A no-contact result for a plate-fluid interaction system in dimension three
Mario Bukal, Igor Kukavica, Linfeng Li, Boris Muha
TL;DR
This work tackles the no-contact problem for a three-dimensional fluid–plate interaction where a viscous incompressible fluid is bounded above by a moving elastic plate. It develops a distance-estimate framework that couples fluid dissipation to plate curvature via a divergence-free test function pair $(\Delta_x \eta, w)$ and a pressure-like quantity, yielding a bound on $\|\eta^{-1}\|_{L^1}$ and, through interpolation, on $\|\eta^{-1}\|_{L^{\infty}}$. Under natural energy bounds and additional regularity, the authors prove a uniform separation from the rigid boundary, i.e., $\sup_{t\in(0,T)} \|\eta^{-1}(t)\|_{L^{\infty}} \le C$, implying no contact on $[0,T]$ and showing this holds even without damping in the plate equation. The approach extends no-contact results from 2D to 3D in a weak-solution setting by avoiding 2D stream-function techniques and leveraging smooth approximations to justify the test-function pair in the weak form. Overall, the paper advances understanding of global-in-time behavior in Navier–Stokes free-boundary problems with elastic upper boundaries.
Abstract
We address the fluid-structure interaction between a viscous incompressible fluid and an elastic plate forming its moving upper boundary in three dimensions. The fluid is described by the incompressible Navier-Stokes equations with a free upper boundary that evolves according to the motion of the structure, coupled via the velocity- and stress-matching conditions. Under the natural energy bounds and additional regularity assumptions on the weak solutions, we prove a non-contact property with a uniform separation of the plate from the rigid boundary. The result does not require damping in the plate equation.