Abstract
We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.
Highlights
The non-homogeneous source term of the beam equation (Tf )2 is the net surface force on the structure which is the resultant of force exerted by the fluid on the structure and the external force pext and it is assumed to be of the following form (Tf )2 = ([−2μD(u) − μId] · nt + P nt) |Γs,t 1 + ηx2 · e2 on ΣsT, (1.5)
Observe that in (1.2) we have considered the initial displacement η(0) of the beam to be zero. This is because we prove the local existence of strong solution of the system (1.2) with the beam displacement η close to the steady state zero
Our interest is to prove the local in time existence of a strong solution to system (1.2)–(1.4)–(1.5) i.e we prove that given a prescribed initial datum (ρ0, u0, η1), there exists a solution of system (1.2)–(1.4)– (1.5) with a certain Sobolev regularity in some time interval (0, T ), provided that the time T is small enough
Summary
Our objective is to study a fluid structure interaction problem in a 2d channel. Concerning the structure we will consider an Euler–Bernoulli damped beam located on a portion of the boundary. In the present article we establish a result on the local in time existence of strong solutions of such a fluid structure interaction problem. To the best of our knowledge, this is the first article dealing with the existence of local in time strong solutions for the complete non-linear model considered here. Let Ω be the domain TL ×(0, 1) ⊂ R2, where TL is the one dimensional torus identified with (0, L) with periodic conditions. For a given function η : Γs × (0, ∞) → (−1, ∞), which will correspond to the displacement of the one dimensional beam, let us denote by Ωt and Γs,t the following sets.
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