On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

1 Introduction

This paper is devoted to reconstructing an unknown structure 𝒮 included in a bounded cavity Ω⊂ℝN (N=2,3) filled by a viscous incompressible fluid. More precisely, we aim to obtain some geometrical information on 𝒮 by measurement on the boundary ∂⁡Ω of Ω. Such a geometrical inverse problem is important in several applied areas such as medicine (foreign bodies in the bloodstream), biology (fishes), naval engineering (submarines), etc.

We assume in what follows that 𝒮 is a compact connected subset of Ω with nonempty interior and that ℱ=Ω∖𝒮 is connected.

In the first part of the article, the fluid equations that we consider are the nonstationary Stokes system

In the above system, (𝒖,p) are the velocity and the pressure of the fluid. Moreover, we have denoted by 𝝈⁢(𝒖,p) the Cauchy stress tensor, which is defined by the Stokes law as

where 𝑰N is the identity matrix of ℳN⁢(ℝ), with ℳN⁢(ℝ) denoting the space of real square matrices of order N, and where 𝑫⁢(𝒖) is the strain tensor defined by

To simplify the writing, we take in this paper the kinematic viscosity of the fluid equal to 1.

The idea is to impose a condition 𝒇 in (1.4) and to measure the corresponding Cauchy force

in order to deduce information on the obstacle 𝒮. Here and in all what follows, 𝒏 denotes the unit outer normal to the fluid domain.

We also consider in this paper the following linear fluid–rigid body system:

Here ℓ and 𝝎 represents respectively the linear and angular velocity of the rigid body. Let us note that in this simplified fluid-rigid body system, the structure domain 𝒮 is fixed. We assume that the density ρ𝒮 of the rigid body is a positive constant. In particular, the mass m and the inertia tensor I0 are defined as follows:

where μ3 denotes the Lebesgue measure in ℝ3 and where 𝑰3 is the 3×3 identity matrix.

In dimension N=2, the above system is slightly modified: ω∈ℝ, I0∈ℝ, equation (1.11) and equation (1.13) write respectively

where

Finally, I0 is defined by

System (1.1)–(1.5) is a linear simplification of the classical Navier–Stokes system

and system (1.8)–(1.15) is a linear simplification of the “full” fluid-rigid body system that can be written in dimension N=2 as

Here 𝒉⁢(t)∈ℝ2 and θ⁢(t)∈ℝ are respectively the center of mass and the orientation of 𝒮⁢(t). In particular, the center of mass of 𝒮0 is located at . We have denoted by 𝑹θ the matrix of rotation of angle θ. Contrary to system (1.8)–(1.15), here the solid is moving (equation (1.33)). Let us emphasize that system (1.8)–(1.15) is important to study system (1.22)–(1.33): for instance, this linear system is used in [36, 37] to prove the existence of strong solutions for system (1.22)–(1.33) with the aid of a fixed point argument. Let us also note that equations (1.26), (1.27) for the rigid body are the Newton laws.

As for the previous systems, the idea is to take some particular choice of 𝒇 and to measure the corresponding Cauchy force given by (1.7) in order to obtain geometrical information on 𝒮⁢(t). However, here there is an important difference: applying 𝒇 at the boundary of ∂⁡Ω makes the rigid body moves through the (unknown) trajectory (𝒉⁢(t),θ⁢(t)). Moreover, with such a boundary condition, it could possible that the rigid body touches ∂⁡Ω and it is not clear what happens after this contact (see [34]).

We also consider a simplification of system (1.22)–(1.33) obtained by assuming that the Reynolds number is very small. In that case, neglecting the inertia forces, the 3D version of system (1.22)–(1.33) can be approximated by

The map A is defined as follows:

This map is related to the vector product by the formula

Let us remark that the above system is not linear since 𝒮⁢(t) is not given. This system is studied in [7] where the identifiability of the rigid body is obtained through the measurement of the Cauchy forces on the boundary. Like system (1.22)–(1.33), the solid moves through the action of 𝒇 on this system.

