Surface tension force on a partially submerged horizontal concave cylinder

Abstract

A horizontal cylinder with a concave cross-section partially submerged in a liquid at a given position may permit multiple menisci around itself. The number and stabilities of the menisci are analysed, and how the menisci change during the processes of gradually hoisting and lowering the cylinder is explained by bifurcation theory. The restoring force on the concave cylinder and the rebounding potential energy (defined as the work done by the restoring force during the whole hoisting process to represent the potential rebounding capacity of a cylinder on water) are also investigated. The results show that, when the radius of the concave arc is smaller than the critical value, the concave cylinder at a given position permits multiple possible menisci. The equilibria form fold bifurcations with the position of the cylinder as the bifurcation parameter, and two successive fold bifurcations can form a one-fold hysteresis loop. The force–distance curve representing the relation between the restoring force and the position of the cylinder also has corresponding hysteresis loops, where the restoring force will jump (i.e. change discontinuously) at the bifurcation points. In contrast to a convex cylinder, a concave cylinder can have different values of the restoring force at the same height because of multiple menisci, and the values depend on whether it is hoisted or lowered. Under the condition of a fixed cross-sectional area, the optimal cross-sectional shape is determined when the maximum rebounding potential energy is reached, and it is close to the shape with the critical concave arc angle for the existence of multiple possible menisci. The cross-sections with concave parts are preferable to circular, laterally planed and corner-concave cross-sections. This paper provides an effective method of enhancing the restoring force and potential rebounding height of a robotic water strider insect or particles on the surface of water.

Exotic flotation theory with new method for floating stability analysis

“…Especially for the concave solid surface (see examples in figures 3 b – d ), the menisci around it are more likely to be unstable and reconfigure spontaneously (Tan et al. 2022). To examine the meniscus stabilities, the direct computation method (Myshkis et al.…”

“…Despite the determination of the exotic floating object, the corresponding exotic flotation phenomenon may not exist in practice due to the instabilities of the menisci around it. Especially for the concave solid surface (see examples in figures 3b-d), the menisci around it are more likely to be unstable and reconfigure spontaneously (Tan et al 2022). To examine the meniscus stabilities, the direct computation method (Myshkis et al 1987;Slobozhanin & Perales 1993;Pesci et al 2015) can be a viable option by solving the eigenvalue problem related to the second variation of the total energy functional, while the direct computation method has been developed further into a critical parameter comparison method (Zhang & Zhou 2020a,b), which enables us to determine the meniscus stabilities with only the physical parameters at the contact point.…”

“…It should be noted that in this study, the right half configuration (of x ≥ 0) is considered, and the curvature of the solid surface convex towards the liquid is defined as positive, possibly leading to a sign difference between (3.3) and the literature (e.g. Zhang & Zhou 2020a; Tan et al 2022).…”

“…Under surface tension effects, two or more possible equilibrium menisci (denoting the non-uniqueness of equilibrium menisci) may exist around the stationary solid of a special shape (e.g. Finn 1988; Tan, Zhang & Zhou 2022). An exotic property that there exists a continuum of distinct menisci in an axisymmetric container with a certain volume of liquid was investigated (Callahan, Concus & Finn 1991; Concus & Finn 1991; Concus, Finn & Weislogel 1999).…”