Strong solutions for the fluid–solid systems in a 2-D domain

Abstract

In this paper, we study the dynamics of a solid body moving in a 2-D fluid flow with slip boundary conditions. The boundaries of the container and the solid belong to C 1,1 . First, we prove existence and uniqueness of a strong solution up to collision. Second, we consider the influence of curvature of the solid boundary on the contact problem under slip boundary conditions. Strong solutions for the fluid-solid systems in a 2-D domain 265 exist and are continuous. For all function u(t, ·) : F (t) → R 2 , denote U (t, x, y) u (t, X(t, x, y)). Then we defineOther function spaces are defined in the same way. Now, we are in position to state the main theorem of this paper. In the first part, we prove existence and uniqueness of strong solutions:loc (0, ∞; L 2 (Ω)) and that they satisfyThen there exists a maximal T 0 > 0 such that (1.1)-(1.7) has a unique solution which satisfies for allwhere U T (F (t)) = L 2 (0, T ; H 2 (F (t))) ∩ C(0, T ; H 1 (F (t))) ∩ H 1 (0, T ; L 2 (F (t))). Moreover, one of the following alternatives holds true:(1) T 0 = +∞, i.e. the solution is global,In the second part, we consider the collision process. We apply the method of [17] to the 2-D case.Here for simplicity, we make the same assumptions as [17]. Let S(t) = R t S 0 be the position of the solid at time t, where R t is a family of affine isometries and Lipschitz in t. More precisely, we shall consider configurations obeying the following assumptions:(A1) The source term f = −ge 2 with a constant g > 0 and e 2 = (0, 1). (A2) Ω is a smooth convex domain of {y > 0}, flat near the origin:(A3) The solid moves along the axis x = 0, that is R t is a vertical translation. (A4) The only possible contact between the solid and the container is at x = 0. More precisely, there exists a positive constant d δ such that dist S(t), ∂Ω \ y = 0, |x| < δ d δ > 0. 266 C. Wang / Strong solutions for the fluid-solid systems in a 2-D domain(A5) Near x = 0, the lower boundary of ∂S(t) is given bywhere x l (t) is the position of the lowest point of the solid.Under the above assumptions, we prove the following theorem. 1.1, the solid will touch the bottom in finite time when α = 2 and will not touch the bottom in finite time when α = 3 or α 4. Theorem 1.2. For any solution which is constructed by TheoremRemark 1.1. To study the collision process, we only need the weak formation of this system which is defined in [16]. Thus, the collision result will be valid for weak solution defined in [16].Remark 1.2. Theorem 1.2 requires to derive the relationship between the drag force and d(∂S(t), ∂Ω). Therefore, we use some special test functions that need to belong to W 2,∞ . Because the coefficient of the test function, α S , contains the curvature of the solid which behaves as |x| α−2 near the origin (see (6.8)), we need to restrict α to {2, 3} ∪ [4, ∞).The plan of the paper is as follows: in Section 2, we introduce the Lagrangian coordinates to reduce the free boundary problem to a fixed domain problem. In Section 3, we study the linear problem in a fixed domain assoc...

Energy Equality and Uniqueness of Weak Solutions of a “Viscous Incompressible Fluid + Rigid Body” System with Navier Slip-with-Friction Conditions in a 2D Bounded Domain

“…where X and Y are defined in (19) and v ε and v F,ε are defined in (21), if we replace f ε and f F,ε by u ε and u F,ε . Observe that U ε ∈ W τ and U ε converges to u in E τ .…”

“…We translate the problem (63)-(72) to an equivalent one on a domain fixed in time. Let X the geometric change of variables associated with S define in (19), following the idea of [7, Section 3] we definẽ v(t, y) = ∇Y (t, X(t, y))v(t, X(t, y)), p(t, y) = p(t, X(t, y)),…”

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

“…Also we mention the article [12] where a local existence up to collisions of a weak solution for a fluid-solid structure was proved. The existence of a strong solution in 2D case was proven by Wang [29].The global in time existence of the weak solution was proven in [2] for a mixed case, when Navier's condition was given on the solid boundary and the non-slip condition on the domain boundary. This result admits the collisions of the solid with the domain boundary (for more detailed discussion concerning influence of boundary conditions on the collision see [13]).…”

Strong solutions in $L^2$ framework for fluid-rigid body interaction problem. Mixed case

“…Proof. We follow Wang verbatim [20]. The difference between Wang´s problem and our problem is that, Wang considered slip boundary conditions on both boundaries and we consider the Mixed case.…”

“…Our article deals with the strong solution of the Mixed case. The existence of strong solution was studied by Takahashi, and Tucsnak [18,19] in the no-slip boundary conditions and in the Slip case by Wang [20] in the 2D case.…”