Hovering of an actively driven fluid-lubricated foil

TL;DR

The paper addresses how an actively driven, soft elastic foil can hover near a wall while bearing load. It develops elastohydrodynamic lubrication theory to couple the foil’s deformation with gap-flow, and uses scaling arguments and two-timescale asymptotics to reveal how forcing can break time-reversal symmetry, producing attraction or repulsion depending on the actuator’s spatial extent. It derives a hovering-height scaling $\tilde{H}_{\rm bv} \sim \tilde{L}^2 (\tilde{\mu}\tilde{\omega}/\tilde{B})^{1/3}$ and a maximum supported weight that scales with $\tilde{F}_a^2/(\tilde{\mu}\tilde{\omega}\tilde{B}^2)^{1/3}$, with higher-order bending modes enabling equilibria at small gaps. Numerical simulations corroborate the theory and align qualitatively with experiments, offering principles for soft robotics and potential insights into biological adhesion phenomena.

Abstract

Inspired by recent experimental observations of a harmonically excited elastic foil hovering near a wall while supporting substantial weight, we develop a theoretical framework that describes the underlying physical effects. Using elastohydrodynamic lubrication theory, we quantify how the dynamic deformation of the soft foil couples to the viscous fluid flow in the intervening gap. Our analysis shows that the soft foil rectifies the reversible forcing, breaking time-reversal symmetry; the relative spatial support of the forcing determines whether the sheet is attracted to or repelled from the wall. A simple scaling law predicts the time-averaged equilibrium hovering height and the maximum weight the sheet can sustain before detaching from the surface. Numerical simulations of the governing equation corroborate our theoretical predictions, are in qualitative agreement with experiments, and might explain the behavior of organisms while providing design principles for soft robotics.

Figures (6)

• Figure 1: Schematic of an elastic sheet of bending rigidity $\tilde{B}$, radius $\tilde{L}$, thickness $\tilde{e}$ and density $\tilde{\rho}_s$ immersed in a fluid of density $\tilde{\rho}$ and viscosity $\tilde{\mu}$. The sheet bends in response to a harmonic normal force $\tilde{F}_a \cos(\tilde{\omega} \tilde{t})$ distributed over an area of radius $\tilde{\ell}$, which drives a flow in the thin gap.
• Figure 2: $(a)$ Schematic of the sheet's deformation based on $\ell$. See also Supplementary Movies S1 and S2 supp. $(b)$ Numerical solutions of \ref{['eq:NS']} and \ref{['eq:forcebalance']} for the time-evolution of the averaged height (symbols) for $\gamma=0.1,~\mathcal{W}=0$. Dashed lines are the asymptotic result \ref{['eq:soft']}. The inset shows a zoom for $\ell=0.5$ that highlights the slow average compared to the fast oscillations (solid line). $(c)$ The function $m(\ell)$ appears in \ref{['eq:soft']} and determines whether the sheet is attracted ($m(\ell)>0$) or repelled $(m(\ell)<0)$ from the wall. The shape of the sheet at $\mathcal{O}({\gamma})$ is $H_{1}(x;\ell)\cos(t)$, characterized in the inset.
• Figure 3: $(a)$ Time-evolution of $h(x=0,t)$ for $\gamma=\Gamma=1$. Dashed lines are the numerical solution of \ref{['eq:equili']}, shaded lines are the numerical solutions of \ref{['eq:NS']} and \ref{['eq:forcebalance']}, with the apparent line thickness coming from the sheet's oscillations.f $(b)$ Comparison between the bifurcation diagram obtained by numerical continuation of \ref{['eq:equili']} (truncating the sum after $N=5$) with numerical results obtained by solving \ref{['eq:NS']} and \ref{['eq:forcebalance']} with $\ell=0.05$ (symbols).
• Figure A1: First three modes $\zeta_i(x)$ in the Galerkin projection.
• Figure A2: Comparison between the bifurcation diagram obtained by numerical continuation of \ref{['eq:equili']} (blue lines) with numerical results obtained from solving \ref{['eq:NS']} and \ref{['eq:forcebalance']} with $\ell=0.05$ for different $\gamma$ (symbols).