work, we consider the dynamical response of a non-linear beam with changing thickness, perturbed in both vertical and axial directions, interacting with a Darcy flow. We explore this fluid-structure interaction problem where the fluid is assumed to be slightly compressible. In an appropriate Sobolev norm, we build an energy functional for the displacement field of the beam and the gradient pressure of the fluid flow. We show that for a class of boundary conditions the energy functional is bounded by the flux of mass through the inlet boundary.


1.
Introduction. The dynamics of non-linear thin structures interacting with fluid flows attracted the attention of many researchers in recent years (see e.g. Refs. [4], [8], [9], [11], [25] and references therein). This problem is inherent to a number of topics in bio-medicine, engineering, geo-sciences, etc. It was observed for many years that ignoring non-linearity in solid part can lead to significant errors in forecasting the dynamics of elastic structures (see Refs. [24], [20]). We want specifically refer on an excellent paper by H. Koch, and I.Lasiecka [17] in which a comprehensive review of Hadamard well-posedness of corresponding IBVP with different homogeneous (zero) boundary conditions and RHS in the system of the equation is presented. These are particular important in exploring the stability of the solid structure subjected to the loads generated by the fluid flow. In turn, the fluid domain is under the impact of the solid body as a moving boundary. The mutual interaction of fluid-structure system is a difficult problem and is far from being fully understood. Biggest progress so far made for solids with non-changing thickness, and incompressible fluid. Fluid-plate interaction between elastic plates and viscous incompressible fluids (described by Navier Stokes or Stokes equations) have also been recently studied in great detail (see e.g. [4], [6], [7], [9], [10], [11], [21] and references therein). Existence of the weak solutions for the coupled systems has been proven in [9], [10]. Existence of solutions to the free boundary fluid-structure interaction system was proven for Navier-Stokes in the fluid part and wave equation for solid part defined in two different but adjacent domains in [18]. This type of model was first introduced in [5] for incompressible fluid, and in [3] for compressible fluid. Long term dynamics of the solutions has been studied in [4], and strong stability estimates have been derived under additional assumptions on the data [6]. In these studies, a more simple plate model is used compared to the von Karman 1134 EUGENIO AULISA, AKIF IBRAGIMOV AND EMINE YASEMEN KAYA-CEKIN model; namely, either in-plane displacements ( [6], [7], [9]) or transverse displacements ( [4], [10], [11]) of the plate are disregarded. Moreover, the fluid domain is in most cases subjected to small deformations and treated as a pure Eulerian system. In [18] most current review of the results was presented in big detail. The biggest progress so far is achieved in numerical studies and is based on the machinery developed to simulate moving boundaries (see e.g. Refs. [2], [23], [22]) and references therein). In the majorities of these studies, • thickness of the plate is assumed to be constant. However, in the present work, we consider a non-linear beam model derived in [13], where the change in the thickness of the beam is taken into account by introducing it into the momentum of inertia under the incompressibility constraint of the solid material. So, even if we use a one dimensional structure for the solid part of our model, considering the change in the thickness of this structure makes this study quite interesting. • either in-plane displacements (Refs. [6], [7], [9]) or transverse displacement (Refs. [4], [10], [11]) of the plate is neglected relative to the other. In our non-linear beam model however, longitudinal and transverse displacements of the neutral axis are fully coupled as a system of equations. • the fluid domain is in most cases subjected to small deformations (and treated as a pure Eulerian system) and, to our knowledge, no fluid has been considered to be compressible. The fluid flow in the present work is described by the linear Darcy's momentum equation but the fluid is considered to be (slightly) compressible.
