小半球水球的共振行為

Abstract

水球與水滴靜態的壓力-體積變化關係相似，兩者動態的振動行為或許也相似，而水球的共振行為或許可由液滴振動理論預測。本研究承襲Chang水球與水滴的研究(On the similarities between the resonance behaviors of water balloons and water drops, Physics of Fluids, vol. 32, no. 12, p. 124113, 2020)，探討夾持角小於90°的水球之共振行為。水球的共振模態可依形狀分為三類：zonal、sectoral、tesseral，其中，Chang並未發現水球的tesseral模態。本研究透過加大水球的底部半徑來增加其慣性力，有效降低產生共振所需的加速度，成功地觸發了三類共振模態，並透過高速攝影拍得模態的側視圖與俯視圖，完整記錄形狀從簡單到複雜的連續模態，並觀察到許多不同的諧振情形。模態的形狀能透過粒子影像測速法(Particle image velocimetry, PIV)等工具進行形狀分析，此方法以灰階圖有效呈現水球的模態形狀。此外，水球不是水滴，水球表面各處的張力分布不同，而液滴只有單一的表面張力。因此，液滴振動理論預測無法直接預測水球的共振頻率，其中又以sectoral模態的預測偏差最為嚴重，而這是前人未發現的現象。對此，本研究提出了一套轉換方法，透過找出模態的作用張力使液滴振動理論能預測水球的共振頻率，而水球表面的張力分布則由超彈性理論求出。這個方法受超彈性理論參數的影響程度低，並能有效連結水球實驗結果與水滴理論，更顯示主導水球振動的作用力或許是沿著緯線方向的張力。

The static pressure-volume relations of water balloons and drops have some similar characteristics. The dynamic resonance behavior of the two may be similar as well, and the drop vibration theory may predict the resonance behavior of the balloon. The hypothesis from the published research(On the similarities between the resonance behaviors of water balloons and water drops, Physics of Fluids, vol. 32, no. 12, p. 124113, 2020) is presented and examined in the study, particularly water balloons with a pinning angle less than 90°. The resonance mode of water balloons can be categorized by shapes: zonal, sectoral, and tesseral. No water balloon tesseral mode is found in the published study. The inertial force of the balloon increases by enlarging the water balloon. Thus, the accelerations trigger resonance modes are decreased effectively. This work presents top and side views of all three categories of modes, including complete sequences of resonance shapes from simple to complex ones. Several resonance types are also observed during the experiments. Mode shapes are presented by grayscale images done by Particle image velocimetry (PIV) and the shape analysis. Besides, water balloons are not drops. The tension of the balloon surface is not the same. In contrast, there is only one value for the drop’s surface tension. Thus, the drop vibration theories cannot predict resonance frequencies of balloons directly. Among all modes compared in the study, the discrepancy between sectoral modes and the predicted value is the largest, which is a new phenomenon. An adjustment method is proposed to solve the problem. The “acting tension” of the mode is proposed and calculated. The tension of the balloon is also calculated by hyperelastic theories. The “acting tension” method is nearly independent of hyperelastic parameters, and it successfully links water balloon experiment results and the drop vibration theory. The “acting tension” theory also indicates that the force that dominates balloon vibration may be the tension along with the circle of latitude.