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arXiv · 2201.07608

Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid -- structure interaction problem

Mario Bukal·Boris Muha
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Abstract

Starting from a nonlinear 2D/1D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, on 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, quantified 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 limit fluid velocity and the pressure can be expressed solely in terms of the solution to the thin-film equation.

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Mario Bukal, Boris Muha. 2022-01-19. Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid -- structure interaction problem. https://doi.org/10.1088/1361-6544/ac7d89

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