Surface waves on a semitoroidal water ring

Abstract

We study the dynamics of surface waves on a semi-toroidal ring of water that is excited by vertical vibration. We create this specific fluid volume by patterning a glass plate with a hydrophobic coating, which confines the fluid to a precise geometric region. To excite the system, the supporting plate is vibrated up and down, thus accelerating and decelerating the fluid ring along its toroidal axis. When the driving acceleration is sufficiently high, the surface develops a standing wave, and at yet larger accelerations, a traveling wave emerges. We also explore frequency dependencies and other geometric shapes of confinement.In 1831, Faraday first observed that surface waves on a vibrated fluid volume oscillate at half the driving frequency [1]. These Faraday waves demonstrate the surface instability caused by strong oscillatory accelerations exerted on the fluid volume. Benjamin & Ursell showed that the amplitude of a surface eigenmode obeys the Mathieu equation [2,3]. It follows that the superposition of multiples of harmonics and subharmonics are also solutions to the Mathieu equation. In the presence of finite viscosity, however, the subharmonic response is dominant [4,5,6].Previous experiments have revealed that patterns of various symmetries, such as striped [7,8], triangular [9], square [10,11,12], and hexagonal [13,14], can be excited on the free surface of a fluid layer. Quasi-one dimensional surface waves have also been studied in both narrow annular and channel geometries [15,16,17,18]. In these experiments, the standing waves often interact with the meniscus that forms at the bounding walls. The contact point and length-scale of this meniscus continually change as a result of the vertical oscillations.Consequently, the meniscus emits waves towards the bulk [16]. In other experiments, to remove this meniscus effect, the contact point is pinned on a sharp edge or brim [19,20,21].

Swimming, flying, and diving behaviors from a unified 2D potential model

“…This condition of a constant velocity potential ( ) along allows us to consider the horizontal surface as the free surface. Similarly, the free surface is modeled as an equipotential line of a velocity potential in many cases (e.g., p. 363 in 27 – 31 ). This fact is useful to describe the case of “(I) Impacting plate” as in Fig.…”