Steady flow of a Navier-Stokes liquid past an elastic body. (English) Zbl 1291.76079
Summary: We perform a mathematical analysis of the steady flow of a viscous liquid, \({\mathcal{L}}\), past a three-dimensional elastic body, \({\mathcal{B}}\). We assume that \({\mathcal{L}}\) fills the whole space exterior to \({\mathcal{B}}\), and that its motion is governed by the Navier-Stokes equations corresponding to non-zero velocity at infinity, \(\mathbf v_{\infty }\). As for \({\mathcal{B}}\), we suppose that it is a St. Venant-Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body, or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady-state fluid-structure problem with the surface of \({\mathcal{B}}\) as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small \(|\mathbf v_{\infty }|\). This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier-Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.
MSC:
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35Q30 | Navier-Stokes equations |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
Keywords:
well-posedness; Sobolev space; fixed-point contraction; existence; uniqueness; Oseen linearizationReferences:
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