Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid–structure interaction problem. (4th August 2022)

• Record Type: Journal Article
• Authors: Bukal, Mario; Muha, Boris
• Is Part Of: Nonlinearity. Volume 35:Number 8(2022)
• Page Start: 4695
• Page End: 4726
• Publication Date: 2022-08-04
• Subjects: thin viscous fluids -- nonlinear fluid–structure interaction -- nonlinear sixth-order thin-film equation; 35M30 -- 35Q30 -- 35Q74 -- 76D05 -- 76D08; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac7d89
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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.