These geometrical inverse problems for fluid systems were already considered in [1] where the authors tackle the problem of recovering the shape and location of a fixed obstacle in a viscous incompressible fluid modeled by the Navier–Stokes system. They show the identifiability of the fixed obstacle: if 𝒇 not identically equal to , then the mapping that associates to 𝒮 the measurement given by (1.7) is one-to-one. They also prove a stability result: if two measurements are close, it implies that the two corresponding obstacles are close. Extensions of this result in the case of a fixed obstacle are obtained in [9] and in [10]. In [2], the authors consider a similar problem in the 2D case and for a fluid modeled by the Stokes system. They develop an integral method in order to recover the structure. The identifiability result of [1] is extended in [7] to the case of a moving rigid body, but only in the case of the stationary Stokes system. In the case of a potential fluid (thus inviscid), one can use, in 2D, complex analysis ([5, 6]) to detect a moving rigid body of particular shape (ball, ellipse) if the fluid fills the exterior of the structure domain.

Numerical aspects are considered in [3]: the authors use shape optimization techniques to detect a fixed obstacle in a viscous incompressible fluid. They prove in particular that the shape Hessian is compact and thus that the problem is ill-posed.

Here we are interested in obtaining geometrical information on the obstacle such as the distance from a fixed point to the obstacle or its convex hull. The problem of finding the distance from a fixed point was considered in [16], in the case of a fixed obstacle in a stationary Stokes fluid. In that study, they use a method based on complex geometrical solutions that was introduced in [35] and that has been applied in several inverse problems ([12, 30, 31, 8, 29, 15], etc.). In order to recover the convex hull of the obstacle, Ikehata introduced the enclosure method and used it in [22, 21, 23], etc. The above references were devoted to works on stationary problems. The case of the heat equation was considered in [13] with the use of complex geometrical solutions and [24, 27, 28] for the enclosure method.

In this work, we consider both methods to deal with nonstationary fluid or fluid-structure systems. More precisely, we use the approach in [27] in order to deal with the nonstationary Stokes system. A first step consists in considering the Laplace transform of the system in order to transform it into a stationary Stokes-type system. Then we show that if (𝒗^α,q^α) is a family of solutions the same (stationary) system but on the whole domain Ω (see (2.1)–(2.2)), then a quantity (see (2.7)) based on the measurement given by (1.7) behaves in similar way as the H1 norm of 𝒗^α on 𝒮 as α goes to ∞ (Theorem 2.1). The idea is then to construct solutions 𝒗^α so that the H1 norm on 𝒮 gives geometrical information on the domain. One of the difficulties in this construction comes from the fact that here the test functions are divergence free. In particular, in the case of the distance of 𝒮 to a point 𝒙0, we need to impose 𝒙0∉ch⁡(Ω) and N=3. These hypothesis are not considered in the case of the heat equation (see [27]).

The above method can not be adapted to the case of nonlinear systems such as (1.17)–(1.21) and (1.22)–(1.33). As a consequence, for these nonlinear systems we use complex geometrical solutions constructed in [16]. This allows us to recover only some partial information, and more precisely, at the contrary to the linear case, we lose one of the inequalities. Nevertheless, these two approaches give some first results in the case of nonstationary fluid systems.

The plan of the paper is the following: in Section 2, we state our main results, for the linear systems and for the nonlinear systems. We recall some preliminaries in Section 3, that allow us to prove our first main result in Section 4: the relation between the measurement and the H1 norm of 𝒗^α on 𝒮, as explained above. Then in Section 5, we construct 𝒗^α in order to recover the convex hull of 𝒮 and in Section 6, we construct 𝒗^α in order to recover the distance from a fixed point to 𝒮. Section 7 is devoted to inverse problems for the nonlinear systems: we use there complex geometrical solutions.

2 Main results

Let us first describe the method used to recover geometric information on the obstacle 𝒮 in the case of the linear systems (1.1)–(1.5) and (1.8)–(1.15).

