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Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid-structure interaction problem. (English) Zbl 1504.35295

Summary: Starting from a nonlinear 2D/1D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, in the vanishing limit of the relative fluid thickness, we rigorously derive a sixth-order thin-film equation describing the dynamics of vertical displacements of the structure. The procedure is essentially based on quantitative energy estimates in terms of the relative fluid thickness and a uniform no-contact result between the structure and the solid substrate. The sixth-order thin-film equation is justified in the sense of strong convergence of rescaled structure displacements to the unique positive classical solution of the thin-film equation. Moreover, the limiting fluid velocity and pressure can be expressed solely in terms of the solution to the thin-film equation.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
76A20 Thin fluid films
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74B20 Nonlinear elasticity
74D10 Nonlinear constitutive equations for materials with memory
35M30 Mixed-type systems of PDEs
35A09 Classical solutions to PDEs
35R35 Free boundary problems for PDEs
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