In this work we primarily investigate mathematical aspects of the dynamics of a one dimensional non-linear beam interacting with a two dimensional potential flow. Having in mind a class of geofluidics problems, we consider the elastic beam as the top boundary of a porous media domain, saturated with one phase slightly compressible fluid. In the current paper we assume that the fluid-structure interaction occurs directly on the boundary between beam and fluid domain. The porous media is represented only as the moving domain of the liquid flow. Although this an idealization of complex processes, it will provide important understanding of the impact of parameters on the dynamics of the coupled system. This article is partially motivated by our previous works Refs. [16], [15], [13], [14], and can be considered as extension of the analysis of two coupled models derived in [14] for fluid part and [13] for solid part. Energy estimates were derived for a nonlinear beam (with constant thickness [15] and changing thickness [13]) perturbed in both the transverse and axial directions and interacting with an incompressible potential flow. In the present work considering a Darcy type potential flow of the compressible fluid makes the problem more difficult in a theoretical and practical point of view. Considering a non-linear Forchheimer model [1] instead of Dsarcy would make the problem even more interesting and challenging but we leave this case to upcoming research.
In this work, we introduce the fluid-beam coupled model and investigate the stability of this system with respect to the inflow (accessible) boundary condition. The difficulty here is that the fluid domain changes as the beam deforms which requires an arbitrary Lagrangian-Eulerian (ALE) formulation for the fluid part. The system is coupled on the top boundary through the pressure exerted from the fluid to the beam. We construct an energy balance equation which relates energy functionals corresponding to both beam and fluid domains with the work done by the external loads on the beam.
Firstly, in Subsection 3.1 we investigate stability of the trivial solution of coupled system without any a priory constraints on the coupled fluid-structure system -this results in Theorem 3.2. Although the obtained result is generic, it does not lead to strong stability of trivial solution because the estimate for energy functional depends on time and may be effective only for vanishing in time input boundary data.
To eliminate this deficiency we impose an extra constraint on the thickness of the beam and geometry of the moving boundary in the Subsection 3.2. In depth analysis of the energy balance equation obtained in Theorem 3.2 results in Theorem 3.5, which states that under some constraints, the coupled fluid-beam energy functional is bound by the flux of mass through the accessible boundary, plus the initial data.
2. System of non-linear beam equations and energy equation. Consider a beam of length L clamped at the end points subjected to some external forces. We enhance the Euler-Bernoulli beam model by considering the axial displacement in addition to the transversal displacement and also by relaxing the simplifying assumption of constant thickness. The following is the system of non-linear equations for this modified beam model (see [13]) Here, are the axial and transversal displacements of a generic point on the beam neutral axis, are all constant physical parameters of the system where C 1 is the damping coefficient, ρ is the density of the beam, A is the area of the beam cross section, E is the Young's modulus, and I 0 is the momentum of inertia in the undeformed configuration, , respectively, are the external axial and transversal applied loads to the beam, is the relative thickness term which is defined as the ratio of the distance of a point from the neutral axis in the deformed configuration to the distance in the undeformed configuration 1 .
Since the beam is considered to be clamped at the end points, the system (1)-(2) satisfies the following boundary conditions For the further analysis, we will make some physical assumptions on the relative thickness term δ. Constraint 1. We will assume that the relative thickness is not vanishing in time, and the rate of the changes of weighted relative thickness of the beam, δ satisfies the following. For some b 0 andC, which will be selected later depending on the parameters of the problem,exist constants 0 < b < b 0 and 0 < C <C such that and there exists positive constant C such that Before we develop a beam-fluid interaction problem, analyzing the non-linear beam system (1)-(2) yields the following energy-balance equation (see [13] for details) where a is any real number such that 0 ≤ a ≤ 1. We will use this energy equation later in the beam-fluid coupled analysis.
3. Beam-fluid interaction. We will now formulate the fluid-beam interaction problem, where the beam is modeled as the top boundary of the 2-dimensional porous media. The beam equations are modeled in a Lagrangian reference domain Γ. The fluid equation is modeled in an moving Euler domainΩ(t), whose reference configuration is given below. Let • Ω be the reference/undeformed configuration of the porous media domain : • The top boundaryΓ(t) is the only moving boundary which deforms according to the the beam displacements. By continuity constraint it satisfies the followinḡ where u, and w are the axial and transversal displacements of a point on the beam reference domain. The moving domainΩ(t) is then the regionx = (x,ȳ) enclosed byΓ(t), Γ 1 , Γ 2 and Γ 3 .
