On the Motion of a Nearly Incompressible Viscous Fluid Containing a Small Rigid Body

We consider the motion of a compressible viscous fluid containing a moving rigid body confined to a planar domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset R^2$$\end{document}. The main result states that the influence of the body on the fluid is negligible if (i) the diameter of the body is small and (ii) the fluid is nearly incompressible (the low Mach number regime). The specific shape of the body as well as the boundary conditions on the fluid–body interface are irrelevant and collisions with the boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} are allowed. The rigid body motion may be enforced externally or governed solely by its interaction with the fluid.


Introduction
There is a vast number of recent studies concerning the motion of a rigid body immersed in/or containing a compressible viscous fluid.We focus on the situation when the body is "small" therefore its influence on the fluid motion is expected to be negligible.By small we mean that the body is contained in a ball with a small radius.The problem is mathematically more challenging in the case of planar (2d) flows, where even small objects may have large capacity.
The motion of a small object immersed in an inviscid (Euler) incompressible fluid is studied by Iftimie, Lopes Filho, and Nussenzveig Lopes [12].Similar problems again in the framework of inviscid fluids have been considered by Glass, Lacave, and Sueur [8], [9].The asymptotic behavior of solutions of the incompressible Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to curve has been studied by Lacave [14].
In the context of viscous Newtonian fluids, the flow around a small rigid obstacle was studied by Iftimie et al. [13].Lacave [15] studies the limit of a viscous fluid flow in the exterior of a thin obstacle shrinking to a curve.
Finally, let us mention results in planar domains, where the body does not influence the flow in the asymptotic limit.Dashti and Robinson [3] consider the viscous fluid-rigid disc system, where the disc is not rotating.Lacave and Takahashi [16] consider a single disk moving under the influence of a viscous fluid.They proved convergence towards the Navier-Stokes equations as the size of the solid tends to zero, its density is constant and the initial data small.Finally, He and Iftimie [10] extend the above result to a general shape of the body and to the initial velocities not necessarily small.
To the best of our knowledge, the problem of negligibility of a small rigid body immersed in a planar viscous compressible fluid is completely open.Bravin and Nečasová [2] addressed the problem in the 3d setting, where the capacity of the object in a suitable Sobolev norm is small enough.

Problem formulation
Neglecting completely the possible thermal effects as well as the external body forces, we consider the isentropic compressible fluid in the low Mach number regime governed by the following system of equations: Navier-Stokes system.
The fluid is confined to a bounded planar domain Ω ⊂ R 2 and the momentum equation where ) The ball B ε,t is the part of the plane containing the rigid object at the time t.Note carefully that, in general, we do not require B ε,t ⊂ Ω.Finally, we impose the no-slip boundary conditions

Main results
Below, we formulate the main hypotheses imposed on the fluid motion.It is convenient to consider the density ̺ = ̺ ε as well as the velocity u = u ε to be defined on the whole physical space (0, T ) × R 2 .Accordingly, we set Throughout the whole text, we assume the following: and (H4) The energy inequality holds for a.a.τ ∈ (0, T ), where P is the pressure potential, and Ω ε,0 ⊂ Ω F ,ε,0 .
Our main result reads as follows: Theorem 1.1.Let Ω ⊂ R 2 be a bounded domain of class C 3 .Let (̺ ε , u ε ) ε>0 satisfy the hypotheses (H1)-(H4).In addition, suppose where min m; 2m γ > 3. (1.16) where u 0 ∈ W 2,∞ (Ω), div x u 0 = 0, u 0 | ∂Ω = 0; (1.17) as ε → 0, where u is the (unique) classical solution of the incompressible Navier-Stokes system The hypotheses (1.15), (1.17) correspond to the well prepared data in the low Mach number limit, cf.Masmoudi [17].Moreover, as u 0 belongs to the class (1.17), the standard maximal regularity theory yields a strong solution of the Navier-Stokes system (1.21), unique in the class see e.g.Gerhardt [7], von Wahl [18].The solution is classical in (0, T ) × Ω as a consequence of the interior regularity estimates.The hypotheses of Theorem 1.1 are satisfied if (̺ ε , u ε ) is a weak solution of the fluid-structure interaction problem of a single rigid body immersed in a viscous compressible fluid in the sense of [5] (see also Desjardins and Esteban [4]) or if the motion of the body is prescribed as in [6].A detailed proof is given in Appendix 5.
The remaining part of the paper is devoted to the proof of Theorem 1.1.Similarly to the purely incompressible setting studied by He and Iftimie [11] (cf. also Lacave and Takahashi [16]), the main problem is the rather weak estimate (1.18) that does not allow for a precise identification of the limit trajectory of the body.In addition, two new difficulties appear in the compressible regime: • Possible fast oscillations of acoustic (gradient) component of the velocity that cannot be a priori excluded even for the well prepared data because of the influence of the rigid body.
• Possible contacts of the body -intersection of the balls B ε,t -with the outer boundary ∂Ω.
To overcome the above mentioned difficulties, we proceed as follows.In Sections 2, 3 we identify the system of equations satisfied by the limit velocity u.Due to the lack of information on ∂ t u ε , the limit of the convective term as well as the kinetic energy is described in terms of the corresponding Young measure.The limit u is therefore a generalized dissipative solution of the incompressible Navier-Stokes system in the sense of [1].In particular, we adapt the approximation of the test functions introduced by He and Iftimie to the geometry of a bounded domain.Finally, in Section 4, apply the weak-strong uniqueness result proved in [1] to conclude that the limit is, in fact, a strong solution of the Navier-Stokes system whereas the associated Young-measure reduces to a parametrized family of Dirac masses.

