Fluid-induced snap-through instability of spherical shells

TL;DR

This work addresses fluid-induced snap-through of a bistable spherical elastic shell in viscous flow at low Reynolds numbers, introducing a minimal elastohydrodynamic system for passive flow control. By combining precision experiments, 1D Koiter-based shell modeling, and 2D FSI simulations, the authors derive a closed-form scaling for the critical Cauchy number $C_Y^{cr}$ that governs snapping as a function of geometry and material parameters, $C_Y^{cr}\simeq 0.1\pi\theta\sin(\theta)\Gamma(\delta)^{-1}\left(\frac{R}{h}\right)$. The results, validated across a wide range of confinement and opening angles, enable a snapping-based valve that abruptly alters hydraulic resistance in coaxial channels, showcasing a robust, fully passive flow-control mechanism. This prototypical fluid-induced instability provides a foundation for soft hydraulics and flow-responsive structures, with potential extensions to networks of memory-enabled valves and decentralized control.

Abstract

We study the snapping instability of a spherical elastic shell induced by a viscous flow, the umbrella flipping problem when life is at low Reynolds numbers. We combine precision desktop-scale experiments, fluid-structure simulations, shell theory, fluid mechanics, and scaling analysis to determine the instability threshold as a function of the geometrical and material parameters of the system. Building on these findings, we devise a snapping-based valve that passively and abruptly alters the hydraulic resistance of a channel, offering robust flow control without active components. Beyond the application, our study presents what we believe to be a prototypical example of fluid-induced elastic instability in viscous flow, providing a foundation for future explorations in soft hydraulics and flow-responsive structures.

Figures (8)

• Figure 1: (a) Experimental setup: an elastic shell of opening angle $\theta$ is installed at the center of a cylindrical channel, with a fully-developed parabolic Hagen-Poiseuille flow investing the elastic shell. (b) Flow around a rigid shell with $R/h = 10^4$, $\theta = 40^\circ$, $\delta = 0.6$: streamlines are superimposed on colormaps of the velocity magnitude (left) and pressure contours are superimposed on colormaps of the pressure (right), normalized by $U_{\textup{max}}$ and $\mu U_{\textup{max}} / R_c$, respectively. (c) Experimental snapshots of a shell undergoing snapping with $R=15$ mm, $h=0.23$ mm, $\theta=30^\circ$, $R_\textup{c}=12.5$ mm, $E=1.1$ MPa, $Q=100$ mL/min, overlaid with the equilibrium shapes obtained via a 1D model with the same $w$ of the snapshots. From (i) to (iv), time increases as the shell deforms slightly up to (iii), before snapping into the final shape (iv).
• Figure 2: (a) Sketch of profile curve of the shell with relevant quantities for the 1D model (left). Deformed shapes for the same axial displacement at the maximum colatitude $w$, until snapping, for fixed $R/h=50$ and $\theta=35^\circ$ (left), $\theta=60^\circ$ (center), $\theta=90^\circ$ (right): 2D FSI for fixed confinement ratio $\delta=0.7$ against 1D model. (b) Normalized equilibrium pressure $\Delta p/E$ as a function of $w$, for different $h/R$ and $\theta=30^\circ,\,45^\circ,\,60^\circ,\,90^\circ$. The maximum in the curve marks the snapping pressure. (c) Snapping pressure $\Delta p_s$ (rescaled with $E\theta$) as a function of $h/R$, for different values of $\theta$ according to the color bar. In the inset: rescaled snapping pressure (normalized by Zoelly's critical pressure and $\theta$) as a function of $\theta$.
• Figure 3: Critical Cauchy number as a function of $F(\theta, \delta, h/R)$: theoretical prediction (black solid line), numerics (colored circles) and experiments (colored diamonds with error bars). The colormap highlights the value of $\theta$ in each experiment and simulation. In the inset, the same plot is reported, but the colormap depicts the confinement ratio. Stars represent critical experimental Cauchy values for the snapping valve in the coaxial channel setup.
• Figure 4: Difference in pressure drop per unit length before and after snapping as a function of the ratio between inner and outer tube radii, nondimensionalized by $\mu U_\textup{max}/R_c^2$: theory (black line), numerics with rigid shells (colored dots) and experiments (diamonds with error bars). Colors represent different values of $\theta$, as indicated by the colorbar. The insets are snapshots of the shell before (bottom) and after (top) snapping, effectively sealing the inner tube.
• Figure S1: Photo of the experimental apparatus comprising the syringe pump, the pressure sensor, the scientific camera with a backlight, and the cylindrical channel. A detail of the VPS shell, the carbon rod, and its support structure are depicted in the top left corner (left). Picture of the 3D printed setup that holds the carbon rod in contact with the north pole of the metal sphere: the VPS polymer is crosslinking (right).