First we consider a family (𝒗^α,q^α)∈C2⁢(Ω¯)×C1⁢(Ω¯) of solutions of a Stokes system

for some domain Ω~⊇Ω and for α>0.

We then consider 𝒇α defined by

with χα∈C∞⁢([0,T]) such that χα⁢(0)=0 and χα⁢(t)>0 in (0,T] and such that

For instance, in what follows, we take

In particular,

We can remark that since 𝒇α is given by (2.3), then it satisfies the condition

The above equation allows us to consider the solution (𝒖α,pα) of the Stokes system (1.1)–(1.5), with the boundary condition

Let us set

We are now in a position to state our first main result.

The above result and the two corollaries below correspond the closure method associated with the evolutionary Stokes system. A general framework for this method in the case of heat type equations is developed in [25]. The first extension of this method to a system of partial differential equation was developed in [26].

The first corollary of Theorem 2.1 corresponds to the reconstruction of the support function h𝒮 of 𝒮. Let us recall that for any subset G of ℝ3, the support function hG of G is defined by

where 𝕊2 is the unit sphere of ℝ3. This function is classically used in the theory of convex sets (see, for instance, [4, p. 26]). In particular, if G is convex,

The second corollary of Theorem 2.1 allows us to obtain the distance d⁢(𝒙0,𝒮) of 𝒮 to a point 𝒙0∉ch⁡(Ω) (the convex hull of Ω).

We have similar results for the linear system (1.8)–(1.15):

The proof of the previous theorem is completely similar to the proof of Theorem 2.1, with the same families constructed in Corollary 2.2 and Corollary 2.3. Therefore, we omit its proof.

In the case of the nonlinear system (1.34)–(1.42), we use a family of solutions (𝒗α,qα)∈C2⁢(Ω¯)×C1⁢(Ω¯) of

for some domain Ω~⊇Ω. Here α>0 is a parameter in the construction of these solutions that eventually goes to ∞. We then consider 𝒇α defined by

As in the linear case, we then consider the solution (𝒖α,pα) of systems (1.34)–(1.42) (respectively (1.22)–(1.33), and (1.17)–(1.21)), with the boundary condition

We set

As explained in the previous section, one difficulty for stating result for this system is that the rigid body can touch ∂⁡Ω. We thus assume that for all regular 𝒇,

Such an hypothesis is satisfied for instance in the case where 𝒮0 and Ω are balls (see [17, 18, 19]).

We fix 𝒙0∉ch⁡(Ω) (the convex hull of Ω) and d>0. Then, we have the following results.

For systems (1.22)–(1.33), and (1.17)–(1.21), we slightly modify the boundary condition by using (𝒗α,qα) depending on time and satisfying (2.10)–(2.11) for all t and we consider the following measurement:

In the case of system (1.22)–(1.33), we need to assume again (2.15) to prevent possible contacts. Again this condition is satisfied for instance in the case where 𝒮0 and Ω are balls (see [17, 18, 19]). It is probably true for other geometries but up to now this has not been proven.

For both systems (1.17)–(1.21) and (1.22)–(1.33), we also impose that N=2 since we are working with regular solutions and for N=3 the existence of global (in time) regular solutions is an open problem. In particular, in the case N=3, one should need to show that the times Tα of existence of the family of solutions (𝒗α,qα) can be chosen uniformly with respect to α.

As explained in the introduction, the above result is only partial since with the other case (as in Theorem 2.6) is not present here. As it appear in the proof, it would imply to prove an estimate on the solutions (𝒖α,pα,ℓα,ωα) for system (1.22)–(1.33).

For simplicity, we suppress in the proofs below the explicit dependence on α in the notation. For example, we write 𝒗^ instead of 𝒗^α.

3 Preliminaries

The above result is quite classical for system (1.1)–(1.5) and is similar for system (1.8)–(1.15). We only give here some ideas of the proof. Note that the particular form of 𝒇 is not needed to obtain the result and the result remains true for more general boundary conditions.