On Vector Fields Describing the 2d Motion of a Rigid Body in a Viscous Fluid and Applications
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Journal of Mathematical Fluid Mechanics
On Vector Fields Describing the 2d Motion of a Rigid Body in a Viscous Fluid and Applications Stathis Filippas and Alkis Tersenov Communicated by D. Gerard-Varet
Abstract. We present some properties of functions in suitable Sobolev spaces which arise naturally in the study of the motion of a rigid body in compressible and incompressible fluid. We relax the regularity assumption of the rigid body by allowing its boundary to be Lipschitz. In the case of a smooth rigid body we obtain a new estimate on the angular velocity. Our results extend and complement related results by V. Starovoitov and moreover we show that they are optimal. As an application we present an example where the rigid body collides with the boundary with non zero speed. Finally, we present a new non collision result concerning a smooth rotating body approaching the boundary, without assuming any special geometry on either the body or the container. Mathematics Subject Classification. 46E35, 35Q30. Keywords. Sobolev spaces, Fluid solid interaction, Estimates on angular speed, Non zero speed collision, Non collision.
1. Introduction Let Ω ⊂ IR2 be a domain, S ⊂ Ω be a bounded connected domain and x∗ ∈ S be a fixed point. By W01,p (S, Ω), p ≥ 1, we denote the vector function space consisting of functions 2 u : Ω → IR2 , u ∈ W01,p (Ω) , such that for a constant vector a∗ ∈ IR2 and a constant ω ∈ IR there holds u(x) = a∗ + ω(x − x∗ )⊥ , f or x ∈ S ,
(1.1)
⊥
with x = (x1 , x2 ), x = (x2 , −x1 ) and u = (u1 , u2 ). Such function spaces arise naturally when studying the motion of a rigid body S inside a fluid region Ω, see e.g., [5,8–12]. In this context u is the velocity field in Ω whereas inside the rigid body the velocity is given by (1.1). This is a consequence of the fact that inside the rigid body the deformation tensor is zero, that is, 1 ∂ui ∂uj + = 0, i, j = 1, 2 . (1.2) Di,j (u) = 2 ∂xj ∂xi The vector fields u for which this holds true, consist of the rigid body vector fields given by (1.1), see e.g., [11,14]. We note that a∗ is the linear velocity of the reference point x∗ and ω is the angular speed of S around x∗ . We are interested both in the static case where the body S touches the boundary ∂Ω as well as in the dynamic case, where S approaches the boundary ∂Ω. When the boundaries ∂Ω and ∂S are C 2 such questions have been studied in [8], whereas the case where the boundaries are C 1,α , 0 < α < 1, have been studied in [11]. 0123456789().: V,-vol
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S. Filippas, and A. Tersenov
JMFM
Suppose that S touches the boundary ∂Ω, that is, dist(S, ∂Ω) = 0. We further assume that u is 2 solenoidal in Ω that is, divu(x) = 0 for x ∈ Ω and moreover u ∈ W01,p (Ω) . It is shown in [11], Theorem 2.1, that if both ∂Ω and ∂S are C 1,α with 0 < α < 1, then 2+α implies a∗ = 0, ω = 0 . p≥ 2α We first show that the above result is optimal, in the sense that for 1 ≤ p < 2+α 2α we construct examples where either ω or a∗ are not zero. Next, we relax the smoothness requirement replacing it by Lipschitz
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