Identifying the limit, the equation of continuity, energy balance
It follows from the hypotheses (1.15), (1.17) that the initial energy on the right-hand side of the energy inequality (1.14) is bounded uniformly for ε → 0. Applying Korn-Poincaré inequality we get, up to a suitable subsequence, Next, ̺ ε satisfies the renormalized equation of continuity (1.11).Moreover, the energy inequality (1.14) yields In particular, we may perform the limit in (1.11) Finally, using the hypotheses (1.17), (1.18) and the property of weak lower semi-continuity of convex functionals, we perform the the limit in the energy inequality obtaining for a.a.τ ∈ (0, T ).Here, C(τ ) ∈ L ∞ (0, T ) is the so called total energy defect defined as (2.4) 3 Identifying the limit, the momentum equation The next and more delicate step is to perform the limit ε → 0 in the momentum equation (1.2).
To eliminate the singular pressure term, we consider the test functions Accordingly, the weak formulation (1.13) gives rise to

Some useful estimates
Note that (3.2) is relevant only on the fluid part ∪ t∈[0,T ] Ω ε,t , where the energy inequality (1.14) yields uniform bounds on the density.This motivates the following decomposition of any measurable functions v: where [v] ess = v1 1 2 ̺≤v≤2̺ .Thanks to the energy inequality (1.14), we get Moreover, by the energy inequality, whence we conclude )), and weakly in L 2 (0, T ; L q (Ω; R 2 )) for any 1 ≤ q < ∞. (3.5) In addition, we also have where, thanks to the energy inequality (1.14), Ωε,τ whence, by interpolation, )) for some r > 1. (3.8) The tensor ̺u ⊗ u ∈ R d×d sym is positively semi-definite and Indeed, for any d ∈ R d : Thus the desired conclusion (3.9) follows from (2.1), (3.4) and weak lower-semicontinuity of convex functions.Finally, as where E is the total energy defect appearing on the left-hand side of the energy inequality (2.3).
As for the residual components, we deduce from the energy inequality Consequently, by Hölder's inequality, and, similarly, for a.a.τ ∈ (0, T ).

Constructing a suitable class of test functions
Our goal is to approximate a test function by a suitable family of admissible test functions (ϕ ε ) ε>0 in (3.2).The test function are obtained following the construction of He and Iftimie [10,11], specifically, The cut-off functions η ε near the disk D(h ε (t), ε) are smooth and satisfy the following properties (see [10,Lemma 3]): where α(ε) is chosen in such a way that As shown in [10, Lemma 5], the functions ϕ ε enjoy the following properties: Unfortunately, the functions ϕ ε do not vanish on ∂Ω unless dist[h(t); ∂Ω] > εα(ε).To remedy this, we consider a convex combination First observe that, similarly to ϕ ε , and Next, we compute the approximation error in the time derivative where the former error term can be controlled in W −1,2 exactly as in He and Iftimie [11] since χ is independent of x.As for the latter, we have where, in accordance with (3.14), the latter error vanishes in W −1,2 for ε → 0 as a consequence of (3.16).