For modeling the balance of fluid momentum through the porous mediaΩ(t), where v f (x, t) is the pore velocity, P (x, t) is the pressure, and Π is equal to the ratio between the the permeability of the porous rock k and the fluid viscosity µ.
Here the porous media is considered to be homogeneous, thus Π is a constant.
In this work, we assume that the fluid is slightly compressible and satisfies the following equation of state where ρ f (x, t) is the density of the fluid, ρ f0 = ρ f (P 0 ) is the density of the fluid at the reference pressure P 0 , and 1 γ is the fluid compressibility. For simplicity we assume P 0 = 0.
Using continuity equation together with the state equation (10), we can easily obtain the following relation for the pressure ∂P ∂t Finally, we consider the following boundary conditions for the porous medium where Q is the flux of mass through the boundary. On the top boundary the coupled system satisfies • the continuity constraint where (x,ȳ) = (x(x, t),ȳ(x, t)) as defined in (8) and n is the unit normal vector ofΓ(t), • and the momentum balance constraint for any test function ω. Here, ρh < q 1 , q 2 > is the force field (ρ and h are the density and initial thickness of the beam), and ds is the infinitesimal arc length onΓ(t).
Assuming that the classical solution of the coupled system exists for all time, we will now analyze the dynamics of the porous media-beam coupled system: 3.1. Estimate of Type 1: No a priori constraint on the deformation and compressibility (a = 0 ). Choosing the test function ω =< u t , w t > in the momentum balance constraint (15) yields Using continuity constraint (14) on the RHS of the equation above, and substituting this result in the RHS of the beam energy equation (7), with a = 0, we obtain the following energy equation for the porous media-beam coupled system whereF Using Green's Theorem, Darcy's equation (9), and the boundary conditions (13), the RHS of the equation (17) can be rewritten as In the following, second term on the RHS of the last equation above will be analyzed.
Lemma 3.1. For any k ≥ 1, the following relation holds Proof. From the equation (12), we have Partial derivative with respect to time in the integral term ∂P k ∂t dĀ can be taken out of the integral as a total derivative in the following way which is an extension of Leibniz's formula for moving domain and can be found in [12]. Using Green's Theorem in the second integral term on the RHS of the equation above yields the following Then the equation (21) can be easily obtained by substituting (24) into (22).
Using the result stated in Lemma 3.1 recursively yields the following Proposition.
Proof. Using Lemma 3.1, it is straight forward to show by induction that where the limit for the classical solution of the coupled system. The infinite sum Combining the two equations (17) and (20), using Proposition 1 and the inlet boundary condition in (13), we can obtain the following result.
or integrating in timẽ where ψ 0 := ψ| ρ f =ρ f 0 and C 0 : ≥ 0 is a constant which depends on the initial data.
From Theorem 3.2, one can get an estimate for the energy functional for all time. As time increases, RHS of the equation (30) can increase in time although ρ f on the inlet is bounded uniformly. In order to obtain more refined estimate, we will impose a priori condition on the domain deformation and on the fluid density.

3.2.
Estimate of Type 2: Under constraints on the deformation (a > 0). If we consider 0 < a < 1 in the beam energy equation (7) , by following the same steps as in the previous section 3.1, we can obtain an energy equation analogous to (29) where and In the following, the integral term aK 1 ρh Γ (t) P u, w · n ds on the RHS of the equation (31) will be analyzed. But in order to deal with this term, we will assume that the following constraint holds.

Constraint 2.
There exists a constant M 1 > 0 such that for any nonnegative weight function f w .

FLUID STRUCTURE INTERACTION PROBLEM 1141
Lemma 3.3. Define < P 1 , P 2 >:= P n . If the constraint 2 is satisfied, then the following estimates hold for all time t ≥ 0 for any positive constants 1 , 2 , and poincare constant C p1 > 0.