Asymptotic limit
The function ϕ ε constructed in Section 3.2 represents a legitimate test function for the momentum balance (3.2).Our goal is to perform the limit ε → 0.

.26)
Step 4: Remaining terms.The final step is to show A direct manipulation reveals

Proof of the main result
Summarizing the results obtained in the preceding section we may infer that limit velocity solves the following problem: for a.a.τ ∈ (0, T ).Here, the tensor R = ̺u ⊗ u − ̺u ⊗ u is positively semi-definite and satisfies (3.10), specifically Consequently, the limit function u is a dissipative solution of the Navier-Stokes system (1.21) in the sense of [1].As the initial velocity is regular, the same problem admits a strong solution in the class (1.22).Thus applying the weak-strong uniqueness result [1, Theorem 2.6.and Remark 2.5] we conclude that u coincides with the strong solution of (1.21).Finally, as the strong solution satisfies the energy equality, its follows from (4.2) that E = 0, and yielding the strong convergence claimed in (1.20).Theorem 1.1 has been proved.

Appendix
Our main result (Theorem 1.1) is valid whenever (̺ ε , u ε ) ε>0 satisfy the hypotheses (H1) -(H4) along with the conditions (1.15)-(1.20).These hypotheses (see (1.10)-(1.14))are satisfied if (̺ ε , u ε ) is a weak solution of the fluid-structure interaction problem of a single rigid body immersed in a viscous compressible fluid in the sense of [5] (see also Desjardins and Esteban [4]) or if the motion of the body is prescribed as in [6].Let the rigid body S ε (t) be a regular, bounded domain and moving inside Ω ⊂ R 2 .The motion of the rigid body is governed by the balance equations for linear and angular momentum.We assume that the fluid domain F ε (t) = Ω \ S ε (t) is filled with a viscous isentropic compressible fluid.Initially, the domain of the rigid body is given by S ε,0 included in the ball B ε,0 and F ε,0 is the domain of the fluid.Let h ε be the position of the centre of mass and β ε be the angle of rotation of the rigid body.The solid domain at time t is given by where R βε is the rotation matrix, defined by The evolution of this fluid-structure system can be described by the following equations the boundary conditions and the initial conditions In the above, the outward unit normal to ∂F ε (t) is denoted by ν ε (t, x).For all x = (x 1 , x 2 ) ∈ R 2 , we denote by x ⊥ , the vector (−x 2 , x 1 ).Moreover, the constants m ε and J ε are the mass and the moment of inertia of the rigid body.We want to state the existence result of the fluid-rigid body interaction system (5.1)-(5.8).To do so, we extend the density and the velocity in the following way: (5.9) (5.10) We have the following existence result for the system (5.1)-(5.8)following [5, Theorem 4.1]: Theorem 5.1.Let Ω ⊂ R 2 be a bounded domain and the pressure p F be given by the isentropic constitutive law Let the initial data (̺ 0 , q 0 ) be defined by (5.10) satisfying (5.12) Then the system (5.1)-(5.8)admits a variational solution (̺ ε , ) in the following sense: ) where P is the pressure potential We can verify the hypotheses (H1)-(H4) and apply Theorem 1.1 under certain conditions to obtain the following result in the framework of fluid-rigid body interaction: Theorem 5.3.Let Ω ⊂ R 2 be a bounded domain of class C 3 and (̺ 0 , q 0 ) satisfy (5.11)-(5.12).Assume that S ε,0 ⊂ B ε,0 , (5.20) • The mass m ε verifies that m ε ε 2 → ∞ as ε → 0. (5.21) u ε → u in L 2 (0, T ; W 1,2 0 (Ω; R 2 )) (5.23) as ε → 0, where u is the (unique) classical solution of the incompressible Navier-Stokes system div x u = 0, ̺∂ t u + ̺div x (u ⊗ u) + ∇ x Π = µ∆ x u, u| ∂Ω = 0, u(0, •) = u 0 (5.24) in (0, T ) × Ω.
Remark 5.4.We want to point out that as observed by He and Iftimie [11], assumption (1.18) holds for the fluid-structure interaction problem if the condition (5.21) satisfies.Observe that the condition (5.21) implies inf ̺ S ε → ∞, where ̺ S ε is the density of the rigid body immersed in the fluid.