Proof. Define then Γ (t) Using Cauchy's inequality, constraint 2 and Poincare inequality, respectively, for the first integral term on the RHS of the equation above, we can obtain the following for any positive constant 1 . Here, C P1 is the Poincare constant. Now consider the second integral term on RHS of the equation (38) − Γ (t) To obtain the estimate (36), we use again Cauchy's inequality, constraint 2 and Poincare inequality, respectively for any positive constant 2 , and Poincare constant C P1 .

EUGENIO AULISA, AKIF IBRAGIMOV AND EMINE YASEMEN KAYA-CEKIN
Now, we will use Lemma 3.3 in equation (31) for the last term in the RHS with which leads to aK 1 ρh Γ (t) P u, w · n ds ≤ aK 1 ρh Using (43) in the equation (31), rearranging some of the terms, and defining the functional G 2 as yields the following inequality Let By Trace Theorem the following inequality for the second integral term on the RHS of the inequality (45) is obtained where C T > 0 is the Trace theorem constant. Then using the inequality above (46) and Poincare-Sobolev inequality, (45) can be written as where Note 2. Choose a such that 0 < a < min Π K1C T M2 , 1 , which implies that M 3 > 0.
Now define the following functions Using the constraint Poincare-Sobolev inequality, and Theorem 3.2 , we can estimate the term P 1 L 1 (Γ(t)) as follows (50) Then we can rewrite the inequality (47) as where M4 := aK1 C 4 p 1

2Π
, and M5 := M5(t) : Note 3. Select in constraint 1 constants b 0 < a andC < (a − b 0 ), then it is not difficult to show that bracket Existence of such t independent follow from assumption (6). Latter implies that M 5 (t) ≥ C M5 > 0 for all t, for some constant C M5 . Define and rewrite the inequality (51) as In the following we assume that the following constraint holds.
Constraint 3. The fluid density satisfies the following constraint for all t and for some positive constant c.
The above constraint reflects the fact that the considered fluid is slightly compressible. It should be noticed that the state equation (10) is valid only for |P | ≤ Cγ, for some small constant C, otherwise the density saturates to a minimum value ρ min for P < −Cγ or a maximum value ρ max for P > Cγ (see [19]). In our analysis however we don't consider these upper and lower bounds, resulting in a density which grows exponentially with the pressure. In order to prove the following result, we assume boundedness of the density growth with a quadratic function of the pressure P . Then the constraint 3 is needed, for mathematical reasons, to satisfy the physics of the problem. Now we can state the following lemma.
Lemma 3.4. Let constraints 1 and 3 are satisfied. Assume that there that and where C M5 is an upper bound for M 5 (t). Then, there exist constants c 1 , c 2 , c ≥ 0 depending on the physical parameters of the porous medium -beam coupled system such that and Proof. Assuming constraint 1 holds, the inequality (58) is obvious. For the inequality (59), observe that each integral term inĨ(t) has an immediate analogous term in G 3 (t) with respect to some constant except the following terms • D 2 2 Γ δ 3 w 2 xx dx: This term can be compared with the term Since δ 3 is a non-increasing function (constraint 1), it is bounded above for all time. On the other hand, by the assumption (56), M 5 − γ ρhρ f 0 Γ1 (ψ − ψ 0 ) ds is bounded below by a positive constant for all time. Therefore, these two terms can be compared with respect to some constant. in G 3 , if the constraint 3 holds. Similarly for the inequality (60), observe that each term in I(t) can be compared with a term in G 3 (t) by the help of the previous result (inequality (59)) except the following term a 2 u t + K 1 u 2 L 2 (Γ) .
But, from classical Poincare and Holder inequalities, the following estimate can be obtained where C P2 is the Poincare constant. The RHS of the inequality above can now be compared with G 3 (t) + G 2 3 (t).